Linear stability analysis of large dynamical systems on random directed graphs

Abstract

We present a linear stability analysis of stationary states (or fixed points) in large dynamical systems defined on random, directed graphs with a prescribed distribution of indegrees and outdegrees. We obtain two remarkable results for such dynamical systems. First, infinitely large systems on directed graphs can be stable even when the degree distribution has unbounded support; this result is surprising since their counterparts on nondirected graphs are unstable when system size is large enough. Second, we show that the phase transition between the stable and unstable phase is universal in the sense that it depends only on a few parameters, such as, the mean degree and a degree correlation coefficient. In addition, in the unstable regime, we characterize the nature of the destabilizing mode, which also exhibits universal features. These results follow from an exact theory for the leading eigenvalue of infinitely large graphs that are locally treelike and oriented, as well as, the right and left eigenvectors associated with the leading eigenvalue. We corroborate analytical results for infinitely large graphs with numerical experiments on random graphs of finite size. We discuss how the presented theory can be extended to graphs with diagonal disorder and to graphs that contain nondirected links. Finally, we discuss the influence of cycles and how they can destabilize large dynamical systems when they induce strong feedback loops.

Localization and Universality of Eigenvectors in Directed Random Graphs

“…In fact, the solutions of Eq. ( 7) are well corroborated by direct diagonalizations of large adjacency matrices [65][66][67]. The analytic results presented below follow from Eq.…”

“…Spectra of infinitely large matrices A.-The spectrum of A has been studied in Refs. [64][65][66][67]. For n → ∞ and c > 1, directed random graphs have a giant strongly connected component [68] and the spectral distribution ρ A ðλÞ ¼ n −1 P n j¼1 δ½λ − λ j ðAÞ of the eigenvalues fλ j ðAÞg n j¼1 is supported on a disk of radius jλ b j ¼ ffiffiffiffiffiffiffiffiffiffi ffi chJ 2 i p centered at the origin of the complex plane.…”

Mean-field theory of vector spin models on networks with arbitrary degree distributions

“…Apart from a few exceptions [83,88,89], the consequences of this interesting mechanism for the breakdown of the central limit theorem to statistical physics remain largely unexplored. We have illustrated its crucial role for the equilibrium of spin models, but one can envisage the far-reaching importance of this mechanism for a variety of problems on networks, such as the nonequilibrium dynamics of spin models [90,91], the storage capacity of neural networks [14], and the stability of large dynamical systems [23].…”

“…Second, seemingly unrelated problems across disciplines can be cast in terms of the unifying framework of spin models on random networks [7,8]. Models of scalar spins on networks have a vast number of applications in a variety of research fields, such as opinion dynamics [9,10], models of socio-economic phenomena [11], artificial neural networks [12][13][14], agent-based models of the market behavior [15][16][17], dynamics of biological neural networks [18][19][20], information theory and computer science [8], sparse random-matrix theory [21,22], and the stability of large dynamical systems [23,24]. Models of vector spins on networks are relevant for the study of synchronization phenomena [25][26][27][28], random lasers [29][30][31], vector spin-glasses (SGs) [32][33][34], and the collective dynamics of swarms [35][36][37][38].…”

Antagonistic interactions can stabilise fixed points in heterogeneous linear dynamical systems

“…Refs. [1][2][3][4][5][6][7]. Although one should be careful in drawing conclusions about the dynamics of nonlinear systems from the study of randomly coupled linear differential equations, random matrix theory has the advantage of providing generic analytical insights about the influence of interactions on linear stability.…”