Equation-free analysis of a dynamically evolving multigraph

TL;DR

This paper demonstrates how equation-free methods, combined with data-mining techniques, can efficiently analyze and accelerate simulations of dynamically evolving multigraph networks, revealing low-dimensional coarse-grained descriptions.

Contribution

It introduces an equation-free framework for multigraph analysis, integrating data-mining to identify effective macroscopic variables for complex network systems.

Findings

Accelerated network simulations via Coarse Projective Integration.

Identification of coarse stationary states using Newton-GMRES.

Data-mining techniques effectively find low-dimensional system descriptions.

Abstract

In order to illustrate the adaptation of traditional continuum numerical techniques to the study of complex network systems, we use the equation-free framework to analyze a dynamically evolving multigraph. This approach is based on coupling short intervals of direct dynamic network simulation with appropriately-defined lifting and restriction operators, mapping the detailed network description to suitable macroscopic (coarse-grained) variables and back. This enables the acceleration of direct simulations through Coarse Projective Integration (CPI), as well as the identification of coarse stationary states via a Newton-GMRES method. We also demonstrate the use of data-mining, both linear (principal component analysis, PCA) and nonlinear (diffusion maps, DMAPS) to determine good macroscopic variables (observables) through which one can coarse-grain the model. These results suggest methods for decreasing simulation times of dynamic real-world systems such as epidemiological network models. Additionally, the data-mining techniques could be applied to a diverse class of problems to search for a succint, low-dimensional description of the system in a small number of variables.