Related papers: Deriving an underlying mechanism for discontinuous…
Explosive Percolation describes the abrupt onset of large-scale connectivity that results from a simple random process designed to delay the onset of the transition on an underlying random network or lattice. Explosive percolation…
We consider inhomogeneous spatial random graphs on the real line. Each vertex carries an i.i.d. weight and edges are drawn such that short edges and edges to vertices with large weights occur with higher probability. This allows the study…
We analyze the component evolution in inhomogeneous random intersection graphs when the average degree is close to 1. As the average degree increases, the size of the largest component in the random intersection graph goes through a phase…
Ever since the Barab\'{a}si-Albert (BA) scale-free network has been proposed, network modeling has been studied intensively in light of the network growth and the preferential attachment (PA). However, numerous real systems are featured…
We introduce a correlated static model and investigate a percolation transition. The model is a modification of the static model and is characterized by assortative degree-degree correlation. As one varies the edge density, the network…
Environmental heterogeneity can drive genetic heterogeneity in expanding populations; mutant strains may emerge that trade overall growth rate for an improved ability to survive in patches that are hostile to the wild type. This…
We study the history-dependent percolation in two dimensions, which evolves in generations from standard bond-percolation configurations through iteratively removing occupied bonds. Extensive simulations are performed for various…
We introduce a model for dynamic networks, where the links or the strengths of the links change over time. We solve the model by mapping dynamic networks to the problem of directed percolation, where the direction corresponds to the…
We study the site-bond percolation on a hierarchical scale-free network, namely, the decorated (2,2)-flower, by using the renormalization group technique. The phase diagram essentially depends on the fraction of occupied sites.…
We introduce a novel stochastic growth process, the record-driven growth process, which originates from the analysis of a class of growing networks in a universal limiting regime. Nodes are added one by one to a network, each node…
We show that for all $d\in \{3,\ldots,n-1\}$ the size of the largest component of a random $d$-regular graph on $n$ vertices around the percolation threshold $p=1/(d-1)$ is $\Theta(n^{2/3})$, with high probability. This extends known…
Can one hear the 'sound' of a growing network? We address the problem of recognizing the topology of evolving biological or social networks. Starting from percolation theory, we analytically prove a linear inverse relationship between two…
Transition points mark qualitative changes in the macroscopic properties of large complex systems. Explosive transitions, exhibiting properties of both continuous and discontinuous phase transitions, have recently been uncovered in network…
Static wireless networks are by now quite well understood mathematically through the random geometric graph model. By contrast, there are relatively few rigorous results on the practically important case of mobile networks, in which the…
We define a general model of stochastically-evolving graphs, namely the \emph{Edge-Uniform Stochastically-Evolving Graphs}. In this model, each possible edge of an underlying general static graph evolves independently being either alive or…
Percolation in complex networks is viewed as both: a process that mimics network degradation and a tool that reveals peculiarities of the underlying network structure. During the course of percolation, networks undergo non-trivial…
Recently much attention has been paid to the study of the robustness of interdependent and multiplex networks and, in particular, networks of networks. The robustness of interdependent networks can be evaluated by the size of a mutually…
We revisit the phase transition for percolation on randomly stretched lattices. Starting with the usual square grid, keep all vertices untouched while erasing edges according as follows: for every integer $i$, the entire column of vertical…
Networks are ubiquitous in diverse real-world systems. Many empirical networks grow as the number of nodes increases with time. Percolation transitions in growing random networks can be of infinite order. However, when the growth of large…
The analysis in this paper helps to explain the formation of growing networks with degree distributions that follow extended exponential or power-law tails. We present a generic model in which edge dynamics are driven by a continuous…