We extend the jigsaw percolation model to analyze graphs where both underlying people and puzzle graphs are Erdös-Rényi random graphs. Let pppl and ppuz denote the probability that an edge exists in the respective people and puzzle graphs and define peff = ppplppuz, the effective probability. We show for constants c1 > 1 and c2 > π /6 and c3 < e −5 if min(pppl, ppuz) > c1 logn/n the critical ef...

In [1] it is shown that the first order theory of almost all generalized Steinhaus graphs is identical to the first order theory of almost all where each generalized Steinhaus graph is given the same probability. A natural probability measure on generalized Steinhaus graphs is obtained by independently assigning a probability of p for each entry in the generating string of the graph. With this ...

The problem of finding birth–death fixation probabilities for configurations of normal andmutants on an N-vertex graph is formulated in terms of a Markov process on the 2N-dimensional state space of possible configurations. Upper and lower bounds on the fixation probability after any given number of iterations of the birth–death process are derived in terms of the transition matrix of this proc...

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We study the Moran process as adapted by Lieberman, Hauert and Nowak. A family of directed graphs is said to be strongly amplifying if the extinction probability tends to 0 when the Moran process is run on graphs in this family. The most-amplifying known family of directed graphs is the family of megastars of Galanis et al. We show that this family is optimal, up to logarithmic factors, since e...

Edge-independent random graphs are a model of random graphs in which each potential edge appears independently with an individual probability. Based on the relative entropy method, we determine the upper and lower bounds for the extremal vertex degrees using the edge probability matrix and its largest eigenvalue. Moreover, an application to random graphs with given expected degree sequences is ...

Measure and category There are two natural ways of saying that a set of countable graphs is “large”. Choose a fixed countable vertex set, and enumerate the pairs of vertices: {x0, y0}, {x1, y1}, . . . There is a probability measure on the set of graphs, obtained by choosing independently with probability 1/2 whether xi and yi are joined, for all i. Now a set of graphs is “large” if it has proba...