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Nonlinear Dynamics of Networks

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The effect of network topology on the stability of discrete state models of genetic control

Edward Ott

University of Maryland

Abstract:   Discrete state networks (in particular, Boolean networks) have been proposed as potientially useful models for genetic control. An important aspect of these networks is their dynamical stability to small perturbations (e.g., we have proposed that instability may be a possible mechanism for causing cancer [1]). Previous approaches to stability have assumed the simplest type of uncorrelated network structure with synchronous update and usually consider two state (Boolean) variables. Real gene networks depart significantly from these idealizing assumptions; e.g., real networks have nontrivial topology significantly different from the previously used random network paradigm. To more directly address real situations, we present a general method[1] for determining the stability of large discrete state networks of logic updated units, allowing for any specified topology, multiple unit states, nonsynchronous update, etc. We find that stability is governed by the maximum eigenvalue of a modified adjacency matrix [1-3], and we test this result by comparison with numerical simulations. We also discuss possible application of our work to experimentally inferred gene networks. References [1] A.Pomerance, E.Ott, M.Girvan, W.Losert, PNAS, vol.106, no.20, pp.8209-8214, May 2009. [2] A.Pomerance, E.Ott, Phys.Rev.E, vol.79, 056111 (2009). [3] A.Pomerance, E.Ott, M.Girvan, arXiv 0910.0509v1 (2009).

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