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The traditional node percolation map of directed networks is reanalyzed in terms of edges. In the percolated phase, edges can mainly organize into five distinct giant connected components, interfaces bridging the communication of nodes in…
We introduce perhaps the simplest models of graph evolution with choice that demonstrate discontinuous percolation transitions and can be analyzed via mathematical evolution equations. These models are local, in the sense that at each step…
We make the first steps towards generalizing the theory of stochastic block models, in the sparse regime, towards a model where the discrete community structure is replaced by an underlying geometry. We consider a geometric random graph…
Binomial random intersection graphs can be used as parsimonious statistical models of large and sparse networks, with one parameter for the average degree and another for transitivity, the tendency of neighbours of a node to be connected.…
In a recent Letter, Yang et al. [Phys. Rev. Lett. 109, 258701 (2012)] introduced the concept of observability transitions: the percolation-like emergence of a macroscopic observable component in graphs in which the state of a fraction of…
We present an exact mathematical framework able to describe site-percolation transitions in real multiplex networks. Specifically, we consider the average percolation diagram valid over an infinite number of random configurations where…
We introduce a model for the formation of social networks, which takes into account the homophily or the tendency of individuals to associate and bond with similar others, and the mechanisms of global and local attachment as well as tie…
The study of time-varying (dynamic) networks (graphs) is of fundamental importance for computer network analytics. Several methods have been proposed to detect the effect of significant structural changes in a time series of graphs. The…
Classical blockmodel is known as the simplest among models of networks with community structure. The model can be also seen as an extremely simply example of interconnected networks. For this reason, it is surprising that the percolation…
Fix a graph $G$ in which every edge is colored in some of $k\ge 2$ colors. Two vertices $u$ and $v$ are CA-connected if $u$ and $v$ may be connected using any subset of $k - 1$ colors. CA-connectivity is an equivalence relation dividing the…
Recently, the percolation transition has been characterized on interacting networks both in presence of interdependent and antagonistic interactions. Here we characterize the phase diagram of the percolation transition in two Poisson…
Finding community structures in networks is important in network science, technology, and applications. To date, most algorithms that aim to find community structures only focus either on unipartite or bipartite networks. A unipartite…
Community detection is a widely-studied unsupervised learning problem in which the task is to group similar entities together based on observed pairwise entity interactions. This problem has applications in diverse domains such as social…
In the new field of financial systemic risk, the network of interbank counterparty relationships can be described as a directed random graph. In "cascade models" of systemic risk, this "skeleton" acts as the medium through which financial…
Community detection in graphs is the problem of finding groups of vertices which are more densely connected than they are to the rest of the graph. This problem has a long history, but it is undergoing a resurgence of interest due to the…
This is a status report on a companion subject to extremal combinatorics, obtained by replacing extremality properties with emergent structure, `phases'. We discuss phases, and phase transitions, in large graphs and large permutations,…
Percolation establishes the connectivity of complex networks and is one of the most fundamental critical phenomena for the study of complex systems. On simple networks, percolation displays a second-order phase transition; on multiplex…
We analyze a minimal model of a growing network. At each time step, a new vertex is added; then, with probability delta, two vertices are chosen uniformly at random and joined by an undirected edge. This process is repeated for t time…
If a vertex $v$ in a graph $G$ has degree larger than the average of the degrees of its neighbors, we call it a groupie in $G$. In the current work, we study the behavior of groupie in random multipartite graphs with the link probability…
Bipartite networks are a common type of network data in which there are two types of vertices, and only vertices of different types can be connected. While bipartite networks exhibit community structure like their unipartite counterparts,…