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We study a new kind of proximity graphs called proportional-edge proximity catch digraphs (PCDs)in a randomized setting. PCDs are a special kind of random catch digraphs that have been developed recently and have applications in statistical…
In this paper we consider the Erd\H{o}s-R\'enyi random graph in the sparse regime in the limit as the number of vertices $n$ tends to infinity. We are interested in what this graph looks like when it contains many triangles, in two…
This work explores the dynamic properties of test particles surrounding a distorted, deformed compact object. The astrophysical motivation was to choose such background, which could constitute a more reasonable model of a real situation…
We study two types of probability measures on the set of integer partitions of $n$ with at most $m$ parts. The first one chooses the random partition with a chance related to its largest part only. We then obtain the limiting distributions…
The distribution of differences of consecutive members of sequences of primes is investigated. A quantitative measure for oscillations among these differences is the curvature of the sequence. If the sequence is not too sparse, then sharp…
We consider vertex percolation on pseudo-random $d-$regular graphs. The previous study by the second author established the existence of phase transition from small components to a linear (in $\frac{n}{d}$) sized component, at…
Graphs naturally appear in several real-world contexts including social networks, the web network, and telecommunication networks. While the analysis and the understanding of graph structures have been a central area of study in algorithm…
We investigate a class of stochastic fragmentation processes involving stable and unstable fragments. We solve analytically for the fragment length density and find that a generic algebraic divergence characterizes its small-size tail.…
We present a detailed analysis of random motions moving in higher spaces with a natural number of velocities. In the case of the so-called minimal random dynamics, under some wide assumptions, we show the joint distribution of the position…
Consider a graph G with n nodes and m edges, which represents a social network, and assume that initially each node is blue or white. In each round, all nodes simultaneously update their color to the most frequent color in their…
In the context of a random walk on an undirected graph, Kemeny's constant can measure the average travel time for a random walk between two randomly chosen vertices. We are interested in graphs that behave counter-intuitively in regard to…
Uncertain graphs are prevalent in several applications including communications systems, biological databases and social networks. The ever increasing size of the underlying data renders both graph storage and query processing extremely…
Families of symmetric simple random walks on Cayley graphs of Abelian groups with a bound on the number of generators are shown to never have sharp cut off in the sense of [1], [3], or [5]. Here convergence to the stationary distribution is…
We study large deviations of the size of the largest connected component in a general class of inhomogeneous random graphs with iid weights, parametrized so that the degree distribution is regularly varying. We derive a large-deviation…
The focus of this thesis is about statistical mechanics on heterogeneous random graphs, i.e. how this heterogeneity affects the cooperative behavior of model systems. It is not intended as a review on it, rather it is showed how this…
Population structure affects the outcome of natural selection. Static population structures can be described by graphs, where individuals occupy the nodes, and interactions occur along the edges. General conditions for evolutionary success…
We study the dynamics of a backtracking procedure capable of proving uncolourability of graphs, and calculate its average running time T for sparse random graphs, as a function of the average degree c and the number of vertices N. The…
The statistical mechanical description of two-dimensional inviscid fluid turbulence is reconsidered. Using this description, we make predictions about turbulent flow in a rapidly rotating laboratory annulus. Measurements on the continuously…
We define a dynamic model of random networks, where new vertices are connected to old ones with a probability proportional to a sublinear function of their degree. We first give a strong limit law for the empirical degree distribution, and…
We study both numerically and analytically what happens to a random graph of average connectivity "alpha" when its leaves and their neighbors are removed iteratively up to the point when no leaf remains. The remnant is made of isolated…