Related papers: Mean-field behavior of the negative-weight percola…
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…
A cellular automaton model is presented for random walkers with biologically motivated interactions favoring local alignment and leading to collective motion or swarming behavior. The degree of alignment is controlled by a sensitivity…
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…
Graph-limit theory focuses on the convergence of sequences of graphs when the number of nodes becomes arbitrarily large. This framework defines a continuous version of graphs allowing for the study of dynamical systems on very large graphs,…
We prove that nearest-neighbor percolation in dimensions $d\geq 11$ displays mean-field behavior by proving that the infrared bound holds, in turn implying the finiteness of the percolation triangle diagram. The finiteness of the triangle…
Hypergraph networks are closer to real life because they can reflect higher-order interactions, so researchers have begun using them to build models for real-world networks. The mean-field approach is the current tool for studying the…
We point out a new mechanism which can lead to mean field type behaviour in nonequilibrium critical phenomena. We demonstrate this mechanism on a two-dimensional model which can be understood as a stochastic and non-conservative version of…
In this note we show that percolation on non-amenable Cayley graphs of high girth has a phase of non-uniqueness, i.e., p_c < p_u. Furthermore, we show that percolation and self-avoiding walk on such graphs have mean-field critical…
We introduce the weighted random graph (WRG) model, which represents the weighted counterpart of the Erdos-Renyi random graph and provides fundamental insights into more complicated weighted networks. We find analytically that the WRG is…
For a complete graph of size $n$, assign each edge an i.i.d. exponential variable with mean $n$. For $\lambda>0$, consider the length of the longest path whose average weight is at most $\lambda$. It was shown by Aldous (1998) that the…
The scaling properties of self-avoiding walks on a d-dimensional diluted lattice at the percolation threshold are analyzed by a field-theoretical renormalization group approach. To this end we reconsider the model of Y. Meir and A. B.…
We develop a general universality technique for establishing metric scaling limits of critical random discrete structures exhibiting mean-field behavior that requires four ingredients: (i) from the barely subcritical regime to the critical…
Percolation theory has become a useful tool for the analysis of large-scale wireless networks. We investigate the fundamental problem of characterizing the critical density $\lambda_c^{(d)}$ for $d$-dimensional Poisson random geometric…
Majority bootstrap percolation is a monotone cellular automata that can be thought of as a model of infection spreading in networks. Starting with an initially infected set, new vertices become infected once more than half of their…
We consider random walk and self-avoiding walk whose 1-step distribution is given by $D$, and oriented percolation whose bond-occupation probability is proportional to $D$. Suppose that $D(x)$ decays as $|x|^{-d-\alpha}$ with $\alpha>0$.…
Bootstrap percolation is an often used model to study the spread of diseases, rumors, and information on sparse random graphs. The percolation process demonstrates a critical value such that the graph is either almost completely affected or…
We present a "black box" proof of mean-field near-critical behaviour for a family of functions on $\mathbb Z^d$ (${d>2}$) satisfying a short list of assumptions. The functions represent two-point functions of a lattice statistical…
The bootstrap percolation (or threshold model) is a dynamic process modelling the propagation of an epidemic on a graph, where inactive vertices become active if their number of active neighbours reach some threshold. We study an…
We review the critical behavior of nonequilibrium systems, such as directed percolation (DP) and branching-annihilating random walks (BARW), which possess phase transitions into absorbing states. After reviewing the bulk scaling behavior of…
In this paper we derive and analyse mean-field models for the dynamics of groups of individuals undergoing a random walk. The random motion of individuals is only influenced by the perceived densities of the different groups present as well…