Let p(n) denote the smallest integer with the property that any graph with n vertices can be covered by p(n) complete bipartite subgraphs. We prove a conjecture of J.-C. Bermond by showing p(n) = n + o(n 11’14+c) for any positive E.

We study a random graph Gn that combines certain aspects of geometric random graphs and preferential attachment graphs. This model yields a graph with power-law degree distribution where the expansion property depends on a tunable parameter of the model. The vertices of Gn are n sequentially generated points x1, x2,..., xn chosen uniformly at random from the unit sphere in R 3. After generating xt, we randomly connect it to m points from those points in x1, x2,..., xt−1. 1

The web graph has been the focus of much recent attention, with several stochastic models proposed to account for its various properties.

Abstract. We consider the random graph model G(w) for a given expected degree sequence w =(w1,w2,...,wn). If the expected average degree is strictly greater than 1, then almost surely the giant component in G of G(w) has volume (i.e., sum of weights of vertices in the giant component) equal to λ0Vol(G)+O ( √ n log3.5 n), where λ0 is the unique nonzero root of the equation n∑ wie i=1 −w n∑ iλ =(1−λ) wi, i=1 and where Vol(G) = ∑ i wi.

ABSTRACT. The Internet’s layered architecture and organizational structure give rise to a number of different topologies, with the lower layers defining more physical and the higher layers more virtual/logical types of connectivity structures. These structures are very different, and successful Internet topology modeling requires annotating the nodes and edges of the corresponding graphs with information that reflects their network-intrinsic meaning. These structures also give rise to different representations of the traffic that traverses the heterogeneous Internet, and a traffic matrix is a compact and succinct description of the traffic exchanges between the nodes in a given connectivity structure. In this paper, we summarize recent advances in Internet research related to (i) inferring and modeling the router-level topologies of individual service providers (i.e., the physical connectivity structure of an ISP, where nodes are routers/switches and links represent physical connections), (ii) estimating the intra-AS traffic matrix when the AS’s router-level topology and routing configuration are known, (iii) inferring and modeling the Internet’s AS-level topology, and (iv) estimating the inter-AS traffic matrix. We will also discuss recent work on Internet connectivity structures that arise at the higher layers in the TCP/IP protocol stack and are more virtual and dynamic; e.g., overlay networks like the WWW graph, where nodes are web pages and edges represent existing hyperlinks, or P2P networks like Gnutella, where nodes represent peers and two peers are connected if they have an active network connection. 1. Introduction. The

We present new generalized copying models for massive self-organizing networks such as the web graph, and analyze their limit behaviour. Our model is motivated by a desire to unify common design elements of the copying models of the web graph, and the partial duplication model for biological networks. In these models, new nodes copy (with some error) the link structure of existing nodes, and a certain number of random links may be added to the new node that can link to any of the existing nodes. In our new models, a function # parameterizes the number of random links, and thereby allows for the analysis of threshold behaviour.

Abstract—One wants to deploy an n-node multi-channel wireless network in an environment that is inaccessible for repairs and/or that contains malicious adversaries. One wants to design the network to be robust in the following strong sense. Even if any set of m < n nodes is disabled, one still wants all of the surviving n − m nodes to be able to communicate with one another. We present a mathematical model for multi-channel wireless network that facilitates the problem of designing such networks with desired properties. We then present a scalable, deterministic design strategy that produces robust networks that are: (a) within a factor of 2 of optimal in size and complexity; (b) power-efficient, in that nodes attempt to communicate with nearer neighbors before trying more remote ones. I. THE ROBUST-NETWORK PROBLEM Advances in technology allow us to deploy increasingly

Abstract. Models of complex networks are generally defined as graph stochastic processes in which edges and vertices are added or deleted over time to simulate the evolution of networks. Here, we define a unifying framework — probabilistic inductive classes of graphs — for formalizing and studying evolution of complex networks. Our definition of probabilistic inductive class of graphs (PICG) extends the standard notion of inductive class of graphs (ICG) by imposing a probability space. A PICG is given by: (1) class B of initial graphs, the basis of PICG, (2) class R of generating rules, each with distinguished left element to which the rule is applied to obtain the right element, (3) probability distribution specifying how the initial graph is chosen from class B, (4) probability distribution specifying how the rules from class R are applied, and, finally, (5) probability distribution specifying how the left elements for every rule in class R are chosen. We point out that many of the existing models of growing networks can be cast as PICGs. We present how the well known model of growing networks — the preferential attachment model — can be studied as PICG. As an illustration we present results regarding the size, order, and degree sequence for PICG models of connected and 2-connected graphs. 1.

Abstract not found