Superelliptical laws for complex networks

All dynamical systems of biological interest—be they food webs, regulation of genes, or contacts between healthy and infectious individuals—have complex network structure. Wigner’s semicircular law and Girko’s circular law describe the eigenvalues of systems whose structure is a fully connected network. However, these laws fail for systems with complex network structure. Here we show that in these cases the eigenvalues are described by superellipses. We also develop a new method to analytically estimate the dominant eigenvalue of complex networks. Eigenvalues and eigenvectors are central to science and engineering. They are used to assess the stability of dynamical systems[1], the synchronization of networks[2], the occurrence of epidemics of infectious diseases[3, 4, 5, 6, 7], and the ranking of web-pages[8]. Two connected results describe the eigenvalue distribution of large matrices with random coefficients: Wigner’s semicircle law[9] (for symmetric matrices), and Girko’s circular law[10] (for asymmetric ones). These laws have been applied in a wide range of disciplines, from number theory[11] to ecology[1]. However, they fall short of describing the eigenvalues of matrices with complex network structure[12]. Such matrices are ubiquitous in the analysis of man-made and biological systems[13, 14]. Here we analyze matrices whose underlying structure is a sparse random network with arbitrary degree distribution. We show that the eigenvalue distributions of these matrices are described by superellipses. We solve the case of symmetric matrices by generalizing Wigner’s law, and extend Girko’s law to matrices with regular-graph structure (where all nodes have the same degree). We also propose a new method to approximate the dominant eigenvalue of a complex network, improving upon current results[15, 16]. Wigner’s semicircle law[9] states that the density of the eigenvalues of a large, symmetric random matrix whose entries are sampled from a normal distribution N (0, σ) is described by a semicircle. Similarly, Girko’s circular law[17, 10, 18] states that the eigenvalues of asymmetric matrices with normally-distributed entries are approximately uniformly distributed in a circle. If there is a nonzero pairwise correlation between the off-diagonal elements of the matrix, the eigenvalues are approximately uniformly distributed in an ellipse[19, 20, 21]. These laws have found applications in a wide array of disciplines, including quantum physics[22], ecology[1] and number theory[11], and were recently proved to be universal[18, 21] – i.e., they hold under very mild conditions on the distribution of the matrix entries. Wigner’s and Girko’s laws hold for matrices whose underlying network structure is a completely connected graph: as s, the size of the graph, goes to infinity, the distribution of the eigenvalues converges to the corresponding law. They also hold for not-completely-connected graphs[18], as long as the number of connections per node sp (where p is the proportion of nonzero entries in the matrix) goes to infinity as s→∞. However, the laws do not describe the eigenvalues of matrices with complex network structure[12]. These networks are typically sparse with a very heterogeneous degree distribution[14]; moreover, for most complex networks, we expect the average number of connections per node to approach a constant as the network grows[12]. In this case, as s → ∞, sp → k. The cap on the average number of connections per node ∗[email protected] 1 ar X iv :1 30 9. 72 75 v2 [ qbi o. PE ] 7 N ov 2 01 3 can arise from spatial or temporal constraints. For example, in an ecological system organisms need to spatiotemporally co-occurr in order to interact. The cap is also evident in empirical data: Facebook counted around 56 million active users in 2008[23], each with an average degree (number of friends) of about 76, and while the number of users grew to more 562 million in 2011, the average degree grew only to 169. Our goal is to extend the circular laws above to the case of large matrices with complex network structure, whose average degree is k s. We show that in such cases the eigenvalue distributions are described by superellipses (|x| n /a + |y| n /b ≤ 1, where x is the real part of an eigenvalue and y is its imaginary part; for n = 2, we recover the equation for an ellipse). For symmetric matrices with normally-distributed entries (the analog of Wigner’s case), the density of the eigenvalues is described by a semi-superellipse – for any degree distribution of the underlying network. For asymmetric matrices (Girko’s case) whose structure is a random k-regular graph (i.e., all nodes have the same degree), we find that the eigenvalues are approximately uniformly distributed in a superellipse. For other network structures, the distribution is still described by superellipses, but is not uniform. Results Symmetric Matrices We analyze s× s matrices with 0 on the diagonal, and off-diagonal pairs (Mij ,Mji) obtained by multiplying the corresponding entries of two matrices (Mij ,Mji) = (Aij , Aji) · (Nij , Nji). A is the adjacency matrix of a random undirected graph—with a given degree distribution—built using the configuration model[24, 14]. The use of the configuration model is important, as it ensures that the networks do not typically have unwanted “secondary structures” (e.g., modules, bipartite or lattice structure) that would affect results. N is a matrix whose off-diagonal pairs (Nij , Nji) are sampled from a bivariate normal distribution (X,Y ), with E[X] = E[Y ] = 0, E[X] = E[Y ] = 1/k, and E[XY ] = ρ/k, where k is the average degree of the network and −1 ≤ ρ ≤ 1 is Pearson’s correlation coefficient. The choice of parameters ensures that for k →∞, we recover the type of matrices studied by Wigner and Girko. Our results also hold for non-normal distributions (e.g., uniform, SI). We start with the analog of Wigner’s case, in which matrices are symmetric (ρ = 1). In this case, all eigenvalues are real, and for k →∞ we recover Wigner’s semicircle probability distribution function: Pr(λ = x) = P (x) = 2 √ (2r)2 − x2 π(2r)2 (1) The variance of this distribution is a function of r: μ2(r) = ∫ xP (x)dx = r. Because the variance of the eigenvalues of a matrix with diagonal zero is Tr(M)/s = ρ = μ2(r), given that in our matrices ρ = 1, then r = 1 and thus the horizontal radius is a = 2r = 2. We next generalize Wigner’s formula to the case where k s, which leads to a semi-superelliptical distribution: P (x) = 2 n √ (2r)n − xn 4(2r)2Γ(1 + 1/n)2Γ(1 + 2/n)−1 (2) where the numerator is the superelliptical equivalent of Wigner’s formulation, and the denominator is the area of a superellipse with a = b = 2r. In this case, we need to solve for two parameters, r and n. Hence, we write equations for the second (μ2(r, n) = ∫ xP (x)dx = Tr(M)/s) and fourth (μ4(r, n) = ∫ xP (x)dx = Tr(M)/s) central moments of the eigenvalue distribution (μ3(r, n) = 0, due to symmetry), thereby obtaining the values of n and r (SI). In Figure 1, we show numerical simulations in which we take a single 5000× 5000 matrix, whose network structure is determined by the average degree k (columns) and a specific algorithm used to construct the degree distribution (rows). In all cases, the density of the eigenvalues is described by a semi-superellipse, which captures the tails especially well. This is important, given the role of dominant eigenvalues in determining the properties of dynamical systems. The distribution tends to underestimate (small k) or overestimate (large k) the number of zeros, especially for very skewed degree distributions – an effect similar to that found for small matrices in Girko’s circular law[18].