Structured Networks and Coarse‐Grained Descriptions

Abstract

This chapter discusses the interplay between structure and dynamics in complex networks. Given a particular network with an endowed dynamics, our goal is to find partitions aligned with the dynamical process acting on top of the network. We thus aim to gain a reduced description of the system that takes into account both its structure and dynamics.In the first part, we introduce the general mathematical setup for the types of dynamics we consider throughout the chapter. We provide two guiding examples, namely consensus dynamics and diffusion processes (random walks), motivating their connection to social network analysis, and provide a brief discussion on the general dynamical framework and its possible extensions.In the second part, we focus on the influence of graph structure on the dynamics taking place on the network, focussing on three concepts that allow us to gain insight into this notion. First, we describe how time scale separation can appear in the dynamics on a network as a consequence of graph structure. Second, we discuss how the presence of particular symmetries in the network give rise to invariant dynamical subspaces that can be precisely described by graph partitions. Third, we show how this dynamical viewpoint can be extended to study dynamics on networks with signed edges, which allow us to discuss connections to concepts in social network analysis, such as structural balance.In the third part, we discuss how to use dynamical processes unfolding on the network to detect meaningful network substructures. We then show how such dynamical measures can be related to seemingly different algorithm for community detection and coarse-graining proposed in the literature. We conclude with a brief summary and highlight interesting open future directions.

How minimizing conflicts could lead to polarization on social media: An agent-based model investigation

“…This is actually expected due to the mathematical properties of the model: a high-tolerance high-volatility scenario replicates the conditions studied in models of consensus dynamics [ 39 , 40 ]. With high tolerance, most users are not pushed by anything but only pulled; and with high volatility they will converge quickly towards the average polarity of their neighbors.…”

Scaling up real networks by geometric branching growth

“…Potential applications that require optimization or control of system size in complex systems are countless. Apart from those explained here, we can mention the assessment of scalability issues in dynamic processes in core functions of real networks, such as in Internet routing Finally, the presence of symmetry has been observed to affect synchronization on networks (63) and the spectrum of eigenvectors of the Laplacian matrix (64), which controls diffusive processes. Future work will be needed to clarify the influence of the evolutionary self-similarity of real networks described in this paper on dynamical processes taking place on them.…”

Modularity and Dynamics on Complex Networks

“…It is well-known that there exists a two-way relationship between dynamics on graphs and the underlying graph structure (Schaub, Delvenne, Lambiotte, & Barahona, 2019). On the one hand, the structure of a network affects dynamical processes taking place on it (M. A.…”

“…In particular, we explain how these concepts are linked with modular network structure in case of a linear dynamics. Our exposition builds on Schaub et al (2019), but provides a substantial amount of additional detail. Time scale separation is the phenomenon when certain state variables of a dynamical system evolve much faster than other state variables.…”