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 tree-like and oriented, as well as, for 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 small cycles and how they can destabilize large dynamical systems when they induce strong enough feedback loops.
Highlights
Scientists use networks to depict the causal interactions between the constituents of large dynamical systems [1,2,3,4,5]
Relating system stability to network topology is important to understand, among others, how systemic risk in financial markets is governed by the topology of the network of liabilities between financial institutions [6,7,8]; how the resilience of an ecosystem to external perturbations depends on the underlying food web of trophic interactions [9,10,11,12,13,14]; and how networks of social interactions determine the spreading of rumours [15,16,17]
Random matrices appear in linear stability analyses of large dynamical systems
Summary
Scientists use networks to depict the causal interactions between the constituents of large dynamical systems [1,2,3,4,5]. To model dynamical systems on large networks, we consider that C is the adjacency matrix of a random, directed graph with a prescribed degree distribution pKin,Kout (k, ) of indegrees Kin and outdegrees Kout. We perform a linear stability analysis of fixed points in dynamical systems defined on random, directed graphs To this aim, we determine the leading eigenvalue λ1(A) of the adjacency matrices A of random, directed graphs with a prescribed degree distribution and with randomly weighted links. We derive in this limit exact expressions for the statistics of the entries of right and left eigenvectors associated with λ1 We use these results to depict a phase diagram for the linear stability of fixed points in dynamical systems defined on large, directed networks. In Appendices E–I, we derive recursion relations for the entries of right and left eigenvectors of random, directed graphs, which are based on the Schur formula
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