Random spanning trees of a graph G are governed by corresponding probability mass distribution (or “law”), μ, defined on the set all G. This paper addresses problem choosing μ in order to utilize edges as “fairly” possible. turns out be equivalent minimizing, with respect expected overlap two independent random sampled law μ. In process, we introduce notion homogeneous graphs. These graphs for ...

The stable matching problem is a prototype model in economics and social sciences where agents act selfishly to optimize their own satisfaction, subject to mutually conflicting constraints. A stable matching is a pairing of adjacent vertices in a graph such that no unpaired vertices prefer each other to their partners under the matching. The problem of finding stable matchings is known as the s...

In this paper, we study randomized consensus processing over general random graphs. At time step k, each node will follow the standard consensus algorithm, or stick to current state by a simple Bernoulli trial with success probability pk. Connectivity-independent and arc-independent graphs are defined, respectively, to capture the fundamental independence of random graph processes with respect ...

In this note we adapt a general result of Riordan [Spanning subgraphs of random graphs, Combinatorics, Probability & Computing 9 (2000), no. 2, 125–148] from random graphs to random r-uniform hypergaphs. We also discuss several spanning structures such as cube-hypergraphs, lattices, spheres and Hamilton cycles in hypergraphs.

In this paper, we propose a new method to compute the barycenter of large weighted graphs endowed with probability measures on their nodes. We suppose that the edge weights are distances between the nodes and that the probability measure on the nodes is related to events observed there. For instance, a graph can represent a subway network: its edge weights are the distance between two stations,...

By a theorem of Volkov (2001) we know that on most graphs, with positive probability, the linearly vertex-reinforced random walk (VRRW) stays within a finite “trapping” subgraph at all large times. The question of whether this tail behavior occurs with probability one is open in general. R. Pemantle (1988) in his thesis proved, via a dynamical system approach, that for a VRRW on any complete gr...

In this paper, we consider problems related to the network reliability problem, restricted to graphs of bounded treewidth. We look at undirected simple graphs with each vertex and edge a number in [0, 1] associated. These graphs model networks in which sites and links can fail, with a given probability, independently of whether other sites or links fail or not. The number in [0, 1] associated t...

Real world graphs have been observed to display a number of surprising properties. These properties include heavy-tails for inand out-degree distributions, small diameters, and a densification law [5]. These features do not arise from the classical Erdos-Renyi random graph model [1]. To address these difficulties, Kronecker Graphs were first introduced in [5] as a new method of generating graph...