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.
Highlights
Hierarchical networks are found ubiquitously across many areas of physics, as well as in social interaction clusters, biological systems, and medicine [1,2,3,4,5,6]
Due to the generic nature of our arguments, we expect the mechanism that we uncover for the archetypal case of Ising networks of the Barabási-Albert type to be relevant for other systems with a strongly hierarchical underlying network structure
To demonstrate that the global mean-field theory approach with a single global order parameter [12] yields a different scaling than Eq (19), we briefly review its derivation here: The probability of two nodes to be connected is expressed by their respective degrees in Eq (3)
Summary
Hierarchical networks are found ubiquitously across many areas of physics, as well as in social interaction clusters, biological systems, and medicine [1,2,3,4,5,6]. Studies of Barabási-Albert-Ising (BAI) networks have been performed in particular using Monte Carlo simulations [10,11] and mean-field theory [12] These studies demonstrated that for finite network sizes (in terms of the number of nodes N) a strong alignment of the Ising degrees of freedom emerges below a specific temperature, resembling the onset of ferromagnetic order in conventional lattice Ising models. The local fields caused by the hubs emerge only when the system orders globally, and are much weaker than predicted by the local mean-field theory Upon taking this effect into account, the TAP approach yields the correct scaling of the transition temperature TT ∼ log(N ), while at the same time it maintains the full information on the network connectivity and explains the hierarchical structure of the local magnetization.
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