The problem of explicitly constructing Ramsey graphs, i.e graphs that do not have a large clique or independent set is considered. We provide an elementary construction of a graph with the property that there is no clique or independent set of t of nodes, while the graph size is t p log log t log log log t . The construction is based on taking the product of all graphs in a distribution that is...

In Chung-Lu random graphs, a classic model for real-world networks, each vertex is equipped with a weight drawn from a power-law distribution (for which we fix an exponent 2 < β < 3), and two vertices form an edge independently with probability proportional to the product of their weights. Modern, more realistic variants of this model also equip each vertex with a random position in a specific ...

cientists building models of the world by necessity abstract away features not directly relevant to their line of inquiry. Furthermore, complete knowledge of relevant features is not generally possible. The mathematical formalism that has proven to be the most successful at simultaneously abstracting the irrelevant, while effectively summarizing incomplete knowledge, is probability theory. Firs...

Recently Karger proposed a new randomized algorithm for nding a minimum cut of an n-vertex graph (weighted or unweighted) with probability (n 2 ). In this paper we present a new probabilistic analysis of Karger's randomized algorithm for a few classes of unweighted graphs. For random graphs whose edges are selected with a given probability p, logn n p 1, we show that the expectation of success ...

The probability that an embedding of a graph in Z 3 is knotted is investigated. For any given graph (embeddable in Z 3) without cut edges, it is shown that this probability approaches 1 at an exponential rate as the number of edges in the embedding goes to innnity. Furthermore, at least for a subset of these graphs, the rate at which the probability approaches 1 does not depend on the particula...

We consider the reliability of graphs for which nodes fail independently of each other with a constant probability 1 p. The reliability of a graph is defined to be the probability that the induced subgraph of surviving nodes is connected. A graph is said to be uniformly best when, for all choices of p, it is most reliable in the class of graphs with the same number of nodes and same number of e...

In majority dynamics, agents located at the vertices of an undirected simple graph update their binary opinions synchronously by adopting those of the majority of their neighbors. On infinite unimodular transitive graphs (e.g., Cayley graphs), when initial opinions are chosen from a distribution that is invariant with respect to the graph automorphism group, we show that the opinion of each age...

A known approach of detecting dense subgraphs (communities) in large sparse graphs involves first computing the probability vectors for short random walks on the graph, and then using these probability vectors to detect the communities, see Latapy and Pons [2005]. In this paper we focus on the first part of such an approach i.e. the computation of the probability vectors for the random walks, a...