We define a percolation problem on the basis of spin configurations of the two-dimensional XY model. Neighboring spins belong to the same percolation cluster if their orientations differ less than a certain threshold called the conducting angle. The percolation properties of this model are studied by means of Monte Carlo simulations and a finite-size scaling analysis. Our simulations show the e...

We reconsider the problem of percolation on an equilibrium random network with degree-degree correlations between nearest-neighboring vertices focusing on critical singularities at a percolation threshold. We obtain criteria for degree-degree correlations to be irrelevant for critical singularities. We present examples of networks in which assortative and disassortative mixing leads to unusual ...

This work deals with bounds on the cost of layout problems for lattice graphs and random lattice graphs. Our main result in this paper is a convergence theorem for the optimal cost of the Minimum Linear Arrangement problem and the Minimum Sum Cut problem, for the case where the underlying graph is obtained through a subcritical site percolation process. This result can be viewed as an analogue ...

We present probabilistic and experimental analysis of induced shortest path one-to-all and one-to-one communication protocols on unreliable mesh networks in this paper. We show that the obtained results are closely related to the results concerning the percolation thresholds studied in physics.

We introduce a new graph parameter called the burning number, inspired by contact processes on graphs such as graph bootstrap percolation, and graph searching paradigms such as Firefighter. The burning number measures the speed of the spread of contagion in a graph; the lower the burning number, the faster the contagion spreads. We provide algorithmic results on the burning number, and prove th...

We develop a real-space renormalization group which renormalizes probabilities directly in the percolation problem. An exact transformation is given in one dimension, and a cluster approach is presented for other lattices. Our results are in excellent agreement with series calculations for the critical percolation concentration p, (both site and bond), and in good agreement for the correlation ...