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We investigate the component sizes of the critical configuration model, as well as the related problem of critical percolation on a supercritical configuration model. We show that, at criticality, the finite third moment assumption on the…
Let $X$ be either $Z^d$ or the points of a Poisson process in $R^d$ of intensity 1. Given parameters $r$ and $p$, join each pair of points of $X$ within distance $r$ independently with probability $p$. This is the simplest case of a…
Almost all network research has been focused on the properties of a single network that does not interact and depends on other networks. In reality, many real-world networks interact with other networks. Here we develop an analytical…
We investigate the dynamic formation of regular random graphs. In our model, we pick a pair of nodes at random and connect them with a link if both of their degrees are smaller than d. Starting with a set of isolated nodes, we repeat this…
We consider bootstrap percolation on uncorrelated complex networks. We obtain the phase diagram for this process with respect to two parameters: $f$, the fraction of vertices initially activated, and $p$, the fraction of undamaged vertices…
Percolation, the formation of a macroscopic connected component, is a key feature in the description of complex networks. The dynamical properties of a variety of systems can be understood in terms of percolation, including the robustness…
Random networks are widely used to model complex networks and research their properties. In order to get a good approximation of complex networks encountered in various disciplines of science, the ability to tune various statistical…
We study the giant component problem slightly above the critical regime for percolation on Poissonian random graphs in the scale-free regime, where the vertex weights and degrees have a diverging second moment. Critical percolation on…
Complex networks display various types of percolation transitions. We show that the degree distribution and the degree-degree correlation alone are not sufficient to describe diverse percolation critical phenomena. This suggests that a…
One of the most influential recent results in network analysis is that many natural networks exhibit a power-law or log-normal degree distribution. This has inspired numerous generative models that match this property. However, more recent…
We study the effects of the degree-degree correlations on the pressure congestion J when we apply a dynamical process on scale free complex networks using the gradient network approach. We find that the pressure congestion for…
We present an analytic formalism describing structural properties of random uncorrelated networks with arbitrary degree distributions. The formalism allows to calculate the main network characteristics like: the position of the phase…
In the infinite configuration network the links between nodes are assigned randomly with the only restriction that the degree distribution has to match a predefined function. This work presents a simple equation that gives for an arbitrary…
The percolation properties of clustered networks are analyzed in detail. In the case of weak clustering, we present an analytical approach that allows to find the critical threshold and the size of the giant component. Numerical simulations…
We investigate the critical phenomena of the degree-ordered percolation (DOP) model on the hierarchical $(u,v)$ flower network. Using the renormalization-group like procedure, we derive the recursion relations for the percolating…
The largest eigenvalue of the matrix describing a network's contact structure is often important in predicting the behavior of dynamical processes. We extend this notion to hypergraphs and motivate the importance of an analogous eigenvalue,…
A wireless communication network is considered where any two nodes are connected if the signal-to-interference ratio (SIR) between them is greater than a threshold. Assuming that the nodes of the wireless network are distributed as a…
In this work, we study the percolation transition and large deviation properties of generalized canonical network ensembles. This new type of random networks might have a very rich complex structure, including high heterogeneous degree…
Percolation is a fundamental concept that brought new understanding on the robustness properties of complex systems. Here we consider percolation on weakly interacting networks, that is, network layers coupled together by much less…
This paper studies networks where all nodes are distributed on a unit square $A\triangleq[(-1/2,1/2)^{2}$ following a Poisson distribution with known density $\rho$ and a pair of nodes separated by an Euclidean distance $x$ are directly…