The unreasonable effectiveness of tree-based theory for networks with clustering

TL;DR

This paper shows that tree-based theoretical models can accurately predict network dynamics in highly clustered networks if the networks are small-world, confirmed through real-world and synthetic network analyses.

Contribution

It demonstrates the surprising effectiveness of tree-based theories for clustered networks under small-world conditions, supported by analytical and numerical evidence.

Findings

Tree-based theories are accurate for highly clustered networks with small mean intervertex distance.

The theory's accuracy depends on the network's small-world property, not just clustering.

Numerical validation on real and synthetic networks confirms the theory's applicability.

Abstract

We demonstrate that a tree-based theory for various dynamical processes yields extremely accurate results for several networks with high levels of clustering. We find that such a theory works well as long as the mean intervertex distance $\ell$ is sufficiently small - i.e., as long as it is close to the value of $\ell$ in a random network with negligible clustering and the same degree-degree correlations. We confirm this hypothesis numerically using real-world networks from various domains and on several classes of synthetic clustered networks. We present analytical calculations that further support our claim that tree-based theories can be accurate for clustered networks provided that the networks are "sufficiently small" worlds.