(515d) Coarse-Graining Dynamical Networks with Intrinsic Heterogeneities

Networks play an important role in many fields ranging from the internet and social networks to biological neural systems and ecological food webs. Understanding the functional role of the network structure in the overall network dynamics is a critical research issue. On top of this, the nodes of many physical networks are inherently different from each other, and this intrinsic heterogeneity also affects the system dynamics. We consider coarse-graining approaches that can lead to reduced, more efficient network descriptions, and here we focus on taking the effect of network heterogeneity into consideration in such model reduction efforts.

We will present three distinct illustrative examples of networks of coupled nonidentical units connected by network structures. The first is a network of Kuramoto [1] oscillators; the eigenvectors of the graph Laplacian (corresponding to their network structure) can be used to coarse-grain the system [2], when the connectivity graph possesses a spectral gap (big jumps in the eigenspectrum of their graph Laplacian). We improve on this coarse model by taking into account the effect of the intrinsic oscillator heterogeneity; our approach is based on the observation of correlations between the states of the network nodes and their intrinsic heterogeneities. We also present two other network problems (one involving opinion propagation on a social network, the second involving epidemiological modeling) where network heterogeneities directly affect the system states. The examples highlight the generality of our approach, which links tools from the study of uncertainty quantification (polynomial chaos [3]) to the reduction of dynamical networks.

We computationally implement our coarse models using the equation-free framework [4], a modeling approach that designs and uses short bursts of detailed simulations to perform system-level computational tasks without requiring explicit coarse-grained equations. We discuss our selection of coarse variables and perform coarse-projective integration, coarse fixed point, coarse limit cycle as well as coarse stability computations.

References

[1]  Y. Kuramoto, Chemical oscillations, waves, and turbulence, Berlin; New York: Springer-Verlag, 1984.

[2]  K. Rajendran, A. C. Tsoumanis, I. G. Kevrekidis, Coarse-Graining the Dynamics of (and on) Networks, AIChE Annual Meeting, Salt Lake City (2010); Abstract 586d

[3]  R. G. Ghanem and P. D. Spanos, Stochastic Finite Elements: A Spectral Approach (Springer-Verlag, New York, 1991).

[4]  G. Kevrekidis, C. W. Gear, and G. Hummer, Equation-free: The computer-aided analysis of complex multiscale systems, AIChE Journal, vol. 50, no. 7, pp. 1346-1355, 2004.