Components in time-varying graphs

TL;DR

This paper extends traditional graph metrics to time-varying graphs, enabling the analysis of dynamic systems and revealing features like variability in node activity patterns that static analysis misses.

Contribution

It introduces a novel approach to define and identify components in time-varying graphs by mapping the problem to maximal cliques in an affine graph, addressing a previously unhandled aspect of dynamic network analysis.

Findings

Time-aware components reveal large variability in node activity.

Static analysis fails to detect fluctuations in individual activity patterns.

Temporal analysis captures important features of real dynamic systems.

Abstract

Real complex systems are inherently time-varying. Thanks to new communication systems and novel technologies, it is today possible to produce and analyze social and biological networks with detailed information on the time of occurrence and duration of each link. However, standard graph metrics introduced so far in complex network theory are mainly suited for static graphs, i.e., graphs in which the links do not change over time, or graphs built from time-varying systems by aggregating all the links as if they were concurrent in time. In this paper, we extend the notion of connectedness, and the definitions of node and graph components, to the case of time-varying graphs, which are represented as time-ordered sequences of graphs defined over a fixed set of nodes. We show that the problem of finding strongly connected components in a time-varying graph can be mapped into the problem of discovering the maximal-cliques in an opportunely constructed static graph, which we name the affine graph. It is therefore an NP-complete problem. As a practical example, we have performed a temporal component analysis of time-varying graphs constructed from three data sets of human interactions. The results show that taking time into account in the definition of graph components allows to capture important features of real systems. In particular, we observe a large variability in the size of node temporal in- and out-components. This is due to intrinsic fluctuations in the activity patterns of individuals, which cannot be detected by static graph analysis.