Flow graphs: interweaving dynamics and structure

TL;DR

This paper introduces flow graphs, a novel integrated network representation embedding dynamics into structure, enabling analysis of complex systems considering both topology and processes simultaneously.

Contribution

It formalizes flow graphs that embed dynamical flows into network links, unifying structure and dynamics analysis in complex systems.

Findings

Flow graphs effectively combine structure and dynamics.

Analysis of linear processes via flow graphs reveals new insights.

Flow graphs can be tailored to specific structural features.

Abstract

The behavior of complex systems is determined not only by the topological organization of their interconnections but also by the dynamical processes taking place among their constituents. A faithful modeling of the dynamics is essential because different dynamical processes may be affected very differently by network topology. A full characterization of such systems thus requires a formalization that encompasses both aspects simultaneously, rather than relying only on the topological adjacency matrix. To achieve this, we introduce the concept of flow graphs, namely weighted networks where dynamical flows are embedded into the link weights. Flow graphs provide an integrated representation of the structure and dynamics of the system, which can then be analyzed with standard tools from network theory. Conversely, a structural network feature of our choice can also be used as the basis for the construction of a flow graph that will then encompass a dynamics biased by such a feature. We illustrate the ideas by focusing on the mathematical properties of generic linear processes on complex networks that can be represented as biased random walks and also explore their dual consensus dynamics.