Extraordinary variability and sharp transitions in a maximally frustrated dynamic network

TL;DR

This paper investigates a dynamic network with two node populations, revealing extraordinary fluctuations and sharp transitions in crosslink density at the population balance point, modeled analytically as an Ising system with long-range interactions.

Contribution

It introduces a minimal model of a dynamic network with introvert and extrovert nodes, demonstrating unprecedented fluctuations at the transition point, and provides an analytical understanding via an Ising model analogy.

Findings

Crosslink density remains near zero or maximum depending on population dominance.

At the critical point, crosslink fraction exhibits large random walk fluctuations.

The system's behavior is analogous to an Ising model with long-range interactions, showing an 'extraordinary transition'.

Abstract

Using Monte Carlo and analytic techniques, we study a minimal dynamic network involving two populations of nodes, characterized by different preferred degrees. Reminiscent of introverts and extroverts in a population, one set of nodes, labeled \textit{introverts} ($I$), prefers fewer contacts (a lower degree) than the other, labeled \textit{extroverts} ($E$). As a starting point, we consider an \textit{extreme} case, in which an $I$ simply cuts one of its links at random when chosen for updating, while an $E$ adds a link to a random unconnected individual (node). The model has only two control parameters, namely, the number of nodes in each group, $N_{I}$ and $N_{E}$). In the steady state, only the number of crosslinks between the two groups fluctuates, with remarkable properties: Its average ($X$) remains very close to 0 for all $N_{I}>N_{E}$ or near its maximum ($\mathcal{N}\equiv N_{I}N_{E}$) if $N_{I}<N_{E}$. At the transition ($N_{I}=N_{E}$), the fraction $X/\mathcal{N}$ wanders across a substantial part of $[0,1]$, much like a pure random walk. Mapping this system to an Ising model with spin-flip dynamics and unusual long-range interactions, we note that such fluctuations are far greater than those displayed in either first or second order transitions of the latter. Thus, we refer to the case here as an `extraordinary transition.' Thanks to the restoration of detailed balance and the existence of a `Hamiltonian,' several qualitative aspects of these remarkable phenomena can be understood analytically.