Related papers: Generating-function approach for bond percolations…
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…
We provide sufficient conditions for a regular graph $G$ of growing degree $d$, guaranteeing a phase transition in its random subgraph $G_p$ similar to that of $G(n,p)$ when $p\cdot d\approx 1$. These conditions capture several well-studied…
The probability distribution for the number of top to bottom spanning clusters in Directed percolation in two and three dimensions appears to be universal and is of the form $P(n) \sim \exp(-\alpha n^2)$. We argue that $\alpha$ is a new…
Percolation threshold of a network is the critical value such that when nodes or edges are randomly selected with probability below the value, the network is fragmented but when the probability is above the value, a giant component…
We describe the effect of power-law initial distributions of clusters on ordinary percolation and its generalizations, specifically, models of explosive percolation processes based on local optimization. These aggregation processes were…
The asymptotic behavior of the percolation threshold $p_c$ and its dependence upon coordination number $z$ is investigated for both site and bond percolation on four-dimensional lattices with compact extended neighborhoods. Simple…
Bootstrap percolation on the random graph $G_{n,p}$ is a process of spread of "activation" on a given realization of the graph with a given number of initially active nodes. At each step those vertices which have not been active but have at…
The growth of two-dimensional lattice bond percolation clusters through a cooperative Achlioptas-type of process, where the choice of which bond to occupy next depends upon the masses of the clusters it connects, is shown to go through an…
Let $G_{n,p}^1$ be a superposition of the random graph $G_{n,p}$ and a one-dimensional lattice: the $n$ vertices are set to be on a ring with fixed edges between the consecutive vertices, and with random independent edges given with…
Dependency links in single-layer networks offer a convenient way of modeling nonlocal percolation effects in networked systems where certain pairs of nodes are only able to function together. We study the percolation properties of the weak…
Using the finite-size scaling, we have investigated the percolation phase transitions of evolving random networks under a generalized Achlioptas process (GAP). During this GAP, the edge with minimum product of two connecting cluster sizes…
We study the behavior of scale-free networks, having connectivity distribution P(k) k^-a, close to the percolation threshold. We show that for networks with 3<a<4, known to undergo a transition at a finite threshold of dilution, the…
We study Mandelbrot's percolation process in dimension $d \geq 2$. The process generates random fractal sets by an iterative procedure which starts by dividing the unit cube $[0,1]^d$ in $N^d$ subcubes, and independently retaining or…
Percolation on a one-dimensional lattice and fractals such as the Sierpinski gasket is typically considered to be trivial because they percolate only at full bond density. By dressing up such lattices with small-world bonds, a novel…
In this paper we introduce a novel description of the equilibrium state of a bond percolation process on random graphs using the exact method of generating functions. This allows us to find the expected size of the giant connected component…
Percolation clusters are random fractals whose geometrical and transport properties can be characterized with the help of probability distribution functions. Using renormalized field theory, we determine the asymptotic form of various of…
We study a generalization of site percolation on a simple cubic lattice, where not only single sites are removed randomly, but also entire parallel columns of sites. We show that typical clusters near the percolation transition are very…
Interdependent networks are more fragile under random attacks than simplex networks, because interlayer dependencies lead to cascading failures and finally to a sudden collapse. This is a hybrid phase transition (HPT), meaning that at the…
We study percolation on the hierarchical lattice of order $N$ where the probability of connection between two points separated by distance $k$ is of the form $c_k/N^{k(1+\delta)},\; \delta >-1$. Since the distance is an ultrametric, there…
We consider long-range percolation on $\mathbb{Z}^d$, where the probability that two vertices at distance $r$ are connected by an edge is given by $p(r)=1-\exp[-\lambda(r)]\in(0,1)$ and the presence or absence of different edges are…