30450351
The chromatic and clique numbers of random scaled sector graphs
Random scaled sector graphs were introduced as a generalization of random geometric graphs to model networks of sensors using optical communication. In the random scaled sector graph model vertices are placed uniformly at random into the [0, 1]2 unit square. Each vertex i is assigned uniformly at random sector Si, of central angle αi, in a circle of radius ri (with vertex i as the origin). An arc is present from vertex i to any vertex j, if j falls in Si. In this work, we study the value of the chromatic number χ(Gn), directed clique number ω(Gn), and undirected clique number ω2 (Gn) for random scaled sector graphs with n vertices, where each vertex spans a sector of α degrees with radius rn = √ln n/n. We prove that for values α π w.h.p., χ(Gn) and ω2 (Gn) are Θ (ln n), being the same for random scaled sector and random geometric graphs, while ω(Gn) is Θ(ln n/ln ln n).
34081065
Connectivity Properties in Random Regular Graphs with Edge Faults
We introduce a new model of random graphs, that of random regular graphs with edge faults (which we denote by ), obtained by selecting the edges of a random member of the set of all regular graphs of degree r independently and with probability p. We can thus represent a communication network in which the links fail independently and with probability f =1-p. In order to deal with this new model, we extend the notion of configurations and the translation lemma between configurations and random regular graphs provided by B. Bollobas, by introducing the concept of random configurations, to account for edge faults, and by providing an extended translation lemma between random configurations and graphs. We investigate important connectivity properties of by estimating the ranges of r, f for which, with high probability, graphs a) are highly connected b) become disconnected and c) admit a giant connected component of small diameter.
10496183
The infection time of graphs
Consider k particles, 1 red and k - 1 white, chasing each other on the nodes of a graph G. If the red one catches one of the white, it "infects" it with its color. The newly red particles are now available to infect more white ones. When is it the case that all white will become red? It turns out that this simple question is an instance of information propagation between random walks and has important applications to mobile computing where a set of mobile hosts acts as an intermediary for the spread of information. 
 
In this paper we model this problem by k concurrent random walks, one corresponding to the red particle and k - 1 to the white ones. The infection time Tk of infecting all the white particles with red color is then a random variable that depends on k, the initial position of the particles, the number of nodes and edges of the graph, as well as on the structure of the graph. 
 
In this work we develop a set of probabilistic tools that we use to obtain upper bounds on the (worst case w.r.t. initial positions of particles) expected value of Tk for general graphs and important special cases. We easily get that an upper bound on the expected value of Tk is the worst case (over all initial positions) expected meeting time m* of two random walks multiplied by Θ(log k). We demonstrate that this is, indeed, a tight bound; i.e. there is a graph G (a special case of the "lollipop" graph), a range of values k < n (such that √n - k = Θ(√n)) and an initial position of particles achieving this bound. 
 
When G is a clique or has nice expansion properties, we prove much smaller bounds for Tk. We have evaluated and validated all our results by large scale experiments which we also present and discuss here. In particular, the experiments demonstrate that our analytical results for these expander graphs are tight.
2840712
The Fourth Moment in Luby's Distribution
Luby (1988) proposed a way to derandomize randomized computations which is based on the construction of a small probability space whose elements are 3-wise independent. In this paper we prove some new properties of Luby's space. More precisely, we analyze the fourth moment and prove an interesting technical property which helps to understand better Luby's distribution. As an application, we study the behavior of random edge cuts in a weighted graph.