(589a) Coarse-Graining Heterogeneous Network Model Dynamics

Networks (graphs) are useful in modeling diverse physical/engineering systems, from the Internet to social interactions, and from epidemiology to gene expression. Simple network models involve a prescribed network structure consisting of identical network nodes; yet real-world networks are usually characterized by heterogeneity, often in multiple node features.

Large scale networks constitute high-dimensional dynamical systems. For analysis and accelerated simulation, we seek reduced descriptions in terms of just a few collective "coarse" variables. Such a more efficient description can be used for coarse-grained system analysis, including coarse projective integration, fixed point, and stability computations.

Coarse-graining heterogeneous networks must take into account the effect of the heterogeneity on the states of the nodes in the network. In previous work [2], we have observed that in problems where the network state becomes quickly correlated with a heterogeneous parameter, one can obtain an efficient reduced description by representing the network states as a polynomial chaos expansion of the heterogeneity [1].

In this work, we consider two distinct types of heterogeneous parameters: (1) parameters that are specific to each node, which we call intrinsic heterogeneities, and (2) properties of the nodes imparted by the network structure, which we call structural heterogeneities. We use a simple epidemiological model with both types of heterogeneity to illustrate this type of coarse graining, in which the node (intrinsic) heterogeneity is the distribution of individuals' ages, and the network heterogeneity is the degree distribution.

References
[1] Roger G. Ghanem and Pol D. Spanos. Stochastic Finite Elements. Courier Dover Publications, 2003.
[2] Karthikeyan Rajendran and Ioannis Kevrekidis. Coarse graining the dynamics of heterogeneous oscillators in
networks with spectral gaps. Physical Review E, 84(3), September 2011.