Network processes on clique-networks with high average degree: the limited effect of higher-order structure

TL;DR

This paper examines how local clique structures influence network processes, finding that for high average degrees, the effect of clustering is minimal, and degree sequences explain most differences in dynamics.

Contribution

The study introduces a random graph model with local clique structures and analytically shows the limited impact of clustering on network processes at high degrees.

Findings

High average degree reduces the influence of local clique structures.

Degree sequence differences explain most behavioral variations in clustered vs. non-clustered networks.

Results extend to different dynamics like the Kuramoto model.

Abstract

In this paper, we investigate the effect of local structures on network processes. We investigate a random graph model that incorporates local clique structures to deviate from the locally tree-like behavior of most standard random graph models. For the process of bond percolation, we derive analytical approximations for large outbreaks and the critical percolation value. Interestingly, these derivations show that when the average degree of a vertex is large, the influence of the deviations from the locally tree-like structure is small. Our simulations show that this insensitivity to local clique structures often already kicks in for networks with average degrees as low as 6. Furthermore, we show that the different behavior of bond percolation on clustered networks compared to tree-like networks that was found in previous works can be almost completely attributed to differences in degree sequences rather than differences in clustering structures. We finally show that these results also extend to completely different types of dynamics, by deriving similar conclusions and simulations for the Kuramoto model on the same types of clustered and non-clustered networks.