We consider the classical push broadcast process on a large class of sparse random multigraphs that includes random power law graphs and multigraphs. Our analysis shows that for every ε > 0, whp O(log n) rounds are sufficient to inform all but an ε-fraction of the vertices. It is not hard to see that, e.g. for random power law graphs, the push process needs whp n rounds to inform all vertices. ...

Notation. We will compute probabilities of certain events. Notation P(A) refers to the probability of event A to occur. We use E to denote event “E exists” for an edge E. Similarly, we use E 6∃ for event “E does not exist”. Therefore, P(E) refers to the probability that E exists. We will consider situations where the algorithm computes the probability of following (or, ‘going along’) a specific...

Riis [Electron. J. Combin., 14(1):R44, 2007] introduced a guessing game for graphs which is equivalent to finding protocols for network coding. In this paper we prove upper and lower bounds for the winning probability of the guessing game on undirected graphs. We find optimal bounds for perfect graphs and minimally imperfect graphs, and present a conjecture relating the exact value for all grap...

Randomized rumor spreading is an efficient protocol to distribute information in networks. Recently, a quasirandom version has been proposed and proven to work equally well on many graphs and better for sparse random graphs. In this work we show three main results for the quasirandom rumor spreading model. We exhibit a natural expansion property for networks which suffices to make quasirandom r...

An important conjecture in percolation theory is that almost surely no infinite cluster exists in critical percolation on any transitive graph for which the critical probability is less than 1. Earlier work has established this for the amenable cases Z and Z for large d, as well as for all non-amenable graphs with unimodular automorphism groups. We show that the conjecture holds for several cla...

In this paper we examine spectral properties of random intersection graphs when the number of vertices is equal to the number of labels. We call this class symmetric random intersection graphs. We examine symmetric random intersection graphs when the probability that a vertex selects a label is close to the connectivity threshold τc. In particular, we examine the size of the second eigenvalue o...

a graph is textit{symmetric}, if its automorphism group is transitive on the set of its arcs. in this paper, we  classifyall the connected cubic symmetric  graphs of order $36p$  and $36p^{2}$, for each prime $p$, of which the proof depends on the classification of finite simple groups.

We are interested in finding bounds for the distant-2 chromatic number of geometric graphs drawn from different models. We consider two undirected models of random graphs: random geometric graphs and random proximity graphs for which sharp connectivity thresholds have been shown. We are interested in a.a.s. connected graphs close just above the connectivity threshold. For such subfamilies of ra...

Interference graphs are used for performance analysis of multiprocessor interconnection networks. In order to model blocked transmissions, nodes can have three states: idle, active or blocked. The resulting steady state probability distribution has a non-product form. Macro states are introduced to calculate performance measures, and the corresponding macro state probability distribution is app...

Recent years have seen a resurgence of interest in graph asymmetry and many of its novel applications ranging from graph compression to discovery of graph dynamics (e.g., age of a node). It has been known for some time that Erdős-Rényi graphs are asymmetric with high probability, but it is also known that real world graphs (web, biological networks) have a significant amount of symmetry. So the...