Yacouba Kaloga

Marion Foare
email: marion.foare@ens-lyon.fr

Nelly Pustelnik

Pablo Jensen

‡ Y Kaloga

Discrete Mumford-Shah on graph for mixing matrix estimation

Keywords: Mumford-Shah, graph, mixing matrice estimation, nonconvex optimisation

come

Introduction

This work focuses on mixing matrix estimation on a graph in the context where the sources are known, but that the mixing matrices differ for each node of the graph. This general setting involves a graph G = (V, E), where E denotes the edges, V the vertice, and |V| the total number of vertices. Our goal is to estimate, for each node n ∈ V, a mixing matrix M n ∈ R P ×Q from the observations z = (z n ) n∈V ∈ R |V| and the sources s = (s n ) n∈V ∈ R |V| such that

(∀n ∈ V) z n = M n s n . (1)
We set M = (M 1 , ..., M |V| ) ∈ R P ×Q×|V| . A specific application of [START_REF] Bioucas-Dias | Hyperspectral unmixing overview: Geometrical, statistical, and sparse regression-based approaches[END_REF], when G models a regular grid, is encountered in hyperspectral unmixing imagery [START_REF] Bioucas-Dias | Hyperspectral unmixing overview: Geometrical, statistical, and sparse regression-based approaches[END_REF], for which recent contributions in the convex setting, involving total variation, have been proposed (see [START_REF] Tiard | Robust linear unmixing with enhanced sparsity[END_REF] and references therein). The framework of problem (1) also fits the challenging question of vote transfer matrix estimation at each polling location, for which no state-of-the-art method is able to provide such a local estimate M n , this problem being especially ill-posed [START_REF] Adam | Ecological inference in the social sciences[END_REF][START_REF] Puig | A cluster analysis of vote transitions[END_REF]. The contributions of this paper are: (i) to encompass the mixing matrix estimation problem into the Mumford-Shah formalism to be able to estimate M n with limited assumptions: two matrices M n and M n should be similar if n and n are neighbor nodes in the graph, except at some locations where sharp behaviors can be captured (Section 2); (ii) a new proximal alternating minimization scheme with convergence guarantees as well as a block version of it, to handle large datasets (Section 3); (iii) experimental validation of the proposed numerical schemes and robustness to the initialization (Section 4).

2 Discrete Mumford-Shah on graph

Generalities on Mumford-Shah

Proposed in the late 80's, the Mumford-Shah (MS) model [START_REF] Mumford | Optimal approximations by piecewise smooth functions and associated variational problems[END_REF] is one of the most studied models in image processing, since it allows both to recover a piecewise smooth approximation of an input image corrupted with white Gaussian noise, and to extract the contours at the same time. Formally, let Ω ⊂ R 2 be a bounded, open set and z : Ω → [0, 1] a corrupted input image, the Mumford-Shah model reads [START_REF] Mumford | Optimal approximations by piecewise smooth functions and associated variational problems[END_REF]:

minimize u,Γ 1 2 Ω (u -z) 2 dx + β Ω\Γ |∇u| 2 dx + λH 1 (Γ ∩ Ω) (2)
In [START_REF] Tiard | Robust linear unmixing with enhanced sparsity[END_REF], the first term is the data-term, ensuring similarities between the noisy image and the estimate u1 , the second term favors solutions having a smooth behavior everywhere except on contours Γ and the third term controls the length of Γ by means of the 1D Hausdorff measure. The parameters β, λ > 0 allows us to adjust the contribution of each term in the estimation. Among the large literature dedicated to the resolution of MS in image processing, we can identify three types of methods. The first class of methods is restricted to piecewise constant approximations such as the Chan-Vese formulation [START_REF] Tony | Active contours without edges[END_REF], the Potts model [START_REF] Potts | Some generalized order-disorder transformations[END_REF][START_REF] Geman | Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images[END_REF][START_REF] Storath | Fast partitioning of vector-valued images[END_REF][START_REF] Kiefer | Iterative Potts minimization for the recovery of signals with discontinuities from indirect measurements-the multivariate case[END_REF] or its convex relaxation known as the ROF model [START_REF] Rudin | Nonlinear total variation based noise removal algorithms[END_REF]. Efficient iterative strategies have been designed and a recent work by Cai and Steidl [START_REF] Cai | Multiclass segmentation by iterated ROF thresholding[END_REF] establishes the link between a thresholded-ROF strategy and the MS model. The second class relies on the Blake-Zisserman model [START_REF] Hohm | An algorithmic framework for Mumford-Shah regularization of inverse problems in imaging[END_REF][START_REF] Weinmann | Iterative Potts and Blake-Zisserman minimization for the recovery of functions with discontinuities from indirect measurements[END_REF][START_REF] Strekalovskiy | Real-time minimization of the piecewise smooth Mumford-Shah functional[END_REF] which handles the piecewise smoothness by considering a different approximation of MS known as the discrete membrane energy. Convergence to local minimizers was proved for the 1D formulation in [START_REF] Weinmann | Iterative Potts and Blake-Zisserman minimization for the recovery of functions with discontinuities from indirect measurements[END_REF]. However, the major drawbacks of the extensions to image processing derived in [START_REF] Hohm | An algorithmic framework for Mumford-Shah regularization of inverse problems in imaging[END_REF] and [START_REF] Strekalovskiy | Real-time minimization of the piecewise smooth Mumford-Shah functional[END_REF] are the weak convergence guarantees, and the lack of flexibility in the data-term choice, whose proximity operator should have a closed form expression. Finally, for handling with the exact MS model, which estimates jointly the contours and the image, we can refer to Ambrosio-Tortorelli alternatives [START_REF] Ambrosio | Approximation of functional depending on jumps by elliptic functional via t-convergence[END_REF][START_REF] Ambrosio | On the approximation of free discontinuity problems[END_REF][START_REF] Foare | Image restoration and segmentation using the Ambrosio-Tortorelli functional and discrete calculus[END_REF], at the price of a huge computational time, or the strategy we proposed in [START_REF] Foare | Semi-Linearized Proximal Alternating Minimization for a Discrete Mumford-Shah Model[END_REF]. Note that in the literature, the extension of MS to graphs relies on a Chan-Vese formulation [START_REF] Grady | The piecewise smooth Mumford-Shah functional on an arbitrary graph[END_REF]. Thus, it is limited to foreground/background segmentation, with weak convergence guarantees due to the structure of the proposed objective function, and cannot handle more complex data-term.

Extension to graphs and mixing matrix estimation

In this work, we extend the Discrete MS-type (D-MS) model proposed in [START_REF] Foare | Semi-Linearized Proximal Alternating Minimization for a Discrete Mumford-Shah Model[END_REF] for image denoising to mixing matrix estimation on graph. We propose to formulate the data-term as

∀M ∈ R P ×Q×|V| L(M ) = n∈V z n -M n s n 2 2 .
The second term aims to promote small variations between two adjacent matrices M n and M n :

S(M , e) = β n∈V n ∈N (n) (1 -e n,n ) 2 M n -M n 2 F
where • F stands for the Frobenius norm, N (n) models the neighborhood of n and where e n,n ∈ R denotes the value of the edge between the nodes n and n . This single penalization ensures e n,n ≡ 1, which cancels this term during the minimization. Thus, another penalization term on e favoring sparsity is required, e.g. a 1 -norm or a quadratic-1 penalization [START_REF] Foare | A new proximal method for joint image restoration and edge detection with the Mumford-Shah model[END_REF]. This penalization over e, controlled by a parameter λ > 0, will be denoted by R. Intuitively, R tends to favor e n,n = 0, except at the locations of strong variations (that is, the contour), where S must vanish, meaning e n,n ≈ 1. Note that the balance between S and R ensures e n,n ∈ [0, 1], since e n,n < 0 increases S while e n,n > 1 increases R. Unlike image denoising tasks, mixing matrix estimation on graph requires additional hard constraints C over M , e.g. a simplex constraint on the columns of each M n . The resulting D-MS model in the context of mixing matrix estimation on graph reads minimize

M∈R P×Q×|V| e∈R |E| Ψ(M , e) := L(M ) + S(M , e) + R(e) + ι C (M ) (3)
where C ⊂ R P ×Q×|V| and ι C denotes an indicator function

2 separable in n, i.e. ι C (M ) = |V| n=1 ι Cn (M n ), where, ∀n ∈ V, C n ⊂ R P ×Q , such that C = C 1 × • • • × C |V| .
This minimization problem is nonconvex with respect to (M , e). In the next section we propose an algorithmic scheme relying on proximal alternating minimization strategy allowing to build a sequence that converges to a critical point.

3 Proposed algorithm

Semi-Linearized Proximal Alternating Method (SL-PAM)

The most encountered strategy to find a solution ( M , e) of ( 3) relies on the Gauss-Seidel scheme (i.e., coordinate descent method), whose convergence guarantees require the minimum of (3) to be uniquely attained at each update of the algorithm, e.g. by assuming the strict convexity with respect to one argument when the other one is fixed. Such a condition being difficult to satisfy in practice, it has been proposed to perform proximal regularization of the Gauss-Seidel scheme. This yields

M [k+1] ∈ arg min M Ψ(M , e [k] ) + µ k 2 M -M [k] 2 F , e [k+1] ∈ arg min e Ψ(M [k+1] , e) + η k 2 e -e [k] 2 2
, where µ k and η k are positive real numbers [START_REF] Auslender | Asymptotic properties of the Fenchel dual functional and applications to decomposition problems[END_REF]. The proof of convergence to a critical point of such a scheme is due to a recent work by Attouch et al. [START_REF] Attouch | Proximal alternating minimization and projection methods for nonconvex problems: An approach based on the Kurdyka-Łojasiewicz inequality[END_REF]. The proof is provided in the nonsmooth and nonconvex setting. The main practical issue is to have a closed form expression of both proximity operators 3 . This has been relaxed by Bolte et al. [START_REF] Bolte | Proximal Alternating Linearized Minimization for nonconvex and nonsmooth problems[END_REF], who derived a proximal alternating linearized scheme (PALM). A hybrid version, named SL-PAM for semi-linearized proximal alternating direction method, has been proposed in [START_REF] Foare | Semi-Linearized Proximal Alternating Minimization for a Discrete Mumford-Shah Model[END_REF], especially adapted to the resolution of the D-MS problem. The iterations of SL-PAM to solve (3) would read

M [k+1] ∈ prox L µ k +ιC M [k] - 1 µ k ∇ 1 S M [k] , e [k] , e [k+1] ∈ prox 1 η k R+S(M [k+1] ,•) e [k] ,
where ∇ 1 denotes the gradient with respect to the first variable. In [START_REF] Foare | Semi-Linearized Proximal Alternating Minimization for a Discrete Mumford-Shah Model[END_REF], a closed form expression has been derived for the update of e while, for the basic image denoising MS, the update of M only involves the proximity operator of L. When dealing with mixing matrix estimation, it is thus of almost importance of either having a closed form expression of the proximity operator involved in the update of M [k+1] or providing a new iterative scheme without such an updating step. However, the proximity operator of a sum of functions is known to have closed form expression for a very limited number of functions (see e.g. [START_REF] Chaux | Nested iterative algorithms for convex constrained image recovery problems[END_REF][START_REF] Yu | On decomposing the proximal map[END_REF][START_REF] Pustelnik | Proximity operator of a sum of functions; application to depth map estimation[END_REF] and references therein), which does not fit to the update of M [k+1] . We thus propose a new proximal alternating scheme.

New algorithmic scheme -SL2-PAM

In the context of mixing matrix estimation on graph considered in this study, the data-term is differentiable with a Lipschitz gradient, we can thus derive the iterations summarized in Algorithm 1,

where P C denotes the projection onto the set C, whose convergence guarantees are provided in Proposition 3.2 and the proof is given in Appendix 6.1.

Algorithm 1 (SL2-PAM) for solving D-MS (3)

Set

M [0] ∈ C and e [0] ∈ R |E| . For k = 0, 1, . . .             -Parameter setting - Let ν(e [k] ) be the Lipschitz constant of ∇ 1 S •, e [k] + ∇L. Set γ > 1 , µ k = γν(e [k]
) and η k > 0. -Variables updates -

M [k+1] ∈ P C M [k] -1 µ k ∇ 1 S M [k] , e [k] -1 µ k ∇L(M [k] ) e [k+1] ∈ prox 1 η k R+S(M [k+1] ,•) e [k] Assumption 3.1 i) Ψ is a Kurdyka-Łojasiewcz function [26, Def. 2.3],
ii) Ψ and R are bounded below; iii) the updating steps of M [k] , e [k] have closed form expressions; iv) (η k ) k∈N is a positive sequence such that the stepsizes η k belong to (η -, η + ) for some positive

η -≤ η + ; v) ∀n ∈ V, ∀k ∈ N, ∇L + ∇ 1 S(•, e [k]
) is globally Lipschitz continuous with moduli ν(e [k] ), and there exists ν -, ν + > 0 such that ν -≤ ν(e [k] ) ≤ ν + .

Proposition 3.2 Assume that the sequence (M [k] , e [k] ) k∈N generated by Algorithm 1 is bounded, and Assumption 3.1. Then this sequence converges to a critical point of Problem (3).

Block updates

In real applications, ν(•) may be very large, essentially due to the large value of the Lipschitz constant of the data-term L, which implies a very small descent stepsize in the first step of Algorithm 1. In order to accelerate the convergence, we suggest to derive a block-coordinate SL2-PAM relying on the separability of Ψ(•, e) on each node n ∈ V. Problem (3) can be equivalently written

Ψ(M , e) := |V| n=1 L n (M n ) + S n (M n , e) + ι Cn (M n ) + R(e) with S n (M n , e) = β n ∈N (n) (1-e n,n ) 2 M n -M n 2 F and L n (M n ) = z n -M n s n 2
2 . The iterations are provided in Algorithm 2, the convergence results is given in Proposition 3.3 and the proof is given in Appendix 6.1. Proposition 3.3 Assume that the sequence (M [k] , e [k] ) k∈N generated by Algorithm 2 is bounded, and Assumption 3.1. Then this sequence converges to a critical point of Problem (3).

Numerical experiments

Context: estimation of the voting transfer matrices

We illustrate the performance of the proposed D-MS model on graph to the estimation of the vote transfer matrices between two elections [START_REF] Foucault | Les reports de voix aux élections départementales de 2015[END_REF]. While in most of the state-of-the-art studies this estimation is performed globally, the challenge here is to provide an estimate of the voting transfer matrices at each location n. In this case, the set of vertices V of the graph G models the polling Algorithm 2 (Block-SL2-PAM) for solving D-MS (3)

For every n ∈ V, set M [0] n ∈ C n and e [0] ∈ R |E| . For k = 0, 1, . . .               -Parameter setting - Let ν n (e [k] ) the Lipschitz constant of ∇ 1 S n •, e [k] + ∇L n . Set γ > 1 , µ k,n = γν n (e [k] ) and η k > 0. -Variables updates - For n = 1, . . . , |V| M [k+1] n ∈ P Cn M [k] n -1 µ k,n ∇ 1 S M [k]
n , e [k] + ∇L(M

[k] n ) e [k+1] ∈ prox 1 η k R+S(M [k+1] ,•) e [k]
locations, and E is the set of edges between to nodes associated with the 8 nearest neighbors. The resulting graph is a directed acyclic graph with |E| = 8|V|. The involved matrices M n are of size P (2) × P (1) where P (1) denotes the number of candidates (including null votes and abstention) competing in the first election E (1) , and P (2) the number in the second one, denoted by E (2) . Hence, s n (resp. z n ) denotes a vector containing the number of votes cast for each candidate at the n-th polling location in election E (1) (resp. E (2) ). The coefficients in the matrices M n denote transfer percentage. Thus, for every n ∈ V, the constraint C n is selected to be a simplex constraint over the columns to impose value between 0 and 1 with a sum equals to 1. The projection P Cn is computed by means of the efficient implementation provided in [START_REF] Condat | Fast projection onto the simplex and the l1 ball[END_REF]. Moreover, in our experiments, R = λ • 1 and the closed form expression of prox 1

η k R+S(M [k] n ,•)
is given in [START_REF] Foare | Semi-Linearized Proximal Alternating Minimization for a Discrete Mumford-Shah Model[END_REF], so that Assumption 3.1 iii) is fulfilled by the proposed functional. This specific choice implies R to be a bounded below, proper, lower semi-continuous, and semi-algebraic function, and so does Ψ. Then it satisfies Assumption 3.1 i)-ii) (see [START_REF] Bolte | Proximal Alternating Linearized Minimization for nonconvex and nonsmooth problems[END_REF]Th. 5.1]), while Assumption 3.1 iv) is ensured by the definition of S provided in Section III.B.

Synthetic data

The graph is built on the polling locations of Lyon (France) leading to n = 283. To simplify the interpretation and accurately measure the performance of the proposed method, we consider synthetic data, with P (1) = 4 and P (2) = 3, and two piecewise smooth regions V (1) and V (2) such that V = V (1) ∪ V (2) , with a frontier in between (see Figure 1). We set the sources (s n ) n∈V at their real value, and generate (z n ) n∈V signals as follows. For region i ∈ {1, 2}, we define two matrices

M (i) top , M (i)
bottom , and we set the ground truth mixing matrix of each polling place n ∈ V (i) as M (i)

n = P C ω n M (i) bottom + (1 -ω n )M (i) top + ε n .
ε n denotes a white Gaussian noise with standard deviation σ, to model uncertainties as in real data, and, ∀n ∈ V, ω n ∈ [0, 1] depends on the latitude l n of the n-th polling place. That is, we divide the interval of all the latitudes into 20 uniformly distributed subsets (L k ) k=1,...,20 , and we set ω n = k 20 if l n ∈ L k . Notice that, in this case, the gradient models the spatial correlation of voting behavior.

Results

State-of-the-art for voting transfer matrices -To the best of our knowledge, it is not possible to provide fair comparisons to the state-of-the-art, because there are no comparable methods capable of dealing with a large number of nodes and recovering matrix gradients. Indeed, one can either encounter estimation of a unique transfer matrix or, at the price of MCMC simulations, provide clusters of vertices sharing the same matrix [START_REF] Puig | A cluster analysis of vote transitions[END_REF], therefore, it cannot recover matrix gradients as consider here. n ∈ B(0, 1)

M [0] n ≡ 1 3 , e [0]
n ∈B(0,1)

M [0]
n ∈ N (0,1), e Good estimation -In Figure 2, we display the results obtained with the Algorithm 2. We performed a grid search on (β, λ) for the Intersection Over Union (IOU) and the MSE 4 . We notice that for a good choice of regularization parameters (β, λ) the proposed approach is able to provide a good estimate of both the edges (white lines) and the matrix coefficients, here (M n,2,3 ) n .

Block versus global -Figures 3(a,b) compare the algorithmic behavior of Algorithms 1 and 2 in terms of time (we observe similar behavior when the plots are displayed w.r.t iterations) for the optimal (β, λ). The block strategy needs twice less time than the original SL2-PAM to achieve the same accuracy, that is,

|Ψ(M [k+1] ,e [k+1] )-Ψ(M [k] ,e [k] )| |Ψ(M [k] ,e [k] )|
< 10 -10 (cf. Figure 3(b)). The more n increases, the more important the gain is. Robustness -We display in Figure 3(c) the evolution of the objective function for Algorithm 2 with 4 MSE = 1 |V|P (1) P (2) n i,j Mn(i, j) -Mn(i, j) 2 F different initialization choices for the optimal (β, λ). We repeat the experiments 10 times, and we display in Figure 3(d) the average distance of the minimizers to the one we obtained in Figure 3(c).

Note that the one used to generate Figure 3(a,b) corresponds to the solid orange plot. Although the proposed method is nonconvex, we observe that, whatever the initialization, it converges to the same value of the objective function, and the minimizers are very similar. Although this analysis does not guarantee the uniqueness of the minimizer, the proposed functional does not seem to get stuck in a non global local minimum.

Conclusion and perspectives

In this work, we propose an extension of the MS model to graph signal processing, with an application to mixing matrix estimation, two new block-coordinate proximal algorithms with convergence guarantees, the comparison of these two schemes (the block strategy outperforms the global one in terms of computational cost) and their robustness in estimating the vote transfer matrices at each polling location, for which we clearly see the efficiency to estimate properly both the matrices M and the transitions e. We are now working on real data with political scientists, to describe the geographical variations of vote matrices and relate these to political characteristics. A Matlab toolbox is available on the Github repository mfoare/DMS-Graph.

6 Appendix , e [k] ), the proof relies on the proof of [21, Prop.2] (see also [START_REF] Bolte | Proximal Alternating Linearized Minimization for nonconvex and nonsmooth problems[END_REF] for more details):

1) A sufficient decrease property: find ρ 1 > 0 such that (∀k ∈ N), ρ1 2 x [k+1] -x [k] 2 ≤ Ψ(x [k] ) -Ψ(x [k+1] );

2) A subgradient lower bound for the iterates gap: assume that (x [k] ) k∈N is bounded and find ρ 2 > 0 such that w [k] ≤ ρ 2 x [k] -x [k-1] , where w [k] ∈ ∂Ψ(x [k] );

3) Kurdyka-Łojasiewicz (KL) property: assume that Ψ is a KL function and prove that (x [k] ) k∈N is a Cauchy sequence.

The proof of steps 1 and 3 follows similar ideas than in [21, Prop.2], with ρ 1 = min n∈I (γ -1)ν - n , η -. Regarding step 2, we prove the following result: Lemma 6.1 Assume that the sequence {x [k] } k∈N generated by Algorithm 2 is bounded. Let k] , e [k] ) -∇L(M [k-1] ) -∇ 2 S(M [k-1] , e [k-1] ) and

A k M := µ k-1,n (M [k-1] n -M [k] n ) n∈I + ∇L(M [k] ) + ∇ 1 S(M [
A k e := η k-1 (e [k-1] -e [k] ). Then (A k M , A k e ) ∈ ∂Ψ(M [k] , e [k] ) and there exists χ > 0 such that

(A k M , A k e ) ≤ A k M + A k e ≤ 2(C + ρ 2 ) x [k-1] -x [k]
, where ρ 2 = max n∈I γν + n + η + .

Figure 1 :

 1 Figure 1: Black (resp. white) nodes model the polling places in region V (1) (resp. V (2) ). M (.) top , M (.)bottom differ only on coefficient (2,3). However, the additive noise and the projection onto C n will change all the coefficients in the generated M (i) n . Hence, the gradient is both non trivial and irregular.

Figure 2 :

 2 Figure 2: Numerical results on synthetic data with σ = 0.1. (a) Ground truth (M n,2,3 ) n and e, (b) Result obtained with the proposed method: estimated ( M n,2,3 ) n values displayed by means of the red-blue color on the graph while estimated e is displayed by a white line between two nodes when the estimated value is 1. The colorbar represents the value of coefficient (2,3) in the local matrices. (c) Jaccard index map. (d) MSE map. The red box denotes the optimal choice for (β, λ) associated to the result displayed in (b).

Figure 3 :

 3 Figure 3: (a) Comparison of Ψ(M [k] , e [k] ) w.r.t the time for SL2-PAM and Block-SL2-PAM. (b) Com-parison of |Ψ(M [k+1] ,e [k+1] )-Ψ(M [k] ,e [k] )| |Ψ(M [k] ,e [k] )|

6. 1

 1 Common proof for Propositions 3.2 and 3.3To prove Proposition 3.2 (resp. Proposition 3.3), we set I = {{1, . . . , |V|}} (resp. I = {{1}, . . . , {|V|}}). Let x[k] = (M[k]

Note that Problem (2) depends on a pair (u, Γ) of a priori independent objects, and requires to work with a relaxed version of (2) in the class of special functions with bounded variations, depending only on u to ensure the existence of û. More details in[START_REF] De Giorgi | Existence theorem for a minimum problem with free discontinuity set[END_REF][START_REF] Maso | A variational method in image segmentation: existence and approximation results[END_REF].

Let H be a finite dimensional Hilbert space. The indicator function of a nonempty closed subset C ⊂ H is denoted ιC and is such that, for every u ∈ H, ιC (u) = 0 if u ∈ C or +∞ otherwise.

Let ϕ : H →] -∞, +∞] be a proper and lower semicontinuous function. Given u ∈ H and γ > 0, the proximal map associated to ϕ is defined by prox γϕ (u) = arg minv∈H ϕ(v) + γ 2 u -v 2 .

* This work is supported by CNRS Défi InFIniTi 2018 Projet OptMatGeo and ANR GRAPHSIP and by the ACADEMICS Grant of the IDEXLYON project of the Université de Lyon -PIA operated by ANR-16-IDEX-0005.

† M.

Proof. The optimality condition for the updating step on M n in Algoritm 2 is given by

where u [k] ∈ ∂ι C (M [k] ). Hence, using the subdifferential property [26, Prop. 2.1] we obtain that ∇L(M [k] )+∇ 1 S(M [k] , e [k] )+u [k] ∈ ∂ 1 Ψ(M [k] , e [k] ). Similarly, we prove that ∇ e S(M [k] , e [k] )+ξ [k] ∈ ∂ e Ψ(M [k] , e [k] ), where ξ [k] ∈ ∂R(e [k] ). Finally, (A k M , A k e ) ∈ ∂Ψ(M [k] , e [k] ). The end of the proof is the same as for [21, Lemma 2].


Sylvain Dejean

Cerege

The role of distance and social networks in the geography of crowdfunding: evidence from France

Keywords: JEL classification: O30, R12, R23 Crowdfunding, economic geography, gravity, social networks

This article aims to estimate the cost of distance in the geographical flow of crowdfunding, and to show how social ties between the 94 French metropolitan regions shape the geography of funding. Our analysis draws upon a unique database provided by the French leader in rewards-based crowdfunding. The main result is that the elasticity of distance remains important (around 0.5), and that social ties between regions determine the flow of funding. Doubling the number of immigrants in a region increases the number of investments by 24% and reduces the impact of distance.

Introduction

In principle, crowdfunding platforms make big promises. Freedom from the need to rely on a few institutional funders not only makes it possible to attract new funding, it also gives entrepreneurs access to information about the potential of a project, reducing uncertainties [START_REF] Viotto Da Cruz | Beyond financing: crowdfunding as an informational mechanism[END_REF]. They can also benefit from community expertise [START_REF] Belleflamme | Crowdfunding: Tapping the right crowd[END_REF], or the wisdom of backers [START_REF] Mollick | Wisdom or Madness? Comparing Crowds with Expert Evaluation in Funding the Arts[END_REF]. Crowdfunding also claims to democratize access to funding [START_REF] Butticè | Crowdfunding to overcome the liability of outsidership: Drivers of immigrant entrepreneurs' fundraising performance[END_REF][START_REF] Mollick | Democratizing Innovation and Capital Access: The Role of Crowdfunding: California Management Review[END_REF] by reducing gender inequalities [START_REF] Barasinska | Is Crowdfunding Different? Evidence on the Relation between Gender and Funding Success from a German Peer-to-Peer Lending Platform[END_REF][START_REF] Marom | Gender Dynamics in Crowdfunding (Kickstarter): Evidence on Entrepreneurs, Investors, Deals and Taste-Based Discrimination[END_REF] or offering new opportunities to fund projects with civic or public purposes [START_REF] Davies | Three provocations for civic crowdfunding[END_REF]. It also challenges the traditional concentration of economic activities and, by convincing a variety of distant investors to fund a project, promises to change the spatial allocation of capital [START_REF] Agrawal | Some Simple Economics of Crowdfunding[END_REF][START_REF] Mollick | The dynamics of crowdfunding: An exploratory study[END_REF].

Informal investors include business angels and venture capitalists, who are an important resource in the early stages of a venture. At this time, a potential investor faces significant information asymmetry that can prevent him or her from accurately assessing the relevance of a project or the trustworthiness of an entrepreneur. By nature, the information needed to match entrepreneurs and the funding capacity of investors exhibits is largely tacit, especially when few signals of reliability are publicly available. The literature has shown the importance of spatial proximity in addressing this issue for the case of venture capitalists and business angels [START_REF] Harrison | Determinants of long-distance investing by business angels in the UK[END_REF][START_REF] Lutz | Importance of spatial proximity between venture capital investors and investees in Germany[END_REF][START_REF] Mason | Informal Sources of Venture Finance[END_REF]. However, the opportunity offered by crowdfunding platforms to increase global awareness of an idea, or invite feedback and comments, forces the social science researcher to reevaluate the role of distance in this new context.

At the same time, crowdfunding embodies a paradox. While, in theory, these platforms enable anyone to reach anyone else, anywhere, at any time, the vast majority of projects rely on closed social networks [START_REF] Agrawal | Crowdfunding: Geography, Social Networks, and the Timing of Investment Decisions[END_REF][START_REF] Agrawal | Some Simple Economics of Crowdfunding[END_REF]. The phenomenon is not new as, typically, friends and family contribute most among informal investors in the early stages of funding [START_REF] Lee | Financing from Family and Friends[END_REF][START_REF] Riding | Business angels and love money investors: segments of the informal market for risk capital[END_REF] and social ties are a reliable way for investors to obtain private information [START_REF] Shane | Network Ties, Reputation, and the Financing of New Ventures[END_REF]. However, its scale, together with the signal it sends to unknown investors, offer an opportunity to stress the precise role played by social networks in the geography of crowdfunding.

Evaluating the role of social networks as a determinant of social capital helps us to understand their importance in shaping the regional development of crowdfunding. Scholars have shown how complex, regional innovation systems rely upon local endowments in the form of social capital [START_REF] Hauser | The Learning Region: The Impact of Social Capital and Weak Ties on Innovation[END_REF]. Dense networks, tacit knowledge, and actor proximity all help to explain why some regions are more successful than others in their ability to foster economic development [START_REF] Rutten | Regional social capital: Embeddedness, innovation networks and regional economic development[END_REF]. This article draws upon an original dataset provided by the French leader in rewards-based crowdfunding to address two objectives. The first is to evaluate the cost of distance in the context of crowdfunding. To achieve this, we use a gravity-like equation model to aggregate individual funding decisions (about 450 000 investments in 13 000 projects representing 19 million euros) in France's 94 metropolitan regions, and evaluate the elasticity of distance for bilateral funding flows between these regions. In this context, information asymmetry cannot be explained by differences in language, means of payment, banking systems or consumption habits, as France is a relatively small country (it is 10 times smaller than the United States, which is the most-studied country) where people share the same culture and institutions. In this very local context, all of the conditions that are needed to make crowdfunding a financial service that reduces the cost of distance, are met.

The second objective, following [START_REF] Agrawal | Crowdfunding: Geography, Social Networks, and the Timing of Investment Decisions[END_REF], [START_REF] Lin | Home Bias in Online Investments: An Empirical Study of an Online Crowdfunding Market[END_REF] and [START_REF] Mollick | The dynamics of crowdfunding: An exploratory study[END_REF], is to study the role of social networks in the bilateral flow of crowdfunding between French regions. To achieve this, we seek to quantify the social ties weaved between the 94 regions, based on the assumption that underlying social networks support inter-regional investments. Drawing upon data from the 2013 Census, we create a variable that describes social bonds between two regions, based on residential migration statistics. Specifically, data describing how many people born in region i(j) live in region j(i) is used as evidence of a set of interpersonal resources that can be mobilized by users of crowdfunding platforms.

The main finding is that the elasticity of distance remains significant (around 0.5) in the context of reward-based crowdfunding platforms. We also find that inter-regional social ties determine the flow of funding (doubling the number of immigrants in a particular region increases the number of investments by 24%), and also significantly reduces or even cancels out (under certain conditions) the impact of geographical distance.

The remainder of this article is structured as follows; first, we present the theoretical background concerning the impact of distance and social networks on crowdfunding. We then describe the gravity equation model and the dataset before turning to our findings. Finally, we conclude with a discussion of the scope and limits of our approach.

Theoretical background

Why distance matters in crowdfunding

Using a dataset of 109 business angels in the United Kingdom, [START_REF] Harrison | Determinants of long-distance investing by business angels in the UK[END_REF] showed that only 24% of their investments could be considered distant, which is consistent with findings from Canada and Finland [START_REF] Lumme | Informal Venture Capital: Investors, Investments and Policy Issues in Finland[END_REF][START_REF] Riding | Informal investors in Canada: the identification of salient characteristics[END_REF]. Results are more nuanced when considering venture capitalists. [START_REF] Fritsch | Does Venture Capital Investment Really Require Spatial Proximity? An Empirical Investigation[END_REF], for instance, studied 85 German financiers, and concluded that the need for spatial proximity was overestimated. However, in the same context, [START_REF] Lutz | Importance of spatial proximity between venture capital investors and investees in Germany[END_REF] showed that spatial proximity proxied by journey time had an impact on the likelihood of closing a financing round between venture capitalists and German portfolio companies.

According to [START_REF] Mason | Informal Sources of Venture Finance[END_REF], various reasons can explain the persistent effect of distance on informal investment. The first concerns awareness of opportunities, meaning that an investor tends to rely on their immediate environment to find business opportunities. The second is related to the ability of investors to trust the entrepreneur; this issue is particularly salient when funding small-scale ventures or individual entrepreneurs, where success relies on the skills and involvement of a few people. Agency costs are a third factor thought to explain the importance of distance [START_REF] Landström | The relationship between private investors and small firms: an agency theory approach[END_REF]. The cost of monitoring investments increases with distance, deterring investors from investing in riskier projects.

The emergence of crowdfunding has expanded opportunities to finance all categories of ventures, but it is of particular interest for the youngest and smallest ones that are neglected by venture capitalists and business angels. The literature cited above argues that informal investors are also sensitive to spatial proximity; therefore, the question of whether crowdfunding is able to reduce the impact of distance depends on their ability to change the way information is acquired and shared by investors and entrepreneurs.

It is clear that crowdfunding offers new opportunities to signal the existence of a project. From a backer's perspective, it makes no difference if an entrepreneur is local or distant -funding requests are considered in terms of their level of interest, while herding behavior is at least as important as geographical considerations. Consequently, crowdfunding is expected to reduce the distance between funders and project developers by raising awareness of remote, small ventures and individual entrepreneurs.

Crowdfunding seems to have a limited effect on agency costs. On the one hand, platforms enable a wide variety of interactions between the project developer and the crowd who funded him or her. News and progress updates, advice and feedback from the crowd, comments, messages, and questions are some of the many ways to control, monitor and align the interests of an investor and the entrepreneur during the development of the venture. However this ability to exercise distant control or influence the entrepreneur is incomplete and, following Strausz, (2017), unlikely to decrease agency costs as such costs have to be absorbed by each backer. As a result, the net effect of crowdfunding platforms is probably very low and consequently, their ability to reduce the cost of distance.

Trust, knowledge, expertise and experience are all forms of tacit knowledge. As such, this is not easily codifiable and partly immune from digitalization or sharing over telecommunication networks. In this context, proximity or even face-to-face interaction is necessary to internalize these externalities [START_REF] Audretsch | R&D Spillovers and the Geography of Innovation and Production[END_REF][START_REF] Storper | Buzz: face-to-face contact and the urban economy[END_REF]. Although crowdfunding platforms provide various ways to collect information about the project founder, such as his/ her previous experience, motivations, and details about the project, tacit knowledge remains important in establishing trust and mutual commitment. [START_REF] Burtch | Cultural Differences and Geography as Determinants of Online Pro-Social Lending[END_REF] highlighted the importance of cultural distance between lenders and borrowers on a peer-lending platform, and underlined that cultural differences also helped to explain the persistent effect of distance. Similarly, [START_REF] Lin | Home Bias in Online Investments: An Empirical Study of an Online Crowdfunding Market[END_REF] investigated the preference for home-based investment on Prosper, a pioneering crowdlending platform. Their work drew upon three experimental designs to understand the reasons underlying the persistence of home bias. They concluded that the effect can only be driven by emotional reasons, which once again highlights the importance of investors and entrepreneurs sharing non-codifiable information. In Australia, [START_REF] Guenther | Is the crowd sensitive to distance?-how investment decisions differ by investor type[END_REF] studied an equity-based crowdfunding platform, and found a negative correlation between distance and a backer's decision to invest. Their results suggest that the behavior of the crowd is similar to that of venture capitalists, despite the opportunities offered by the platform.

Our study focuses on reward-based crowdfunding. Many of the projects found on these platforms are dedicated to cultural activities [START_REF] Mollick | The dynamics of crowdfunding: An exploratory study[END_REF], and shared norms, culture or history between backers and project owners undoubtedly reinforce the role of non-codifiable information. Consequently, we expect to find little reduction in the cost of distance in this context.

The role of social networks in crowdfunding

Crowdfunding platforms are the first to recognize that relying on relatives to finance a project is a necessary and sometimes sufficient condition. Consequently, Ulule, the French leader in rewardbased crowdfunding has separated potential investors into three circles1 . The first two are family and friends, and friends of friends, while the third consists of unknown people. Ulule states, "Most small and medium-sized projects can be financed exclusively from your first and second network circles" or, put another way, the social network of the project developer is a key resource in fundraising. [START_REF] Agrawal | Crowdfunding: Geography, Social Networks, and the Timing of Investment Decisions[END_REF] demonstrated the importance of social networks on funding decisions in a study of the pioneering crowdfunding platform SellABand. Their investigation of the timing of investments, and investors' characteristics found that distant funders use early investments by locals as a signal of quality. They also showed that most local investors are friends and family. After controlling for the social networks of founders, distance no longer influenced the decision to invest.

The importance of the social network structure of entrepreneurs is also emphasized in [START_REF] Zheng | The role of multidimensional social capital in crowdfunding: A comparative study in China and US[END_REF], who demonstrated that the number of social network ties has a positive impact on a funding campaign, both in the United States and China. This result echoes the work of [START_REF] Mollick | The dynamics of crowdfunding: An exploratory study[END_REF] who found that the chance of success increases with the size of the social network. The role of social ties does not end in the preexisting social networks of project developers -they can also be built within platforms by ongoing interactions with its community [START_REF] Butticè | Serial Crowdfunding, Social Capital, and Project Success[END_REF]. Based on a large sample of projects on Kickstarter, [START_REF] Colombo | Internal Social Capital and the Attraction of Early Contributions in Crowdfunding[END_REF] showed that so-called 'internal' social capital allows entrepreneurs to attract early backers. Finally, research by [START_REF] Giudici | Reward-based crowdfunding of entrepreneurial projects: the effect of local altruism and localized social capital on proponents' success[END_REF] found that the characteristics of a geographical area, such as local altruism or local social capital positively impacted funds raised from the crowd of internet users.

However, with the exception of [START_REF] Agrawal | Crowdfunding: Geography, Social Networks, and the Timing of Investment Decisions[END_REF], little research has addressed the question of the impact of the social networks of project owners on the geography of crowdfunding. The question is challenging, as we need to measure the extent to which people who live in different geographical areas share the same social networks. Ideally, we would have access to data about each backer's relationship with the entrepreneur, as in [START_REF] Agrawal | Crowdfunding: Geography, Social Networks, and the Timing of Investment Decisions[END_REF]. But the latter study only looked at a couple of projects, and the method is impossible to generalize to the 13000 projects in our study. Instead, and inspired by the literature on knowledge diffusion [START_REF] Agrawal | How do spatial and social proximity influence knowledge flows? Evidence from patent data[END_REF][START_REF] Lissoni | International migration and innovation diffusion: an eclectic survey[END_REF] we used residential migration as a proxy for inter-regional social ties. The social networks of inventors, researchers or workers can support very different exchanges at very different geographical levels, from the flow of knowledge [START_REF] Agrawal | Gone but not forgotten: knowledge flows, labor mobility, and enduring social relationships[END_REF], to international technological transfers [START_REF] Kerr | Ethnic Scientific Communities and International Technology Diffusion[END_REF], and even inter-regional trade [START_REF] Combes | The trade-creating effects of business and social networks: evidence from France[END_REF]. Reward-based crowdfunding reflects individual entrepreneurship or small ventures led by a few individuals, and the crowd of backers is not, by definition, made up of finance professionals. Using a measure of inter-regional migration provides a picture of social connections and, thus, distant funding opportunities.

All of the studies presented above demonstrate the importance of entrepreneurs' social capital and, especially, the role played by the resources that constitute their social networks. If, as we think, we are able to proxy social ties between inhabitants of the 94 French metropolitan regions, we expect that they will shape the bilateral flow of funding and, consequently, reduce the impact of geographical distance.

The gravity model

We adopted a modified version of the gravity equation model to study the role of distance and the impact of social networks on the flow of crowdfunding between French regions. Initially developed to study the determinants of international trade [START_REF] Anderson | A Theoretical Foundation for the Gravity Equation[END_REF][START_REF] Bergstrand | The Gravity Equation in International Trade: Some Microeconomic Foundations and Empirical Evidence[END_REF] gravity models have been used to study exchanges in different contexts, including e-commerce [START_REF] Blum | Does the internet defy the law of gravity[END_REF][START_REF] Hortaçsu | The Geography of Trade in Online Transactions: Evidence from eBay and MercadoLibre[END_REF][START_REF] Lendle | There Goes Gravity: How Ebay Reduces Trade Costs[END_REF] and, more recently, lending-based crowdfunding [START_REF] Burtch | Cultural Differences and Geography as Determinants of Online Pro-Social Lending[END_REF]. Like Newton's theory of gravity, the equation considers that the relationship between two entities is proportional to their size and inversely related to the distance between them. In this article, we consider the flow of crowdfunding from region j to region i. Funds are aggregated individual investments in different projects.

The model is specified as follows: ln(𝐹 𝑖𝑗 + 1) = 𝛼 1 -𝛼 2 ln(𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑖𝑗 ) + 𝛼 3 ln(𝑊𝑒𝑎𝑙𝑡ℎ 𝑖 × Wealth 𝑗 ) + 𝛼 4 ln(𝑑𝑒𝑛𝑠𝑖𝑡𝑦 𝑖 × 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 𝑗 ) + 𝛼 5 ln (𝑐𝑟𝑒𝑎𝑡𝑖𝑣 𝑖 × 𝑐𝑟𝑒𝑎𝑡𝑖𝑣 𝑗 ) + 𝛼 6 ln(𝑚𝑖𝑔𝑟𝑜𝑟𝑖 𝑖𝑗 ) + 𝛼 7 ln(𝑚𝑖𝑔𝑟𝑑𝑒𝑠𝑡 𝑗𝑖 ) + 𝜀 𝑖𝑗 (eq1

)
Where Fij is the number of investments from region j to region i. In this first specification, we control for economic variables that are assumed to impact the propensity of a region to generate online exchanges via a crowdfunding platform. The variable WEALTHi(j) is a proxy for the size of the economy and wealth of the population of region i(j). It is calculated by multiplying median disposable income by the region's population. The variable DENSITYi(j) is a measure of the population per unit area (km 2 ) in the region i(j)and captures differences in urbanization. Many rewards-based crowdfunding platforms specialize in cultural projects [START_REF] Mollick | The dynamics of crowdfunding: An exploratory study[END_REF] and, to account for this, we include the proportion of the active population employed in the cultural and arts sectors as a proxy for local skills in the creative industries (CREATIVi and CREATIVj). This variable also represents an opportunity to indirectly account for the impact of the creative classes [START_REF] Florida | Cities and the Creative Class[END_REF]) on the regional success of crowdfunding.

Our strategy of taking into account the impact of social networks is based on [START_REF] Combes | The trade-creating effects of business and social networks: evidence from France[END_REF], who estimate the economic effect of migrants in metropolitan French regions. We define MIGRORIij as the number of people living in region i but born in region j, respectively, while MIGRDESTji are those who live in region j but were born in region i. Although both variables define the social networks that link different French regions, they have potentially different implications. MIGRORIij are immigrants: in the context of our work, this variable reflects the ability of project founders to mobilize the social network in his or her region of origin. Inversely, MIGRDESTji are emigrants: potential investors do so because of the social ties with their region of origin.

In a second step, we consider a fixed-effects model that takes into account all region-specific characteristics that may influence the flow of funding. As region-specific variables are dropped from the estimation, the model takes the following form:

ln(𝐹 𝑖𝑗 + 1) = 𝛼 1 + 𝛼 2 ln(𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑖𝑗 ) + 𝜂 𝑖 + 𝜂 𝑗 + 𝜀 𝑖𝑗 𝑒𝑞(2)
Where 𝜂 𝑖 𝑎𝑛𝑑 𝜂 𝑗 are unobserved characteristics of regions i and j.

We estimate these models by ordinary least squares (OLS) using a standard log-linearized equation. The drawback of this method is that estimates are inefficient in the presence of heteroscedasticity and consider observations when the dependent variable equals zero. We deal with the latter issue by using ln(𝐹 𝑖𝑗 + 1) as the dependent variable, although the actual number of zeros in the sample is moderate (10%). The issue of heteroscedasticity is discussed in different robustness tests, and presented in section 5.

Data

Estimates of the model described above are based on a database provided by Ulule, the French (and European) leader in rewards-based crowdfunding. Since its launch in 2010, the platform has funded more than 17 000 projects with a value of more than €78 million. For the present article, the platform provided us with full details for the period 2012-2015; these 15 000 projects represent one million investments with a value of €35 million. Non-French projects and funds were discarded from the dataset, together with aberrant observations, leaving 14 800 projects funded by 732 686 contributions.

An important point to note is that geographical identification is not mandatory for subscription to the platform, neither for the founder of a project nor for backers who decide to invest. Nevertheless, it was available for 87% (12 887)2 of projects and 62% (452 850) of contributions, with a total value of €19 million. We are aware that this is a potential limitation; however, it is only a concern if there is selection bias in the location of funders. This issue, which is discussed in section 6, seems to have very limited consequences for our results.

Users who do declare their location provide a postal code. The two first digits of this code identify the region in France. This information is used to construct the dependent variable, namely the number of investments provided by backers to project founders, for the period 2012-2015. As Table 1b shows, mean inter-regional contributions were 33.3. Standard errors were high, suggesting strong heterogeneity in the distribution (for instance, in 9.7% of regions, bilateral funding equaled zero). Intra-regional investment (when the backer and founder live in the same region) is an important contributor to total funding, representing 28% of the total number of investments and 29% of the value.

The French département is our unit of analysis. To simplify the analysis, and because this geographical unit is specific to France, here we refer to it as a "region". There are 94 such units. The département is an administrative division of France; it was created in 1790 and, as [START_REF] Combes | The trade-creating effects of business and social networks: evidence from France[END_REF] highlight, this spatial delimitation is not neutral and shapes economic activities and social networks at the national level.

The key variables for our demonstration are two representative, regional social networks based on data from the 2013 Census provided by the National Institute of Statistics. Specifically, we drew upon the Residential migration: location of city of residence and city of prior residence database. This dataset provides details of 19 million French residents at the time of the census and their region of birth. Each individual is weighted according to his or her weight in the overall population. These data made it possible to build a matrix of inter-regional migration based on estimates of the number of people born in region i(j) and living in region j(i) in 2013. The legitimacy of using these variables as a proxy for the geographical pattern of social networks lies in their ability to provide a picture of the stock of social ties that individuals in a particular region have accumulated. Obviously, only considering the region of birth and current residence does not provide a full picture, as people may have lived in many different regions between their birth and 2013, and can also have spent little time in either of the two regions. However, we do, at least, know that a social connection exists and, given the size of the sample (and its adjustment over the whole population), we can be confident in the ability of our matrix to provide, on average, a faithful picture of regional migration.

According to the descriptive statistics given in Table 1b, and consistent with [START_REF] Combes | The trade-creating effects of business and social networks: evidence from France[END_REF], the average percentage of migrants, per region, was 0.4% and the percentage of the population still living in the region where they were born was 51.4%3 .

Figure 1a to 1c show spatial patterns of the main variables. These figures show the geographical distribution of crowdfunding, immigrants and emigrants for three French regions, selected to highlight different patterns. The Rhône-Alpes region (Fig. 1a) hosts the regional metropolis of Lyon, which is the third-biggest city in France. The Paris region (Fig. 1b) highlights a disproportionate number of crowdfunding projects, while the Finistère region (Fig. 1c) is the westernmost region in France with a strong cultural identity. Overall, Figure 1 shows a clear correlation between migration and crowdfunding flows in the three regions. Inward funding appears, at first sight, to be less concentrated around the region of destination than the flow of migrants, which appears to more closely related to spatial proximity. This higher dispersion of crowdfunding flows seems to be partly driven by the biggest cities in France (Toulouse, Bordeaux and Marseille), which concentrate a high number of potential backers. A variety of migratory patterns are seen for different regions (this is especially true for Rhône-Alpes), suggesting the existence of distinct social networks and thus a potential differential impact on bilateral funding.

Regional population, disposable income and population density figures were supplied by the French Institute of Statistics for the year 2015. Finally, the distance between two regions was estimated as the geodesic distance in kilometers between the biggest cities in each region.

Findings

Table 2 shows the results of OLS estimates of regional bilateral crowdfunding excluding intra-regional investments, making this specification closer to the canonical gravity model. Columns (1) and ( 2) estimate the model without the effect of social networks. The elasticity of distance is around 0.5, suggesting that doubling the distance between two regions divides the number of investments by two. At first sight, this finding seems important, given the factors presented in the introductionnotably the lack of shipping costs or significant cultural differences between French regions. As a comparison, using international trade data collected from E-bay, [START_REF] Lendle | There Goes Gravity: How Ebay Reduces Trade Costs[END_REF] found elasticity of between 0.35 and 0.5 (depending on their specification), while Hortaçsu et al. ( 2009) used a comparable sample (but only considered intra-country trade), and found much lower elasticity of between 0.05 and 0.1. This result confirms the existence of important information-related costs in the choice of funding a project, and the inability of the platform to disseminate this information online.

Introducing variables related to regional migration (to account for the existence of regional social networks) significantly reduced the elasticity of distance. This provides support for the idea that social ties are a means to convey the trust needed to generate contributions in crowdfunding projects. Columns ( 3) and ( 4) show the result when adding immigrant and emigrant networks.

Although the elasticity of distance decreases sharply, the network of immigrants (i.e. those who live in the region where the project is proposed and were born in the region where funds originate) has a stronger positive impact on bilateral investments than the network of emigrants. In other words, the impact of social networks is higher when the flow of funding follows the migration of individuals. Interestingly when both networks (immigrants and emigrants) are included in the regression (column 5) the effect of distance almost disappears and is no longer significant.

Table 3 incorporates intra-regional funding by adding the variable HOME (a dummy which equals one when the funder and the founder are in the same region) in the estimate. The elasticity of distance is close to that shown in Table 2 and approaches zero when taking into account the impact of social networks. The most striking result is the magnitude of the home effect. In the more complete specifications (columns 2 to 5) being in the same region increases the number of funders by 300-400%. This 'primacy' given to the very close proximity of funders and project developers suggests a strong non-linearity in the effect of distance, even when controlling for the existence of social networks. Being located in the same region undoubtedly increases the probability of direct or indirect social interactions with the project developer, reducing uncertainly about the viability of the project. It also increases the probability of local impacts (e.g. opening a bar, a service activity or promoting a local project with high social value). Once the regional border is crossed, distance has a very limited impact when controlling for potential social relationships between funders and project developers.

The other dependent variables likely to have an impact have the expected effect whatever the specification. The number of funders between two regions increases with the wealth of the population, the density of the region, and the percentage of employees working in the arts and creative industries.

Table 4 shows the result of the fixed-effects specification. Introducing dummies for region of origin and destination captures all specific effects. The results are consistent with the other specifications. Doubling the number of immigrants increases the number of funders by 25%, compared to 17% when doubling the number of emigrants. The effect of distance remains (negatively) significant with an elasticity of -0.05.

As projects were divided into 15 categories, we were able to estimate our model for each category.

Table 5 provides the estimate for some of these categories, notably the biggest in terms of volume of projects. It should be noted that categories were self-determined by the owner of the project who was able to associate a tag proposed by the platform with his/ her project. The results suggest that elasticities of distance as well as the magnitude of the effect of social networks vary within categories. Although it is difficult to draw any general conclusions for individual categories, some results can be highlighted. The impact of social networks seems to be more important for the 'charity' and 'music' categories than for the 'comics' or 'stage' categories, suggesting that the former are more dependent on social ties than the latter. We also note that funding in the charity and music categories is more impacted by the emigrant than the immigrant social network. A clear home bias is seen for the stage category, and the elasticity of distance is high. This finding may be explained by the nature of these projects, which are produced and consumed locally. Another interesting result appears in column 6 of Table 5 where we only consider projects concerning pre-ordered goods (mainly comics, music albums or craft items). In this case, both distance and social networks have a very limited effect on the flow of funding, which can be explained by the fact that these pre-ordered goods are standardized or, at least, ready to be produced. Here, there is less dependence on trust, and information asymmetry has less impact, meaning that for these categories of 'less' informationrelated goods, the cost of distance may be significantly reduced.

Although we should be cautious about these interpretations, especially for categories that only contain a few hundred projects, the results suggest that the characteristics of projects are consistent with different geographical patterns.

Robustness

Table 6 presents estimates based on the amount of funding as the dependent variable; Aij is the amount transferred from region j to region i. Columns 1 and 2 can be compared respectively with columns 1 and 5 in Table 3 and column 4 in Table 4. The results show that considering the amount of funding, rather than the number of funders does not change the result, and only marginally changes elasticities. The main difference is that the coefficient associated with the social network is higher, and the log of distance remains non-significant even in the fixed-effect specification.

The impact of the region of Paris is a concern and a potential source of heteroscedasticity. The predominance of the capital of the French economy is well known. Paris and its surroundings represent 3.4% of the total population, but concern 11.7% of the projects in our dataset. It is also where employment in the creative industries is concentrated (11.4% of the active population compared to the national mean of 1.5%). Seventeen percent of funding goes to Paris and 11% originates from it. For economic reasons, the region is very attractive for workers. Therefore, we estimated the gravity model excluding central Paris and its surrounding Ile-de-France region4 , to ensure the robustness of our main findings. The results are presented in Table 7. It is clear that they are comparable with those given in Table 3 despite slight differences. Notably, the effect of distance remains significant after the introduction of regional social tie variables and the home effect is less significant, which confirm the attraction of Paris and its surrounding regions. However, the coefficient associated with the social network variable is unchanged.

Conclusion and discussion

Our results are some of the first to address the spatial consequences of crowdfunding. Estimating the effect of distance on regional flows, using a gravity equation model revealed that distance plays an important role. This result contrasts with the context of 'local' crowdfunding, which seems to favor the disappearance of the distance effect. The introduction of variables to account for the existence of social networks between French regions showed that potential ties between the inhabitants of two regions not only increased the number of funders and the amount of funding, but also significantly lessened the negative impact of distance. Before discussing the implications of our main findings, we first discuss the potential limitations of our results.

It is clear that factors other than social networks between regions can explain the correlation between the migration of individuals and the flow of crowdfunding. People seek job opportunities and thus choose to relocate to areas where economic activity is growing. These are also the areas where ideas and projects emerge, making them a natural destination for funding. However, the fact that our result is robust to fixed-effects model limits the possibility that omitted variables or endogeneity would weaken our findings.

As mentioned in section 3, we were only able to consider 67% of the database's contents. This is because users do not have to declare their location to make a contribution. Does this have an impact on our results? Is it evidence of a selection bias (i.e. is a user who declares his/ her location different from another who does not)? A way to control for this potential bias is to compare the two subsamples using a common attribute. The distribution of contributions appears to be a good candidate. A comparison of the means and standard deviation of these two samples (with and without declared localization) is presented in Table 8 and reveals no significant difference in means, but a significant difference in standard errors.

However, although the standard deviation is lower for the subsample used in this article, it seems unlikely that it hides a significant difference in the nature, motivation or location of contributions.

Another way to highlight differences with and without localization samples is to use the tag or category associated with each project. Here, the main categories (music, film and video, stage, publishing) are present in the same proportions in the two subsamples. The only difference comes from the under-representation of contributions made to projects categorized as charity (respectively 14% and 19% of the sample with and without declared localization) and an over representation of contributions made to games and comics (90% of investments in these two categories are geographically localized). Nevertheless, as these categories only represent 10% of all projects and there is no difference in the mean amount contributed between the two subsamples, we can be confident that our results are only weakly affected by the non-systematic declaration of localization.

As the introduction to this article highlights, the rise of crowdfunding is concomitant with various promises. Among them is the hope of the democratization of access to funding. A corollary is that the place where a project is developed matters less in a world where crowdfunding exists. Our results tend to mitigate this assertion: funding flows are higher between wealthy regions and nothing suggests that crowdfunding platforms increase opportunities for less-dynamic areas. Like the migration of populations, financial exchanges partly reflect the geography of social networks. The fact that two regions have a high level of bilateral contributions when they share a migration history suggests that platform users are looking for projects with which they have an emotional connection. Social ties, but also a regional attachment or a distant relationship with a particular community can drive these emotions. Some links between regions are obviously due to a shared cultural history or geographical proximity, but others can be the consequence of a historical industrial relationship or even the existence of infrastructure that facilitates inter-regional mobility. All of this information can help to understand the motivations that drive funding between two regions and, consequently, how crowdfunding helps to create new geographical access to capital.

We must also keep in mind that the rewards-based crowdfunding is mainly aimed at 'cultural' projects, i.e. those where information asymmetry is consubstantial to their nature. Digital platforms have clearly fluidized exchanges and, in our case, financial contributions. However, for many information-intensive interactions, it is likely that they fail to deeply modify opportunities for market protagonists. When asking the crowd to fund a project, it seems more accurate to claim that crowdfunding platforms have helped to concretize latent interactions, rather than creating and extending the field of possibilities.

Figure 1a :

 1a Figure 1a: Geographical patterns of funding and migration in the Rhône-Alpes region

Table 2 :

 2 OLS estimation of bilateral regional funding

	VARIABLE	Ln(Fij)	Ln(Fij)	Ln(Fij)	Ln(Fij)	Ln(Fij)
		(1)	(2)	(3)	(4)	(5)
	Ln(DISTANCE)	-0.520*** -0.470*** -0.204*** -0.286*** -0.0293
		(0.0161)	(0.0162)	(0.0190)	(0.0201)	(0.0229)
	Ln(WEALTHi * WEALTHj) 1.013*** 0.795*** 0.586*** 0.652*** 0.450***
		(0.00904) (0.0140)	(0.0173)	(0.0186)	(0.0207)
	Ln(DENSITYi * DENSITYj)		0.0701*** 0.0703*** 0.0703*** 0.0705***
			(0.00935) (0.00910) (0.00906) (0.00894)
	Ln(CREATIVi * CREATIVj)		0.353*** 0.321*** 0.331*** 0.300***
			(0.0157)	(0.0155)	(0.0155)	(0.0153)
	Ln(MIGRORIij)			0.240***		0.237***
				(0.0127)		(0.0123)
	Ln(MIGRDESTji)				0.165*** 0.161***
					(0.0135)	(0.0129)
	Constant	-41.41*** -32.49*** -26.00*** -28.02*** -21.76***
		(0.429)	(0.576)	(0.658)	(0.700)	(0.751)
	Observations	8,742	8,742	8,742	8,742	8,742
	R-squared	0.627	0.656	0.670	0.663	0.677
	Robust standard errors in parentheses
		*** p<0.01, ** p<0.05, * p<0.1

Table 3 :

 3 OLS estimation of bilateral regional funding with intra-regional investment

	VARIABLE	Ln(Fij)	Ln(Fij)	Ln(Fij)	Ln(Fij)	Ln(Fij)
		(1)	(2)	(3)	(4)	(5)
	Ln(DISTANCE)	-0.520*** -0.471*** -0.197*** -0.279*** -0.0207
		(0.0161)	(0.0161)	(0.0190)	(0.0201)	(0.0227)
	Ln(WEALTHi * WEALTHj) 1.006*** 0.794*** 0.581*** 0.645*** 0.444***
		(0.00900) (0.0140)	(0.0172)	(0.0184)	(0.0204)
	Ln(DENSITYi * DENSITYj)		0.0654*** 0.0661*** 0.0659*** 0.0666***
			(0.00947) (0.00913) (0.00912) (0.00892)
	Ln(CREATIVi * CREATIVj)		0.353*** 0.321*** 0.330*** 0.299***
			(0.0156)	(0.0153)	(0.0153)	(0.0151)
	Ln(MIGRORIij)			0.247***		0.241***
				(0.0127)		(0.0122)
	Ln(MIGRDESTji)				0.173*** 0.165***
					(0.0134)	(0.0128)
	HOME	0.996*** 1.283*** 1.460*** 1.407*** 1.575***
		(0.131)	(0.132)	(0.117)	(0.120)	(0.111)
	Constant	-41.06*** -32.37*** -25.79*** -27.76*** -21.55***
		(0.428)	(0.575)	(0.652)	(0.692)	(0.739)
	Observations	8,836	8,836	8,836	8,836	8,836
	R-squared	0.653	0.680	0.694	0.687	0.701
	Robust standard errors in parentheses
		*** p<0.01, ** p<0.05, * p<0.1

Table 4 :

 4 Fixed-effects estimation of bilateral regional funding

		Without intra-regional	With intra-regional
		funding	funding
	VARIABLE	Ln(Fij)	Ln(Fij)	Ln(Fij)	Ln(Fij)
		(1)	(2)	(3)	(4)
	Ln(DISTANCE) -0.564*** -0.0611***	-0.551*** -0.0476***
		(0.0132)	(0.0164)	(0.0141)	(0.0174)
	Ln(MIGRORIij)		0.251***		0.257***
			(0.0214)		(0.0215)
	Ln(MIGRDESTji)		0.164***		0.170***
			(0.0219)		(0.0220)
	HOME			0.932***	1.304***
				(0.143)	(0.0914)
	Constant	8.097***	2.593***	8.031***	2.412***
		(0.116)	(0.194)	(0.120)	(0.198)
	Fixed effects	YES	YES	YES	YES
	Observations	8,742	8,742	8,836	8,836
	R-squared	0.864	0.880	0.870	0.888
		Robust standard errors in parentheses
		*** p<0.01, ** p<0.05, * p<0.1

Table 5 :

 5 Estimation of bilateral funding for different categories of projects

		Ln(Fij)	Ln(Fij)	Ln(Fij)	Ln(Fij)	Ln(Fij)	Ln(Fij)
	VARIABLES	Comics	Video	Music	Charity	Stage	Pre-order
		(1)	(2)	(3)	(4)	(5)	(6)
	Ln(DISTANCE)	-0.109*** -0.0966*** -0.222*** -0.0621*** -0.153*** -0.0129
		(0.0291)	(0.0290)	(0.0217)	(0.0239)	(0.0222)	(0.0300)
	Ln(WEALTHi * WEALTHj) 0.317***	0.133***	0.154***	0.220***	0.0193	0.118***
		(0.0221)	(0.0241)	(0.0199)	(0.0198)	(0.0167)	(0.0219)
	Ln(DENSITYi * DENSITYj) 0.114***	0.187*** 0.0842*** 0.0757*** 0.105*** 0.114***
		(0.0121)	(0.0118)	(0.00916) (0.00967) (0.00869) (0.0118)
	Ln(CREATIVi * CREATIVj) 0.134***	0.177***	0.216***	0.248***	0.195*** 0.152***
		(0.0155)	(0.0161)	(0.0131)	(0.0139)	(0.0108)	(0.0157)
	Ln(MIGRORIij)	0.0284** 0.0952*** 0.0769*** 0.105*** 0.0392***	0.0205
		(0.0119)	(0.0139)	(0.0113)	(0.0118)	(0.00910) (0.0127)
	Ln(MIGRDESTji)	-0.00191 0.0812*** 0.160***	0.129*** 0.0525*** 0.0609***
		(0.0124)	(0.0135)	(0.0111)	(0.0111)	(0.00893) (0.0121)
	HOME	0.817***	1.403***	1.041***	1.733***	1.464*** 0.923***
		(0.187)	(0.163)	(0.151)	(0.144)	(0.176)	(0.202)
	Constant	-14.68*** -7.826*** -7.487*** -11.29*** -1.370** -6.580***
		(0.791)	(0.852)	(0.713)	(0.707)	(0.587)	(0.775)
	Observations	8,836	8,836	8,836	8,836	8,836	8,836
	R-squared	0.388	0.461	0.576	0.557	0.487	0.287
	Number of projects	293	2312	1725	2542	1006	262
		Robust standard errors in parentheses
		*** p<0.01, ** p<0.05, * p<0.1

Table 6 :

 6 Estimation with the amount of bilateral funding as the dependent variable

	VARIABLE	Ln(Aij)	Ln(Aij)	Ln(Aij)
		(1)	(2)	(3)
	Ln(DISTANCE)	-0.744*** -0.0191	0.00794
		(0.0272)	(0.0407) (0.0345)
	Ln(WEALTHi * WEALTHj) 1.500*** 0.794***
		(0.0167)	(0.0359)
	Ln(DENSITYi * DENSITYj)		-0.00343
			(0.0143)
	Ln(CREATIVi * CREATIVj)		0.423***
			(0.0304)
	Ln(MIGRORIij)		0.367*** 0.458***
			(0.0230) (0.0495)
	Ln(MIGRDESTji)		0.245*** 0.234***
			(0.0238) (0.0503)
	HOME	0.489**	1.205*** 0.902***
		(0.222)	(0.181)	(0.172)
	Constant	-59.45*** -35.43*** 5.162***
		(0.787)	(1.323)	(0.373)
	Fixed-effects	NO	NO	YES
	Observations	8,836	8,836	8,836
	R-squared	0.537	0.571	0.759
	Robust standard errors in parentheses
	*** p<0.01, ** p<0.05, * p<0.1

Table 7 :

 7 OLS estimation of bilateral regional funding excluding the Ile-de-France region

	VARIABLE	Ln(Fij)	Ln(Fij)	Ln(Fij)
		(1)	(2)	(3)
	Ln(DISTANCE)	-0.624*** -0.593*** -0.109***
		(0.0190)	(0.0189)	(0.0327)
	Ln(WEALTHi * WEALTHj) 0.956*** 0.900*** 0.564***
		(0.0102)	(0.0204)	(0.0284)
	Ln(DENSITYi * DENSITYj)		-0.0350	-0.0336
			(0.0214)	(0.0210)
	Ln(CREATIVi * CREATIVj)		0.281*** 0.259***
			(0.0172)	(0.0168)
	Ln(MIGRORIij)			0.241***
				(0.0139)
	Ln(MIGRDESTji)			0.128***
				(0.0146)
	HOME	0.538*** 0.716*** 1.372***
		(0.138)	(0.136)	(0.129)
	Constant	-38.20*** -35.60*** -25.40***
		(0.464)	(0.787)	(0.992)
	Observations	7,396	7,396	7,396
	R-squared	0.609	0.622	0.640
	Robust standard errors in parentheses
	*** p<0.01, ** p<0.05, * p<0.1

Table 8 :

 8 Comparison of subsamples with and without location of contributors

	Group	Obs.	Mean	Std. Err. Std. Dev.
	Without location 279836 46.03	.20	106.2
	With location 452850 45.93	.13	91.3
	Combined	732686 45.971 .113708	97.33
	Difference		.1068415 .2340347
		* Significant at 1%

https://community.ulule.com/topics/-3-crowfunding-circles-881/

Descriptive statistics of the 12 887 crowdfunding campaigns are available in Table1a.

[START_REF] Combes | The trade-creating effects of business and social networks: evidence from France[END_REF] studied worker migration and found average percentages of 0.5% and 52.6% respectively.

This region is composed of 8 French départements.


est destinée au dépôt

Weak bond strength between successive layers in extrusion-based additive manufacturing

Emmanuel

Weak bond strength between successive layers in extrusion-based additive manufacturing

Emmanuel Keita 1 , Hela Bessaies-Bey², Wenqiang Zuo 1 , Patrick Belin 1 and Nicolas Roussel 1

1 : Laboratoire Navier IFSTTAR/CNRS/ENPC, Université Paris-Est 2 : Laboratoire CPDM IFSTTAR, Université Paris-Est e-mail : emmanuel.keita@ifsttar.fr Weak bond strength is often reported to result from a low level of intermixing between successive layers. It is often concluded that there seems to exist an upper limit either on structuration rate or on resting time between layers, above which thixotropy is at the origin of so-called "cold joints". In parallel to intermixing issues, our results show that the superficial drying of the resting layer seems to also contribute to weaker interfaces bonds. We plot in the figure below the relative interface strength measured using a splitting test as a function of the resting time between layers for an interface protected from drying and an interface exposed to drying. The material is a W/C = 0.2 thixotropic mortar, for which the structuration rate A_thix was measured to be above 0.4Pa/s. We see that, even after 5 hours rest between layers, the interface strength is still higher than 90% of the reference interface strength if the material is protected from drying. However, if exposed to drying, the interface strength drops by 50% in a matter of a couple tens of minutes. As printable materials are often accelerated in order to improve their ability to quickly build up a structure, we can expect some increase in temperature resulting from this increase in chemical activity. This shall, in turn, increase drying rate and make these printable materials extremely sensitive to the external environment (temperature and humidity) or to the so-called printing environment.

Références

[1] Emmanuel Keita, Hela Bessaies-Bey, Wenqiang Zuo, Patrick Belin and Nicolas Roussel, « Weak bond strength between successive layers in extrusion-based additive manufacturing: measurement and physical origin ». Submitted

DiXite3dPrint

  Keita, Hela Bessaies-Bey, Wenqiang Zuo, Patrick Belin, Nicolas Roussel To cite this version: Emmanuel Keita, Hela Bessaies-Bey, Wenqiang Zuo, Patrick Belin, Nicolas Roussel. Weak bond strength between successive layers in extrusion-based additive manufacturing. DiXite3dPrint : Fabrication Additive pour la Construction. Quelle Actualité Nationale ?, École des Ponts ParisTech, Jan 2019, Champs-sur-Marne, France. hal-02119419