Nucleation during phase transitions in random networks

Complex networks have been mostly characterized from the point of view of the degree distribution of their nodes and a few other motifs (or modules), with a special attention to triangles and cliques. The most exotic phenomena have been observed when the exponent $\gamma$ of the associated power law degree-distribution is sufficiently small. In particular, a zero percolation threshold takes place for $\gamma<3$, and an anomalous critical behavior sets in for $\gamma<5$. In th...

We extend the standard scale-free network model to include a ``triad formation step''. We analyze the geometric properties of networks generated by this algorithm both analytically and by numerical calculations, and find that our model possesses the same characteristics as the standard scale-free networks like the power-law degree distribution and the small average geodesic length, but with the high-clustering at the same time. In our model, the clustering coefficient is also...

We review the recent fast progress in statistical physics of evolving networks. Interest has focused mainly on the structural properties of random complex networks in communications, biology, social sciences and economics. A number of giant artificial networks of such a kind came into existence recently. This opens a wide field for the study of their topology, evolution, and complex processes occurring in them. Such networks possess a rich set of scaling properties. A number ...

Phase transitions in equilibrium and nonequilibrium systems play a major role in the natural sciences. In dynamical networks, phase transitions organize qualitative changes in the collective behavior of coupled dynamical units. Adaptive dynamical networks feature a connectivity structure that changes over time, co-evolving with the nodes' dynamical state. In this Letter, we show the emergence of two distinct first-order nonequilibrium phase transitions in a finite size adapti...

We consider the edge-triangle model, a two-parameter family of exponential random graphs in which dependence between edges is introduced through triangles. In the so-called replica symmetric regime, the limiting free energy exists together with a complete characterization of the phase diagram of the model. We borrow tools from statistical mechanics to obtain limit theorems for the edge density. First, we investigate the asymptotic distribution of this quantity, as the graph s...

We study random subgraphs of an arbitrary finite connected transitive graph $\mathbb G$ obtained by independently deleting edges with probability $1-p$. Let $V$ be the number of vertices in $\mathbb G$, and let $\Omega$ be their degree. We define the critical threshold $p_c=p_c(\mathbb G,\lambda)$ to be the value of $p$ for which the expected cluster size of a fixed vertex attains the value $\lambda V^{1/3}$, where $\lambda$ is fixed and positive. We show that for any such mo...

Ensembles of networks are used as null models in many applications. However, simple null models often show much less clustering than their real-world counterparts. In this paper, we study a model where clustering is enhanced by means of a fugacity term as in the Strauss (or "triangle") model, but where the degree sequence is strictly preserved -- thus maintaining the quenched heterogeneity of nodes found in the original degree sequence. Similar models had been proposed previo...

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...

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 giant $k$-core --- maximal connected subgraph of a network where each node has at least $k$ neighbors --- is important in the study of phase transitions and in applications of network theory. Unlike Erd\H{o}s-R\'enyi graphs and other random networks where $k$-cores emerge discontinuously for $k\ge 3$, we show that transitive linking (or triadic closure) leads to 3-cores emerging through single or double phase transitions of both discontinuous and continuous nature. We als...