Hypergraph entropy is an information theoretic functional on a hypergraph with a probability distribution on its vertex set. It is sub-additive with respect to the union of hypergraphs. In case of simple graphs, exact additivity for the entropy of a graph and its complement with respect to every probability distribution on the vertex set gives a characterization of perfect graphs. Here we inves...

This research focuses on analyzing selection amplifiers in population genetics. Since the structure of a population graph can influence selection, this paper focuses on finding undirected graphs that can amplify selection. To clearly demonstrate the idea, the paper will briefly discuss the basic Moran model at first. Then it will analyze the star graph as an example of selection amplifiers that...

We construct a finitely generated group G without the Liouville property such that the return probability of a random walk satisfies p2n(e, e) & e−n 1/2+o(1) . This shows that the constant 1/2 in a recent theorem by Saloff-Coste and Zheng, saying that return probability exponent less than 1/2 implies the Liouville property, cannot be improved. Our construction is based on permutational wreath p...

Upper and lower bounds on the error probability of linear codes under maximum-likelihood (ML) decoding are shortly surveyed and applied to ensembles of codes on graphs. For upper bounds, focus is put on Gallager bounding techniques and their relation to a variety of other reported bounds. Within the class of lower bounds, we address de Caen’s based bounds and their improvements, sphere-packing ...

We show that the number of labeled cubic planar graphs on n vertices with n even is asymptotically αnρn!, where ρ . = 3.13259 and α are analytic constants. We show also that the chromatic number of a random cubic planar graph that is chosen uniformly at random among all the labeled cubic planar graphs on n vertices is three with probability tending to e /4! . = 0.999568, and is four with probab...

We study the two most common types of percolation process on a sparse random graph with a given degree sequence. Namely, we examine first a bond percolation process where the edges of the graph are retained with probability p and afterwards we focus on site percolation where the vertices are retained with probability p. We establish critical values for p above which a giant component emerges in...

This paper presents an effective method for task scheduling in heterogeneous distributed systems. Its objective is to minimize the last task’s finish time and to maximize the system reliability probability. The optimum is carried out through a non-domination sort genetic algorithm. The experimental results based on both randomly generated graphs and the graphs of some real applications showed t...

We study maximum likelihood estimation in Gaussian graphical models from a geometric point of view. An algebraic elimination criterion allows us to find exact lower bounds on the number of observations needed to ensure that the maximum likelihood estimator (MLE) exists with probability one. This is applied to bipartite graphs, grids and colored graphs. We also study the ML degree, and we presen...

We show two Conductance-like theorems for mixing time of finite non-reversible non-lazy Markov Chains, i.e. when the Markov kernel is neither self-adjoint nor positive definite. The first holds for walks with small holding probability, while the second theorem holds even for walks with no holding probability. These are used to derive two canonical path theorems for such non-reversible non-lazy ...

Bounds for the expected return probability of the delayed random walk on finite clusters of an invariant percolation on transitive unimodular graphs are derived. They are particularly suited for the case of critical Bernoulli percolation and the associated heavy-tailed cluster size distributions. The upper bound relies on the fact that cartesian products of finite graphs with cycles of a certai...