Related papers: Ordinary Percolation with Discontinuous Transition…
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…
The basic notion of percolation in physics assumes the emergence of a giant connected (percolation) cluster in a large disordered system when the density of connections exceeds some critical value. Until recently, the percolation phase…
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…
The Gaussian model of discontinuous percolation, recently introduced by Ara\'ujo and Herrmann [Phys. Rev. Lett., 105, 035701 (2010)], is numerically investigated in three dimensions, disclosing a discontinuous transition. For the…
We show that only considering the largest cluster suffices to obtain a first-order percolation transition. As opposed to previous realizations of explosive percolation our models obtain Gaussian cluster distributions and compact clusters as…
Percolation has long served as a model for diverse phenomena and systems. The percolation transition, that is, the formation of a giant cluster on a macroscopic scale, is known as one of the most robust continuous transitions. Recently,…
Transition out of a topological phase is typically characterized by discontinuous changes in topological invariants along with bulk gap closings. However, as a clean system is geometrically punctured, it is natural to ask the fate of an…
We present an exact solution of percolation in a generalized class of Watts-Strogatz graphs defined on a 1-dimensional underlying lattice. We find a non-classical critical point in the limit of the number of long-range bonds in the system…
Three-dimensional bond or site percolation theory on a lattice can be interpreted as a gauge theory in which the Wilson loops are viewed as counters of topological linking with random clusters. Beyond the percolation threshold large Wilson…
We study the percolation phase transition in hierarchical scale-free nets. Depending on the method of construction, the nets can be fractal or small-world (the diameter grows either algebraically or logarithmically with the net size),…
We introduce a bond percolation procedure on a $D$-dimensional lattice where two neighbouring sites are connected by $N$ channels, each operated by valves at both ends. Out of a total of $N$, randomly chosen $n$ valves are open at every…
The physics of $k$-core percolation pertains to those systems whose constituents require a minimum number of $k$ connections to each other in order to participate in any clustering phenomenon. Examples of such a phenomenon range from…
We consider a percolation process in which $k$ points separated by a distance proportional to system size $L$ simultaneously connect together ($k>1$), or a single point at the center of a system connects to the boundary ($k=1$), through…
We introduce a cluster growth process that provides a clear connection between equilibrium statistical mechanics and an explosive percolation model similar to the one recently proposed by Achlioptas et al. [Science 323, 1453 (2009)]. We…
Percolation refers to the emergence of a giant connected cluster in a disordered system when the number of connections between nodes exceeds a critical value. The percolation phase transitions were believed to be continuous until recently…
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…
Percolation is perhaps the simplest example of a process exhibiting a phase transition and one of the most studied phenomena in statistical physics. The percolation transition is continuous if sites/bonds are occupied independently with the…
Global physical properties of random media change qualitatively at a percolation threshold, where isolated clusters merge to form one infinite connected component. The precise knowledge of percolation thresholds is thus of paramount…
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…
The bond-percolation properties of the Hanoi networks are analyzed with the renormalization group. Unlike scale-free networks, they are meant to provide an analytically tractable interpolation between finite dimensional, lattice-based…