(586d) Coarse-Graining the Dynamics of (and on) Networks

Networks play an important role in many scientific areas ranging from chemistry to ecology and to the social sciences. In dynamical systems that involve networks, coarse-graining approaches are essential in helping us understand the interplay between the structure and the dynamics of networks. In this work, we propose and implement two different coarse-graining approaches for reducing dynamical network problems. To illustrate these approaches, we consider two dynamical network examples involving distinct types of network dynamics: ?dynamics of a network' and ?dynamics on a static network'. Our first example falls under the former category. It is a dynamical model of social networks (Marsili, Vega-Redondo et al. 2004), in which the addition of an edge to the graph, e.g. an acquaintance of a new friend under the perspective of the social networks, and removal of an edge, e.g. the termination of a bond, follow simple probabilistic rules. As there is not yet a systematic framework in order to identify the coarse-grained observables of the system we use numerous temporal simulations and a thorough examination of the model to derive coarse-grained variables in an ad-hoc manner. The second example that we consider is the dynamical evolution of nonidentical oscillators connected by a static network structure. The dynamics of the individual oscillators are defined by the Kuramoto model (Kuramoto 1984). For this problem, we use the components of the vector of phase angles along the first few eigenvectors of the graph Laplacian (corresponding to the network that defines the connectivity between the oscillators) as coarse variables. This choice for the set of coarse variables is found to be appropriate if the underlying network has a gap in the eigenvalue spectrum of the corresponding graph Laplacian, suggesting that the graph dynamics may be low-dimensional). Our illustrative network topology satisfied this condition. For both these illustrative examples, we computationally solve the coarse-grained model using the equation-free framework (Kevrekidis, Gear et al. 2004) that circumvents the derivation of explicit coarse-grained equations. Thus, the bulk of the burden of coarse-graining falls on the selection of suitable coarse variables that represent the dynamics. We perform coarse-projective integration, coarse fixed point and coarse limit cycle computations to illustrate our coarse graining approaches.

References

Kevrekidis, I. G., C. W. Gear, et al. (2004). "Equation-free: The computer-aided analysis of complex multiscale systems." Aiche Journal 50(7): 1346-1355.

Kuramoto, Y. (1984). Chemical oscillations, waves, and turbulence. Berlin ; New York, Springer-Verlag.

Marsili, M., F. Vega-Redondo, et al. (2004). "The rise and fall of a networked society: A formal model." Proceedings of the National Academy of Sciences of the United States of America 101(6): 1439-1442.