David Ellison
email: david.ellison2@rmit.edu.au

Bertrand Jouve
email: bertrand.jouve@univ-tlse2.fr

Lewi

Lewi Stone
email: lewi.stone@rmit.edu.au

Switching Checkerboards in (0, 1)-Matrices

Keywords: Graph theory, binary matrices, spectral radius, graph modifications, Bruhat order (05C90)

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Switching Checkerboards in (0, 1)-Matrices

Introduction

In the study of diffusion processes along a network, the largest eigenvalue of the adjacency matrix, called the spectral radius of the network and denoted by λ 1 , plays a critical role. For instance, in epidemic models, having a reproduction number R 0 greater or smaller than 1/λ 1 determines whether the epidemic will die out or keep spreading. Also, in the problem of synchronisation of coupled oscillators on a network, a key threshold criterion for the stability of the synchronised solution is based on λ 1 . The largest eigenvalue similarly affects other percolation processes such as wildfires or rumors spreading in social networks and is also of interest in the study of random matrices.

The parameter which has the greatest impact on the value of λ 1 is the degree distribution of the network. Indeed, some lower bounds and approximations of λ 1 are entirely determined by the degree distribution [START_REF] Restrepo | Approximating the largest eigenvalue of network adjacency matrices[END_REF]. Such approximations can be very precise for the vast majority of cases, but tend to become wildly inaccurate in extreme cases of networks with very particular topological properties. In this article, we consider the degree distribution fixed in order to analyse how the topology of the network affects λ 1 . The study of matrices with fixed degree distribution has applications to epidemic models [START_REF] Coupechoux | How clustering affects epidemics in random networks[END_REF], synchronisation problems [START_REF] Mcgraw | Clustering and the synchronization of oscillator networks[END_REF], ecology [START_REF] Brualdi | Nested species subsets, gaps, and discrepancy[END_REF], LT codes [START_REF] Jiang | A degree distribution optimization algorithm for image transmission[END_REF] and many other areas. Once the degree distribution is fixed, the degree assortativity is the most important parameter of real networks [START_REF] Xulvi-Brunet | Reshuffling scale-free networks: From random to assortative[END_REF][START_REF] Newman | Mixing patterns in networks[END_REF].

When working with a fixed degree distribution, we rely on a basic operation which consists in switching the 1s and 0s in a 2 × 2 submatrix of the adjacency matrix of the form 1 0 0 1 or 0 1 1 0 , called a checkerboard. Note that these submatrices can be selected from any two rows and columns, not necessarily adjacent, in the adjacency matrix. Any such switching amounts to rewiring two edges in a way that leaves the degree distribution invariant. This allows us to visualise the set of binary matrices with a given degree distribution as the vertex set of a graph where two matrices are adjacent if they differ by one switch. It was shown in [START_REF] Ryser | Combinatorial properties of matrices of zeros and ones[END_REF] that this graph of matrices is connected.

In this paper, we call the two above submatrices positive and negative checkerboards. This polarisation of checkerboards defines an orientation of the edges of the graph of matrices, which represents an order known as the secondary Bruhat order in the literature [START_REF] Brualdi | More on the Bruhat order for (0, 1)-matrices[END_REF][START_REF] Fernandes | Extremal matrices for the Bruhatgraph order[END_REF][START_REF] Brualdi | The bruhat shadow of a permutation matrix, Mathematical Papers in Honour of Eduardo Marques de Sá[END_REF]. In the case of adjacency matrices, we evidence links between secondary Bruhat order, spectral radius and second Zagreb index. In Section 1, we introduce the basic definitions surrounding checkerboards. Then, in Section 2, we show that the oriented graph of matrices is a directed acyclic graph and discuss its sources and sinks as well as when A can be reached from A by following an oriented path. Finally, in Section 3, we describe how the second Zagreb index, an approximation of λ 1 , evolves throughout the graph of matrices. We then show that λ 1 reaches its global maximum at a sink of the graph of matrices and describe the evolution of λ 1 through simulations. The proofs of two theorems from Section 2 have been placed in the appendix for the sake of improving readability. In order to maximise the generality of our results, we have considered non-symmetric binary matrices with fixed row and column sums in all sections except Section 3, where we consider only adjacency matrices. Note that a non-symmetric matrix A can be interpreted as a characteristic submatrix of M = 0 A A t 0 , where M is the adjacency matrix of a bipartite graph.

Positive and Negative Switches

Given a (0, 1)-matrix A, a checkerboard of A is a 2 × 2 submatrix of A of the form either 1 0 0 1 or 0 1 1 0 . A checkerboard found on rows i and j and columns k and l, with i < j and k < l, is said to have coordinates (i, j, k, l). A unitary checkerboard is a checkerboard selected from adjacent rows and adjacent columns and has coordinates (i, i + 1, k, k + 1). Switching a checkerboard refers to replacing one form with the other (also called interchange or swap in the literature). Note that row and column sums are invariant under this operation.

The following result was shown by Ryser in 1957:

Theorem 1 (Ryser, [START_REF] Ryser | Combinatorial properties of matrices of zeros and ones[END_REF]). Given two matrices A and A in the class of (0, 1)matrices having specified row and column sums, one can pass from A to A by a finite sequence of switches.

This property makes switching checkerboards an essential tool for studying classes of matrices with fixed row and column sums as well as classes of graphs with fixed degree distributions. Applying successive random switches to a binary matrix creates a Markov chain which visits all possible matrices with the same row and column sums. This Markov chain can then be used to sample randomly matrices with a fixed degree distribution [START_REF] Artzy-Randrup | Generating uniformly distributed random networks[END_REF].

In this paper, we attribute a sign to checkerboards and switches: a checkerboard of the form 1 0 0 1 is said to be positive while a checkerboard of the form 0 1 1 0 is said to be negative. A positive switch corresponds to replacing a negative checkerboard with a positive one while a negative switch is the reverse. (Note that positive switches are called forward switches in [START_REF] Fernandes | Extremal matrices for the Bruhatgraph order[END_REF] and L 2 → I 2 interchanges in [START_REF] Brualdi | More on the Bruhat order for (0, 1)-matrices[END_REF].) We define a switching matrix C i,j,k,l , with i < j and k < l, as having coefficients c ik = c jl = 1, c il = c kj = -1 and 0 elsewhere. Thus, a positive (resp. negative) switch of coordinates (i, j, k, l) corresponds to adding (resp. subtracting) C i,j,k,l . Note that C i,j,k,l = i≤p<j,k≤q<l C p,p+1,q,q+1 ; i.e. any switching matrix is a sum of unitary switching matrices. Note also that the unitary switching matrices are linearly independent and form a basis of the space of zero-sum (for rows and columns) matrices.

Oriented Graph of Matrices

Sources and Sinks

Given vectors R ∈ N p and C ∈ N q , let M(R, C) denote the set of binary matrices with row and column sums R and C. We assume M(R, C) = ∅ (i.e. the conditions of the Gale-Ryser theorem are satisfied). Let G(R, C) denote the oriented graph with vertex set M(R, C) and there is an arc from A to A if A can be obtained from A with a positive switch.

As mentioned in the introduction, the undirected graph underlying G(R, C), called interchange graph in [START_REF] Brualdi | Matrices of zeros and ones with fixed row and column sum vectors[END_REF], is well-studied in the literature. It is known to be connected (Theorem 1). In [START_REF] Artzy-Randrup | Generating uniformly distributed random networks[END_REF], multiple algorithms which generate random networks with a fixed degree distribution are based on this graph. Our purpose here, however, is to focus on properties which derive from the orientation. Proposition 1. The graph G(R, C) is acyclic.

Proof. For A ∈ M(R, C), let I(A) = i,j ija ij . Let A be obtained from A with a positive switch of coordinates (i, j, k, l) where i < j and k < l; i.e there is an arc from A to A in G(R, C). Then,

I(A ) -I(A) = I(C i,j,k,l ) = (i -j)(k -l) > 0.
Since I increases along arcs, and therefore along directed paths, G(R, C) is acyclic.

Recall that directed acyclic graphs have sources and sinks which are vertices with zero in-degree and out-degree, respectively. In G(R, C), sources and sinks are, respectively, matrices with no positive checkerboards and no negative checkerboards. The graph G(R, C) also represents a partial order on M(R, C), known as the secondary Bruhat order in the literature. The sources and sinks of G(R, C), which are minimal and maximal elements of the secondary Bruhat order, have been studied in [START_REF] Brualdi | More on the Bruhat order for (0, 1)-matrices[END_REF][START_REF] Fernandes | Extremal matrices for the Bruhatgraph order[END_REF] in the case of regular graphs.

A matrix A ∈ M(R, C) is said to be nested (resp. anti-nested) if the sequence 01 (resp. 10) does not appear in any row or column; i.e if the 1s occur before the 0s (resp. 0s before 1s) in every row and column. Nested graphs have important applications, in particular in ecology [START_REF] Brualdi | Nested species subsets, gaps, and discrepancy[END_REF]. For our purpose, the nested case is both trivial and theoretically important, as indicated by the following proposition and its corollary. Proof. If A is nested, the sequence 01 does not appear in any row or column of A. Hence, there is no checkerboard in A.

Conversely, if A is not nested, there is a row or column containing the sequence 01. Say it is a row; since the column with the 0 has degree at least equal to that with the 1, there is another row containing 10 in the same two columns. Hence, there is a checkerboard. Corollary 1. The set M(R, C) is a singleton if and only if it includes a matrix which becomes nested after reordering its rows and columns by non-increasing degree.

Proof. This result derives from Proposition 2 by using Theorem 1 and the fact that the number of checkerboards in a matrix is invariant under row and column permutation.

We define a zebra as a matrix in M(R, C) which is the sum of two matrices, one nested and one anti-nested. The name zebra refers to the three stripes formed by the 1s and 0s in the matrix, as shown in Example 1. We say that a zebra is split vertically (resp. horizontally) if no column (resp. row) has 1s from both the nested and anti-nested parts (see Example 1). In other words, a split zebra can be split along a vertical (or horizontal) axis so that the left (resp. top) half is nested while the right (resp. bottom) half is anti-nested. Note that the two "halves" need not be of equal size. Also, we define an anti-zebra as the complement of the vertical reflection of a zebra; i.e. b ij = 1 -a n-i,j (see Example 1).

Example 1 (Zebras and Anti-zebras). The matrices A and A are zebras. In A, the nested and anti-nested parts overlap horizontally and vertically, so A is not split. Meanwhile, A is a horizontally split zebra, where the top three rows are nested and the bottom three anti-nested. Matrices B and B are antizebras which are the complement of the vertical reflection of A and A , so B is horizontally split and B is not split.

A =         1 1 1 1 1 0 1 1 1 1 0 0 1 1 0 0 0 1 1 0 0 0 0 1 1 0 0 0 1 1 0 0 1 1 1 1         and B =         1 1 0 0 0 0 0 1 1 1 0 0 0 1 1 1 1 0 0 0 1 1 1 0 0 0 0 0 1 1 0 0 0 0 0 1         A =         1 1 1 1 1 0 1 1 1 1 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 1 1 1 1         and B =         1 1 0 0 0 0 1 1 1 1 0 0 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 0 1 1 0 0 0 0 0 1        
Like nested graphs, zebras and anti-zebras can be characterised by the absence of certain submatrices: Claim 1. If we exclude full (resp. empty) rows and columns, zebras (resp. antizebras) are exactly the matrices without

0 1 * 0 or 0 * 1 0 (resp. * 1 1 0 or 0 1 1 *
) submatrices, where * denotes either 0 or 1.

Proof. The absence of those submatrices derives from the geometry of the zebra. The converse, which is not needed to prove the theorem, we leave as an exercise.

Note that the forbidden submatrices include negative checkerboards; thus zebras and anti-zebras form sinks of G(R, C). Proof. This derives from the geometry of the split zebra and anti-zebra.

Theorem 2. If M(R, C) contains a split zebra or a split anti-zebra, then that split zebra or anti-zebra is the only element of M(R, C) without negative checkerboards; i.e it is the unique sink in G(R, C).

(Proof in the appendix) Corollary 2. Let B ∈ M(R, C) be a split zebra or a split anti-zebra. For all A ∈ M(R, C), B can be reached from A via a sequence of positive switches.

Corollary 3. If M(R, C) contains the complement of a split zebra or a split anti-zebra, then it is the unique source in G(R, C).
Proof. The complement of a unique sink is a unique source.

Adapting Ryser's Theorem to the Oriented Graph of Matrices

It follows from Theorem 1 that the undirected version of G(R, C) is connected. Having introduced an orientation naturally raises the question of when can a matrix A be reached from matrix A via a sequence of positive switches. The remainder of this section is devoted to answering this question. We will denote by A → A that there is a directed path from A to A in G(R, C).

Proposition 3. Let R ∈ N p , C ∈ N q , let A, A ∈ M(R, C) and let M = A -A. If A → A , then M is a sum of unitary switching matrices; i.e. (i) There is a unique T = [t ik ] ∈ M p-1,q-1 (N): M = i,k t ik C i,i+1,k,k+1 .
Proof. Each positive switch corresponds to adding a switching matrix and each switching matrix is a sum of unitary switching matrices. The unicity of T follows from the linear independence of the unitary switching matrices.

Remark 1.

• Note that T is uniquely determined by M, even when multiple switching sequences lead from A to A (see Example 2).

• Given M = A -A which satisfies condition (i), if we switch a negative checkerboard in A to positive, then the coefficients of T inside a rectangle corresponding to the checkerboard are all decreased by 1. The new matrix M still satisfies condition (i) if and only if the coefficients of T all remain non-negative. Reaching A via successive switches corresponds to reducing T to 0 in this fashion. (See examples below for more details.)

We denote by (i) the necessary condition given in Proposition 3. Unfortunately, condition (i) is not sufficient to ensure A → A , as shown in Example 3.

Example 2. Let A =   0 0 1 1 0 0 1 1 0   and A =   1 0 0 0 1 0 1 0 1   . There are two directed paths from A to A : A →   1 0 0 0 0 1 1 1 0   → A and A →   0 1 0 1 0 0 1 0 1   →
A . In the first case, we add the switching matrices

C 1,2,1,3 =   1 0 -1 -1 0 1 0 0 0   followed by C 2,3,2,3 =   0 0 0 0 1 -1 0 -1 1   . In the second case, we add C 1,3,2,3 =   0 1 -1 0 0 0 0 -1 1   and C 1,2,1,2 =   1 -1 0 -1 1 0 0 0 0   . We have M = A -A =   1 0 -1 -1 1 0 0 -1 1   = C 1,2,1,3 +C 2,3,2,3 = C 1,3,2,3 +C 1,2,1,2 .
In both of these sums, the first switching matrix can be split into two unitary switching matrices. So M is the sum of three unitary switching matrices:

M = C 1,2,1,2 +C 1,2,2,3 +C 2,3,2,3 . Thus M satisfies condition (i) with T = 1 1 0 1 . Example 3. Let A =     0 0 0 1 1 1 0 1 1 0 1 1 1 0 0 0     and A =     1 0 0 0 1 0 1 1 1 1 0 1 0 0 0 1     . We have M = A -A =     1 0 0 -1 0 -1 1 0 0 1 -1 0 -1 0 0 1     , which satisfies condition (i) with T =   1 1 1 1 0 1 1 1 1   .
Matrix A has only one negative checkerboard, with coordinates (1, 4, 1, 4). After switching it to positive, reaching A now requires a negative switch of coordinates (2, 3, 2, 3). Indeed, switching that checkerboard decrements all coefficients of T , leaving a -1 in the centre. Hence A → A .

While the matrix M is the sum of eight unitary switching matrices, and can be written as a sum of (not all unitary) switching matrices in many ways, none of these sums includes C 1,4,1,4 . So none of the switches appearing in these sums is feasible in A.

Lemma 1. Let A, A ∈ M(R, C) and M = A -A such that there exists

T ∈ M p-1,q-1 (N) such that M = i,k t ik C i,i+1,k,k+1 (i). Matrix T is now extended via t ij = 0 if i = 0 or p, or if j = 0 or q.
Then, for all i, j, we have:

m ij = t ij + t i-1,j-1 -t i,j-1 -t i-1,j .
Proof. Each switching matrix has four non-zero coefficients. So in condition (i), exactly four terms in the sum contribute to m ij (see Figure 1). If M satisfies (i), we create a (p -1) × (q -1) grid, with the coefficients of T inside the cells and the coefficients of M at the corners (see Figure 1). Each cell corresponds to a unitary switching matrix and each coefficient of T indicates how many times its cell needs to be switched in order to go from A to A . Let m = max t ik . For 1 ≤ i ≤ m, we define P i (M) as the shape formed by grouping the cells of the grid with coefficients at least i (see Example 4). Note that P i (M) is formed by one or several polyominoes. We recall that a polyomino is a shape formed by a finite number of orthogonally connected cells in a square grid. A polyomino is simply connected if it is delimited by a simple loop, i.e. if it has no hole.

m i,j-1 m i,j m i,j+1 m i-1,j-1 m i-1,j m i-1,j+1 m i+1,j-1 m i+1,j m i+1,j+1 t i,j-1 t i-1,j-1 t i,j t i-1,j m ij = t ij + t i-1,j-1 -t i,j-1 -t i-1,j
Example 4. Let A =     0 0 1 1 0 0 1 1 1 1 0 0 1 1 0 0     and A =     1 1 0 0 1 1 0 0 0 0 1 1 0 0 1 1     . We have M = A -A =     1 1 -1 -1 1 1 -1 -1 -1 -1 1 1 -1 -1 1 1     , which satisfies condition (i) with T =   1 2 1 2 4 2 1 2 1   . Thus, P 1 (M) is a 3 × 3 square, P 2 (M) is an X-shaped
pentomino (see Figure 2) and P 3 (M) and P 4 (M) are monominoes surrounding only the central cell. Note that we have A → A and going from A to A requires at minimum four switches.

-1 Lemma 2. If P i (M) contains two diagonally adjacent cells, then at least one of the two common neighbouring cells is also in P i (M).

-1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 1 2 2 4 1 2 1 2 1
Proof. Say the cells of coordinates (i, j) and (i -1, j -1) are in P i (M) and the cells of coordinates (i -1, j) and (i, j -1) are not in P i (M) (or vice versa). Then,

t ij , t i-1,j-1 ≥ i (resp. ≤ i -1) and t i-1,j , t i,j-1 ≤ i -1 (resp. ≥ i). It follows from Lemma 1 that m ij ≥ 2 (resp. ≤ -2
). Since M = A -A and A and A have binary coefficients, the coefficients of M are in {-1, 0, 1}. This is impossible.

Remark 2. This means that P i (M) cannot contain two polyominoes connected by a corner, and there is no distinction between connected and orthogonally connected components of P i (M).

Theorem 3. Let R ∈ N p , C ∈ N q , let A, A ∈ M(R, C) and let M = A -A.
If M satisfies the following conditions:

(i) ∃T ∈ M p-1,q-1 (N): M = i,k t ik C i,i+1,k,k+1 , (ii) For all i, each connected component of P i (M) is simply connected, (iii) |i -i| ≤ 1 and |k -k| ≤ 1 ⇒ |t i k -t ik | ≤ 1, then A → A .
(Proof in the appendix)

Note that the third condition says that the coefficients inside orthogonally or diagonally adjacent cells must be equal or successive integers. The authors have yet to find an instance where A → A when condition (iii) in Theorem 3 is not satisfied. Also, it seems that if for some i, P i (M) is not simply connected, it should be possible to create a counter-example, like in Example 3. Combining these two observations yields the following conjecture:

Conjecture 1. Let R ∈ N p , C ∈ N q and let M ∈ M p,q ({-1, 0, 1}). We have A → A for all A, A ∈ M(R, C) such that A -A = M if and only if M satisfies: (i) ∃T ∈ M p-1,q-1 (N): M = i,k t ik C i,i+1,k,k+1 , (ii
) For all i, each connected component of P i (M) is simply connected.

Spectral Radius and Second Zagreb index

Extrema

In this section, we consider the class of adjacency matrices of simple graphs with fixed degree distribution D. We investigate how the topology of simple graphs affects their spectral radius. To this purpose, we analyse the effect of successive positive checkerboard switches on the spectral radius λ 1 as well as on the second Zagreb index M 2 , which was used in [START_REF] Abdo | Estimating the spectral radius of a graph by the second zagreb index[END_REF] to create an approximation of λ 1 :

Z 2 = M2
m , where m is the number of edges. Amongst all the existing approximations for λ 1 , we chose to focus on Z 2 because we are able to map very precisely how it varies throughout the set of adjacency matrices of fixed degree distribution. In contrast, most other commonly used approximations are determined by the degree distribution, meaning that they are invariant under checkerboard switching. [START_REF] Pastor-Satorras | Eigenvector localization in real networks and its implications for epidemic spreading[END_REF][START_REF] Restrepo | Approximating the largest eigenvalue of network adjacency matrices[END_REF].

Let G = (V, E) be a simple graph with |V | = n and |E| = m and adjacency matrix A. Let d i denote the degree of vertex i ∈ V . The spectral radius of G, denoted by λ 1 (G) or λ 1 (A), is the largest eigenvalue of A. The first and second Zagreb indices are defined as M 1 = i∈V d 2 i and M 2 = ij∈E d i d j and we have Z 1 = M1 n and Z 2 = M2 m . Note that Z 1 is the quadratic average over V of the degrees, while Z 2 is the quadratic average over E of d i d j . It is well-known that λ 1 ≥ Z 1 , with equality if G is a regular graph [START_REF] Elphick | On the relations between the zagreb indices, clique numbers and walks in graphs[END_REF]. Thus, it is the heterogeneity of D that allows λ 1 and Z 2 to vary among the class of graphs with degree distribution D. Also, for a fixed degree distribution, M 2 is proportional to the degree assortativity coefficient r, which is the standard Pearson coefficient for correlation between the degrees [START_REF] Li | Supplemental material: The s(g)-metric and assortativity [a supplement to" towards a theory of scalefree graphs: Definition, properties, and implications[END_REF]:

r = M 2 -( n i=1 1 2 d 2 i ) 2 /m n i=1 1 2 d 3 i -( n i=1 1 2 d 2 i ) 2 /m .
Let D ∈ N n be the degree distribution of a simple graph. We now consider the class M(D) of adjacency matrices of simple graphs of order n with fixed degree distribution D; i.e. symmetric binary matrices with zeroes on the diagonal and row and column sums D. Due to the absence of loops, we only consider, in this section, checkerboards with no coefficient on the diagonal. Also, given the symmetry of the matrices, checkerboards always come in symmetric pairs which are always switched together. In terms of graphs, a checkerboard corresponds to a 4-cycle of alternating edges and non-edges. Switching a checkerboard means switching the edges and non-edges. We define the symmetric switching matrix

C ijkl = C ijkl + C t ijkl .
Note that we now require the coordinates (i, j, k, l) to be all different. The distinction between positive and negative checkerboards for graphs is dependent on an ordering of the vertices. We will always sort the vertices by non-increasing degree; i.e D is non-increasing. A positive (resp. negative) switch of coordinates (i, j, k, l) now corresponds to adding (resp. subtracting) C ijkl to the adjacency matrix. We define G(D) as the directed graph with vertex set M(D) and an arc joins A to A when A can be obtained from A by a positive switch. Proof. The connectedness is shown in [START_REF] Berge | of monographie universitaire de mathématique[END_REF]. The acyclicity follows from the acyclicity in the asymmetric case. Lemma 3. Let A, A ∈ M(D) be such that A can be obtained from A by a positive switch of coordinates (i, j, k, l). We have

M 2 (A ) -M 2 (A) = (d i - d j )(d k -d l ).
Proof. A is obtained from A by adding the edges ik and jl and removing il and kj.

So M 2 (A ) = M 2 (A) + d i d k + d j d l -d i d l -d j d k .
Proposition 5. M 2 and Z 2 are non-decreasing along arcs of G(D).

Proof. This follows from Lemma 3 and D being non-decreasing.

Note that M 2 and Z 2 are constant only along arcs of G(D) which correspond to a checkerboard of coordinates (i, j, k, l) where

d i = d j or d k = d l .
Corollary 4. The sources and sinks of G(D) are, respectively, local minima and maxima of M 2 and Z 2 .

The above results give a clear picture of the variations of M 2 and Z 2 along G(D). Unfortunately, the variations of λ 1 are not quite so neatly organised. The following lemma gives a lower bound to the effect of a single positive switch on the spectral radius: Lemma 4. Let A, A ∈ M(D) such that A can be obtained from A by a sequence of positive switches of coordinates (i p , j p , k p , l p ). Let X denote the normalised principal eigenvector of A. We have

λ 1 (A ) -λ 1 (A) ≥ 2 p (x ip -x jp )(x kp -x lp ).
Proof. Let X denote the normalised principal eigenvector of A . Recall that

λ 1 (A) = X t AX = max x∈R n ,||x||=1 x t Ax,
where || || is the L 2 -norm. We have:

λ 1 (A ) -λ 1 (A) = X t A X -X t AX ≥ X t A X -X t AX = X t p C ip,jp,kp,lp X = 2 p (x ip -x jp )(x kp -x lp ).
Unfortunately, the coefficients of the principal eigenvector are not always in the same order as the degrees. They are, however, strongly correlated [START_REF] Li | Correlation between centrality metrics and their application to the opinion model[END_REF]; so λ 1 increases along most arcs of G(D). In fact, they are perfectly correlated in the case where λ 1 is maximum: Proposition 6. If A realises the maximum of λ 1 over M(D), then the coefficients of the normalised principal eigenvector X of A are in the same order as the degrees; i.e.

d i > d j ⇒ x i ≥ x j .
Proof. Assume we have d i > d j and x i < x j . We will show that A does not maximise λ 1 . Let E be a set of d i -d j vertices adjacent to i and not j, with j / ∈ E. Let A be obtained from A by replacing the edges between i and E with edges between j and E. Note that all the non-zero coefficients of A -A are in rows and columns i and j. We have:

λ 1 (A ) -λ 1 (A) ≥ X t A X -X t AX = 2(x j -x i ) k∈E x k ≥ 0.
Consider the case where 2(x j -x i ) k∈E x k = 0. According to the Perron-Frobenius theorem for non-negative matrices, X has non-negative coefficients. Thus, since

x j > x i , ∀k ∈ E, x k = 0. Let k ∈ E, then l a kl x l = λ 1 (A)x k = 0.
Since all terms in l a kl x l are non-negative and k is adjacent to i, we deduce that x i = 0. Similarly, any vertex l in the connected component of i has x l = 0.

Recall that if there are multiple connected components, the spectral radius of A is the maximum of the spectral radii of the connected components. Also, the principal eigenvector has zero coefficients on those connected components with spectral radii less than the maximum. Since x j > 0, λ 1 (A) is the spectral radius of the connected component of j, to which i does not belong. Thus, the connected component of j in A is a proper sub-graph of that in A . So λ 1 (A ) > λ 1 (A) in this case and in all cases.

The degree distribution of A is different from that of A. We resolve this by creating A , which is obtained from A by switching the i-th and j-th rows and the i-th and j-th columns. A has the same row and column sums as A and we have λ 1 (A ) = λ 1 (A ) > λ 1 (A). Thus, A does not maximise λ 1 .

From this, we deduce the following: Theorem 4. The maximum of λ 1 is reached at a sink of G(D).

Proof. Let A maximise λ 1 over M(D). For each set of same-degree vertices, we can reorder the rows and columns of A so that within these sets, the coefficients of the principal eigenvector are non-increasing. This operation does not affect λ 1 . It follows from Proposition 6 that all the coefficients of X are now nonincreasing. Let A be obtained from A by a sequence of positive switches of coordinates (i p , j p , k p , l p ). It follows from Lemma 4 that

λ 1 (A ) -λ 1 (A) ≥ 2 p (x ip -x jp )(x kp -x lp ) ≥ 0.
Since λ 1 (A) is maximum, any matrix A ∈ M(D) such that A → A also maximises λ 1 , including the sinks that can be reached from A.

Switching Algorithm Simulations

While Theorem 4 tells us that at least one sink of G(D) realises the global maximum of λ 1 , it gives no indications for identifying which sink to aim for when G(D) contains several, as is usually the case. In this section, we showcase how applying successive random switches to Erdős-Rényi and Small-World graphs, after ordering its vertices by degree, leads to an increase in λ 1 , which is significant when the edge density is low. Instead of using a Monte-Carlo method as in the Xulvi-Brunet Sokolov algorithm [START_REF] Xulvi-Brunet | Reshuffling scale-free networks: From random to assortative[END_REF], we only apply positive switches selected randomly in the adjacency matrix. Simulations that begin with an Erdős-Rényi graph E(N, p) with vertices ordered by degree are shown in Figure 3 and Small World graphs appear in Figure 4. In both cases, the end result of the switching process is very close to a zebra when p > 0.5 and very close to an anti-zebra when p < 0.5. When p = 0.5, the end point is simultaneously almost a zebra and almost an anti-zebra. In the Erdős-Rényi case with p = 0.2, the spectral radius increases by more than 15%, which is very impressive for a fixed degree distribution. Conversely, when p = 0.7, the spectral radius increases only by a few decimal points.

Conclusion

Our analysis of the oriented graph of matrices has given us the tools needed to study how the topology of graphs impacts their spectral radius λ 1 . We have shown that the global maximum of λ 1 is reached for a matrix without negative checkerboards. Our simulations show that successively switching negative checkerboards to positive can yield a high increase in λ 1 , especially for sparse graphs. forms a negative checkerboard in A together with the middle three coefficients of our sequence; and thus A is not a sink. Assume now that it is a 1 in A .

The first two terms of our sequence are respectively 1 and 0 in A. According to Claim 2, there is no 1 0 1 submatrix in A. So the coefficient at A is 0 in A and 1 in M. Thus, we can shorten our cycle by replacing the middle three coefficients of our sequence with this 1. The resulting cycle has one fewer left turns. By repeating this operation, we obtain a cycle with no left turn.

Let us now consider the corresponding cycle in A . It satisfies the following properties:

• It alternates between 0s and 1s, going from 0 to 1 horizontally and 1 to 0 vertically.

• It always turns to the right.

• All vertical segments cross the split.

If this cycle self-intersects, it forms one of the patterns shown in Figure 6 (possibly rotated by 180 • ). In case a) (resp. b)), if the coefficient located at intersection point A is a 0 (resp 1), it forms a negative checkerboard with the second, third and fourth (resp. fourth, fifth and sixth) terms of the sequence. Else, if the coefficient located at B is a 0 (resp. 1), A, B and the fifth and sixth terms (resp. second and third terms) form a negative checkerboard. Else, if B is a 1 (resp. 0), the sequence can be shortened by replacing the five middle terms with B. This removes the intersection at A while maintaining the three properties of the cycle. By repeating this operation, we obtain a cycle with no left turn and no intersection; i.e. a negative checkerboard. Thus A must have a negative checkerboard. The case of vertically split zebras follows by symmetry along the main diagonal. The vertical reflection of a unique sink is a unique source; and the complement of a unique source is a unique sink. Thus, the same property holds if A is a split anti-zebra.

Proof of Theorem 3

Theorem. Let R ∈ N p , C ∈ N q , let A, A ∈ M(R, C) and let M = A -A. If M satisfies the following conditions: (i) ∃T ∈ M p-1,q-1 (N): M = i,k t ik C i,i+1,k,k+1 , (ii) For all i, each connected component of P i (M) is simply connected, (iii) |i -i| ≤ 1 and |k -k| ≤ 1 ⇒ |t i k -t ik | ≤ 1, then A → A .
In order to prove Theorem 3, we will first prove that one of the shapes described in Figure 7 must appear on the contour of any simply connected polyomino. We will then show that where this shape appears on P m (M), we can locate a checkerboard to switch. Repeating the operation will create a directed path from A to A . Lemma 5. On the contour of any simply connected polyomino, there appears at least one of the three motifs described in Figure 7.

Proof. Let P be a simply connected polyomino. If P is a rectangle, we have motif 1. We will now assume that P is not a rectangle.

We define left (resp. right) corners of P as corners where the contour of P, when followed clockwise, turns left (resp. right). Note that P is locally non-convex (resp. convex) around left (resp. right) corners.

For x, y ∈ P, we define an orthogonal path from x to y as a line joining x and y consisting of only horizontal and vertical segments included inside P. We 2) In motif 2, the two vertical sides must have equal length; i.e. AB = CD. Also, the inside of rectangle ABCD must be entirely included in the polyomino.

The lengths are variable and the motif may be rotated.

3) In motif 3, the rightmost side is longer than AB. The inside of rectangle ABCD must be entirely included in the polyomino. The lengths are variable and the motif may be rotated or reflected. denote by OP (x, y) the set of orthogonal paths joining x and y. We then define the length of an orthogonal path L as the pair (N L , l L ), where N L is the number of segments of L and l L is the length of the last segment. We define d(x, y), the distance between x and y, as the length of the minimal orthogonal path, going by lexicographic order (ie. d(x, y) = (N 0 , l 0 ) with N 0 = min L∈OP (x,y) N L and l 0 = min L∈OP (x,y);N L =N0 l L ). 1Since P is not a rectangle, it is not convex; so we may choose a point x 0 ∈ P such that P is not starred in x 0 . The set of points in P which maximise the distance to x 0 comprises one or several segments from the contour of P. Let B and C be the ends of one such segment. Both B and C must be corners of P, else points on (B, C) outside of [B, C] would be at an equal or greater distance from x 0 . Let A and D be the other two corners adjacent to B and C, respectively. B and C must be right corners, otherwise points inside of [AB] or [CD] would be at a greater distance from x 0 . The rectangle ABCD must be included inside of P: if a part of ABCD was outside P, since P is simply connected, the minimal orthogonal path from x 0 to B would need to go around one side of the missing part, and reaching the other side from x 0 would require a path with more turns.

We may assume w.l.o.g that AB ≤ CD. Then, A must be a left corner, else reaching [AB] from x 0 would require one more turn than [BC]. If AB = CD, D must similarly be a left turn, and we have the second motif. If AB < CD, we have the third motif.

We may now prove Theorem 3.

Proof of Theorem 3. Let A, A and M satisfy the conditions of Theorem 3; let T satisfy (i) and let m = max t ik . It follows from applying Lemma 5 to a connected component of P m (M) that there is a rectangle ABCD corresponding to one of the three motifs described in Figure 7 included inside P m (M). Since m is the maximum coefficient, all cells in P m (M) have coefficient m. It follows from (iii) and Remark 2 that all cells orthogonally or diagonally adjacent to P m (M) have coefficient m -1. From Lemma 1, we have m ij = t ij + t i-1,j-1t i,j-1 -t i-1,j , where t ij = 0 if i = 0 or p or j = 0 or q. For a right (resp. left) corner of P m (M), three (resp. one) of the incident cells have coefficient m -1 and one (resp. three) has coefficient m. Thus coefficients of M located at corners of P m (M) must be 1 or -1. More precisely, regardless of the orientation of ABCD, the coefficients of M located at the corners are 1 for the top left and bottom right corners and -1 for the top right and bottom left, except for D in motif 3 which has a 0.

A 1 in M means a 0 in A and 1 in A ; a -1 in M is the reverse; and a 0 in M means either two 0s or two 1s in A and A . So, a 1 -1 -1 1 submatrix in M means there is a negative checkerboard in A and a positive checkerboard in A . If any one of the four coefficients is replaced by 0, then there is either a negative checkerboard in A or a positive checkerboard in A . Switching that checkerboard, either in A or in A , results in a new instance where ABCD has been cropped from P m (M). Thus, the new instance still satisfies the three conditions of Theorem 3. Repeating the process terminates with M = 0, as t ik is reduced at each step. This constructs a directed path from A to A in G(R, C).

Proposition 2 .

 2 If R and C are non-increasing, then A ∈ M(R, C) is nested if and only if it has no checkerboards.

Claim 2 .

 2 Split zebras, split anti-zebras and their complements include at most one of the following vectors as submatrix: 1 0 1 , 0 1 0 , particular, horizontally split zebras have no 1 0 1 ,

Figure 1 :

 1 Figure 1: Relation between coefficients of M and T

Figure 2 :

 2 Figure 2: Polyomino P 2 (M) of Example 4. Coefficients of M are placed at the intersections of the grid and coefficients of T are inside the cells.

Proposition 4 .

 4 The graph G(D) is acyclic and the underlying undirected graph is connected.

Figure 3 :

 3 Figure 3: Spectral radius (plain) and Z 2 (dotted) as functions of the number of positive switches, starting from Erdős-Rényi graphs with N = 100 vertices. From left to right, the edge density p ranges from 0.2 to 0.7. Both the initial (Erdős-Rényi with vertices ordered by degree) and the final matrices are shown in each case.

Figure 4 :

 4 Figure 4: Same as Figure 3 with Small-World graphs obtained from a 10 × 10 regular grid after a random rewiring of 10% of the edges.

Figure 5 :

 5 Figure 5: Section of a cycle with a left turn. The dotted line indicates the horizontal split of the zebra.

Figure 6 :

 6 Figure 6: Section of a self-intersecting cycle with no left turn. The dotted line indicates the horizontal split of the zebra.

1 )

 1 Motif 1 occurs when the entire polyomino is a rectangle.

Figure 7 :

 7 Figure 7: Motifs from the contour of a polyomino

Note that d is not a distance in the usual sense, as its co-domain is not R + . Yet, while d : (x, y) → N 0 +

π arctan l 0 defines an actual distance, d is more practical for our purposes.

Acknowledgements

We would like to thank Cristophe Crespelle for a fruitful research session in Bergen, where Example 3 was discovered and Theorem 3 was intuited, and the LabEx SMS ANR-11-LABEX-0066 for providing mobility funding.

Acknowledgement: This work is partially supported by Victorian Government through Victorian Higher Educations State Investment Fund.

Appendix: Proofs of Theorems 2 and 3

Proof of Theorem 2

Theorem. If M(R, C) contains a split zebra or a split anti-zebra, then that split zebra or anti-zebra is the only element of M(R, C) without negative checkerboards; i.e it is the unique sink in G(R, C).

Proof of theorem 2. Let A ∈ M(R, C) be a zebra with a horizontal split (i.e. the top half is nested and the bottom anti-nested) and let A ∈ M(R, C), A = A. We will show that A has a negative checkerboard. Let M = A -A. Since A and A have the same row and column sums, M has as many 1s and -1s in each row and column. We can thus choose in each row and column of M a matching associating each 1 to a -1. Let us now choose a -1 in M as a starting point for the sequence defined by the following rules: a -1 is followed by its paired 1 in the same row; a 1 is followed by its paired -1 in the same column. Since the matrix M is finite, the sequence must form a cycle. A 1-11 sub-sequence in M corresponds to a 010 sequence in A, with the first 0 in the same column as the 1 and the second 0 in the same row. Thus, it follows from Claim 1 that at a -1, the cycle must form a right turn. Also, after turning right (resp. left) at a 1 in M (0 in A), the following -1 is located in the same column on the other side (resp. the same side) of the split in A.

If the cycle turns left at every 1 (and right at every -1), it forms an infinite staircase pattern, which is impossible. Assume that there is at least one left turn at a 1. There must be a left turn at a 1 followed by a right turn at the next 1 somewhere in the cycle. This yields a -11-11-1 sequence with a left turn at the first 1 and a right turn at the second, as shown in Figure 5. Note that the first four digits are located on the same side of the split while the final -1 is on the other side. We now consider the coefficient located at the intersection point A of the row containing the first two digits and the column with the last two. Note that A must be positioned between those last two digits due to the final -1 being on the other side of the split. If that coefficient is a 0 in A , then it


Angelo Furno

Bertrand Jouve
email: bertrand.jouve@univ-tlse2.fr.

Bruno Revelli

Paul Rochet

Alix Rigal

Najla Touati

Impact of the COVID-19 pandemic on bike-sharing use in Toulouse

Urban areas have been dramatically confronted with the rapid spread of the COVID-19 pandemic. As one of the most noticeable consequences of the pandemic, people have quickly reconsidered their travel options to minimize infection risk. Many studies on the Bike Sharing System (BSS) of several cities have shown that, in this context, cycling appears as a resilient, safe and reliable mobility option. In this paper, we analyze the spatio-temporal effects of the pandemic on BSS in the french city of Toulouse using Exploratory Factor Analysis (EFA). We discuss how this approach enables to highlight different aspects of the change in usage of BSS in the four years surrounding the pandemic, both from a geographical and temporal point of view. We use Origin/Destination data for the four years from 2019 (pre-pandemic), 2020 and part of 2021 (pandemic), and 2022 (post-pandemic). Our results confirm that cycling increased during the pandemic, with rush times remaining exactly the same as during the pre-pandemic year, before returning to the previous standards around 2022 after a notable transitional year 2021. The EFA approach enables to show the differences in these effects in the city center and the periphery of Toulouse.

Introduction

The rapid spread of the COVID-19 pandemic has brought unprecedented challenges and threatens normal human life and global public health. Since the first case of COVID-19 identified in December 2019 in the city of Wuhan in China, measures have been taken worldwide and at different scales to stop the spread of the virus. They concern of course the field of health (research and administration of vaccines, rapid identification of new clusters, etc.), but also transportation (limitation of international flights, punctual bans on inter-regional transportation, etc.) as human mobility has significantly contributed to virus propagation. In urban areas, many people have voluntarily chosen to avoid using public transport, especially due to difficulties in respect barrier measures when traveling. In this context, individuals have often re-evaluated their travel options towards more-isolated modes such as private cars, personal or shared bicycles [BSH + 20, Buc20]. Considerations of health risks, travel flexibility, traffic reduction, or the desire to spend less money in times of economic crisis entered into individual decisions about transportation. Under these circumstances, several observations show that cycling can be an interesting choice [PCZZ20, FLBA19, SGG + 18, SZQR22], and, for some, can help the economy to recover quickly. In many cities, public authorities are implementing temporary facilities, strategies derived from "tactical" urban planning, to accompany this increase in cycling [START_REF] Vidalenc | S'inspirer de l'urbanisme tactique pour adapter les villes à la distanciation physique[END_REF][START_REF] Silva | Tactical urbanism: Towards an evolutionary cities' approach?[END_REF]. Bicycle is a largely flexible, inexpensive and easily adaptable mode of urban transport to a wide range of situations, compared to, for example, public transport, whose adaptation is often a longer-term challenge [START_REF] Gkiotsalitis | Public transport planning adaption under the covid-19 pandemic crisis: literature review of research needs and directions[END_REF].

More than three years after the beginning of the pandemic, COVID-19 and its variants no longer constitutes a public health emergency of international concern but are still present throughout the world and the efforts are now more focused on prevention, control and longer-term sustainable management of the disease. Besides the fact that the emergence of new pandemics is a scenario now considered highly probable by the scientific community, feedback from COVID-19 suggests that vaccination alone is inefficient to eradicate such a virus and a fine understanding of the parameters of its diffusion is a key factor [IHN + 23]. Under these circumstances, it is natural to question the dynamics of bicycle use in urban areas since the pandemic. The effects of the pandemic on urban mobility meet more gradual transformations that have been underway for several decades now, resulting from the awareness of the ecological emergency [SSS + 22]. These transformations tend to favor public transport and active or "soft" mobilities [START_REF] Wheeler | Planning for sustainability in European cities: A review of practice in leading cities[END_REF][START_REF] Rojas | Built environment, transport, and covid-19: a review[END_REF]. Among these, the bicycle has the advantage over public transport of allowing a certain physical activity, a greater freedom of use, an even greater reduction of environmental impact [START_REF] Zhang | Shared bicycle distribution connected to subway line considering citizens' morning peak social characteristics for urban low-carbon development[END_REF], and a relatively low cost deployment. On the other hand, its use is now largely dependent on weather, topography and urban environment (speed moderation and bicycle facilities [START_REF] Stefania Bergantino | Influencing factors for potential bike-sharing users: An empirical analysis during the covid-19 pandemic[END_REF]). The development of cycling has been encouraged by the public authorities by setting up Bike Sharing Systems (BSS), adapted infrastructures and, more recently and in some countries, financial incentives for the purchase or repair of bicycles. These policies are part of a new paradigm that advocates sustainable, multimodal mobility (as opposed to the single-mode, infrastructure-focused paradigms of the automobile and public transport). With an interesting flexibility in multimodal trips, BSS uses are strongly related to the connectivity to metro and rail stations [START_REF] Böcker | Bike sharing use in conjunction to public transport: Exploring spatiotemporal, age and gender dimensions in oslo, norway[END_REF] in a noticeably different manner in urban center and suburban areas [WLG + 21, LJFS20]. However, when possible, door-to-door trips are favoured [START_REF] Frère | Les usages des services de transports personnalisés: de nouvelles routines pour une mobilité durable? Espace populations sociétés[END_REF].

Cycling thus appears in several respects to be a resilient and reliable urban mobility option in the short and long term, compatible with both the health crisis as we know it and the objectives of sustainable development. Since mobility habits are difficult to change [START_REF] Rocci | Comment rompre avec l'habitude?. les programmes d'accompagnement au changement de comportements de mobilité[END_REF][START_REF] Suzana R Moro | Barriers to bicycle sharing systems implementation: analysis of two unsuccessful pss[END_REF][START_REF] Büchel | Covid-19 as a window of opportunity for cycling: Evidence from the first wave[END_REF], taking advantage of a change in behavior during the current health crisis, of a cognitive appropriation of cycling sometimes facilitated by the economic situation, to sustainably transform urban mobility practices and habits is an opportunity and a challenge for many cities. The development of BSS is an effective response by public authorities to increase bicycle use in urban areas [DeM09, SGZ12, HVWM10, PDH10]. For example, in the year following the introduction of BSS in the city of Lyon, a 44% increase in bicycle trips [START_REF] Bührmann | New seamless mobility services: Public bicycles[END_REF] was observed, with 96% new users who had not previously cycled in Lyon's city center. Similarly, in Paris, cycling increased by 70% with the launch of the Vélib BSS [START_REF] Nadal | Vélib one year later[END_REF][START_REF] Shaheen | Bikesharing in europe, the americas, and asia: past, present, and future[END_REF]. While effective in the short term, several studies have shown that the development of bike sharing also contributes to a sustainable increase in the cycling population [START_REF] Demaio | Bike-sharing: History, impacts, models of provision, and future[END_REF] and that BSS usages are related to bike usages as a whole [START_REF] Sylwia Pazdan | Impact of environment on bicycle travel demand-assessment using bikeshare system data[END_REF]. Today, a little more than half a century after the first bike-sharing system was implemented in 1965 in Amsterdam, BSS exist in many cities around the world with growing success [SGZ10, MYVOH20, EU20] and often with quite similar technological solutions. BSS appeared in France with a pioneer initiative in La Rochelle in 1976 [START_REF] Huré | La rochelle, france, and the invention of bike sharing public policy in the 1970s[END_REF] and a fully automated solution (probably the first in the world) was implemented in Rennes in 1998. But, in France, the generation of BSS as we have it today (with the need for each user to register a personal account) was firstly introduced in Lyon in 2005, followed by Paris and Toulouse in 2007.

The success of BSS all over the world has given rise to a rapid increase in the scientific research on these systems. There have been several literature review papers summarizing the research done in this field [START_REF] Eren | A review on bike-sharing: The factors affecting bike-sharing demand[END_REF][START_REF] João | The motivations for using bike sharing during the covid-19 pandemic: Insights from lisbon[END_REF][START_REF] Fishman | Bike Share[END_REF][START_REF] Médard | Bicycle Sharing[END_REF]. The effect of the pandemic on BSS usage has been widely studied, with data collected from cities all over the world. Some studies use operator data, either Origin/Destination or simply traffic data. Among these are studies comparing five cities in the USA [START_REF] Ilderim | Change of bike-share usage in five cities of united states during covid-19[END_REF], about New-York [TL20, WN21, PCZZ20], Chicago [START_REF] Hu | Examining spatiotemporal changing patterns of bike-sharing usage during covid-19 pandemic[END_REF], Lisbon [START_REF] João | The motivations for using bike sharing during the covid-19 pandemic: Insights from lisbon[END_REF], London [START_REF] Li | Impacts of covid-19 on the usage of public bicycle share in london[END_REF], Seoul [START_REF] Sung | Causal impacts of the coronavirus-19 pandemic on daily ridership of public bicycle sharing in seoul[END_REF], Singapore [START_REF] Song | Where are public bikes? the decline of dockless bike-sharing supply in singapore and its resulting impact on ridership activities[END_REF], Zurich [START_REF] Li | Understanding the variations of micro-mobility behavior before and during covid-19 pandemic period[END_REF], Beijing [CGXJ20, SCB + 21], Kosice Slovakia [START_REF] Kubal'ák | The bike-sharing system in slovakia and the impact of covid-19 on this shared mobility service in a selected city[END_REF], or Valencia [SPCAG + 23]. Several other studies are based on online or on-site surveys in Tessaloniki in Greece [START_REF] Nikiforiadis | Assessing the Impact of COVID-19 on Bike-Sharing Usage: The Case of Thessaloniki, Greece[END_REF], Lisbon [START_REF] Teixeira | The effects of covid-19 on female and male bike sharing users: Insights from lisbon's gira[END_REF] in Portugal, San Antonio in the USA [START_REF] Jobe | Bike share responses to covid-19[END_REF], in several Italian towns [START_REF] Stefania Bergantino | Influencing factors for potential bike-sharing users: An empirical analysis during the covid-19 pandemic[END_REF], distributed in several European towns [START_REF] Monterde-I Bort | Mobility patterns and mode choice preferences during the covid-19 situation[END_REF], or with participants from all over the world [BSH + 20, BLP + 21, CSS + 20].

In a recent paper, [SZQR22] [TSMeS23] summarizes recent progress in bike-share studies related to COVID-19. They classify the issues into three main domains. The first one focuses on spatio-temporal changing patterns of bike-share usages before and after the pandemic crisis, mainly with data analysis methods applied on BSS data. The second one is more about comparing different modes of transport. The third one is centered on user behaviors via survey analysis. [START_REF] Heydari | Effect of the covid-19 pandemic on bike-sharing demand and hire time: Evidence from santander cycles in london[END_REF] provides an interesting summary of the diversity of published results.

In the present paper, we focus on the spatio-temporal aspect of BSS, a topic on which numerous recent papers have emerged. With a Bayesian time-series model on monthly aggregated data, [START_REF] Heydari | Effect of the covid-19 pandemic on bike-sharing demand and hire time: Evidence from santander cycles in london[END_REF] estimates the impact of the pandemic on both trip duration and number of trips in London. Confirming previous results, in (e.g., [START_REF] Li | Impacts of covid-19 on the usage of public bicycle share in london[END_REF][START_REF] Wang | Bikeshare and subway ridership changes during the covid-19 pandemic in new york city[END_REF]), the authors observe that trips are longer than expected during and after the spring 2020 lockdown. [START_REF] Song | Spatiotemporal evolving patterns of bike-share mobility networks and their associations with land-use conditions before and after the covid-19 outbreak[END_REF] uses an original graph theoretical approach for comparing flows of bikes during 4 COVID-relevant period in Singapore. The proposed approach gives an appropriate way to measure the increase of local trips observed during and after the lockdown. [PCZZ20] also uses a graph theoretical approach on New-York BSS data showing the reconfiguration of BSS trips on Manhattan Island during the lockdown. [START_REF] Ilderim | Change of bike-share usage in five cities of united states during covid-19[END_REF] leverages a General Linear Model (GLM) to formalize the association of the month-to-month change of bike-share activity within a geographical tract and its distance from the CBD. Using weekly aggregated data for 6 town in the USA and 6 months, they confirm previous results that the distance to the CBD is positively correlated to BSS usage [FIEEG + 14, WLSH15, NSG16, YHTC20] and, more, explains significantly well the dynamics of changing due to COVID-19: CBD areas are more responsive to the pandemic with usage declining faster but also recovering faster.

To precisely evaluate the impact of the lockdown (and lockdown ease) on the BSS usage in London, [START_REF] Li | Impacts of covid-19 on the usage of public bicycle share in london[END_REF] crosses two models: a segmented regression model and a Bayesian structural time-series model, both on aggregated data at daily level for the period from January 2019 to June 2020. Rainy conditions and BSS 2019 trips are included in the regression model as binary covariates. The authors distinguish the morning and evening peaks from the rest of the day and three levels of travel durations (short, middle, long). Their main findings are that the number of long trips increases during the lockdown and lockdown ease, which is not observed for short trips, and that the lockdown ease has few effects on the morning peak which remains low all along the lockdown. Moreover, immediate effects on BSS usages are more important on highly infected boroughs than in low infected one. With a similar goal of analyzing traffic-bike change during the lockdown, [START_REF] Büchel | Covid-19 as a window of opportunity for cycling: Evidence from the first wave[END_REF] uses data collected through 13 and 20 bike counters respectively in Basel and Zurich (Swiss). The daily cycling traffic is estimated by Random Forests, separately for the working-days and the weekends. Both models depend on weather, hour of the day and whether the day is a holiday (school or public). Among the main findings, the daily traffic during the lockdown is acknowledged to be more complicated than just a mix between pre-lockdown working and weekend days traffic patterns. It is also found that the bike traffic during the second COVID wave follows the same pattern observed during the first one. After the first lockdown, bike usage increases more in Zurich than in Basel although cycling culture is much better in Basel than in Zurich.

The study reported in [START_REF] Hu | Examining spatiotemporal changing patterns of bike-sharing usage during covid-19 pandemic[END_REF] is very similar to ours. Instead of a log-linear regression model, the authors used Generalized Additive Models (GAMs) for comparing a regular BSS usage between March and July 2019 in Chicago with a pandemic BSS usage for the same period in 2020. They control consider weather conditions, seasonality and holidays effects. Like previous studies, they observe that trips are longer in 2020 than in 2019, and bike-sharing is more resilient than other transport modes. They also show that pandemic impact on bike sharing has a differential impact depending on socio-demographic patterns around each dock-station.

[SCB + 21] proposes a novel method to calculate bikers' trajectories and estimate the environmental benefits during COVID-19. The exact route taken by the cyclists is not trivial to infer. With the same method as the one proposed in [SCB + 21] for Beijing data, we used OSMnx and OpenStreeMap to calculate shortest paths between origins and destinations for Toulouse BSS data. Checking the "bike" option in OpenStreeMap has been observed to be a bad solution, leading to many highly improbable speeds. Checking the "pedestrian" option proved instead to be a much better approach. It seems to indicate that Toulouse BSS riders are taking shortcuts, not necessarily referenced as cycle routes. In a still ongoing survey in Toulouse, we have asked 125 riders to describe their route and the results show that 88% match well with the "pedestrian" option of OSMnx but the remaining 12% does not follow clear logics. Because of this difficulty, in the following we will not use any hypotheses on the route.

In this paper, we propose a spatio-temporal analysis of COVID-19 effect on BSS usage in Toulouse, France which was among the first cities in the world to implement automated BSS and, therefore, where the use of BSS is well established. For this purpose, we consider 2019 as a baseline year against which we can compare corresponding periods in the following years 2020, 2021 and 2022. To our knowledge, this is the first study of this type on French cities. In contrast to many previous papers, we use several level of data aggregation (week, day, 10 minutes) to improve the accuracy of the results. After describing the data (Section 3), the following section (Section 4) provides general statistics and shows that the number of bikes leaving or returning a dock-station is very well modeled by a log-linear function of time variables and of the rain level. Section 5 uses matrix factorization to highlight the presence of latent explanatory factors, which can for instance depend on the spatial distribution of the stations on the territory, and study their dynamics before, during and after the COVID pandemic. Calculations were made using R software and python language, and maps were produced with QGIS software.

Data description and context

In Toulouse, a dock-based Bike-Sharing System (BSS) has been installed by JC Decaux in 2007 who shared the anonymous data of all the Origin/Destination trips for the years 2019 to 2022, excluding service operations. Each row of the dataset includes departure and arrival times at a one-minute precision as well as the origin and destination stations. The year 2019 will be considered as a "normal" year, as opposed to 2020 when France faced the onslaught of the first wave of COVID-19. For each year, a record of the dataset contain the positioning and timing information of locking and unlocking of bikes, excluding that of rebalancing operations. Because data are anonymized, it is not possible to follow a user in his/her daily use of BSS. We removed trips with duration less than or equal to 2 min with identical origin and destination, or with a duration more than 12 hours regardless of origin or destination. Following these filters, the data contain 287 different dock-stations with no change between 2019 and 2022 and a total number of trips of 3.83 million in 2019, 2.93 million in 2020 respectively (corresponding to a 23% decrease), 3.45 million in 2021 and 3.95 million in 2022. As expected, the density of dock-stations is higher in the city center (Figure 1). The traffic change between 2019 and 2020 is not homogeneous among the dock-stations: while the city center shows a high decrease of uses, more periphery and residential dock-stations are less impacted and can even show an increasing usage (Figure 1). These observations are consistent with [CGXJ20, HXLZ21, Kim21] observations in Beijin, Chicago and Seoul respectively, but in contrast with [HCC + 21] results in Nanjing (China).

Weather data from the Blagnac station (closest to Toulouse city-center) were provided by Météo-France at a hourly basis, but are only available for the years 2019 and 2020. The "rain level" variable has the highest negative correlation with BSS usages, which is consistent with previous findings [START_REF] Bean | How does weather affect bikeshare use? a comparative analysis of forty cities across climate zones[END_REF].

Regarding the development of the pandemic, the dynamics of intensive care hospitalizations1 peaked at the end of March 2020, followed by a strong during the 3 summer months (June-July-August) and then rised again at the end of September to reach the peak of the second wave around November 10, 2020. We provide below a short timeline of the main dates of policy interventions in France relative to the pandemic situation. The proposed model, in the spirit of [START_REF] Hu | Examining spatiotemporal changing patterns of bike-sharing usage during covid-19 pandemic[END_REF], is as follows

BSS ≈ f (hour of day, type of day) × g(week) × h(holidays) × ℓ(level of precipitation), (1)
The functions f , g, h and ℓ are estimated non-parametrically via a linear model on the logarithm of BSS, with all variables treated as factors. The amount of precipitation was only available for the years 2019 and 2020 and, while it is highly significant in the model these two years, it only accounts for less than 0.5% of explained variance. This variable is removed from the model for the years 2021 and 2022.

According to the model, holiday induces a mean decrease of 25.65% in BSS use. Similarly, the weather is responsible for a 23.1% decrease in BSS use for a Medium level of precipitation and for a 26.2% in case of a High level, the two coefficients being highly significant. Nevertheless, the overall periodic time components (time of the day, type of day and week) accounts for more than 96% of explained variance, compared to less than 1% for holidays and precipitations.

Since the data are measured every 10 minutes, the estimation of the first function f accounts for 144 parameters associated to the typical BSS use distribution during a working day (one for every 10 minute segment) and 144 parameters for the distribution over a weekend day. The remaining parameters correspond to the weekly evolution along the year (53 values) represented by the function g, an additional parameter adjusting for the 11 holidays of the year and 3 parameters for the three levels of precipitations. Accounting for identifiability, we reach a total of 343 parameters for 52217 degrees of freedom for 2019, 2021 and 2022, and 52361 for the leap year 2020. The R 2 in the linear model varies between 0.97 and 0.99 for the four years considered. Nearly all parameters are highly significant (p-value < 10 -16 ) with the very few exceptions corresponding to night values with very low BSS use. Figure 2 show the values of the function f of Equation (1) for working days and weekend days, as fitted by the model for each year. The very high R 2 in each model suggests that these distributions are representative of the patterns of BSS use over the course of one day throughout the year, after correcting for holidays (and meteo conditions for 2019 and 2020). These curves are normalized so as to have a unit underneath area, so as to represent the distribution of BSS use over the day, independently of the total amount of BSS use, which is represented in Figure (4) as a weekly variation throughout the year.

BSS use distribution during a working day in Toulouse

These distributions strongly suggest that the changes in BSS use caused by the pandemic took two years to go back to original, both for working days and weekends. Indeed, after a transition during the year 2021, whose pattern goes even further away than the year 2020 of the pandemic, the year 2022 returns to a nearly exact replica of 2019. To quantify the proximity of the different periods of time, we use the L 1 distance between each pair of functions (normalized so as to have an area under curve of one). We measure a L 1 distance of around 0.02 between the years 2019 and 2022 compared to 0.10 between 2019 and 2021 and 0.17 between 2019 and 2022. Similarly for weekend days, the L 1 distance to the year 2019 preceding the pandemic are around 0.13 for 2020, 0.23 for 2021 while only 0.03 for 2022.

For working days, the four visible traffic peaks occur at nearly the exact same time each year, namely around 8:45 AM, 12:15 PM, 1:45 PM and 6:15 PM. We observe nonetheless a change in behavior in 2020, with a small decrease of the 8:45 AM peak compensated by a higher and more spread out use during the late morning and the beginning of the afternoon compared to 2019. This change is amplified in 2021 before going back to the pre-pandemic standard in 2022.

The decrease of BSS use in the morning rush during the pandemic situation was observed in previous studies but often more pronounced [HXLZ21, BMC22, KKH21, LZHA20]. In these papers, the morning peak completely disappears during the pandemic, contrary as what we observe in Toulouse when focusing on the lockdown periods, see Figure 3. The relative increase of BSS use we observe around noon exists in some of these papers but a normalization of the data would be necessary to highlight them. What is surprising, however, is the absence of two peaks around noon in all the previous studies. It may reflect different habits, or just the fact that these studies consider data with a one-hour precision, whose effect is to aggregate the two peaks. In spite of the high quality of fit, Figure 3 shows high variations of BSS use over the course of a day in 2020 during the two lockdowns. The left graph show that the heavy restrictions of the first lockdown in France almost completely obliterated the BSS use. Nevertheless, three of the four peaks are still discernible and occur at the exact same times, around 8:45 AM and 6:15 PM. The impact of the second lockdown was much less noticeable than the first one with a similar day pattern as usual and only a moderate decrease in volume. It is interesting to notice that, with or without COVID, the morning peak for working-days is, for this period of the year, preceded by a smaller one around 8 AM. This small peak, somehow present in the March-May period in 2019, reveals a bimodal distribution which probably deserves to be studied in more detail. Curves are normalized so that the total use over the year is equal to the observed value. The lockdown periods in 2020 are represented in gray shading (17 march to 11 may and 30 october to 15 december).

Figure 4 represents the weekly components of the model of Eq. (1) for the use of shared bikes in Toulouse for each year between 2019 and 2022. Similarly as for the hourly component within a day, these differ from the raw data of weekly use in that the weather and holidays effects of 2019 and 2020 are corrected, even though they account for less than 1% in the explained variance. As for the hourly usage, the year 2021 appears as a transitional period before returning in 2022 to a pattern very similar to 2019. A refined model allowing different weekly behaviors for working-days and weekends (i.e. with a component g(week, type of day) instead of g(week) in Eq. ( 1)) was tested. The additional parameters in this model were not statistically significant, which suggests that the weekly variations in BSS traffic on working-days and weekends were roughly proportional.

Figure 4 allows the comparison with two other types of urban transport, private cars and Public-Transport. Please note that for FCD data, all the urban territory is under consideration which contains the urban highways around the city. The first observation is that the summer period is clearly less impacted. Public transport's resilience to the pandemic is particularly bad since it remains largely underused until autumn 2020, with patterns for the weeks following the first lockdown which are similar to those during the second lockdown. The decrease is about 40% in 2020 compared to 2019, which is similar for example to observations in Shenzhen that show a decrease of 34% [START_REF] Rojas | Built environment, transport, and covid-19: a review[END_REF]. With a latency of one month as for BSS, the road-traffic volumes (as represented by the observed fleet of vehicles in the available FCD data) return to stable levels (out of summer) better than those of Public-Transport and around 85% of the 2019 levels. The order highlighted here regarding the impact of the pandemic on the three modes of transportation is consistent with previous findings [START_REF] Monterde-I Bort | Mobility patterns and mode choice preferences during the covid-19 situation[END_REF].

Main spatio-temporal patterns of hourly BSS uses

In this section, we use Exploratory Factor Analysis (EFA) [START_REF] Jolliffe | Principal Component Analysis and Factor Analysis[END_REF] to investigate the spatio-temporal effects of the pandemic on BSS usages. EFA is an unsupervised data compression technique that allows learning from data a small number of latent factors that meaningfully summarize the covariance matrix of the observed features (i.e., variables). In the model, the variables correspond to the p = 287 dock-stations in Toulouse, for which we observe the amount of trips that begin at a given station in a given one-hour slot. Therefore, for each station, we have a total of q = 35087 = 24 × (365 + 366 + 365 + 365) observations. Each one hour time slot represents a population sample point for which all variables are observed. Unlike the previous analysis in which we have distinguished among working-days, weekend-days or periods of the year on the basis of prior knowledge of relevant factors, we now assume to not have such preliminary knowledge, and thus aim to learn directly from the data the factors that can best explain existing similar temporal patterns between the different stations.

In the fundamental equation of the EFA model, observed variables are expressed as linear combinations of a given (fixed) number of unknown common factors, plus an additive term that can be considered as noise, or, in EFA terminology, as the uniqueness associated to each observed variable. Based on such principles, EFA fundamentally differs from Principal Component Analysis [START_REF] Joost | Common factor analysis versus principal component analysis: a comparison of loadings by means of simulations[END_REF] in the sense that factors are constructed by assuming that the covariance of the observed variables could be decomposed into two parts, i.e., a common and a unique one, while in PCA all the variance in the observed variables is analyzed and used for the determination of the components. Hence, EFA focuses on the interpretation of the relationships among the variables and on identifying the latent factors that explain such relationships as noisily observed in the data covariance matrix.

Following the usual EFA terminology, we call (common) factors the set of K latent "variables" that enters the construction of the observed variables and factor loading matrix the set of numerical relationships describing how much each factor explains each (observed) variable. The fundamental equation of factor analysis is

X = ΛF + U
where X is the p-vector of observed variables, F is the K-vector of unknown factors, Λ is the unknown p × K matrix of factor loadings and U is the unknown p-vector of error variables (uniquenesses) associated to each of the p observed variable. Because of the normalization, the loadings can be interpreted as the correlations between the associated factor and the observed variables. More details on the EFA model, the solving procedure to compute factor loadings and uniquenesses, as well as approaches to select the number of K factors and to estimate factor scores can be found in the well-established literature on the subject [START_REF] Mulaik | Foundations of Factor Analysis[END_REF][START_REF] Andrew | A first course in factor analysis[END_REF][START_REF] Karl | Structural analysis of covariance and correlation matrices[END_REF].

Factors extraction was performed using the Minimum Residuals (MINRES) approach [START_REF] Harry | Factor analysis by minimizing residuals (minres)[END_REF]. The working principle of MINRES is to minimize the sum of off-diagonal squared residuals of the correlation matrices of X. An important design choice concerns the number of common factors that EFA should target. We rely on Parallel Analysis [START_REF] John L Horn | A rationale and test for the number of factors in factor analysis[END_REF], which uses the eigenvalues of the observations correlation matrix as rough estimates of the actual common factors. The presence of common factors shall induce large eigenvalues: the number of factors is set to the lowest rank above which all data eigenvalues are larger than those from the uncorrelated variables. Finally, once a set of factors is extracted, it is usual to perform a rotation in order to produce a more interpretable and simplified solution [START_REF] Abdi | Encyclopedia for Research Methods for the Social Sciences[END_REF]. Varimax rotation over BSS data worked best in that respect. It is an orthogonal rotation (which thus keeps factors uncorrelated) that maximizes the sum of the variance of the squared loadings.

In this study, Parallel Analysis yielded 23 factors sorted in decreasing order of explained variance. These account for 41.5% of the total variance, of which 82% is due to the first three factors. Lower-ranked factors are either restricted to a handful of stations or concentrated in a very small time period (e.g. a particular day). The following analysis has been performed using the departures data only. A similar one on the arrivals has been conducted, producing identical conclusions. Therefore, only the data from departures are presented to avoid redundancy.

The three main factors obtained from the EFA model, as well as the sixth factor, exhibit a strong periodic daily behavior that can be associated to a particular time of the day, as shown in Figures 5 and7.

is the only factor that shows significant scores for this morning peak and for a large number of stations, as seen in Figure 6. This factor is mainly associated with departures from residential areas. The second factor, which accounts for ≈ 11.8%, appears to be representative of the afternoon BSS use in relation to work places during the five working-days. Contrary to the morning peak, the afternoon traffic seems to have disappeared completely during the few months following the first lockdown. This phenomenon is less pronounced during the period following the second lockdown. The second factor corresponding to the afternoon traffic is the only one that scores positively for workingdays and negatively for weekend-days. The high loadings are located almost exclusively in the city-center and the university areas, highlighting the traffic from people going back home from work. The third factor, visible in Figure 7 (left), is highly representative of the night-time activity in Toulouse. It shows daily peaks during the evenings and early nights, which are more pronounced in the middle of the week (say from Wednesday to Saturday) compared to the other days, while the loading map in Figure 8 pinpoints bike stations nearby areas with high concentration of bars, restaurants and other leisure activities such as the place du Capitole, place Saint Georges, place Esquirol or place Saint-Pierre. The curfew imposed between 9PM and 6AM starting from mid-october is clearly visible. In particular, for these last two weeks of October, we observe a peak of traffic between 8PM and 9PM, just before the curfew, probably due to people using bikes to come back home. The sixth factor is concentrated in the three stations Saint Aubin Camichel, Victor Hugo and Colombette where the sunday marketplace may explain the peak in activity in these locations. The marketplace was closed during the first lockdown but remained allowed during the second lockdown, as is clearly visible in Figure 7.

Finally, the forth and fifth factors are concentrated in a few stations and reveal a change in activity during the considered period, due to e.g. renovations of the Matabiau train station that lasted until the end of 2021.

Conclusion

In this paper, we investigate the evolution of Bike-Sharing System (BSS) usages in Toulouse during the COVID-19 pandemic situation. Firstly, we compare the evolution of the amount of rental bikes during the four-year period between 2019 and 2022, with a 10-minutes time granularity. In this analysis, data from all the dock-stations have been aggregated, thus ignoring any spatial considerations. The results show that these time series can be very well explained as the product of two main effects: the annual tendency and the daily evolution. Our multiplicative regression model shows very good fitting scores, taking into account schoolholidays and precipitations. Secondly, we use Exploratory Factor Analysis (EFA) to derive spatio-temporal patterns in the BSS usage over the four years. In the data, dock-station spatial granularity is preserved, thus permitting to observe and interpret, via the retrieved latent factors, the dynamics at the level of dock-stations instead of the entire network as in the first study. EFA highlights five and six latent variables, globally explaining around 50% of the variance.

Road traffic and Public-Transport were much more impacted than BSS. None of them appears to recover the pre-COVID level. After the first lockdown, with the same delay of about 1 month and until the second lockdown, with the exception of summer holidays, the road-traffic stabilizes at approximately 85% of the 2019 values for both towns. The traffic flow of public transport has the same dynamic in both cities, recovering a maximum of 60% of its value after the first lockdown and 75% of its value after summer holidays. This is consistent with previous findings in other cities. We found that trips duration significantly increased during the pandemic period for working-days but not for weekend-days where the reverse occurs.

The Explanatory Factor Analysis exhibited four strong periodic behaviors in BSS usage. The morning peak activity associated with departures from residential areas to work, appears as the main factor in the analysis, followed by the afternoon traffic present in areas with high density of business and activity centers in a large portion of the Toulouse city center. The Toulouse night-time activity arises as the third stronger component of the analysis. Although the periodic nature of the time component, whether it concerns the daily, weekly or yearly evolution, are not included as preliminary knowledge in the EFA model, it does appear very distinctively in each factor and with high discrepancy from one factor to another. This shows the relevance of the methodology on this particular dataset.

Our results globally show that BSS appeared as a resilient mobility alternative during the pandemic situation. After a notable transitional year in 2021, BSS usage seemingly returned to the pre-pandemic standards in 2022.

Figure 1 :

 1 Figure 1: Locations of bike-sharing docks in Toulouse. Each station is colored according to the percentage of traffic increase/decrease from 2019 to 2020. To avoid bias in case a station is out of order for a given period, the corresponding periods of both 2019 and 2020 were dismissed in the calculation of the percentage.

4

  r t i a l r e -o p e n i n g o f s t o r e Time modeling for citywide bike sharing use BSS use is measured every 10 minutes over the course of the years 2019 to 2022 yielding a sample of size 365 × 144 = 52560 for 2019, 2021 and 2022 and 52704 for the leap year 2020.The data consist of the raw number of trips over the whole city initiated in each 10 minute segment, without any pre-processing. Temporal bike usage is modeled as a multiplicative function of time and amount of precipitation. The time component accounts for the hour of day (among the 144 10 min segments per day), the type of day (working or weekend day), the week of the year (1 to 53) and holidays. The precipitation variable is divided into three levels according to the duration of rain within the hour: Low (less than 20 min), Medium (between 21 and 40 min) and High (more than 41 min).

Figure 2 :

 2 Figure 2: Distribution of the BSS use over the course of a working-day and weekend-day in Toulouse, as fitted by the model.

Figure 3 :

 3 Figure 3: Comparison of the average BSS use during a working-day Toulouse during the two 2020 lockdown periods.

Figure 4 :

 4 Figure 4: Weekly evolution of BSS use over the course of the years 2019 to 2022 in Toulouse as fitted by the model.

  apr may jun jul aug sep oct nov dec

Figure 6 :

 6 Figure 6: Geographical view of the loadings associated to the morning peak of the first factor (left) and the afternoon peak of the second factor (right).

Figure 7 :

 7 Figure 7: Visual representation of the third factor (left) highlighting the Toulouse night-life, and the sixth factor (right) mainly concentrated on Sundays around noon.

Figure 8 :

 8 Figure 8: Geographical view of the loadings for the third factor (left) which depicts the Toulouse night-life, and the sixth factor (right) representative of the Sunday activity.

Figure 10 :

 10 Figure 10: Geographical view of the the forth (left) and fifth (right) factor with a loading distribution concentrated in two or three stations, including Matabiau train station in both cases.

https://www.data.gouv.fr/fr/datasets/donnees-hospitalieres-relatives-a-lepidemie-de-covid-19/

2019-Sep 2019-Oct 2019-Nov 2019-Dec 2020-Jan 2020-Feb 2020-Mar 2020-Apr 2020-May 2020-Jun 2020-Jul 2020-Aug 2020-Sep 2020-Oct 2020-Nov 2020-Dec 2021-Jan 2021-Feb 2021-Mar 2021-Apr 2021-May 2021-Jun 2021-Jul 2021-Aug 2021-Sep 2021-Oct 2021-Nov 2021-Dec 2022-Jan 2022-Feb 2022-Mar 2022-Apr 2022-May 2022-Jun 2022-Jul 2022-Aug 2022-Sep 2022-Oct 2022-Nov 2022-Dec 1

2019-Jan2019-Feb 2019-Mar 2019-Apr 2019-May 2019-Jun 2019-Jul 2019-Aug 2019-Sep 2019-Oct 2019-Nov 2019-Dec 2020-Jan 2020-Feb 2020-Mar 2020-Apr 2020-May 2020-Jun 2020-Jul 2020-Aug 2020-Sep 2020-Oct 2020-Nov 2020-Dec 2021-Jan 2021-Feb 2021-Mar 2021-Apr 2021-May 2021-Jun 2021-Jul 2021-Aug 2021-Sep 2021-Oct 2021-Nov 2021-Dec 2022-Jan 2022-Feb 2022-Mar 2022-Apr 2022-May 2022-Jun 2022-Jul 2022-Aug 2022-Sep 2022-Oct 2022-Nov 2022-Dec

2019-Jul 2019-Aug 2019-Sep 2019-Oct 2019-Nov 2019-Dec 2020-Jan 2020-Feb 2020-Mar 2020-Apr 2020-May 2020-Jun 2020-Jul 2020-Aug 2020-Sep 2020-Oct 2020-Nov 2020-Dec 2021-Jan 2021-Feb 2021-Mar 2021-Apr 2021-May 2021-Jun 2021-Jul 2021-Aug 2021-Sep 2021-Oct 2021-Nov 2021-Dec 2022-Jan 2022-Feb 2022-Mar 2022-Apr 2022-May 2022-Jun 2022-Jul 2022-Aug 2022-Sep 2022-Oct 2022-Nov 2022-Dec 1

2019-Jan2019-Feb 2019-Mar 2019-Apr 2019-May 2019-Jun 2019-Jul 2019-Aug 2019-Sep 2019-Oct 2019-Nov 2019-Dec 2020-Jan 2020-Feb 2020-Mar 2020-Apr 2020-May 2020-Jun 2020-Jul 2020-Aug 2020-Sep 2020-Oct 2020-Nov 2020-Dec 2021-Jan 2021-Feb 2021-Mar 2021-Apr 2021-May 2021-Jun 2021-Jul 2021-Aug 2021-Sep 2021-Oct 2021-Nov 2021-Dec 2022-Jan 2022-Feb 2022-Mar 2022-Apr 2022-May 2022-Jun 2022-Jul 2022-Aug 2022-Sep 2022-Oct 2022-Nov 2022-Dec 2

2019-Jul 2019-Aug 2019-Sep 2019-Oct 2019-Nov 2019-Dec 2020-Jan 2020-Feb 2020-Mar 2020-Apr 2020-May 2020-Jun 2020-Jul 2020-Aug 2020-Sep 2020-Oct 2020-Nov 2020-Dec 2021-Jan 2021-Feb 2021-Mar 2021-Apr 2021-May 2021-Jun 2021-Jul 2021-Aug 2021-Sep 2021-Oct 2021-Nov 2021-Dec 2022-Jan 2022-Feb 2022-Mar 2022-Apr 2022-May 2022-Jun 2022-Jul 2022-Aug 2022-Sep 2022-Oct 2022-Nov 2022-Dec

† This research is party funded by the ANR projects ANR-21-COVR-0027-01 and the LABEX SMS ANR-11-LABEX-0066.

The first factor, which accounts for around 14.4% of the total explained variance, displays a dominant morning peak for working-days during the four-year period with the sole exception of the first lockdown. It


est destinée au dépôt

Multicentre validation of a EUCAST method for the antifungal susceptibility testing of microconidia-forming dermatophytes

Maiken Cavling Arendrup, Karin Meinike Jørgensen, Jesus Guinea, Katrien Lagrou, Erja Chryssanthou, Marie-Pierre Hayette, Francesco Barchiesi, Cornelia Lass-Flörl, Petr Hamal, Eric Dannaoui, et al.

To cite this version:

Maiken Cavling Arendrup, Karin Meinike Jørgensen, Jesus Guinea, Katrien Lagrou Results: 100%/96% (voriconazole) and 84%/84% (itraconazole) MIC determinations fell within the QC ranges for the two QC strains, respectively, and 96%/92% terbinafine MICs fell in a 0.25-1 mg/L 3 twofold-dilution range suggesting a high interlaboratory reproducibility. Across the six methods, the number of terbinafine MEs varied from 2 (2.6%) to 5 (6.6%) for T. rubrum and between 0 and 2 (2.0%) for T. interdigitale (lowest for the CC-method (2.6%-4.4%/ 0-1% for T. rubrum/T. interdigitale). The difference between the modes for the wt and mutant population were ≥7 two-fold-dilutions in all cases (Table ). If excluding a I121M/V237I T. rubrum mutant, and two mixed T. interdigitale strains, the number of VMEs were CC visual: T. rubrum: 1/77 (1.3%), CC spec-90%: 3/68 (4.4%) and CC spec-50%: 1/76 (1.3%), and none for T. interdigitale. The activity of voriconazole, itraconazole and amorolfine were quite uniform against T. rubrum and T. interdigitale, but unacceptably wide MIC ranges were found for the visual and spec-90% inhibition methods for itraconazole (data not shown).

Conclusions:

Although none of the laboratories perform dermatophyte testing at a regular basis an acceptable interlaboratory agreement and good separation between SQLE wt and mutants were found, suggesting a robust performance of the proposed method.

03950328 ECCMID 2020 Abstract Multicentre validation of a EUCAST method for the antifungal susceptibility testing of microconidia-forming Dermatophytes.

  Terbinafine resistance is increasingly reported in Trichophyton rubrum and

	Maiken Cavling Arendrup	1,2,3 , Karin Meinike Jørgensen	1 , Jesus Guinea	3,4 , Katrien Lagrou	5 , Erja
	Chryssanthou	6 , Marie-Pierre Hayette	7 , Franchesco Barchiesi	8 , Cornelia Lass Flörl	9 , Petr Hamal 10 , Eric
	Dannaoui 11 , Anuradha Chowdhary 12 , Joseph Meletiadis
	Trichophyton interdigitale rendering susceptibility testing important particularly in non-responding
	cases. We performed a multicentre evaluation of a recently proposed modified EUCAST method
	implementing medium supplemented with chloramphenicol and cycloheximide (CC) to avoid
	contamination.

, Erja Chryssanthou, et al.. Multicentre validation of a EUCAST method for the antifungal susceptibility testing of microconidia-forming dermatophytes. Journal of Antimicrobial Chemotherapy, 2020, 75 (7), pp.1807-1819. 10.1093/JAC/DKAA111. hal-13 Background: Materials/methods: A blinded panel of wild-type and squalene epoxidase (SQLE) target gene mutant T. rubrum and T. interdigitale strains were distributed to 10 European laboratories. Susceptibility to terbinafine, itraconazole, voriconazole and amorolfine) were performed according to the E.Def 9.3.1 method with and without addition of chloramphenicol and cycloheximide (final concentrations 50 mg/L and 300 mg/L, respectively). Plates were incubated at 25 °C (one laboratory used 30 °C) for 5-7 days until sufficient growth. MICs were determined visually (ignoring trailing growth for itraconazole) and spectrophotometrically with 90% and 50% endpoints yielding a total of 7,829 MICs. A. flavus ATCC 204304 and A. flavus CNM-CM1813 were included as controls.


Pandit Kulkarni's Astrological

Michelle Pfeffer

David Zeitlyn

Pandit About

Kulkarni

Emmanuel Grimaud

ASTRO-SKELETONICS: ABOUT

In India, astrology is called jyotish (light). The contexts in which such light is necessary to compensate for our lack of vision are so varied that a full life wouldn't be enough to document its techniques and forms, past or present. From one consultation to another astrologers help their clients to cultivate an astral matrix, invisible yet luminous. Astrologers reveal the movements of this dense, bright matrix, and study its whims, its intimate rhythms in order to understand, optimize and multiply the possibilities of action. Within an astral matrix which has its own periodicity, they help identify the friendly influences as well as the blockages and conflicts due to hostile influences. An 'interactive decision theory' expands radically when the stars are included, recognised here as partners in their own right with whom one must deal. The astral matrix provides a whole set of behavioral entities with which one must co-ordinate for a given time in order to increase the chance of a gain (or from which one must protect oneself).

In India, it is possible to cast the astrological chart of almost any object on earth: human, animal, temple, house, project and even a country, an institution, or the nation. Nothing escapes the temperamental choreography of the stars.

Astrologers may be consulted to make decisions based on prior knowledge or to neutralize the influence of deleterious planets, for which there are always remedies (rituals, etc.), according to the astrologer's prescriptions. Much more than this: some are true researchers, exploring new techniques, resurrecting old ones and thus renewing their discipline through contact with new tools or media. Such is the case of the astrologer Pandit Kulkarni (1930Kulkarni ( -2019)).

The son of an astrologer himself, Pandit Kulkarni worked for many years as a draughtsman on archaeological excavations. He was known for his reconstructions of human skulls and his geological drawings. His landscapes showing the rocks beneath were animated by a unique inner tension, giving the impression of cosmic and telluric forces in action. He also drew portraits of people, sometimes even without having seen them, based on their birth charts. He left behind hundreds of portraits that are the traces of a fluid research process, on the borders of archaeology, physical anthropology, art and divination. His clientele was mainly made up of women, including sex workers and, actresses, but politicians, mafiosi and gangsters also followed his predictions. His secret wish was to compose an astrological encyclopaedia of mankind. The Pandit became probably one of the greatest portraitists that modern India has ever known, but his work has been largely ignored.

The Pandit used terms such as astro-morphology, astro-skeletonics and robot-portrait to talk about his work. He drew parallels between the morphology of rocks, landscapes, human evolution and the ballet of the stars. Above all, he offered people an innovative predictive support: robot-portrait integrated astral influences right into the face. The face, the skull and the horoscope were considered the photographic negatives of each other, as the skeleton was seen as being tied to the movement of the planets. There is a difference between the working portraits kept by the Pandit and the ones made for his clients, drawn during the consultation. The portraits made for the clients follow the same format, with the face on the front and the underlying skull on the back of the page. The working portraits kept by the astrologer display the skull and face on the same page and are usually accompanied by more personal and often poetic comments, sometimes written in the future tense when they are about the person's past or switched to the past tense when giving predictions for the future.

As well as working with clients in person the Pandit also made conjectural horoscopes based on famous figures or faces reconstructed from cranial remains. This could involve studying the faces of historical figures when portraits survived but whose birth chart was unknown. Yet he also reconstructed faces (associated with possible horoscopes) from archaeological skulls, changing his method depending which factor was unknown (face, skull or horoscope).

"No one can see himself as he is, even in a mirror. He meets only his inverted image," he said. Astrology, on the other hand, can. But what does it reveal? About a woman with a 'good reputation in intellectual circles', he wrote: "Fortune, Jupiter and Venus are in high position in the northwest quadrant. As a result, she will only achieve well-being after 24 years. After June 2002, Rahu will be in Gemini and will destroy the enemy completely and Saturn in Pisces will finish defeating her enemy... Because of the influence of Jupiter and Venus, she enjoys a rare beauty, a magnificent figure and body. Ketu in Pisces gives her dense hair and large eyes. The god of education and the enemy god are in the position to influence knowledge. The position of the Husband is aligned with that of Knowledge in the horoscope. She is likely to marry someone she meets at her workplace. Her boss will also be her husband's boss. However, before that, she will make a serious mistake and to correct this mistake, she will want to get married. But in the meantime, she will have been cheated because the shadow of Death will already be looming over her body."

For the Pandit each face was a pluriverse, a battlefield where entities-divinities-planets confront each other, a zone of conflicting and passionate tension crossed by multipolar influences, never the play of a single power. Any concretion, whether a face or a landscape, perseveres in its "fragile cohesion" between fluctuating forces in tension. Thus a man wanted by the police gives rise to the following portrait: "Man on the run, date 28-10-1993. The god of death, Venus, is opposite the god of Intelligence and he emerges in Virgo placed in low position. The consequence of this configuration is that his mind has been filled with a desire for emancipation but his enemies have taken over his body." About a woman, known to stir up the passions of men: "Date of birth, 19 November 1977 dawn, 6:15 Birth and marriage horoscope: Arrogant beauty. Pretentious woman with eyes capable of arousing violent male passions. Nature could only have given birth to such a creature in a moment of distraction. The Body is influenced by the Sun and Budh, the Intelligence, influences her face, the suppleness of her structure and her fragile cohesion. Hence also her bright milk complexion of magnetic power, her strong bones and the solid structure of her muscles influenced by Budh.

As for her beauty, she owes it to Venus and the influence of Libra."

The portrait operates a rise in the presence of astral forces by offering a paralyzed image of the person, freezing the grip of these forces. Bringing the subject face to face with pure forces, the portrait makes visible the inescapable current that makes a man "a gun", the instrument of an interstellar war happening beyond his consciousness. Thus, the face of a gangster is commented on as follows :

"A good portrait is one that makes you feel the danger of a person and the risk you take when you approach him. Look at the face of this merchant. He is frightening and as dangerous as a hydrogen bomb. An explosive man!" This is very different from nineteenth century phrenology in which the inclinations and passions are buried inside the cranium. The Pandit, by contrast, relocates these inclinations from the 'depths of the human soul' (as Gall would say 1810) to the depths of the cosmos. The deeper one goes into the interiority of a being, the more one opens up to the life of the cosmos.

How does astrology connect resonances to the realm of passions? In Kulkarni's astro-morphology each trait (laksana) includes several possible interpretations that increase the possibilities of 'sounding right'. Astrology then becomes a ballistics of passions through the medium of portraiture. A person is always a configuration, the relationship of relationships, a constellation. A body is unique from an astrological point of view, because it is located at the meeting point of forces that sometimes attract, repel or neutralize each other. By 'passion', we must understand a particular form of crystallization: humans continue to circle but at the point of falling out of orbital movements which, when they encounter living matter, create passion.

It is worth noting that the reasons people consulted the Pandit were varied and that he used the birth chart, a facial examination and a palpation of the skull in combination. Clients could contemplate their deepest tendencies in portraits. The astral portrait, once seen by the subject, was intended to gradually replace their own self-image. Superimposing the skull and face then completed the astromorphosis, putting the person back into the field of the astral forces at work, revealing to them the cosmic drama that was being played out within and through them. If each portrait was destined to appear in an astral encyclopaedia of evolution, alongside rocks, plants and great apes, Kulkarni's drawings published in L'Etrange Encyclopédie du docteur K. (2014) are the only remaining testimonies of the stellar history of mankind he had in mind.