Zbigniew Palka

On the number of vertices of given degree in a random graph

This note can be treated as a supplement to a paper written by Bollobas which was devoted to the vertices of a given degree in a random graph. We determine some values of the edge probability p for which the number of vertices of a given degree of a random graph G ∈ (n, p) asymptotically has a normal distribution.

A Random Graph Model of Mobile Wireless Networks

Abstract We propose a time-invariable, integrated model of wireless networks working in hostile environment with obstructions and other factors causing interference. We present a new probabilistic model well suit for analysis of distributed algorithms requiring synchronous communication. The model is based on a random graph with a specific edge probability distribution, which is related to the radio wave propagation characteristics.

On the order of the largest induced tree in a random graph

Abstract Consider a random graph K(n, p) with n labeled vertices in which the edges are chosen independently and with a probability p. Let Tn(p) be the order of the largest induced tree in K(n, p). Among other results it is shown, using an algorithmic approach, that if p=(c log n)/n, where c ≥ e is a constant, then for any fixed e > 0 1 c −e log log n log n n n (p) 2 c +e log log n log n almost surely.

Maximal induced trees in sparse random graphs

A study of the orders of maximal induced trees in a random graph G p with small edge probability p is given. In particular, it is shown that the giant component of almost every G p , where p = c/n and c > 1 is a constant, contains only very small maximal trees (that are of a specific type) and very large maximal trees. The presented results provide an elementary proof of a conjecture from [3] that was confirmed recently in [4] and [5].

Extreme degrees in random graphs

Let G* be a simple undirected graph on n labeled vertices. A general approach to the investigation of the probability distribution of extreme degrees in a random subgraph of G* is given. As an example of the application of the method, we consider the case when G* is a complete bipartite graph.

On a method for random graphs

Abstract In this paper we examine a method for establishing an almost sure existence of a subgraph of a random graph with a given subgraph property. Since the method has been abused in the literature, we state some conditions under which it can be safely used. As an illustration we apply the method to induced cycles, maximal induced trees and arbitrary subgraphs of a random graph K n , p .

Random subgraphs of the n-cycle

Abstract Let RC( p ) denote the random subgraph of the n -cycle C obtained by selecting or rejecting each of the lines of C with independent probability p or q = 1− p , respectively. By definition RC( p has the same point set as C . The number of lines N in RC( p ) is clearly seen to be a random variable having a binomial probability distribution. The expected value, variance, distribution, and some asymptotic distributions of X j , the number of points in RC( p ) having degree j ( j =0,1,2) are determined. A component of RC( p ) is called big if its order is greater than [ n 2 ]. The probability that RC( p ) will contain a big component is derived. From this it is shown how different choices of p (as a function of n ) effect this probability as n goes to infinity.

In-degree sequence in a general model of a random digraph

A general model of a random digraph D(n,P) is considered. Based on a precise estimate of the asymptotic behaviour of the distribution function of the binomial law, a problem of the distribution of extreme in-degrees of D(n,P) is discussed.

Rulers and slaves in a random social group

LetD(n, d) be a digraph chosen at random from a family of alld-out-regular digraphs onn points. LetK(n, p) be a simple graph onn points in which each edge appears independently with probabilityp. A relationship between the properties of extreme in-degrees ofD(n,d) and extreme degrees ofK(n, p) in the case whend = (n − 1)p = o(n) is presented. A sociological interpretation is also provided.

On the communication complexity of Bar-Yehuda, Goldreich and Itai's randomized broadcasting algorithm

The algorithm by Bar-Yehuda, Goldreich and Itai is one of the best known randomized broadcast algorithms for radio networks. Its probability of success and time complexity are nearly optimal. We propose a modification of this algorithm, which decreases the communication complexity, preserving other properties. Moreover, we show that the local communication complexity of the modified algorithm is deterministic.