Stability of structured random matrices

Hastings, Harold M.; Juhasz, Ference; Schreiber, Micheline A.

doi:10.1098/rspb.1992.0108

Cited by 5 publications

(1 citation statement)

References 10 publications

(15 reference statements)

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“…We show that, essentially, the problem can be reduced to analyzing the maximal eigenvalue of a correlated random matrix, as discussed in the previous section [46,47,48].…”

Section: A Theoretical Remarkmentioning

confidence: 99%

Synchronization in networks with random interactions: Theory and applications

Feng

Jirsa²,

Ding³

2006

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Synchronization is an emergent property in networks of interacting dynamical elements. Here we review some recent results on synchronization in randomly coupled networks. Asymptotical behavior of random matrices is summarized and its impact on the synchronization of network dynamics is presented. Robert May's results on the stability of equilibrium points in linear dynamics are first extended to nonlinear systems where the synchronized dynamics can be periodic or chaotic and then to systems with time delayed coupling. Finally, applications of our results to neuroscience, in particular, networks of Hodgkin-Huxley neurons, are included.Behaviour and structures of interacting dynamical systems are complex. Most research on complex networks has been focused so far on networks with deterministic interactions. In this paper we study the behaviour of networks with random interactions. We extend Robert May's results on the stability of equilibrium points in linear dynamics to nonlinear systems where the synchronized dynamics can be periodic or chaotic and then to systems with time delayed coupling. Results are then applied to networks in Neuroscience.

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“…We show that, essentially, the problem can be reduced to analyzing the maximal eigenvalue of a correlated random matrix, as discussed in the previous section [46,47,48].…”

Section: A Theoretical Remarkmentioning

confidence: 99%

Synchronization in networks with random interactions: Theory and applications

Feng

Jirsa²,

Ding³

2006

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Evolutionary constraints on the complexity of genetic regulatory networks allow predictions of the total number of genetic interactions

Campos-González

Freyre-González

2018

Preprint

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Genetic regulatory networks (GRNs) have been widely studied, yet there is a lack of understanding with regards to the final size and properties of these networks, mainly due to no network is currently complete. In this study, we analyzed the distribution of GRN structural properties across a large set of distinct prokaryotic organisms and found a set of constrained characteristics such as network density and number of regulators. Our results allowed us to estimate the number of interactions that complete networks would have, a valuable insight that could aid in the daunting task of network curation, prediction, and validation. Using state-of-the-art statistical approaches, we also provided new evidence to settle a previously stated controversy that raised the possibility of complete biological networks being random. Therefore, attributing the observed scale-free properties to an artifact emerging from the sampling process during network discovery. Furthermore, we identified a set of properties that enabled us to assess the consistency of the connectivity distribution for various GRNs against different alternative statistical distributions. Our results favor the hypothesis that highly connected nodes (hubs) are not a consequence of network incompleteness. Finally, an interaction coverage computed for the GRNs as a proxy for completeness revealed that high-throughput based reconstructions of GRNs could yield biased networks with a low average clustering coefficient, showing that classical targeted discovery of interactions is still needed.

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Modular networks emerge from multiconstraint optimization

Pan

Sinha

2007

Phys. Rev. E

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Modular structure is ubiquitous among complex networks. We note that most such systems are subject to multiple structural and functional constraints, e.g., minimizing the average path length and the total number of links, while maximizing robustness against perturbations in node activity. We show that the optimal networks satisfying these three constraints are characterized by the existence of multiple subnetworks (modules) sparsely connected to each other. In addition, these modules have distinct hubs, resulting in an overall heterogeneous degree distribution.

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Stability of structured random matrices

Cited by 5 publications

References 10 publications

Synchronization in networks with random interactions: Theory and applications

Synchronization in networks with random interactions: Theory and applications

Evolutionary constraints on the complexity of genetic regulatory networks allow predictions of the total number of genetic interactions

Modular networks emerge from multiconstraint optimization

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