Diameter Of Random Graphs Research Articles

Случайные графы, модели и генераторы безмасштабных графов

In this paper various models of random graphs describing real networks arising in different research fields: biology, computer science, engineering, sociology are considered. Particular attention is paid to models describing social networks. We start with observation of classical Erdos-Renyi model and basic facts concerning properties of random graphs depending on the value p of the probability of appearance each edge in a graph. Then we observe so-called scale-free model of random graphs proposed by Barabassi and Albert and some its generalizations. Some mathematical results about properties of random graphs in such models are described. For one such model of Bollobas-Riordan we describe results concerning the diameter of random graph, the number of verticies of given degree (power-law distribution) and other different properties. This class of models can be characterized by the property that so-called power law constant cannot be chosen in advance and has one fixed value. For example in Bollobas-Riordan model this constant is 3. Finely we observe more general models of random graphs such as Aiello-Chung-Lu model where the power law constant can be chosen as a parameter. In such models there are interesting results concerning evolution properties of random graphs depending on the value of parameter of power law constant. The main example of such property is the size of maximum clique in random graph which we consider in this paper.

The Diameter of Sparse Random Graphs

We consider the diameter of a random graph G(n,p) for various ranges of p close to the phase transition point for connectivity. For a disconnected graph G, we use the convention that the diameter of G is the maximum diameter of its connected components. We show that almost surely the diameter of random graph G(n,p) is close to lognlog(np) if np→∞. Moreover if nplogn=c>8, then the diameter of G(n,p) is concentrated on two values. In general, if nplogn=c>c0, the diameter is concentrated on at most 2⌊1/c0⌋+4 values. We also proved that the diameter of G(n,p) is almost surely equal to the diameter of its giant component if np>3.6.

On Shortest Paths in Graphs with Random Weights

We consider the shortest paths between all pairs of nodes in a directed or undirected complete graph with edge lengths which are uniformly and independently distributed in [0, 1]. We show that die longest of these paths is bounded by c log n/n almost surely, where c is a constant and n is the number of nodes. Our bound is the best possible up to a constant. We apply this result to some well-known problems and obtain several algorithmic improvements over existing results. Our results hold with obvious modifications to random (as opposed to complete) graphs and to any distribution of weights whose density is positive and bounded from below at a neighborhood of zero. As a corollary of our proof we get a new result concerning the diameter of random graphs.

The diameter of random graphs

Extending some recent theorems of Klee and Larman, we prove rather sharp results about the diameter of a random graph. Among others we show that ifd=d(n)⩾3d = d(n) \geqslant 3andm=m(n)m = m(n)satisfy(log⁡n)/d−3log⁡log⁡n→∞(\log n)/d - 3\,\log \log n \to \infty,2d−1md/nd+1−log⁡n→∞{2^{d - 1}}{m^d}/{n^{d + 1}} - \log n \to \inftyanddd−2md−1/nd−log⁡n→−∞{d^{d - 2}}{m^{d - 1}}/{n^d} - \log n \to - \inftythen almost every graph withnnlabelled vertices andmmedges has diameterdd.