| Most network systems possess large- or mesoscale structures, which are not captured by local measures such as, e.g. degree and subgraph statistics. Many topological network models, as well as mean field analysis of dynamical processes on networks neglect such features. Stochastic blockmodels provide a straightforward generalization of the usual random graph ensembles, with which such large scale properties can be arbitrarily described. Furthermore, it enables the extension of mean field analysis of dynamical processes to incorporate such large scale structures, resulting in a very versatile framework for connecting dynamical properties with network structure. In this talk I provide a short introduction to stochastic blockmodels, and illustrate their use in the analysis of a paradigmatic Boolean dynamics based on majority functions, meant to describe systems which are robust against noise, such as gene regulation. A model for the evolution of such systems is also proposed, where networks with more robust dynamics survive with greater probability. By mapping the evolutionary process into a Gibbs ensemble, and making use of a general analytical expression for entropy of stochastic blockmodel ensembles, the free energy of the system is minimized, and its equilibrium properties are obtained. The analysis reveals a topological phase transition at a specific value of selective pressure, where a "segregated core" topology emerges, characterized by the existence of a smaller subset of the nodes which regulate the entire system; a feature which is also found in real gene regulatory systems. This example highlights the necessity of incorporating large scale properties into network models in order to determine what topological features are most relevant to a given dynamical process, and which topologies are more likely to occur when a given behavior is desired. References ---------- Tiago P. Peixoto, Entropy of stochastic blockmodel ensembles, arXiv: 1112.6028 Tiago P. Peixoto, The behavior of noise-resilient Boolean networks with diverse topologies, J. Stat. Mech. P01006 (2012) Tiago P. Peixoto, Emergence of robustness against noise: A structural phase transition in evolved models of gene regulatory networks, arXiv:1108.4341 |
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