Global hierarchy vs. local structure: spurious self-feedback in scale-free networks

TL;DR

This paper reveals that mean-field theory's apparent accuracy in predicting the onset of order in scale-free networks is due to a cancellation of self-feedback effects at hubs, highlighting the importance of hierarchical structure in network dynamics.

Contribution

It uncovers the spurious self-feedback effect in mean-field theory and explains the distinct roles of hubs in local versus global order in scale-free networks.

Findings

Higher order interactions cancel self-feedback on hubs.

Hubs are crucial for the onset of local and global order.

Mean-field theory's reliability is due to a cancellation effect.

Abstract

Networks with fat-tailed degree distributions are omnipresent across many scientific disciplines. Such systems are characterized by so-called hubs, specific nodes with high numbers of connections to other nodes. By this property, they are expected to be key to the collective network behavior, e.g., in Ising models on such complex topologies. This applies in particular to the transition into a globally ordered network state, which thereby proceeds in a hierarchical fashion, and with a non-trivial local structure. Standard mean-field theory of Ising models on scale-free networks underrates the presence of the hubs, while nevertheless providing remarkably reliable estimates for the onset of global order. Here, we expose that a spurious self-feedback effect, inherent to mean-field theory, underlies this apparent paradox. More specifically, we demonstrate that higher order interaction effects precisely cancel the self-feedback on the hubs, and we expose the importance of hubs for the distinct onset of local versus global order in the network. Due to the generic nature of our arguments, we expect the mechanism that we uncover for the archetypal case of Ising networks of the Barab\'asi-Albert type to be also relevant for other systems with a strongly hierarchical underlying network structure.