Konstantin Avrachenkov

Laura Cottatellucci

Arun Kadavankandy

Spectral Properties of Random Matrices for Stochastic Block Model

Keywords: Spectral Graph Theory, Random Graphs, Random Matrices, Stochastic Block Model * Eurecom, France Carectéristiques Spectrales des Matrices Aléatoires Theorie spectrale de graphes, Matrices Aléatoires, Graphes Aléatoires, Stochastic Block Model

We consider an extension of Erdős-Rényi graph known in literature as Stochastic Block Model (SBM). We analyze the limiting empirical distribution of the eigenvalues of the adjacency matrix of SBM. We derive a fixed point equation for the Stieltjes transform of the limiting eigenvalue empirical distribution function (e.d.f.), concentration results on both the support of the limiting e.s.f. and the extremal eigenvalues outside the support of the limiting e.d.f. Additionally, we derive analogous results for the normalized Laplacian matrix and discuss potential applications of the general results in epidemics and random walks.

Introduction

Systems consisting of a huge number of interacting entities are often represented by complex networks to capture the essence of their interactions. They are used to model systems from the most disparate fields: interactions among atoms in matter, biological molecules, nodes of the Internet networks, documents in the web, social networks, connectivity of users in a wireless network, etc. Due to the typical gigantic dimensions of the systems targeted in this field, it is essential to gain understanding and master the system via few fundamental parameters which are able to characterize the macroscopic features of the system.

One important approach to study complex networks is based on the theory of random graphs. The first natural random graph model of complex networks is Erdős-Rényi graph [START_REF] Erdős | On random graphs[END_REF] where edges between nodes appear with equal probabilities. This model has many appealing analytical properties but unfortunately does not model important properties of many real complex networks. In particular, the Erdős-Rényi graph fails in describing community structures in complex networks. To mitigate this shortcoming the more refined Stochastic Block Model (SBM) has been introduced, first studied in [START_REF] Condon | Algorithms for graph partitioning on the planted partition model[END_REF] to the best of our knowledge. In SBM, the nodes are divided into subsets (communities) such that nodes within a given community have a link between them with a higher probability than nodes from different communities.

Random graphs can generate a variety of random matrices, e.g. adjacency matrix, standard Laplacian, normalized Laplacian. The spectral properties of those random matrices are fundamental tools to predict and analyze the complex network behavior, the performance of random algorithms, e.g. searching algorithms, on the network, but also to design algorithms. For example, the convergence properties of a random walk on a graph are dependent on the gap between the first and the second largest eigenvalues of its normalized Laplacian [START_REF] Chung | Spectral graph theory[END_REF]. This is of particular significance in random search algorithms deployed widely for information retrieval in big data systems. The community detection problem relies on the properties of several of the largest eigenvalues of the adjacency matrix and their corresponding eigenvectors, e.g. [START_REF] Van Mieghem | Graph spectra for complex networks[END_REF]. The cost of epidemic spread is characterized by the spectral properties of the adjacency matrix [START_REF] Bose | The cost of an epidemic over a complex network: A random matrix approach[END_REF].

Recently, there has been a stream of works on SBM in community detection problems [START_REF] Decelle | Asymptotic analysis of the stochastic block model for modular networks and its algorithmic applications[END_REF][START_REF] Heimlicher | Community detection in the labelled stochastic block model[END_REF][START_REF] Nadakuditi | Graph spectra and the detectability of community structure in networks[END_REF]. In [START_REF] Decelle | Asymptotic analysis of the stochastic block model for modular networks and its algorithmic applications[END_REF], the authors investigate detectability of communities in a SBM graph by analyzing phase transition in the spectrum of the adjacency matrix using methods from statistical physics. The authors of [START_REF] Heimlicher | Community detection in the labelled stochastic block model[END_REF] analyze a similar problem in the context of labeled stochastic matrices with two communities and provide theoretical evidence for detectability thresholds in [START_REF] Decelle | Asymptotic analysis of the stochastic block model for modular networks and its algorithmic applications[END_REF]. There, the nodes are randomly categorized into communities, with symmetric probabilities and the goal is to detect the correct community to which a node belongs. All these works focus on the case of diluted graphs, i.e. when the expected degrees are bounded irrespectively of the network size. In [START_REF] Nadakuditi | Graph spectra and the detectability of community structure in networks[END_REF] the authors also study the detectability problem in the two community case. There, the scaling law of probabilities is not clearly defined. The case of dense graphs has been studied in [START_REF] Fishkind | Consistent adjacency-spectral partitioning for the stochastic block model when the model parameters are unknown[END_REF] for M communities.

In this contribution, we analyze the limiting empirical distribution of the eigenvalues of the adjacency matrix of SBM. We find that the Stieltjes transform of the limiting distribution satisfies a fixed point equation and provide an explicit expression in the case of symmetric communities. Furthermore, we obtain tight bound on the support of the asymptotic spectrum, and concentration bounds on the extremal eigenvalues. Additionally, we derive the asymptotic spectrum of the normalized Laplacian matrix in the dense regime of edge probabilities and subsequently discuss potential applications of the general results in epidemics and random walks.

In comparison to the previous works on SBM [START_REF] Decelle | Asymptotic analysis of the stochastic block model for modular networks and its algorithmic applications[END_REF][START_REF] Fishkind | Consistent adjacency-spectral partitioning for the stochastic block model when the model parameters are unknown[END_REF][START_REF] Heimlicher | Community detection in the labelled stochastic block model[END_REF][START_REF] Nadakuditi | Graph spectra and the detectability of community structure in networks[END_REF] we consider the dense and sparse regimes when the expected degrees scale not slower than log c (n), for some c ≥ 4.

In contrast to the classical Erdős-Rényi graph, the limiting eigenvalue e.d.f. of SBM adjacency matrix does not follow the semicircular law anymore and there are M largest eigenvalues outside the support of the e.d.f. The concentration results on the extreme eigenvalues become more complex. Specifically, the result in [START_REF] Vu | Spectral norm of random matrices[END_REF], if directly applied, leads to a weaker bound on the edge of the e.d.f. support than the one we derive here. The authors of [START_REF] Decelle | Asymptotic analysis of the stochastic block model for modular networks and its algorithmic applications[END_REF][START_REF] Heimlicher | Community detection in the labelled stochastic block model[END_REF][START_REF] Nadakuditi | Graph spectra and the detectability of community structure in networks[END_REF] consider only the case of two communities and their work does not include the study the limiting e.d.f. In [START_REF] Fishkind | Consistent adjacency-spectral partitioning for the stochastic block model when the model parameters are unknown[END_REF] a general M community SBM is analyzed in the quite dense regime, i.e. when the average degree is of the order n/log(n).

Mathematical Notation and Definitions

Throughout this paper, 1 C is a C-dimensional column vector with unit entries and

J C = 1 C 1 T
C is a square matrix whose entries are all equal to 1. The notations f n = O(g n ) and f n = o(g n ) mean that lim n→∞ |fn| |gn| ≤ c for some constant c > 0 and lim n→∞ fn gn = 0, respectively. We say that f (x) dominates g(x) asymptotically and we write f (n) ∈ ω(g(n)) if asymptotically for n → +∞, |f (n)|> k|g(n)| for any constant k > 0. By δ y (x) we denote the Kronecker delta function equal to one when x = y and zero everywhere else and the ceiling function that maps a real number to the smallest following integer is denoted by • . The indicator function of a subset A of a set X is denoted by χ A (x) and, for any x ∈ X , χ A (x) = 1 if x ∈ A and χ A (x) = 0 otherwise. The field of complex numbers is represented by C, the operator (•) maps a complex number onto its imaginary part and C + denotes the open half space of complex numbers with nonnegative imaginary part, i.e., z ∈ C + , iif (z) > 0. Given a random variable X and a distribution D, the notation X ∼ D indicates that the random variable X follows the probability distribution D and P X (x) denotes its cumulative distribution function.

Given an n × n Hermitian matrix H, we index its eigenvalues in nonincreasing order and denote by λ i (H) the i-th eigenvalue of H, i.e. λ 1 (H) ≥ λ 2 (H) ≥ λ 3 (H)... ≥ λ n (H). The operator • 2 denotes the Euclidian norm of a vector when the argument is a vector and spectral norm 1 when the argument is a matrix, i.e.

H 2 = sup x 2≤1, y 2≤1 x T Hy. (1)
The empirical spectral distribution (e.s.d.) of a Hermitian matrix H, is defined as:

F H (x, n) = 1 n n i=1 δ λi(H) (x) (2)
An important tool that is used in Random Matrix Analysis is the Stieltjes transform. It is extensively used to study various properties of the limiting spectral distribution such as the limiting shape and speed of convergence. Refer to [START_REF] Bai | Methodologies in spectral analysis of large dimensional random matrices, a review[END_REF] for more details. The Stieltjes transform s F (z) of a probability distribution F (x) is defined as the Riemann-Stieltjes integral

s F (z) = dF (x) x -z (3)
for z ∈ C + . 1 The spectral norm coincides also with the induced 2-norm.

Inria 3 Stochastic Block Model and its Representations

We consider a complex network with n nodes and M communities Ω m , for m = 1, . . . , M, of equal sizes K = n/M , which is assumed to be an integer. This complex network is described by an undirected random graph referred to as Stochastic Block Model (SBM) with M blocks, one for each community. If two nodes belong to different communities, then there is an edge between them with probability p 0 (n). Given two nodes belonging to the same community Ω m , there exists an edge between them with probability p m (n), 1 ≤ m ≤ M . Throughout this paper, for the sake of conciseness, we adopt the short notation p m for the probabilities p m (n), keeping in mind that the dependence on n is implicit. For a random graph as defined above, we can define a number of related random matrices whose spectral characteristics are relevant to capture related properties of the network. In this work we focus on two classes of random matrices for the SBM: the adjacency matrix and the normalized Laplacian matrix.

SBM
   A ij = A ji ∼ B(p m ), if i, j ∈ Ω m A ij = A ji ∼ B(p 0 ), if i ∈ Ω and j ∈ Ω m , = m. (4)
We implicitly assume that the diagonal elements of the matrix A are randomly distributed according to a given Bernoulli probability distribution. This corresponds to the assumption that the random graph has cycles of unit length with a certain probability. There are definitions of complex networks that do not admit cycles of unit length, which corresponds to matrices A with diagonal elements deterministically equal to zero. It is worth noting that the results on the asymptotic spectrums of adjacency matrices in this contribution hold for both these definitions under the assumptions made.

For further studies, it is convenient to normalize the matrix A by a scaling factor2 γ(n) in general depending on n such that the support of the limiting eigenvalue distribution function stay finite and positive. Then, we consider the normalized SBM adjacency matrix A = γ(n)A and we express it as the sum of a deterministic matrix A equal to its expectation and a random matrix with zero mean random entries A, i.e.

A = A + A. ( 5
)
Consistently with the definitions in (4) and (5), A, the expectation of matrix A, is a finite rank matrix of the following form:

A = P ⊗ J K (6)
being

P the M × M P = γ(n)       p 1 p 0 . . . p 0 p 0 p 2 . . . p 0 . . . . . . . . . p 0 . . . . . . p M       . (7)
In general, for p m = p 0 and m = 1, . . . , M, the matrix P has rank M and thus, also A has rank M.

The random centered SBM adjacency matrix is also a symmetric matrix whose elements follow the distributions

C(p m , γ) = γ(1 -p m ), w.p. p m ; -γp m , w.p. 1 -p m ; m = 0, 1, . . . , M, (8)
having zero mean and variance σ 2 m = γ 2 (1 -p m )p m . Consistently, with the definitions in ( 4) and ( 5)

A ij = A ji ∼ C(p m , γ) if i, j ∈ Ω m A ij = A ji ∼ C(p 0 , γ) if i ∈ Ω and m ∈ Ω m with = m. (9)
It is worth noting that the entries of this random matrix depend of the matrix size.

Random SBM normalized Laplacian matrix L Let us define the random variable

D i = n j=1
A ij [START_REF] Decelle | Asymptotic analysis of the stochastic block model for modular networks and its algorithmic applications[END_REF] corresponding to the degree of node i. Then, the symmetric SBM normalized Laplacian matrix L is defined as

L ij = L ji =        1 - A ii D i , if i = j; - A ij D i D j , otherwise. (11)
4 Useful Existing Results

Erdős Rényi Graphs and Wigner matrices

A random graph where all the pairs of nodes have equal probability p(n) of having an edge, independently of the presence of other edges is well-known as Erdős-Rényi (ER) graph. It is straightforward to verify that an SBM graph with M = 1, corresponding to a complex network with a single community, reduces to an ER graph. As for random SBM graphs, we can consider representations of random ER graphs by classes of random matrices. In this paper, we focus on random ER adjacency matrices A ER . The upper diagonal elements of the Hermitian matrix A ER are independent and identically distributed (iid) according to B(p(n)), a Bernoulli distribution with parameter p(n), i.e.

A ER ij = A ER ji ∼ B(p(n)).
As for random SBM adjacency matrices,

Inria

we consider a matrix A ER normalized by the scalar γ

(n) = ( np(n)(1 -p(n))) -1 , i.e. A ER = γ(n)A ER ,

and decompose it as

A ER = A ER + A ER ,
where A ER = γ(n)p(n)J n and the centered ER adjacency matrix is given by

A ER ij = A ER ji ∼ C(p(n), ( np(n)(1 -p(n))) -1 ) ∀i ≥ j, i = 1, . . . n. (12)
The parameter p(n) depends on the network size n and, thus, also the average degree of a node i

d i,av = E    j A i,j    = np(n). (13)
Based on the average node degree d i,av , the ER graphs are classified as dense, if

d i,av = O(n), sparse if d i,av = o(n) and d i,av → ∞, and diluted if d i,av = O(1) [6] .
Closely related to the centered ER adjacency matrix is the Wigner matrix defined as a Hermitian matrix W whose upper diagonal entries are zero mean independent random variables with variance equal to σ 2 .

It is worth noting that the centered ER adjacency matrix is a special case of Wigner matrices with a well defined distribution

C(p(n), ( np(n)(1 -p(n))) -1
)) equal for all the entries.

The properties of the eigenvalue spectrum of both Wigner matrices and ER adjacency matrices have been intensively studied. In this section, we recall the results on the limiting spectral distributions and the spectral norms of these random matrices. Defined the empirical spectral distribution (e.s.d.) of a Hermitian matrix H of size n as in (2), the limiting spectral distribution F H (x) is the deterministic limiting distribution, if it exists, of the random e.d.f. as the size of the matrix H tends to infinity. The spectral norm of a matrix is defined in (1).

Limiting e.s.d. of Centered ER Adjacency Matrices

A key role in the convergence of the e.s.d. of large random Hermitian matrices is played by the following assumption.

Assumption 1 [15, 16, Chapter 1] The Hermitian matrix H with zero mean independent upper diagonal entries H ij of variance σ 2 ij such that lim n→+∞ sup i,j=1,...n σ 2 ij = 0 satisfies the Lindeberg's condition, i.e. for any δ > 0

lim n→∞ max i=1,...n n j=1 |x|>δ x 2 dP Hij (x) = 0. (14)
This assumption essentially implies that the tails of the distributions characterizing the random variables H ij diminish as n → ∞. Under such an assumption, the sequence of the e.s.d. converges weakly to a limiting eigenvalue distribution in the almost sure sense as stated by the following proposition.

Proposition 1 [15,16, Chapter 1] The Wigner matrix W with zero mean independent random entries W ij satisfies Assumption (1) and additionally, all the equal variances satisfy σ 2 i,j = σ 2 /n with 0 < σ 2 < +∞. Then, the sequence of the e.s.d. converges weakly to a the Wigner semicircle law in the almost sure sense, i.e. for any bounded continuous function f

f (x)F Wn (x)dx a.s. --→ f (x)µ sc (x, σ 2 )dx, RR n°8703
where F Wn (x) denotes an e.d.f. of the Wigner matrix of size n and µ sc (x, σ 2 ) is the Wigner semicircular distribution with parameter σ 2 given by

µ sc (x, σ 2 ) = 1 2πσ 2 (4σ 2 -x 2 ) + . ( 15
)
This result can be immediately specialized to normalized centered ER adjacency matrices A ER . Since for the matrix A ER it holds σ 2 ij = n -1 , for i, j = 1, . . . n, the conditions of Proposition (1) are satisfied if the limit ( 14) holds, i.e. if for any τ > 0

lim n→+∞ (1 -p)χ 1 -p ≥ τ np(1 -p) + pχ p ≥ τ np(1 -p) = 0. ( 16
)
It is straightforward to verify that this condition is equivalent to the condition p ≥ (τ 2 n + 1) -1 for any τ > 0. Then, we can state the following corollary.

Corollary 1 Let us consider a normalized centered ER adjacency matrices A ER with p(n) ∈ ω(n -1 ) as n → ∞. Then, the sequence of the e.s.d. converges weakly to a the Wigner semicircle law in the almost sure sense, i.e. for any bounded continuous function

f f (x)F A ER (x)dx a.s.
--→ f (x)µ sc (x, 1)dx.

Then, whether the e.s.d. of a centered ER adjacency matrix converges to a semi-circle distribution depends the convergence rate of p(n) to zero as n → +∞. It is difficult to state any results on the limiting eigenvalue e.d.f. when p(n) = c n because for this probability, Assumption (1) does not hold. It is known that for diluted graphs, there is no explicit expression for the limiting eigenvalue e.d.f. but it displays atoms [START_REF] Bordenave | Resolvent of large random graphs[END_REF]. For this reason in the following we limit our attention to probabilities p(n) such that p(n) ∈ ω(n -1 ), i.e. for the dense and sparse regime. This is tantamount to stating that np n (1 -p n ) → ∞ [START_REF] Ding | Spectral distributions of adjacency and laplacian matrices of random graphs[END_REF].

Spectral Norm of the Centered ER Adjacency Matrix

Let us observe that, if the multiplicity of an eigenvalue does not scale with n, the definition of the e.d.f. implies that, in the limit for n → +∞, the eigenvalue e.s.d. is not able to capture the existence of this eigenvalue in the spectrum matrix. Then, Corollary (1) can only provide a lower bound of the spectral norm of the normalized centered ER adjacency matrix A ER . Hence, it is important to find an upper bound on the spectral norm of A ER to better understand its spectral properties. The following result comes in handy.

Lemma 1 [START_REF] Vu | Spectral norm of random matrices[END_REF] Let W be a Wigner matrix with independent random elements W ij , i, j = 1, . . . n having zero mean and variance at most σ 2 (n). If the entries are bounded by K(n) and there exist

a constant C such that σ(n) ≥ C n -1/2 K(n) log 2 (n), then there exists a constant C such that almost surely W 2 ≤ 2σ(n) √ n + C(K(n)σ(n)) 1/2 n 1/4 log(n). (17)
By applying Lemma (1) to the normalized centered adjacency matrix A ER we obtain the following concentration result.

Inria

Corollary 2 Let us consider the normalized centered adjacency matrix A ER . If the probability p(n) satisfies the inequality p(n) ≥ C log 4 (n)n -1 for some constant C > 0, then there exists a constant C > 0 such that almost surely

A ER 2 ≤ 2 + C 4 1 -p(n) np(n) log n. ( 18
)
Proof: From the definition of

A ER it results σ = n -1/2 . Then, condition σ ≥ C * n -1/2 K log 2 (n) in Lemma 1 implies K ≤ (C * log 2 n) -1 .
Additionally, the bound on the elements

A ER ij implies 1 -p n(1 -p)p ≤ K. Thus, 1 -p np ≤ K ≤ (C * log 2 n) -1 . (19)
Then, K exists if

1 -p np ≤ (C * log 2 n) -1 or if p satisfies the more stringent constraint p ≥ C n -1 log 4 n,
where C is a constant depending on C * . Inequality ( 18) is obtained from (17

) setting K = 1 -p np .
Let us observe that for spectral norm of matrix A ER is lower bounded by the extreme of the support of the limiting e.s.d. F A ER (x). Additionally, if for any δ > 0, p > δn -1 log 4 n, then the spectral norm is concentrated around the extreme of the support of F A ER (x).

Spectrum of the Normalized ER Adjacency Matrix

In the previous Section (4.2) and (4.3) we focused on the spectral properties of the normalized centered ER adjacency matrix A ER while in this section we analyze the spectral properties of the normalized ER adjacency matrix A ER and the effect of the mean component A ER on it. The following lemma plays a key role to establish a fundamental relation between the eigenvalue e.d.f. F A ER (x) studied in the previous sections and F A ER (x).

Lemma 2 [START_REF] Bai | Spectral analysis of large dimensional random matrices[END_REF] If F A (x), F B (x) are the eigenvalue e.d.f. of A , B, Hermitian matrices of size n, then

|F A (x) -F B (x)|≤ rank(A -B) n .
We recall that A ER = A ER -A ER has unit rank for any n. Then, asymptotically for n → ∞, the limiting eigenvalue distribution of the matrix A ER converges to the semicircular law as the limiting eigenvalue distribution of the matrix A ER . As for A ER , the limiting eigenvalue distribution of the matrix A ER provides only a lower bound for the spectral norm that requires independent study. The spectral norm of the two matrices A ER and A ER are different, because the largest eigenvalue changes when a unit rank matrix is added to a Hermitian matrix. From Bauer-Fike theorem for Hermitian matrices [START_REF] Saad | Numerical methods for large eigenvalue problems[END_REF], we have

|λ i ( A ER ) -λ i (A ER )|≤ || A ER || 2 (20)
for

1 ≤ i ≤ n.
Let us note that λ 1 (A ER ) = nγ(n)p n and λ i (A ER ) = 0 for i ≥ 2. Then, from [START_REF] Van Mieghem | Graph spectra for complex networks[END_REF], asymptotically as n → +∞, A ER 2 a.s.

→ 2. Thus, we get the following concentration result for the largest eigenvalue of the perturbed matrix

A ER λ 1 ( A ER ) - np(n) 1 -p(n) ≤ 2. ( 21
)
We notice that, for dense and sparse networks,

np(n) 1 -p(n) 2.
Hence the above result implies that λ 1 ( A ER ) → nγ(n)p n . Thus, the following lemma holds.

Lemma 3 [START_REF] Ding | Spectral distributions of adjacency and laplacian matrices of random graphs[END_REF] Let A ER satisfies the conditions of Corollary 2. Then,

λ n 1 ( A ER ) n→∞,a.s. ------→ np(n) 1 -p(n) ,
i.e., the largest eigenvalue of A ER tends to the largest eigenvalue of the mean matrix A ER as n → ∞.

Limiting Spectral Distribution of Centered Hermitian Matrices

In this section we present a useful existing result on centered symmetric matrices H with independent upper diagonal entries whose distributions have in general different variances. This result provides the Stieltjes transform of the limiting eigenvalue distribution as n → +∞ as solution of a nonlinear system of n equation.

s n (z) = dS H n (x) x -z = 1 n n i=1 c i (z, n),
where c i (z, n) is the unique solution to the system of equations

c i (z, n) =        -zI -δ pl n s=1 c s (z, n)σ 2 sl n p,l=1   -1      ii , i = 1, 2, ..n
in the class of analytic functions In this section we consider the normalized centered SBM adjacency matrix A with γ(n) = (np * (1 -p * )) -1 where p * = max m=1,...M p m . Additionally, we assume that all the probabilities p m scales at the same rate, i.e. lim n→+∞ pi pj = c ij for some c ij > 0, and p m (n) ∈ ω(n -1 ) as n → +∞. Then, it is straightforward to verify that the conditions of Proposition 2 are satisfied and the following corollary holds.

A = { (z) (c i (z, n)) > 0, (z) = 0}.
Corollary 3 Let A be the normalized centered SBM adjacency matrix defined in [START_REF] Chung | Spectral graph theory[END_REF] with

γ(n) = (np * (1 -p * )) -1 and p * = max m=1,...M p m . If p m (n) ∈ ω(n -1 ) and p m (n) = O(p 0 (n))
for all m = 1, . . . M, then, almost surely, the eigenvalue e.d.f. converge weakly to a distribution function whose Stieltjes transform is given by

s(z) = M m=1 c m (z) (22)
c m (z) being the unique solution to the system of equation

c m (z) = -1/M z + ς m c m (z) + ς 0 =m c (z) , (23)
with ς = lim n→+∞ p (1 -p ) p * (1 -p * )
, that satisfies the conditions

(c (z)) (z) > 0 for z > 0. (24)
The above result implies that in general the asymptotic eigenvalue e.d.f. of an SBM is not a semicircular law any longer.

Spectrum of the Full Adjacency Matrix

The result above gives the spectrum of the matrix A. Let us recall the definition of matrix A as given in (5)

A = A + A.
Using Lemma 2 on the finite rank perturbation, we deduce that the asymptotic eigenvalue e.d.f. of A and A are the same. However, their spectra differ in the extreme eigenvalues.

Extreme Eigenvalue of Adjacency Matrix

Using the Bauer-Fike Theorem on perturbation of eigenvalues of Hermitian matrices, for matrices A, A, and A, we have

|λ i ( A) -λ i (A)|≤ A 2 .
This is useful in getting asymptotic characterization of the M largest eigenvalues of A. Since A has M non-zero eigenvalues, this result says that the first M largest eigenvalues of A are concentrated around these eigenvalues, within an error of the order of the spectral norm of A.

The other eigenvalues of A are below the spectral norm of A, and hence they fall inside the continous part of the limiting e.d.f. To use this result, we need a bound on the spectral norm of the centered SBM adjacency matrix A. We strengthen the result in Theorem 1.4 in [START_REF] Vu | Spectral norm of random matrices[END_REF] to derive a tighter bound on the spectral norm of A.

Theorem 1 Let A be a normalized centered SBM adjacency matrix defined in ( 9) and satisfying the same conditions as in Corollary 3. Additionally, p 0 (n) satisfies the inequality p 0 (n) ≥ C n -1 log 4 n for some constant C > 0 and sup p0(n)(1-p0(n)) p * (n)(1-p * (n)) is bounded. Then, there exists a constant C > 0 such that almost surely

A 2 ≤ 2 M -1 (1 + (M -1)ς 0 ) + C 4 1 -p 0 (n) np 0 (n) log(n)
with ς 0 defined as in Corollary 3.

Proof: Much of the proof follows that of Theorem 1.3 in [START_REF] Vu | Spectral norm of random matrices[END_REF]. We first bound the spectral norm of the unnormalized centered adjacency matrix A = A/γ(n). At the end of the proof, we use this bound to derive a bound on the spectral norm of A, which gives us the result. We use the idea that spectral norm, which is the largest dominating eigenvalue in absolute value, can be bounded by the trace of the matrix raised to an even exponent, and that the larger the exponent, the sharper the bound:

|| A || 2k 2 = max 1≤i≤n |λ i ( A )| 2k ≤ ( n i=1 |λ i ( A )| 2k ) = (tr( A ) 2k ).
The method of proof is the popular moment method orginated by Füredi and Komlós, which bounds the moments of the e.s.d. of the matrix by bounding the expected trace of the matrix using combinatorial arguments [START_REF] Füredi | The eigenvalues of random symmetric matrices[END_REF].

Once we obtain a bound on the expected spectral norm, we can use Markov inequality, to bound the tail probabilities.

Pr{|| A || 2 ≥ λ} ≤ E|| A || 2k 2 λ 2k ≤ Etr( A ) 2k λ 2k (25)
If A is a standard Wigner matrix, for fixed k, the right hand side of the above equation is n times the 2k th moment of the empirical spectral distribution, which by Semicircle law tends to C k . Therefore, for such matrices, if k were chosen to be a fixed number independent of n, the right hand side tends to infinity for large n, making it rather useless. Therefore we choose k to be a function of n. The idea is that when k is a slowly increasing function of n, the semicircle law still holds, and since C k ≤ 4 k , the upper bound tends to 0, for any λ ≥ 2. Here, we extend this idea to Wigner matrix displaying community structure. Next, we need to find a bound on the quantity Etr( A ) 2k . To do this we expand the trace as a summation of expectation over cycles of length 2k of vertices in the set {1, 2, 3, ...n}

Etr( A ) 2k = E i1,i2,i3,....i 2k X i1,i2 X i2,i3 ....X i 2k ,i1 ,
where {i 1 , i 2 ...i 2k , i 1 } form a cycle over edges such that i j ∈ {1, 2, 3, ...n}, for each 1 ≤ j ≤ 2k. Each edge {i j-1 , i j } corresponds to a random variable X ij-1,ij .

We can partition the graphs based on the number of unique vertices that appear in the graph, called the weight of the graph, denoted by t, 1 ≤ t ≤ 2k. We can represent the original graph on 2k edges and 2k vertices equivalently by using a condensed undirected connected graph on t vertices. An edge exists in this graph if and only if it exists in the original graph and if it exists more than once in the original graph(walk), then this edge has a weight equal to the number of times this edge is traversed by the walk. Since each such random variable is zero mean and since Inria each variable is independent, if an undirected edge {i j-1 , i j } has a weight equal to 1, that is it only appears once in the walk, then the whole term becomes zero. So we need only consider the contribution of those graphs that have every edge appearing at least twice.

By virtue of independence and zero mean property, if t is greater than k + 1, and because the number of edges required for connectivity is greater than or equal to t + 1, there must at least be k + 1 edges in the graph. Since the total number of edges is 2k in the walk, this means there exists an edge that appears only once, making the contribution of such a term zero. Hence we must have 1 ≤ t ≤ k + 1. Thus we can bound the quantity of interest as below:

Etr( A ) 2k ≤ k+1 t=1 G∈G t,n,2k EX G ,
where G t,n represents a set of graphs on t vertices drawn from {1, 2, ..n} with 2k edges.

An edge e = {i j-1 , i j } such as described above, is called an innovation edge, if the vertex i j is such that i j ∈ {i 1 , i 2 , ...i j-1 }, i.e., it appears for the first time in e. The other edges in the graph either overlap the innovation edges or are interconnections between two vertices that already exist in the graph. Since in our case the random variables are bounded in absolute value by 1( since they are Bernoulli), the contribution over all the edges other than the innovation edges can be bounded by 1. For any graph on t vertices, there must be exactly t -1 innovation edges and each edge must have at least weight 2. The contribution to the expectation of each such edge would have a weight of at most σ 2 i , 0 ≤ i ≤ M depending on whether the edge is between two communities or within some community i. This bound is exact if each such edge has a weight two; if it has weight more than two, then this is an upper bound. Then, by independence, the contribution of the group of edges can be bounded by the product. Therefore, the following is true:

|Ea i1,i2 a i2,i3 ...a i 2k ,i1 |≤ ( max 1≤i≤M σ 2 i ) (t-1-i) (σ 2 0 ) i ,
where i are integers such that 0 ≤ i ≤ t -1. This corresponds to choosing out of t edges, i edges such that the vertices of those edges belong to two different communities. Once we choose such i edges we need to choose the communities from which the corresponding vertices emerge. For convenience, we can assume the vertices have a preferred ordering. The first such vertex can then be chosen from any of the M communities. Once such a community is chosen, if the next edge is upper bounded by max i σ 2 i , the next vertex of the edge can be chosen only in 1 way, because this corresponds to an edge belonging to the same community. If the edge is bounded by σ 2 o , then the community to which the next vertex belongs can be chosen in at most M -1 ways, since this edge corresponds to an edge between communities. Corresponding to each selection of a community to which the edge can belong, the vertices can be chosen in (n/M ) t ways.

Therefore, we can finally bound the full term as follows:

Etr A 2k ≤ k+1 t=1 M (n/M ) t t-1 i=0 t -1 i (M -1) (t-1-i) (σ 2 0 ) (t-1-i) ( max 1≤j≤m σ 2 j ) i W (k, t).
The inner summation on the variable i turns out to be the Binomial expansion. W (k, t) is the number of equivalent graphs of t fixed vertices with 2k edges and is related to the number of the Catalan number C t . We use a bound on this quantity which is available in [START_REF] Vu | Spectral norm of random matrices[END_REF]:

W (k, t) ≤ 2k 2p -2 p N 2 k+N +1 (N + 2) N ,
where N = 2k -2(t -1). Finally, we get:

Etr A 2k ≤ k+1 t=0 n t (σ 2 n) t-1 W (k, t),
where,

σ 2 = 1 M (max i σ 2 i + (M -1)σ 2 o ). (26)
As in [START_REF] Vu | Spectral norm of random matrices[END_REF] it can be shown that when 2k = aσ 1/2 n 1/4 , for some a, the term within the summation is bounded by a geometric series with growth factor 1/2. Using this fact we finally obtain:

Etr( A ) 2k ≤ 2n(2σ √ n) 2k .
Substituting the above in equation ( 25), and using λ = 2σ √ n + C(σ) 1/2 n 1/4 log(n) we have:

Pr{|| A || 2 ≥ 2σ √ n + C(σ) 1/2 n 1/4 log(n)} ≤ 2n 2σ √ n 2σ √ n + C(σ) 1/2 n 1/4 log(n) 2k = 2n(1 - Cσ 1/2 n 1/4 log(n) 2σ √ n + Cσ 1/2 n 1/4 log(n) ) 2k ≤ 2n(1 - Cσ 1/2 n 1/4 log(n) 3σ √ n ) 2k ≤ 2n exp(- c log(n)k 3 √ σn 1/4 ) = 2n exp(-ca log(n)/3),
where the second inequality above follows from the assumption that σ ≥ C n -1/2 log 2 (n), the third inequality because e -x ≥ 1 -x, and the last equality by the form of k. Now since the right hand side is summable in n for appropriate constants c and a, by Borel-Cantelli Lemma [4], we have:

|| A ||≤ 2σ √ n + C(σ) 1/2 n 1/4 log(n) a.s.
for σ ≥ C Kn -1/2 log 2 (n), where K = 1. This is the same identity as in [START_REF] Vu | Spectral norm of random matrices[END_REF] except with the appropriate definition of σ from (26).

In the above, since A is unnormalized, we were able to upper bound each variable by K = 1. It is easy to redo the above proof for the matrix A, where because of the normalization, we have the upper bound on the matrix entries, K =

1-p0 √ np * (1-p * )
. One can easily see that the condition σ ≥ KC n -1/2 log 2 (n) holds when p 0 ≥ C n -1 log 4 (n), and the bound on A becomes:

|| A|| 2 ≤ 2σ √ n + C(Kσ) 1/2 n 1/4 log(n) ≤ 2 1 M (1 + (M -1)ς 0 ) + C 1 -p 0 np 0 (1 -p 0 ) 1/2 p 0 (1 -p 0 ) p * (1 -p * ) 1/4 (M -1)p 0 (1 -p 0 ) + p * (1 -p * ) np * (1 -p * ) 1/4 n 1/4 log(n) ≤ 2 1 M (1 + (M -1)ς 0 ) + C n 1/4 1 -p 0 p 0 1/4 log(n),

Inria

where we used the fact that sup n p0(1-p0) p * (1-p * ) is finite, and C is an arbitrary constant.

Eigenvalues of the Mean Matrix

By the result above on the spectral norm of the zero mean matrix, we know that the largest eigenvalue of the matrix is somewhere close to the edge of the spectrum. But when the mean matrix is added to this matrix, the largest eigenvalue escapes the bounded spectrum. Namely, since the mean matrix has rank M , by interlacing inequalities on the sum of two Hermitian matrices, we can see that there are exactly M eigenvalues outside the bounded spectrum. Recall that A = A + A By the Bauer-Fike Theorem we have

|λ i ( A) -λ i (A)|≤ A . (27)
From Theorem 1 we have that asymptotically almost surely A ≤ 2ς + δ with ς = M -1 (1 + (M -1)ς 0 ) and δ → 0. For i > M , λ i (A) = 0. Therefore, we see that λ i ( A) for i > M lies below the spectral norm of A.

Eigenvalues of A

Let the eigenvalues of P in (6) be µ i , 1 ≤ i ≤ M . They depend on the probabilities p i , 0 ≤ i ≤ M and the following relationship holds between the eigenvalues of A and µ i .

Lemma 4 Under the conditions in Proposition 1 the M eigenvalues of A, outside the continuous spectrum of A are given as:

|λ i ( A) -µ i |≤ 2ς + δ (28) for 1 ≤ i ≤ M .
Thus, we conclude that asymptotically, the M largest eigenvalues of the adjacency matrix converge to those of the mean matrix almost surely.

To complete this argument, we need the approximate locations of µ i 's. By Gershgorin disc theorem [START_REF] Saad | Numerical methods for large eigenvalue problems[END_REF], the µ i 's should satisfy:

µ i M γ(n)n -p i ≤ p 0 (M -1). (29)
Note on special case of Symmetric SBM :

When the probabilities p 1 = p 2 = p 3 .. . . . = p M = p * , we can significantly simplify the equations (23) and achieve useful insight into the shape of the spectrum. In this case we have that ς m = 1 for m = 1, 2, ..M and the fixed point equation ( 23) becomes:

M c m (z) = -1 z + c m (z) + ς 0 =m c (z)
.

We see that the equations are symmetric, hence, by uniqueness property of Corollary 3, we must have that c 1 (z) = c 2 (z) = c 3 (z) = . . . = c M (z) = c(z), and s(z) = M c(z), which leads to

s(z) = -1 z + (1+(M -1)ς0) M s(z)
.

Consequently, the following holds.

Lemma 5 The distribution of matrix 1 2 √ nP is given by:

c i = -1/2 z + σ i c i + σ 0 c j , (33)
for i, j = 1, 2 and i, j = 2, 1 respectively. where

σ 1 = σ 2 1 (p1+p0) 2 , σ 2 = σ 2 2 (p2+p0) 2 , σ 0 = σ 2 0 (p0+p1)(p0+p2)
and the limiting distribution has a spectrum whose Stieltjes transform is given by c(z

) = c 1 (z) + c 2 (z). Since √ n 2 L = √ n 2 -
√ n 2 P , its distribution has a bulk component that lies around √ n/2, with an approximate width of 2 max(σ 1 , σ 2 ) + σ 0 . This matrix also has an eigenvalue at 0, by the property of Laplacian.

In the two-community case, it can be seen from simulations that there exists one more eigenvalue outside the bulk, which remains to be properly characterized.

Proof of the lemma: We follow the steps in the proof of Theorem 1.1 in [START_REF] Bordanave | Spectrum of large random reversible markov chains: Two examples[END_REF]. The first step is a form of uniform strong law of large numbers called Kolomogorov-Marcinkiewicz-Zygmund strong law of large numbers. Since the elements of the matrix A are independent and have finite fourth moments from Lemma 2.3 in [START_REF] Bordanave | Spectrum of large random reversible markov chains: Two examples[END_REF] we have the following as true:

n j=1 A ij = n 2 (p 1 + p 0 + δ (1) i ), (34)
where

max i |δ 1 i |= o(1) for 1 ≤ i ≤ n/2 and n j=1 A ij = n 2 (p 2 + p 0 + δ (2) i ), (35)
where max i |δ 2 i |= o(1) for 1 + n/2 ≤ i ≤ n. From equations (34) and (35) we have uniform convergence for 1 ≤ i ≤ n, with the error, max i (δ

(1) i , δ (2)
i ) = o(1):

D i = n j=1 A ij = n 2 (p k + p 0 ) + i , (36)
where

p k = p 1 if i ∈ Ω 1 and p k = p 2 if i ∈ Ω 2 , and max i | i |= = o(1) uniformly.
Next step is to use Hoffman-Wielandt inequality [START_REF] Anderson | An Introduction to Random Matrices[END_REF] to bound the error between the e.s.d. of

A √ n and
√ n 2 P . We have using Hoffman-Wielandt inequality and bound on Stieltjes transforms found in [START_REF] Bai | Methodologies in spectral analysis of large dimensional random matrices, a review[END_REF]:

|s F 1 √ n A (z) -s F √ n 2 P (z)|≤ c n z ij | √ n 2 P ij - 1 √ n A ij | 2 ,
where z ∈ C + . For any i, j, we have

√ n 2 P ij = √ n 2 A ij D i D j = √ n 2
A ij n/2(p k + p 0 + i )n/2(p l + p 0 + j ) = A ij n(p k + p 0 )(p l + p 0 ) (1 + O( i ))(1 + O( j )) = A ij √ n (1 + O( )), (37)
where i ∈ Ω k and j ∈ Ω l and is infinitesimally small. The last equality follows because i and j tend to 0 uniformly for large n. Thus by Hoffman Wielandt inequality we have:

|s F 1 √ n A (z) -s F √ n 2 P (z)|≤ c n 2 z ij |A ij | 2 O( 2 ) → 0 a.s.
The last relation follows from the strong law of large numbers on variables A ij and since = o(1). Hence we have the result.

Example Application: Epidemic Spreading

In this section, we discuss an important potential application of the result we derived above for adjacency matrices, namely, in the topic of epidemic spreading. We refer to the recent paper [START_REF] Bose | The cost of an epidemic over a complex network: A random matrix approach[END_REF].

Inria

In this work, the authors study an epidemic process over a random network of nodes. The spread of the epidemic from one node to another is governed by the Random network, i.e., a node can only infect another if there exists an edge between the two nodes. They present a concise result delineating the relationship between the expected cost of the epidemic per node denoted by C D (n)(disease cost) [START_REF] Bose | The cost of an epidemic over a complex network: A random matrix approach[END_REF], and the largest eigenvalue of the modified adjacency matrix; namely,

C D (n) ≤ αc d 1 -λ 1 (M ) , (38)
where M = (1 -δ)I + βA is the matrix which governs the dynamics of the system [START_REF] Bose | The cost of an epidemic over a complex network: A random matrix approach[END_REF], with β being the probability of infection, δ is the probability of recovery of any node, and c d is the cost parameter. We direct the reader to the original paper for more details. A is as usual the adjacency matrix of the random graph.

We examine the epidemic spread on a graph which follows SBM with M communities. We know that in this case λ 1 (A) → n/M µ 1 as n → ∞ a.s. under certain conditions. Also by (29) we have that µ 1 ≤ p 1 + (M -1)p 0 , therefore we obtain:

λ 1 (M ) = (1 -δ) + βλ 1 (A) ≤ 1 -δ + β(n/M µ 1 ). This yields, C D (n) ≤ αc d δ -βn/M (p 1 + (M -1)p 0 ) . (39)
If p 1 p i , for i ≥ 2, then we can venture to say that this bound is tight, and that the community with the largest edge probability governs the disease cost.

Numerical Results

In this section we provide simulation results to demonstrate the results obtained above. More specifically, we corroborate our results on the spectrum of adjacency matrix by comparing the spectrum obtained by simulating a 2-community SBM with the distribution obtained by inverting the Stieltjes transform, which is an explicit solution of the simultaneous equations (23). In the simulations, we use a matrix of size n = 10 4 . For a 2-community system, the solution amounts to solving explicitly the resulting quartic equation and choosing the solution branch that satisfies the conditions (24). The inverse relationship between the limiting e.s.d. and the Stieltjes transform thus obtained, is given by the well known Stieltjes inversion formula:

f (x) = lim y→0 s F (x + √ -1y)/π, (40)
where f (x) is the p.d.f. corresponding to the c.d.f. F (x), whenever the limit exists.

Figure 2 shows the histogram of normalized adjacency matrix 1 √ n A and compares it to the theoretical spectrum obtained as above for n = 10 4 , and several values of edge probabilities.

In the second part of this section we turn our attention to the extremal eigenvalues of the adjacency matrix for a 3-community SBM of size n = 999. Over several independent runs, we get values of the top 4 eigenvalues of the matrix A, for 0.3 ≤ p 1 ≤ 0.48, 0.15 ≤ p 2 ≤ 0.33, 0.08 ≤ p 3 ≤ 0.26 and 0.03 ≤ p 0 ≤ 0.031, randomly picked. We note that as expected in (Figure 3), there are three eigenvalues outside the bulk, which agree very well with the expected values, i.e., the non-zero eigenvalues of A. In addition, it can also be seen that the upper bound in Theorem 1 is remarkably tight for the simulated probabilities.

Next, we consider the spectrum of the normalized Laplacian matrix. In fact we consider the spectrum of the following matrix which we denote L, which is given by: L = √ n/2 -√ n/2L. By assertion its spectrum is given by the solution of (33). We explicitly solve this equation for a two-community model, and compare it to numerically obtained results for a matrix whose size is n = 999 for various values of the probabilities p 1 , p 2 and p 0 (Fig. 4).

RR n°8703

Conclusion

In this work we studied in detail the spectra of adjacency and normalized Laplacian matrices of an SBM with M communities. In particular, we analyzed the limiting empirical distribution of the eigenvalues of the adjacency matrix of SBM. We find that the Stieltjes transform of the limiting distribution satisfies a fixed point equation and provide an explicit expression in the case of symmetric communities. Furthermore, we obtained a tight bound on the support of the asymptotic spectrum, and concentration bounds on the extremal eigenvalues.

As future work we plan to analyze the structure of the eigenvectors and develop more detailed applications to graph clustering and network sampling. It will be also interesting to consider SBM models where sizes of communities are not uniform.

Acknowledgement

This work was partly funded by the French Government (National Research Agency, ANR) through the "Investments for the Future" Program reference #ANR-11-LABX-0031-01.

Ce travail a bénéficié d'une aide de l'Etat gérée par l'Agence Nationale de la Recherche au titre du programme «Investissements d'Avenir »portant la référence : ANR-11-LABX-0031-01.

Inria

Inria 5

 5 Results for Adjacency Matrix of M community Model5.1 Finding the Spectrum of Centered Adjacency Matrix

Figure 2 :

 2 Figure 2: Comparison plot between empirically obtained spectrum (bar graph), and explicit solution(line) of 2-community SBM adjacency matrix

Figure 3 :

 3 Figure 3: Extremal eigenvalues of 3-community SBM normalized matrix compared to expected values.

Figure 4 : 15 6

 415 Figure 4: Histogram of 2-community L for various edge probabilities compared to theoretical spectrum

RR n°8703

We use the short notation γ when it is not necessary to emphasize the dependency on n.RR n°8703

 15), with σ 2 ≡ (1+(M -1)ς0)

M

. Thus, we see that in the symmetric scenario, the spectrum of the adjacency matrix becomes a semicircle law, and the upper bound in Theorem 1 becomes exact. Similarly, the eigenvalues of the mean matrix become:

and

for i = 2, 3, . . . , M . Thus, from Lemma 4, the largest eigenvalue of the A converges to (30) above, and the next M -1 largest eigenvalues converge to (31), i.e., the second largest eigenvalue of the adjacency matrix has multiplicity M -1.

In Figure 1 we show diagramatically the general form of the asymptotic histogram of the scaled adjacency matrix of an SBM, with approximate locations of its various components all probabilities scale as log c (n) for some c > 4. We observe that under these conditions, there is sufficient separation between the continuous part of the spectrum and the discrete extremal eigenvalues.

Spectral Distribution of Normalized Laplacian Matrix

We recall that the normalized Laplacian Matrix is given by

For the sake of simplicity, we consider the case of two blocks, i.e., M = 2, and probabilities p i , 0 ≤ i ≤ 2 that are not dependent on the size of the matrix n.

Let P = D -1/2 AD -1/2 . We show that asymptotically, the e.s.d. of the matrix 1 2 √ nP converges to the e.s.d. of the matrix 1 √ n A , defined as

A ij / (p 1 + p 0 )(p 2 + p 0 ) otherwise Inria


Gautier Verhille

Ruslan Khalilov

Nicolas Plihon

Peter Frick

Jean-François Pinton

Transition from hydrodynamic turbulence to magnetohydrodynamic turbulence in von Kármán flows

The influence of an externally applied magnetic field on flow turbulence is investigated in liquid-gallium von Kármán (VK) swirling flows. Time-resolved measurements concern global variables (such as the flow power consumption) and local recordings of the induced magnetic field are made. From these measurements, an effective Reynolds number is introduced as Rm eff = Rm(1α √ N ), so as to take into account the influence of the interaction parameter N . This effective magnetic Reynolds number leads to unified scalings for both global variables and the locally induced magnetic field. In addition, when the flow rotation axis is perpendicular to the direction of the applied magnetic field, significant flow and induced magnetic field fluctuations are observed at low interaction parameter values, but corresponding to an Alfvèn speed v A of the order of the fluid velocity fluctuations u rms . This strong increase in the flow fluctuations is attributed to chaotic changes between hydrodynamic and magnetohydrodynamic velocity profiles.

Introduction

The understanding of how flows of electrically conducting fluids develop when a significant magnetic field is imposed is of both fundamental and technical importance. Such situations are common in practical applications such as electromagnetic pumping of conducting fluids, control of flow motions during metal casting or crystallization processes or confinement of thermonuclear plasmas in magnetic confinement devices. Magnetic fields are also ubiquitous in astrophysical bodies, such as galaxies, stars and planets, where the dynamo instability converts parts of the kinetic energy of flowing conducting fluids into magnetic energy. The influence of magnetic fields on astrophysical flows is thus also a generic feature (it can be observed for instance through the influence of the Earth's magnetosphere on the solar wind).

In the incompressible magnetohydrodynamics (MHD) approximation, the dynamics is governed by the coupled set of Navier-Stokes (NS) and induction equations:

ρ(∂ t u + (u • ∇)u) = -∇p + ρν∆u + j × B ,
(1.1)

∂ t B = ∇ × (u × B) + λ∆B , (1.2) ∇ • u = 0 (1.3) ∇ • B = 0 (1.4)
where u(r, t), p(r, t), B(r, t) are the velocity, pressure and magnetic fields in a fluid with (constant) density ρ, kinematic viscosity ν and magnetic diffusivity λ = 1/µ 0 σ, where σ is the electrical conductivity of the fluid and µ 0 the magnetic permeability; the current density stems from the Maxwell-Ampère equation, µ 0 j(r, t) = ∇ × B(r, t). The nature of the problem is usually governed by values of the main dimensionless parameters defined as a function of characteristic velocity U , length scale L of the flow and typical value of the magnetic field B 0 from the analysis of the NS and induction equations. The kinetic Reynolds number Re = U L/ν compares the amplitude of the inertial term to the viscous term in the NS equation. In the limit of large kinetic Reynolds number one defines an interaction parameter N = σLB 2 0 /ρU which compares the Lorentz force to the inertial term in the N.S equation (in the low-Reynolds-number limit, the Hartmann number Ha 2 = N Re, -which compares the Lorentz force to the viscous term -is used). The magnetic Reynolds number Rm = U L/λ compares the induction term to the diffusion term in the induction equation. The ratio of the magnetic Reynolds number to the kinetic Reynolds number defines the magnetic Prandtl number P m = ν/λ, which compares the diffusive temporal/length scales of the velocity and magnetic fields and depends only on the physical properties of the fluid.

Among the various situations in the (Re, Rm, N, P m) parameter space, some asymptotic limits have been studied in details. The case of flow with moderate kinetic Reynolds number and large interaction parameter number has been extensively studied for industrial applications. As explained and verified experimentally by [START_REF] Sommeria | Why, how and when, MHD-turbulence becomes twodimensional[END_REF], [START_REF] Moreau | MHD turbulence at the laboratory scale: established ideas and new challenges[END_REF] and [START_REF] Eckert | MHD turbulence measurements in a sodium channel flow exposed to a transverse magnetic field[END_REF], the Lorentz force acts as an anisotropic diffusivity which damps the velocity field fluctuations along the applied magnetic field and hence drives the flow toward a two-dimensional structure. The issue of exact twodimensional versus weak three-dimensional features has recently been addressed experimentally by [START_REF] Klein | Appearance of three dimensionality in wall-bounded MHD flows[END_REF] and numerically by [START_REF] Mück | Three-dimensional MHD flows in rectangular ducts with internal obstacles[END_REF]. Another limit, relevant in several astrophysical situations, corresponds to large kinetic Reynolds numbers onto which a weak magnetic field is applied. In this case the interaction parameter is small, the Lorentz force does not influence significantly the hydrodynamics and the situation is reduced to the advection of a passive vector field by a (prescribed) turbulent velocity field. Several features have then been derived in the framework of the Kolmogorov theory of turbulence, as for instance the spatial spectrum of magnetic energy, B 2 (k) ∝ k -11/3 -see for instance [START_REF] Moffatt | The amplification of a weak applied magnetic field by turbulence in fluids of moderate conductivity[END_REF] -verified experimentally in [START_REF] Odier | Advection of a magnetic field by a turbulent swirling flow[END_REF]. An open question, currently debated, is the possibility that the flow turbulence induces an effective magnetic diffusivity which may become much larger than the molecular value, cf. [START_REF] Frick | Direct measurement of effective magnetic diffusivity in turbulent flow of liquid sodium[END_REF]. The astrophysical limit of large values of the kinetic and magnetic Reynolds numbers has also received much attention during the last few decades. Among these studies, the understanding of the dynamo instability has benefited from joint theoretical, numerical and experimental efforts.

In the context of liquid-metal dynamos, especially relevant for planetary dynamo studies, a number of recent studies [START_REF] Gailitis | Magnetic field saturation in the Riga dynamo experiment[END_REF][START_REF] Müller | Experiments at a two-scale dynamo test facility[END_REF][START_REF] Monchaux | The von Kármán sodium experiment: turbulent dynamical dynamos[END_REF]) explored a regime in the limit of large kinetic Reynolds numbers and moderate (to high) interaction parameter values. Dynamo action has so far been observed in three experiments using liquid-sodium as the working fluid: the Riga experiment, based on a Ponomarenko screw flow [START_REF] Ponomarenko | Theory of the hydromagnetic generator[END_REF]; the Karlsruhe experiment using an array of like-sign helical motions as proposed by G.O. Roberts [START_REF] Roberts | Dynamo action of fluid motions with two-dimensional periodicity[END_REF] ; and the von Kármán sodium (VKS) experiment based on swirling motions, as studied numerically by Dudley and James [START_REF] Dudley | Time-dependent kinematic dynamos with stationary flows[END_REF]. In liquid-sodium flows, owing to the very low value of the magnetic Prandtl number P m, even moderate values of Rm are associated with very high Re values, i.e. fully developed turbulence. In this context, how the self-generated dynamo field acts on the velocity field so as to saturate the growth of the (supercritical) instability is an open problem. In the Riga experiment, a significant reduction of the rotational velocity of the jet was observed, as shown in [START_REF] Gailitis | Magnetic field saturation in the Riga dynamo experiment[END_REF] and [START_REF] Kenjeres | Tackling complex turbulent flows with transient RANS[END_REF]. In the same manner the dynamo field in the Karlsruhe experiment led to a measurable braking of the fluid in the pipe system [START_REF] Müller | Experiments at a two-scale dynamo test facility[END_REF]). In the case of the VKS dynamo [START_REF] Monchaux | The von Kármán sodium experiment: turbulent dynamical dynamos[END_REF], the coupled (u, B) evolution has not yet been measured and the interplay between magnetic and velocity fields is yet to be clarified.

It is the purpose of the study presented here to investigate how a fully developed turbulent flow interacts with a strong applied magnetic field. Investigated flows are of von Kármán type, generated in a cylinder by the rotation of impellers, and hence of the same kind as the VKS dynamo flow, as reported for instance in [START_REF] Monchaux | The von Kármán sodium experiment: turbulent dynamical dynamos[END_REF]. Two applied magnetic field geometries have been studied: the applied magnetic field may be perpendicular or parallel to the flow rotation axis. Details of the experimental setup are given in section 2. The variation of a global quantity, namely the power injected into the flow in order to keep the driving impellers at a fixed rotation rate, with the intensity of the applied magnetic field is then discussed in section 3. We show that it varies linearly with the interaction parameter N . This scaling is confirmed in section 4 by the analysis of the local induction measurements. In section 5, we detail a noteworthy finding of our study: in the transition from a purely hydrodynamic regime (for the lowest N values) to a fully "magnetic" state (at high N , for which turbulence is damped), an increase in the flow fluctuations has been observed. This increase is mainly seen in the long-time dynamics of the flow, and was observed when the influence of the magnetic field balances the influence of the flow rotation.

Experimental set-up

The liquid-gallium flow is generated by the rotation of two impellers at each end of a cylindrical vessel. The vessel is sketched in figure 1(a) and has a radius R c of 97 mm and a length of 323 mm. The impellers have a diameter equal to 2R = 165 mm and are fitted to a set of eight straight blades with height 10 mm. All parts used in this setup are machined from non-magnetic stainless-steel. The two impellers are separated by a distance H = 203 mm. They are independently driven by two AC-motors which provide a constant rotation rate in the interval (|F 1 |, |F 2 |) ∈ [0.5, 25] Hz. When constrained to operate at constant rotation speed, the motors electric drives deliver an analogue output proportional to the current in the motor, as an image of the applied mechanical torque.

Liquid-gallium is cooled by water circulation located behind the driving impellers; the experiments are run at a constant temperature between 40 • C and 48 • C. Liquid gallium has density ρ = 6.09 × 10 3 kg.m -3 , electrical conductivity σ = 3.68 × 10 6 Ω -1 .m -1 , hence a magnetic diffusivity λ = 1/µ 0 σ = 0.29 m 2 .s -1 . Its kinematic viscosity is ν = 3.1 × 10 -7 m 2 .s -1 . In the present study, the flow is driven either by both impellers rotating at equal speed, in the same or opposite direction, or by the rotation of only one disk. In any configuration, the integral kinematic and magnetic Reynolds numbers are defined as Re = 2πR 2 F /ν and Rm = 2πR 2 F/λ where F is the rotation rate (in Hertz) and R the radius of the driving impeller(s). Rm values up to 5 are achieved, with corresponding Re in excess of 10 6 . The liquid-gallium flows generated are thus fully turbulent, with high levels of fluctuations, depending on the topology of the time-averaged flows. Detailed experimental investigations of time-averaged flow topologies, fluctuation levels and characteristics may be found in [START_REF] Ravelet | Supercritical transition to turbulence in an inertially driven von Kármán closed flow[END_REF].

Three flow topologies were considered in this study, depending on the relative values of F 1 and F 2 . These flows are well documented from water experiments [START_REF] Ravelet | Supercritical transition to turbulence in an inertially driven von Kármán closed flow[END_REF] and their time-averaged features are briefly recalled here. The 'counter-rotating' flow is defined as corresponding to the case of the two impellers rotating in opposite directions at the same speed (i.e. F 1 = F 2 with the notation of figure 1(a)). In this case, the mean flow is made up of two toroidal cells rotating in opposite directions and two poloidal cells due to the centrifugal forces localized close to the impellers, as sketched in figure 1(b). The 'counter-rotating' flow is a cylindrical equivalent of the s2t2 flow introduced by [START_REF] Dudley | Time-dependent kinematic dynamos with stationary flows[END_REF]. Slow dynamics of the flow (as compared to the period of impellers' rotation) was shown to be linked to the evolution of the shear layer localized in the mid-plane in de la [START_REF] De La Torre | Slow dynamics in a turbulent von Kármán swirling flow[END_REF]. If only one impeller rotates at the frequency F and the other one is kept at rest, the mean flow is made up of one single toroidal cell and one poloidal cell as shown in figure 1(c). We refer to this flow topology as the "one-disk" flow which corresponds to the s1t1 flow introduced by [START_REF] Dudley | Time-dependent kinematic dynamos with stationary flows[END_REF]. Finally, when the two impellers rotate in the same direction and at the same rotation rate, the mean flow is made up of one toroidal cell and two poloidal cells, as shown in figure 1(d). This flow topology is referred to as the 'corotating' flow (i.e. F 1 = -F 2 with the notation of figure 1(a)), and is of the s2t1 type introduced by [START_REF] Dudley | Time-dependent kinematic dynamos with stationary flows[END_REF].

A set of two water-cooled electrical coils can be arranged in an Helmholtz-like configuration to produce an applied magnetic field B A either aligned with the axis of the impellers (along the z-axis) or perpendicular to it (along the x-axis). The current in the coils is controlled and adjusted between 0 and 150 A by a Drusch 30 kW power supply. The resulting applied magnetic field B A has an intensity which can reach up to 2000 G. Measurements show that the applied magnetic field is homogeneous within 10 % over the vessel volume. As for the Reynolds numbers, an integral interaction parameter can be defined as N = σ(B A ) 2 /ρ2πF . Its value ranges between 0 and 0.5 depending on the operational conditions. The maximal value of the interaction parameter is achieved for F =2 Hz. Below this rotation frequency the flow is not fully turbulent even in the absence of an externally applied magnetic field.

Magnetic measurements are obtained either with a homemade Hall-sensors array (Sentron CSA-1V), or with a commercial Bell Gaussmeter, both probing one component of the magnetic field. The bandwidth of these sensors is over 150 Hz so both time averages and fluctuations of the induced magnetic field B I are probed. The spatial evolution of the induced magnetic field is probed by an Hall sensor array at eight positions along a radius of the vessel. For every configuration in this study, the magnetic probes lie in the mid-plane at θ = π/2, i.e. parallel to the y-axis (see figure 1a). Magnetic field data and electrical signals from the motor drives are recorded using a National Instrument PXI-4472 digitizer (resolution of 23 bits) at a sampling rate equal to 1 kHz.

Global behavior: injected power

In this section, the influence of the applied magnetic field B A on the global power budget of turbulent flows is addressed. We note that, as torques and rotation rates of the impellers are conjugate variables, it is not possible to keep both at constant value. Thus, for each run, the rotation rates of the driving impellers are kept constant and the time evolutions of the driving torques are recorded. The time-averaged injected power is then computed as the product of the driving rotation rate times the time-averaged driving torque, summed for the two impellers. Figure 2 shows the time series for the injected power in the counter-rotating flow case, at F =10 Hz, for two values of the transverse applied magnetic field: B A = 100 G and 1550 G. As expected, under the increased action of Lorentz force, one clearly observes an increase of the mean power consumption and a decrease in the fluctuations.

Hereafter, we focus on the evolution of the mean power consumption. Let us first consider the hydrodynamic regimes, i.e. when B A = 0. The injected power grows as the third power of the rotation rate F of the impellers, reflecting the turbulent nature of the flow. This scaling is indeed obtained in the limit of very large kinetic Reynolds numbers, when the flow power consumption is independent of the viscosity of the fluid. In turbulent flows, the power dissipated by the flow is equal to P H = M , where M ρR 3 is the mass of fluid set in motion and = u 3 rms /R is the injected power per unit mass in the turbulent cascade. The injected power is thus proportional to R 2 u 3 rms ∝ F 3 . In order to compare our measurements with previous studies, let us introduce a dimensionless power number (per motor) K P , as K P = P H /(2ρR 5 (2πF ) 3 ), following [START_REF] Ravelet | Toward an experimental von Kármán dynamo: numerical studies for an optimized design[END_REF]. Measured values are K P = 10 -2 for the corotating flow, K P = 1.31 × 10 -2 for the one-disk flow, and K P = 3.57 × 10 -2 for the counter-rotating flow. These differences are linked to the intensity of the large-scale shearing motions in the flow, and resulting velocity fluctuations -as shown by [START_REF] Marié | Experimental measurement of the scale-by-scale momentum transport budget in a turbulent shear flow[END_REF]. The measured K P are in close accordance with measurements in water flows in similar conditions [START_REF] Ravelet | Bifurcations globales hydrodynamiques et magnétohydrodynamiques dans un écoulement de von Kármán turbulent[END_REF].

Let us now consider the case of a transverse applied magnetic field, i.e. the magnetic field is parallel to the x-axis and perpendicular to the axis of rotation of the impellers. (see figure 3). When increasing the intensity of the applied magnetic field, a first observation is that, in all configurations, the power injection at a constant rotation rate increases (cf. figure 3a-c). This increase can be ascribed to the Joule dissipation resulting from the induced electrical currents (or equivalently to the Lorentz force trying to slow down the flow); the total power dissipation results from viscous friction and Joule heating. However, the evolution of the injected power remains proportional to F 3 even for the highest applied magnetic field in each configuration. This suggests that the power consumption is still dominated by turbulent dissipation, with a modification of the large scale flow as detailed in section 5.

For the three configurations, at first-order, the total injected power P T varies linearly with the interaction parameter N , as displayed in figure 3(d-f). This scaling will now be explained using dimensional analysis. As was pointed out previously, in the presence of a magnetic field, the nature of the problem involves a coupling between the velocity and the magnetic fields (u, B). In the remainder of the paper, the uncoupled hydrodynamic fields will be denoted (U, 0) in the absence of an applied magnetic field, while the magnetohydrodynamic fields will be denoted (U , B) in the presence of an applied magnetic field B A -with B = B A + B I . On average, the injected power in a turbulent flow is equal to that dissipated. The total dissipation P T in presence of an applied field is the sum of two terms: the turbulent dissipation P H (U , B) and the Joule dissipation P J (U , B) depending on both the velocity field and the magnetic field. As in the experiments the interaction parameter is moderate, one can assume, to first-order, that the turbulent dissipation is not modified by the Lorentz force:

P H (U , B) ∼ P H (U, 0) ∼ ρR 2 u 3 rms (3.1)
This assumption holds if the turbulent cascade is not significantly modified, as is confirmed by the spectra of the induced magnetic field presented later in section 5 and also as reported in [START_REF] Alemany | Influence of an external magnetic field on homogeneous MHD turbulence[END_REF]. The Joule dissipation is defined as P J (U , B) = V j 2 /σdτ ∼ j 2 L 3 /σ, V being the flow volume. As often observed in gallium flows (see for instance [START_REF] Bourgoin | Induction mechanism in von Kármán swirling flows of liquid gallium[END_REF]), the induced magnetic field is small compared to the applied magnetic field (B I /B A ∼ 0.5) and, again in a first approximation, one assumes that the current density j scales as j ∼ σU B A . So the Joule dissipation can be written as:

F =6 Hz F =8 Hz F =10 Hz F =12 Hz F =14 Hz F =16 Hz F =18 Hz F =20 Hz
F =6 Hz F =8 Hz F =10 Hz F =12 Hz F =14 Hz F =16 Hz F =18 Hz F =20 Hz 0 0.1 0.2 0.3 0.4 0 0.5 1 1.5 2 2.5 3 3.5 4 N P [kW] F =6 Hz F =8 Hz F =10 Hz F =12 Hz F =14 Hz F =16 Hz F =18 Hz F =20 Hz
P [kW] B A =310 G B A =510 G B A =610 G B A =780 G B A =940 G B A =1130 G B A =1230 G B A =1400 G B A =1550 G B A =100 G 0 5 10 15 20 0 0.5 1 1.5 2 2.5 3 3.5 4 F [Hz] P [kW] B A =300 G B A =440 G B A =630 G B A =770 G B A =930 G B A =1080 G B A =1260 G B A =1380 G B A =1580 G B A =160 G
P [kW] B A =300 G B A =470 G B A =650 G B A =770 G B A =920 G B A =1080 G B A =1240 G B A =1440 G B A =1550 G B A =160 G F 1 F 2 2 F 1 F F 2 =0 F 1 (a) (b) (c) (d) (e) (f)
P J (U , B) ∼ P J (U , B A ) ∼ σU 2 B A 2 R 3 . (3.2)
At this order, one also assumes that the typical velocity (and the turbulent fluctuations) do not vary with B A -this approximation, certainly valid at low N values, deteriorates as the applied magnetic field increases. So, by taking U = u rms , the Joule dissipation can be estimated as:

P J (U , B) ∼ P J (U, B A ) ∼ σu 2 rms B A 2 R 3 = σ B A 2 R ρu rms • ρu 3 rms R 2 ∝ N P H (U, 0) , (3.3)
so that one expects, at first-order, the net power delivered by the motors to scale as where γ is a dimensionless constant which depends on the configuration. Figure 3(d-f)

P T (U , B) = P H (U, 0) + P J (U, B A ) = P H (U, 0) [1 + γN ] , (3.4
B A = 130 G B A = 380 G B A = 640 G B A = 920 G B A = 1120 G B A = 1410 G B A = 1580 G 0 0.1 0.2 0.3 0.4 0 1 2 3 4 5 6 7 8 N P [kW] F =6 Hz F =8 Hz z F =10 H z F =12 H z F =14 H z F =16 H z F =18 H z F =20 H (a) (b)
shows that, at first-order, this scaling is in good agreement with the measurements over the range of explored magnetic field intensities. However, note that the coefficient γ is not constant as dimensionally predicted, but grows non linearly with the rotation rate of the impellers. A saturation of the power injection is also observed for the one-disk configuration. This traces back to our assumption that the net magnetic field in the flow is essentially the applied one. If, in the estimation of the induced current, one includes the induced magnetic field B I , j ∼ σU (B A + B I ), with B I = F(U, B A ), one introduces a dependence of γ on Rm and N .

Let us now finally consider the case where the applied magnetic field is parallel to the axis of rotation, the two disks rotating in opposite directions and at the same speed. The induction processes with an axial applied magnetic field for other types of flow are much weaker and have not been systematically studied. The evolution of the injected power as a function of the impeller rotation rate, for increasing values of the applied magnetic field, is displayed in figure 4(a). As for a transverse applied magnetic field, the evolution of the injected power is not strongly affected by the amplitude of the applied magnetic field.

Here the evolution as a function of the interaction parameter (figure 4(b)) is not linear: for small interaction parameter, the injected power decreases as the applied magnetic field increases; whereas for the highest interaction parameters, the power increases with the applied magnetic field. In this configuration, it is known that a strong applied magnetic field tends to decrease the Kelvin-Helmholtz instability which occurs in the mid-plane of a von Kármán flow (see for instance [START_REF] Biskamp | Magnetohydrodynamic turbulence[END_REF]). This laminarization of the shear layer tends to decrease the turbulent energy dissipated in the flow. This effect might be stronger than the dissipation linked to the Joule effect for low interaction parameter and thus lead to a decrease of the total injected power at the lowest N values.

To conclude this section, let us return to our interpretation of the linear scaling of P (N ). At first-order, this argument, based on dimensional analysis, is valid as long as the energy transfer in the turbulent cascade scales as u 3 rms / . However, this simple argument is not sufficient to fully understand the evolution of the injected power which is strongly linked to the topology and the dynamics of the flow. Thus, this global analysis has to be supplemented by an inspection of the evolution of the local velocity gradients. For low-Rm flows, magnetic measurements provide this kind of information, as detailed in [START_REF] Volk | Fluctuation of magnetic induction in von Kármán swirling flows[END_REF].

Local dynamics : induced magnetic field

Experimental results

The evolution of a local variable, namely the induced magnetic field B I , as a function of the interaction parameter N is investigated for the different configurations studied. In this section, all measurements are performed in the mid-plane, at r ∼ 0.36R along the y-axis.

As for the global variable, let us first consider the configurations with a transverse applied magnetic field. Before discussing the effect of large N values on the induced magnetic field, the basic features of the time-averaged induction processes at low N values are briefly recalled here for the three flows. For the counter-rotating flow, the main induced component at the measurement location is along the z-axis. This induction process is linked to the differential rotation of the toroidal cells and the electrical boundary condition of the flow, as described in [START_REF] Bourgoin | Induction mechanism in von Kármán swirling flows of liquid gallium[END_REF]. The induced magnetic field, normalized by the value of the applied magnetic field, was shown to vary linearly with Rm. For the one-disk flow, the flow is strongly helical, and the induced magnetic field in the axial direction is generated by Parker's stretch-and-twist mechanism introduced in Parker (1955); the normalized induced magnetic field varies quadratically with Rm at low N values, as observed experimentally in [START_REF] Pétrélis | Nonlinear magnetic induction by helical motion in a liquid sodium turbulent flow[END_REF]. Finally in the case of the corotating flow, the strong coherent vortex generated in the center of the flow vessel causes expulsion of the transverse magnetic field, as detailed in [START_REF] Moffatt | Magnetic field generation in electrically conducting fluid[END_REF]; this effect is tracked by measurements of the radial component of the induced magnetic field [START_REF] Odier | Magnetic induction by coherent vortex motion[END_REF]; the normalized induced magnetic field varies linearly with Rm at small Rm values, before showing saturation when the magnetic expulsion from the vortex becomes significant.

The evolution of the time-averaged value of the induced magnetic field, normalized by the value of the applied magnetic field, is displayed in figure 5(a-c) as a function of the magnetic Reynolds number, for increasing values of the applied magnetic field. For small values of the applied magnetic field, the normalized induced magnetic field varies with the magnetic Reynolds number as described above. The situation changes significantly as the intensity of the applied magnetic field is increased. The amplitude of the induced magnetic field decreases for all flows, and the induction mechanisms become more complex, as illustrated in the case of the counter-rotating flow: the evolution with Rm is no longer linear and changes noticeably with B A (note in figure 5(a) the significant changes even for quite moderate values of the applied magnetic field).

However, it will now be shown that measurements collapse onto a master curve for each configuration. In a previous study of the influence of large magnetic field on turbulent flows, [START_REF] Brito | Experimental study of a geostrophic vortex of gallium in a transverse magnetic field[END_REF] have shown that a unified behavior can be derived when using the (modified) value of the flow velocity in the definition of an effective magnetic Reynolds number. This is indeed what is observed in the present case when using

Rm eff = Rm(1 -α √ N )
, where α is a dimensionless constant which depends only on the configuration. The discussion on the derivation of such an effective magnetic Reynolds is left to subsection 4.2. The corresponding rescaled curves are shown in the bottom panels of figure 5(d-f) with α = 3.8 for the counter-rotating flow, α = 0.7 for the one-disk flow and α = 1.15 for the corotating flow. For each configuration, the collapse is obtained with one adjustable parameter, the coefficient α, which does not depend on the amplitude of Let us now consider the configuration with an applied magnetic parallel to the axis of rotation of the impellers, with the counter-rotating flow. The leading-order induction process, at the measurement location, is the so-called ω-effect, which induces an azimuthal magnetic field from the strong differential velocity of the toroidal cells [START_REF] Bourgoin | Induction mechanism in von Kármán swirling flows of liquid gallium[END_REF]; at low N values, the normalized induced magnetic field varies linearly with Rm. Its evolution as a function of the magnetic Reynolds number is displayed in figure 6(a) for increasing values of the applied magnetic field. Similarly to the transverse configurations, the induced magnetic field is reduced at large N values, and the effective magnetic Reynolds number Rm eff ∝ Rm(1α √ N ) is also observed to collapse the data, as shown in figure 6(b) with α = 0.5. A similar study in the axial configuration has been reported in [START_REF] Gallet | Influence of an external magnetic field on forced turbulence in a swirling flow of liquid metal[END_REF]. In that paper, it is shown that the induced magnetic field B I normalized by RmB A only depends on the interaction parameter N . It is possible to fit these data using the scaling we propose with α = 0.56, which is close to the value measured in the present work. A direct comparison of the behavior of the small-scale fluctuations is less straightforward. [START_REF] Gallet | Influence of an external magnetic field on forced turbulence in a swirling flow of liquid metal[END_REF] report velocity fluctuations using a bandpass potential probe and observed that the spectrum is mainly modified at high frequencies, i.e. for turbulent scales. In our case, the spectrum of the induced magnetic field reported in section 5 shows that the dominant effect of the magnetic field lies in the dynamics of the largescales.

F 1 F 2 B I r 2 F 1 F B I z F 2 =0 B I z F 1 0 1 2 3 4 5 0 0.1 0.2 0.3 0.4 0.5 B A =300 G B A =460 G B A =640 G B A =760 G B A =920 G B A =1080 G B A =1240 G B A =1420 G B A =1540 G B A =160 G Rm B I /B A
.08 B A =340 G B A =500 G B A =620 G B A =780 G B A =920 G B A =1120 G B A =1260 G B A =1400 G B A =1540 G B A =160 G Rm B /B A
Rm B I /B A B A =300 G B A =440 G B A =620 G B A =780 G B A =920 G B A =1080 G B A =1240 G B A =1380 G B A =1580 G B A =160 G (a) (b) (c) (d) (e) (f)

Rm

B I θ /B A z B A = 130 G B A = 380 G B A = 640 G B A = 920 G B A = 1120 G B A = 1410 G B A = 1580 G -4 -2 0 2 4 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 Rm 1 -0.5 √ N B I θ /B A z B A = 130 G B A = 380 G B A = 640 G B A = 920 G B A = 1120 G B A = 1410 G B A = 1580 G (a) (b)

Discussion and estimation of the Joule dissipation

In this subsection, based on power budget estimates, we first give an explanation for the expression of the effective magnetic Reynolds number, Rm eff = Rm(1α √ N ), then we discuss the consistency of our approach by evaluating the Joule dissipation in the system.

Let us now return to the flow power consumption, P T (U , B) = P H (U , B) + P J (U , B) addressed in section 3. Since no direct measure of the changes in the velocity field as the applied magnetic field is imposed is available, and since the velocity field and the magnetic field are coupled, any approach to determine the relative importance of the Joule dissipation compared to the viscous dissipation is based on a series of approximations. In section 3, the lowest order was considered, for which (i) the induced magnetic field is neglected in the Joule dissipation and (ii) the Lorentz force is neglected and the hydrodynamic dissipation is estimated assuming an unperturbed velocity field. In that case we showed that P J (U, B A ) ∝ N P H (U, 0) and that

P T (U , B) ∼ P T (U, B A ) = P H (U, 0) [1 + γN ] (4.1)
which is in good agreement with the power consumption data, but which also implies that B I /B A ∝ Rm. The ratio of the induced magnetic field to the applied magnetic field should scale linearly with Rm, which is in contradiction with our observations. To account for the observed evolution of B I /B A with Rm and N and its rescaling using the effective magnetic Reynolds number Rm(1-α √ N ), the next-order approximation should be considered: the modification of the velocity field under the action of the Lorentz force has to be estimated. As the interaction parameter is moderate, one can assume that the modification of velocity field v by the Lorentz force is small: v = uu u where u and u are the velocity fields with and without an applied magnetic field. Let us now assume that (i) the induced field is neglected in the Joule dissipation as in section 3, and (ii) the hydrodynamic dissipation is estimated from the modified velocity field U :

P T (U , B) ∼ P T (U , B A ) = P H (U, 0) + P H (V, 0) + P J (U, B A ) (4.2)
which implies that u and v are not correlated. The observed linearity of the injected power as a function of the interaction parameter suggests that the hydrodynamic dissipation linked to the velocity field v also scales linearly with N : P H (V, 0) ∝ N P H (U, 0). Writing P H (V, 0) as ρR 3 V 2 /τ with τ the characteristic energy transfer time still being of the order of U/R, one gets

P H (V, 0) ∼ ρR 2 V 2 U ∝ N P H = N ρR 2 U 3 (4.3)
The velocity perturbation is thus estimated as V ∝ U √ N . Using the modified velocity U' = U+V in the estimation of an effective magnetic Reynolds number as was introduced in [START_REF] Brito | Experimental study of a geostrophic vortex of gallium in a transverse magnetic field[END_REF], indeed leads to an effective magnetic Reynolds number defined as Rm(1α √ N ), as proposed in section 4.1. Finally, as a complementary approach and to the same order of approximation as for equation 4.2, one can assume that (i) the induced magnetic field has to be taken into account for the Joule power dissipation and (ii) that the Lorentz force is neglected:

P T (U , B) ∼ P T (U, B) = P H (U, 0) + P J (U, B A + B I ) (4.4)
The magnetic induction measurements are now used to improve the estimation of the Joule power dissipation. For each flow configuration, the experimental curves B I /B A are fitted by a function g(Rm eff ) with Rm eff = Rm(1α √ N ). Joule dissipation is then expressed as

P J (U, B A + B I ) ∼ σU 2 (B I + B A ) 2 R 3 = C J N P H (1 + g(Rm eff )) 2 (4.5)
where C J is a dimensionless constant. The net power consumption can then be computed according to equation 4.4. These estimations are displayed as crosses (×) on figures 3 and 4. Good agreement is found for C J = (1/2) 3 for the one-disk flow and the counter-rotating flow with a transverse applied magnetic field, and C J = (1/3) 3 for other configurations (this may be thought of as a change in the spatial extent of the electrical currents). The discrepancies for the largest N values (10-20 % of the injected power) may be due to the dissipation associated with the dynamics of the large scale.

To conclude this section, an investigation of the evolution of the induced magnetic field with the interaction parameter shows that the use of an effective magnetic Reynolds number Rm eff ∝ Rm(1α √ N ) is compatible with the global power dissipation scalings derived in section 3 and collapses the measurements for all investigated configurations. Although the power dissipation scaling is based on the assumption that the flow is only marginally modified, magnetic induction measurements show that the details of the flow topology can be locally strongly modified by the Lorentz force (refer to the dramatic modification of the induced magnetic field in figure 5(a)). A direct measurement of the velocity field is not yet available. However, a detailed study on the evolution of the Transition from hydrodynamic to magnetohydrodynamic turbulence 13 induced magnetic field in the counter-rotating case as a function of N would illustrate the action of the Lorentz force on the flow.

Transition from hydrodynamic to magnetohydrodynamic flows

We investigate in this section the details of how the transition from a situation where the magnetic fields plays the role of a passive vector to a situation where the magnetic field has a leading influence on the flow dynamics. In this problem, geometry is of prime importance. We will focus on the configuration where the applied magnetic field is perpendicular to the rotation axis of the counter-rotating flow of the von-Kármán flow. The rationale is that very large magnetic fields or rotation rates will each act to two-dimensionalize the flow in planes perpendicular to its direction. This competition is expected to, and does, generate non trivial dynamics [START_REF] Zikanov | Direct numerical simulation of forced MHD turbulence at low magnetic Reynolds number[END_REF], as presented here.

As in the previous section, all magnetic measurements are made in the mid-plane, at r ∼ 0.36R along the y-axis, the applied magnetic field is parallel to the x-axis and the two impellers generate a counter-rotating flow by rotating at F = 12 Hz in opposite directions. As we want to study the transition, we limit ourselves to interaction parameter smaller than 0.1 (B A ∼ 1200 G). In the range 0.1 < N < 0.27, no flow modification has been observed.

Figure 7(a) shows the evolution of the time-averaged induced magnetic field as a function of the interaction parameter. For small applied magnetic field, the induced magnetic field is proportional to the square root of the interaction parameter, which is equivalent to the induced magnetic field being proportional to the applied magnetic field, as expected at low Rm, for which the induction equation is dominated by the balance between the stretching of field lines and diffusion, i.e. λ∆B I ∼ B A • ∇u. We refer to this regime as an "hydrodynamic" regime, because the main features are accounted for when one computes the induced magnetic field from an purely hydrodynamic velocity field (i.e. with vanishing influence of the Lorentz force in the NS equation). At higher magnetic fields, the √ N scaling is no longer observed, and the induced magnetic field eventually decreases with increasing N values. This regime where the flow is strongly influenced by the magnetic field will be referred to as a magnetohydrodynamic (MHD) regime. The transition to this regime can be estimated in the following manner. From induction measurements, let us now redefine an effective magnetic Reynolds number as Rm ind eff ≡ B I /B A . Then, our experimental observation is that the MHD regime is reached when N > Rm ind eff , as shown in figure 7(a). Physical insight is gained when expressing the interaction parameter as

N = σRB A 2 ρu rms = µσRu rms • B A 2 ρµu 2 rms ∼ Rm ind eff • v A u rms 2 , (5.1)
where v A is the Alfvèn velocity. In this context, the transition to the MHD regime happens when the Alfvèn velocity becomes of the order of the hydrodynamics velocity fluctuations (i.e. N ∼ Rm ind eff ). This is indeed observed in figure 7.a for N 0.02. For instance, for F = 12 Hz, the maximal velocity is of the order of U = 2πRF ∼ 7 m/s with fluctuations of about half that value (in the pure hydrodynamic situation); the Alfvèn speed for B A = 1 kG is v A ∼ 1 m/s. The flow fluctuations locally excite Alfvèn waves which are evanescent in liquid metals (in conditions where the applied magnetic field is about 1 kG). The Joule dissipation linked to the local current is quite effective at damping the motion.

The fluctuations of the induced magnetic field display the same behavior as shown ). In the hydrodynamic regime, the lowfrequency spectral index of the induced magnetic field PSD is around -1/2 [START_REF] Volk | Fluctuation of magnetic induction in von Kármán swirling flows[END_REF]). In the transition regime, the increase of the low-frequency fluctuations lead to a spectral index reaching -1. In the MHD regime, there is a dramatic damping of lowfrequency fluctuations. On the other hand, the high-frequency turbulent fluctuations are not modified at large applied magnetic field. For the three regimes, the spectral slope index is compatible with the turbulent f -11/3 scaling. The probability density functions (p.d.f.), displayed in figure 9(b), are also strongly modified when increasing the interaction parameter. The variance and the flatness are increased in the transition regime, before decreasing at larger values of the interaction parameter. For the three regimes, the deviation from a Gaussian disappears when highpass filtering the signal (cut-off frequency f c = F = 12 Hz -not shown).

C B I (τ ) B A =150 G B A =400 G B A =570 G B A =670 G B A =970 G 0
Such long correlation times and slow evolutions in turbulent flows have been previously reported in situations where the flow topology changes slowly in time between several preferential states [START_REF] Cortet | Experimental evidence of a phase transition in a closed turbulent flow[END_REF]. As an attempt to test further this possibility of preferential states, we use an array of Hall probes to record the induced magnetic field profile along a radial direction in the vessel, as introduced in [START_REF] Volk | Fluctuation of magnetic induction in von Kármán swirling flows[END_REF]. For the three previous values of the applied magnetic field B A , we show in figure 10 the spatial evolution of the normalized magnetic field. The time-averaged spatial profile is displayed by filled black symbols embedded in a grey area which represents the standard deviation variations around the mean profile. Extrema are displayed by open and grey symbol for the furthest below and above the mean profile. The mean profile is defined as the time-averaged profile over 180 seconds, i.e. 1800 flow integral turnover times, which corresponds to 1300 magnetic diffusion times. Averages over 1 s are referred to as instantaneous profiles (10 turn-over times or 7 magnetic diffusion time).

In the hydrodynamic regime, the induction profile is identical to previous measure- ments by [START_REF] Volk | Fluctuation of magnetic induction in von Kármán swirling flows[END_REF], which peaks at about 3R/4 in the mid-plane. In the MHD regime, the shape is drastically different: the induced magnetic field no longer varies appreciably in the radial direction. In the transition regime, the induced profile is observed to change chaotically between a configuration which lies close the hydrodynamics case and a configuration close to the MHD case. The long-time dynamics of the transitions, the existence of hysteresis and the nature of the bifurcation between these two regimes are currently under investigation.

Concluding remarks

Our study has focused on essential effects of a magnetic field applied on a turbulent flow, as it evolves from a situation for which the magnetic field is a passive tracer to a situation where the Lorentz force is dominant. We have observed that, at leading order, the flow power consumption increases linearly with the interaction parameter N . A firstorder of approximation for the power consumption leads to the introduction of an effective magnetic Reynolds number Rm eff = Rm(1α N ). The effective magnetic Reynolds number, introduced from global measurements, also collapses local magnetic induction measurements. Finally, the very good agreement between the estimate of the Joule dissipation from induced magnetic field measurements at the same order of approximation and the measured power consumption showed the consistency of our approach. When the magnetic field is perpendicular to the axis of rotation of the VK flow, a refined study on the modification of the induced magnetic field at a given flow driving exhibits a transition from a hydrodynamic regime to an MHD regime as N exceeds about 0.05. In the MHD regime the large scale dynamics is significantly changed, essentially corresponding to a slow down of the flow under the action of the Lorentz force. The small-scale turbulent fluctuations are not significantly modified; this happens at larger values of the interaction parameter. In the transition regime from the hydrodynamic to the MHD regimes, strong increase in the flow fluctuations was observed. This corresponds to topological changes as the flow alternates chaotically between the original hydrodynamic mean flow profile and one dominated by MHD forces. This type of increased intermittency in the largescales may be generic in situations where the large scale vorticity and magnetic fields are perpendicular. Solving the details of this complex MHD dynamics requires highresolution measurements of the evolution of the local velocity field; such measurements are progressively becoming available, and will be implemented in future studies.

Figure 1 .

 1 Figure 1. (a) Geometry of the experimental setup (see details in the text), and coordinate system used throughout the text; the magnetic probe shaft is along the y-axis. (b) Counter-rotating (s2t2 ) flow : two counter-rotating toroidal cells (solid arrows) and two poloidal cells (dashed arrows). (c) One-disk (s1t1 ) flow : one toroidal cell and one poloidal cell. (d) Corotating (s2t1 ) flow: one toroidal cell and two poloidal cells.

Figure 2 .

 2 Figure2. Time evolution of the power consumption, measured from the output of the motors' electric power supplies. Counter-rotating case at 10 Hz with an applied magnetic field B A = 100 G (solid line) and B A = 1550 G (dashed line).

Figure 3 .

 3 Figure3. Evolution of the injected power as a function of the rotation rate F (top panels (a) to (c)) and as a function of the interaction parameter N (bottom panels (d) to (f)): measurements ( ) and estimation (×), see section 4.2. The applied magnetic field is perpendicular to the axis of rotation. The three investigated flows are: counter-rotating flow (a,d), one-disk flow (b,e) and corotating flow(c,f)

Figure 4 .

 4 Figure 4. Evolution of the injected power for the counter-rotating flow with an axial applied field as a function of the rotation rate (a) and the interaction parameter (b): measurements (•), estimation (×).

Figure 5 .

 5 Figure 5. Evolution of the induced magnetic B I (straight arrows as shown in the sketch of the experimental setup in figure 1) as a function of the magnetic Reynolds number F for increasing values of the transverse applied magnetic field B A (a-c), and as a function of an effective magnetic Reynolds number Rm(1 -α √ N ) (d-f). Three distinct induction processes are probed for the three investigated flows (see text for details): counter-rotating flow (a-d), one-disk flow (b-e) and corotating flow (c-f).

Figure 6 .

 6 Figure 6. Evolution of the azimuthal induced magnetic field from an axial applied magnetic field with a counter-rotating flow, as a function (a) of the magnetic Reynolds number Rm and (b) of an effective magnetic Reynolds number Rm(1 -α √ N ).

Figure 7 .

 7 Figure 7. Evolution of the axial component of the induced magnetic field measured in the mid-plane as a function of N : (a) time-averaged value; (b) standard deviation. The disks rotate at 12 Hz in opposite directions and the applied magnetic field is transverse. Insets show value normalized by the applied magnetic field. The solid line corresponds to a fit of the experimental measurements in the hydrodynamic regime and the dashed line corresponds to N = B I /B A .

Figure 8 .Figure 9 .

 89 Figure 8. (a) Correlation function C B I (τ ) of the induced magnetic field for different applied magnetic fields. (b) Correlation time of the induced magnetic field as a function of the interaction parameter. Point with number correspond to values at which the correlation function evolution is displayed in (a).

Figure 10 .

 10 Figure 10. Instantaneous profiles of the axial induced magnetic field in the mid-plane: a) B A = 150 G, hydrodynamic regime, b) B A = 520 G, transitory regime, c) B A = 1200 G, MHD regime. The black symbols correspond to the time averaged induction profile. The furthest profiles from the mean values are represented by dashed symbols (greater than the mean) and empty symbols (smaller than the mean). The gray area corresponds to the standard deviation of the profile.

Acknowledgements

We acknowledge the technical support of M. Moulin. We thank an anonymous referee for remarks that raised the discussion of section 4.2. This work was supported by contract ANR-08-BLAN-0039-02 and CNRS-RFBR PICS Grant No. 07-01-92160