Ensemble inequivalence and phase transitions in unlabeled networks

TL;DR

A first-order phase transition is discovered in the canonical ensemble of random unlabeled networks with a prescribed average number of links, causing the nonconcavity of microcanonical entropy to cause the celebrated percolation transition in labeled networks to be absent.

Abstract

We discover a first-order phase transition in the canonical ensemble of random unlabeled networks with a prescribed average number of links. The transition is caused by the nonconcavity of microcanonical entropy. Above the critical point coinciding with the graph symmetry phase transition, the canonical and microcanonical ensembles are equivalent and have a well-behaved thermodynamic limit. Below the critical point, the ensemble equivalence is broken, and the canonical ensemble is a mixture of phases: empty networks and networks with average degrees diverging logarithmically with the network size. As a consequence, networks with bounded average degrees do not survive in the thermodynamic limit, decaying into the empty phase. The celebrated percolation transition in labeled networks is thus absent in unlabeled networks. In view of these differences between labeled and unlabeled ensembles, the question of which one should be used as a null model of different real-world networks cannot be ignored.

Figures (2)

• Figure 1: Nonconcavity of the microcanonical entropy:(a) The microcanonical entropy $S_{nm}$ of the unlabeled Erdős-Rényi model is shown as a function of the graph density $m/N$ for $n=50$, juxtaposed against the microcanonical entropy of the labeled Erdős-Rényi model, pulled down by $\log n!$. The inset zooms onto the critical point at $m_c/N=\log (n)/(n-1)$, where the unlabeled entropy derivative is near its maximum. (b) The discrete derivatives $\Delta S_{nm}=S_{n,m+1}-S_{nm}$ of unlabeled entropy are shown as functions of the graph density rescaled by the critical value $m/m_c$ for different values of $n$. The dots are the numerical maxima of the derivatives corresponding to the entropy inflection points.
• Figure 2: Broken equivalence and phase transitions:(a) The middle (red) curve shows the unnormalized distribution $p_{nm}Z_n$ for $n=50$ and the value of $q=q_{ce}\approx0.13$, which is such that the expected number of links $\left\langle{m}\right\rangle=\sum_m mp_{nm}\approx88$ matches the critical point maximizing the entropy derivative for the same $n$ in Fig. \ref{['figmicroen']}(b) ($\left\langle{m}\right\rangle/m_c=m/m_c\approx0.90$). The other two curves show the same distribution for two values of $q$ slightly above and below this critical value. The vertical dotted lines indicate the corresponding values of $\left\langle{m}\right\rangle$, while the black dots are the local maxima of the distributions. (b) The rescaled values of $\left\langle{m}\right\rangle$ are shown as functions of the rescaled Boltzmann factor $q$ for different values of $n$, along with the theoretical thermodynamic limit in \ref{['Minf']}.