Phase transitions in a complex network

Within the conventional statistical physics framework, we study critical phenomena in a class of configuration network models with hidden variables controlling links between pairs of nodes. We find analytical expressions for the average node degree, the expected number of edges, and the Landau and Helmholtz free energies, as a function of the temperature and number of nodes. We show that the network's temperature is a parameter that controls the average node degree in the who...

In this work we study the entropy of the Gibbs state corresponding to a graph. The Gibbs state is obtained from the Laplacian, normalized Laplacian or adjacency matrices associated with a graph. We calculated the entropy of the Gibbs state for a few classes of graphs and studied their behavior with changing graph order and temperature. We illustrate our analytical results with numerical simulations for Erd\H{o}s-R\'enyi, Watts-Strogatz, Barab\'asi-Albert and Chung-Lu graph mo...

Preferential attachment is a central paradigm in the theory of complex networks. In this contribution we consider various generalizations of preferential attachment including for example node removal and edge rewiring. We demonstrate that generalized preferential attachment networks can undergo a topological phase transition. This transition separates networks having a power-law tail degree distribution from those with an exponential tail. The appearance of the phase transiti...

The unconstrained exponential family of random graphs assumes no prior knowledge of the graph before sampling, but it is natural to consider situations where partial information about the graph is known, for example the total number of edges. What does a typical random graph look like, if drawn from an exponential model subject to such constraints? Will there be a similar phase transition phenomenon (as one varies the parameters) as that which occurs in the unconstrained expo...

Due to their conceptual and mathematical simplicity, Erd\"os-R\'enyi or classical random graphs remain as a fundamental paradigm to model complex interacting systems in several areas. Although condensation phenomena have been widely considered in complex network theory, the condensation of degrees has hitherto eluded a careful study. Here we show that the degree statistics of the classical random graph model undergoes a first-order phase transition between a Poisson-like dist...

We analyze maximum entropy random graph ensembles with constrained degrees, drawn from arbitrary degree distributions, and a tuneable number of 3-loops (triangles). We find that such ensembles generally exhibit two transitions, a clustering and a shattering transition, separating three distinct regimes. At the clustering transition, the graphs change from typically having only isolated loops to forming loop clusters. At the shattering transition the graphs break up into exten...

In this paper, we reveal the relationship between entropy rate and the congestion in complex network and solve it analytically for special cases. Finding maximizing entropy rate will lead to an improvement of traffic efficiency, we propose a method to mitigate congestion by allocating limited traffic capacity to the nodes in network rationally. Different from former strategies, our method only requires local and observable information of network, and is low-cost and widely ap...

We introduce a mapping between graphs and pure quantum bipartite states and show that the associated entanglement entropy conveys non-trivial information about the structure of the graph. Our primary goal is to investigate the family of random graphs known as complex networks. In the case of classical random graphs we derive an analytic expression for the averaged entanglement entropy $\bar S$ while for general complex networks we rely on numerics. For large number of nodes $...

We introduce an evolving network model in which a new node attaches to a randomly selected target node and also to each of its neighbors with probability $p$. The resulting network is sparse for $p<\frac{1}{2}$ and dense (average degree increasing with number of nodes $N$) for $p\geq \frac{1}{2}$. In the dense regime, individual networks realizations built by this copying mechanism are disparate and not self-averaging. Further, there is an infinite sequence of structural anom...

We generalize the original majority-vote (MV) model from two states to arbitrary $p$ states and study the order-disorder phase transitions in such a $p$-state MV model on complex networks. By extensive Monte Carlo simulations and a mean-field theory, we show that for $p\geq3$ the order of phase transition is essentially different from a continuous second-order phase transition in the original two-state MV model. Instead, for $p\geq3$ the model displays a discontinuous first-o...