Related papers: Dynamics of k-core percolation in a random graph
The problem of vertex coloring in random graphs is studied using methods of statistical physics and probability. Our analytical results are compared to those obtained by exact enumeration and Monte-Carlo simulations. We critically discuss…
We introduce the heterogeneous-$k$-core, which generalizes the $k$-core, and contrast it with bootstrap percolation. Vertices have a threshold $k_i$ which may be different at each vertex. If a vertex has less than $k_i$ neighbors it is…
We compute the joint large deviation rate functional in the limit of large time for the current flowing through the edges of a finite graph on which a boundary-driven system of stochastic particles evolves with zero-range dynamics.This…
Previous work showed that the collective activity of large neuronal networks can be tamed to remain near its critical point by a feedback control that maximizes the temporal correlations of the mean-field fluctuations. Since such…
Graph bootstrap percolation is a discrete-time process capturing the spread of a virus on the edges of $K_n$. Given an initial set $G\subseteq K_n$ of infected edges, the transmission of the virus is governed by a fixed graph $H$: in each…
Graph coloring is one of the most famous computational problems with applications in a wide range of areas such as planning and scheduling, resource allocation, and pattern matching. So far coloring problems are mostly studied on static…
We study the structure and the dynamics in the formation of irreversible gels by means of molecular dynamics simulation of a model system where the gelation transition is due to the random percolation of permanent bonds between neighboring…
Information processing in the brain crucially depends on encoding properties of single neurons, with particular relevance of the spike-generation mechanism. The latter hinges upon the bifurcation type at the transition point between resting…
Consider a critical Erd\"os-R\'enyi random graph: $n$ is the number of vertices, each one of the $\binom{n}{2}$ possible edges is kept in the graph independently from the others with probability $n^{-1}+\lambda n^{-4/3}$, $\lambda$ being a…
Bootstrap percolation in (random) graphs is a contagion dynamics among a set of vertices with certain threshold levels. The process is started by a set of initially infected vertices, and an initially uninfected vertex with threshold $k$…
We describe the critical window for percolation in the universality class of sparse growing random graphs. In our models, vertices arrive sequentially and connect independently to each earlier vertex $v$ with probability proportional to a…
We propose a bond-percolation model intended to describe the consumption, and eventual exhaustion, of resources in transport networks. Edges forming minimum-length paths connecting demanded origin-destination nodes are removed if below a…
Contraction of an edge merges its end points into a new vertex which is adjacent to each neighbor of the end points of the edge. An edge in a $k$-connected graph is {\em contractible} if its contraction does not result in a graph of lower…
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…
Let $k \ge 3$ be a fixed integer. We exactly determine the asymptotic distribution of $\ln Z_k(G(n,m))$, where $Z_k(G(n,m))$ is the number of $k$-colourings of the random graph $G(n,m)$. A crucial observation to this aim is that the…
We study directed random graphs (random graphs whose edges are directed) as they evolve in discrete time by the addition of nodes and edges. For two distinct evolution strategies, one that forces the graph to a condition of near acyclicity…
A set of discrete individual points located in an embedding continuum space can be seen as percolating or non-percolating, depending on the radius of the discs/spheres associated with each of them. This problem is relevant in theoretical…
Building on previous work by Cameron et al. in [3], we give a recurrence for computing the number of acyclic orientations of complete $k$-partite graphs, which can be implemented to obtain a dynamic programming algorithm running in time…
Practical methods for quantitative analysis of radial and angular coordinates of leafy organs of vascular plants are presented and applied to published phyllotactic patterns of various real systems from young leaves on a shoot tip to…
The edge-degeneracy model is an exponential random graph model that uses the graph degeneracy, a measure of the graph's connection density, and number of edges in a graph as its sufficient statistics. We show this model is relatively…