Unraveling complex networks under the prism of dynamical processes: relations between structure and dynamics

Abstract

We consider relations of structure and dynamics in complex networks. Firstly, a dynamical perspective on the problem of community detection is developed: how to partition a graph into sets of nodes which have stronger relations to each other than to other nodes in the network. We show how several approaches to this problem can be re-interpreted from a dynamical perspective. It is demonstrated how this perspective can circumvent limitations of commonly used, structure based community detection methods such as Modularity or the map-equation, which are prone to over-partition communities of large effective diameter. Secondly, we present graph-theoretical measures to quantify edge-to-edge relations, inspired by the notion of flow redistribution induced by edge failures. We demonstrate how our measures can reveal the dynamical interplay between the edges in a network, including potentially non-local interactions. We showcase the general applicability of our edge-centric measures through analyses of several example systems from different areas. Finally, relations between structure and dynamics are discussed in the context of neural networks. We show how the topology of networks of leaky-integrate-and-fire neurons can be changed such that a “slow-dynamics” arises, in which groups of neurons vary their firing rates coherently, and discuss how this is reflected in spectrum of the network’s coupling matrix. We further consider the problem of detecting cell assemblies, groups of neurons which share a more similar temporal activity pattern when compared to members of other groups, in time series of neural firing events. Using a biophysically inspired pairwise coupling measure we can infer a functional network from the data, and map the task of finding cell assemblies onto a community detection problem, which can be solved within our dynamical framework.