7 research outputs found
ASYMPTOTIC PROPERTIES OF A RANDOM GRAPH WITH DUPLICATIONS
We deal with a random graph model evolving in discrete time steps by duplicating and deleting the edges of randomly chosen vertices. We prove the existence of an almost surely asymptotic degree distribution, with stretched exponential decay; more precisely, the proportion of vertices of degree d tends to some positive number c(d) > 0 almost surely as the number of steps goes to infinity, and c(d) similar to (e pi)(1/2)d(1/4)e(-2)root d holds as d -> infinity
Asymptotic degree distribution of a duplication-deletion random graph model
We study a discrete-time duplication-deletion random graph model and analyse
its asymptotic degree distribution. The random graphs consists of disjoint
cliques. In each time step either a new vertex is brought in with probability
and attached to an existing clique, chosen with probability
proportional to the clique size, or all the edges of a random vertex are
deleted with probability . We prove almost sure convergence of the
asymptotic degree distribution and find its exact values in terms of a
hypergeometric integral, expressed in terms of the parameter . In the regime
we show that the degree sequence decays exponentially at rate
, whereas it satisfies a power-law with exponent
if . At the threshold the
degree sequence lies between a power-law and exponential decay.Comment: 1 figur
Topological properties of P.A. random graphs with edge-step functions
In this work we investigate a preferential attachment model whose parameter
is a function that drives the asymptotic proportion
between the numbers of vertices and edges of the graph. We investigate
topological features of the graphs, proving general bounds for the diameter and
the clique number. Our results regarding the diameter are sharp when is a
regularly varying function at infinity with strictly negative index of regular
variation . For this particular class, we prove a characterization for
the diameter that depends only on . More specifically, we prove that
the diameter of such graphs is of order with high probability,
although its vertex set order goes to infinity polynomially. Sharp results for
the diameter for a wide class of slowly varying functions are also obtained.
The almost sure convergence for the properly normalized logarithm of the clique
number of the graphs generated by slowly varying functions is also proved
The non-equilibrium statistical physics of stochastic search, foraging and clustering
This dissertation explores two themes central to the field of non-equilibrium statistical physics. The first is centered around the use of random walks, first-passage processes, and Brownian motion to model basic stochastic search processes found in biology and ecological systems. The second is centered around clustered networks: how clustering modifies the nature of transition in the appearance of various graph motifs and their use in modeling social networks.
In the first part of this dissertation, we start by investigating properties of intermediate crossings of Brownian paths. We develop simple analytical tools to obtain probability distributions of intermediate crossing positions and intermediate crossing times of Brownian paths. We find that the distribution of intermediate crossing times can be unimodal or bimodal. Next, we develop analytical and numerical methods to solve a system of diffusive searchers which are reset to the origin at stochastic or periodic intervals. We obtain the optimal criteria to search for a fixed target in one, two and three dimensions. For these two systems, we also develop efficient ways to simulate Brownian paths, where the simulation kernel makes maximal use of first-passage ideas. Finally we develop a model to understand foraging in a resource-rich environment. Specifically, we investigate the role of greed on the lifetime of a diffusive forager. This lifetime shows non-monotonic dependence on greed in one and two dimensions, and surprisingly, a peak for negative greed in 1d.
In the second part of this dissertation, we develop simple models to capture the non-tree-like (clustering) aspects of random networks that arise in the real world. By 'clustered networks', we specifically mean networks where the probability of links between neighbors of a node (i.e., 'friends of friends') is positive. We discuss three simple and related models. We find a series of transitions in the density of graph motifs such as triangles (3-cliques), 4-cliques etc as a function of the clustering probability. We also find that giant 3-cores emerge through first- or second-order, or even mixed transitions in clustered networks
ASYMPTOTIC PROPERTIES OF A RANDOM GRAPH WITH DUPLICATIONS
We deal with a random graph model evolving in discrete time steps by duplicating and deleting the edges of randomly chosen vertices. We prove the existence of an almost surely asymptotic degree distribution, with stretched exponential decay; more precisely, the proportion of vertices of degree d tends to some positive number c(d) > 0 almost surely as the number of steps goes to infinity, and c(d) similar to (e pi)(1/2)d(1/4)e(-2)root d holds as d -> infinity