Structural Transitions in Dense Networks

We study a mean field model of a complex network, focusing on edge and triangle densities. Our first result is the derivation of a variational characterization of the entropy density, compatible with the infinite node limit. We then determine the optimizing graphs for small triangle density and a range of edge density, though we can only prove they are local, not global, maxima of the entropy density. With this assumption we then prove that the resulting entropy density must ...

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 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 ...

In this study we introduce and analyze the statistical structural properties of a model of growing networks which may be relevant to social networks. At each step a new node is added which selects 'k' possible partners from the existing network and joins them with probability delta by undirected edges. The 'activity' of the node ends here; it will get new partners only if it is selected by a newcomer. The model produces an infinite-order phase transition when a giant componen...

We study the evolution of graphs densifying by adding edges: Two vertices are chosen randomly, and an edge is (i) established if each vertex belongs to a tree; (ii) established with probability $p$ if only one vertex belongs to a tree; (iii) an attempt fails if both vertices belong to unicyclic components. Emerging simple random graphs contain only trees and unicycles. In the thermodynamic limit of an infinite number of vertices, the fraction of vertices in unicycles undergoe...

We show, through local estimates and simulation, that if one constrains simple graphs by their densities $\varepsilon$ of edges and $\tau$ of triangles, then asymptotically (in the number of vertices) for over $95\%$ of the possible range of those densities there is a well-defined typical graph, and it has a very simple structure: the vertices are decomposed into two subsets $V_1$ and $V_2$ of fixed relative size $c$ and $1-c$, and there are well-defined probabilities of edge...

I start by reviewing some basic properties of random graphs. I then consider the role of random walks in complex networks and show how they may be used to explain why so many long tailed distributions are found in real data sets. The key idea is that in many cases the process involves copying of properties of near neighbours in the network and this is a type of short random walk which in turn produce a natural preferential attachment mechanism. Applying this to networks of fi...

Real-world networks are rarely static. Recently, there has been increasing interest in both network growth and network densification, in which the number of edges scales superlinearly with the number of nodes. Less studied but equally important, however, are scaling laws of higher-order cliques, which can drive clustering and network redundancy. In this paper, we study how cliques grow with network size, by analyzing several empirical networks from emails to Wikipedia interac...

We analyze the 3-parameter family of random networks which are uniform on networks with fixed number of edges, triangles, and nodes (between 33 and 66). We find precursors of phase transitions which are known to be present in the asymptotic node regime as the edge and triangle numbers are varied, and focus on one of the discontinuous ones. By use of a natural edge flip dynamics we determine nucleation barriers as a random network crosses the transition, in analogy to the proc...

We study a general set of models of social network evolution and dynamics. The models consist of both a dynamics on the network and evolution of the network. Links are formed preferentially between 'similar' nodes, where the similarity is defined by the particular process taking place on the network. The interplay between the two processes produces phase transitions and hysteresis, as seen using numerical simulations for three specific processes. We obtain analytic results us...