Random graph theory is used to examine the “small-world phenomenon”– any two strangers are connected through a short chain of mutual acquaintances. We will show that for certain families of random graphs with given expected degrees, the average distance is almost surely of order log n / log ˜ d where ˜ d is the weighted average of the sum of squares of the expected degrees. Of particular interest are power law random graphs in which the number of vertices of degree k is proportional to 1/k β for some fixed exponent β. For the case of β> 3, we prove that the average distance of the power law graphs is almost surely of order log n / log ˜ d. However, many Internet, social, and citation networks are power law graphs with exponents in the range 2 < β < 3 for which the power law random graphs have average distance almost surely of order log log n, but have diameter of order log n (provided having some mild constraints for the average distance and maximum degree). In particular, these graphs contain a dense subgraph, that we call the core, having n c / log log n vertices. Almost all vertices are within distance log log n of the core although there are vertices at distance log n from the core.

A large body of work has been devoted to identifying community structure in networks. A community is often though of as a set of nodes that has more connections between its members than to the remainder of the network. In this paper, we characterize as a function of size the statistical and structural properties of such sets of nodes. We define the network community profile plot, which characterizes the “best ” possible community—according to the conductance measure—over a wide range of size scales, and we study over 70 large sparse real-world networks taken from a wide range of application domains. Our results suggest a significantly more refined picture of community structure in large real-world networks than has been appreciated previously. Our most striking finding is that in nearly every network dataset we examined, we observe tight but almost trivial communities at very small scales, and at larger size scales, the best possible communities gradually “blend in ” with the rest of the network and thus become less “community-like.” This behavior is not explained, even at a qualitative level, by any of the commonly-used network generation models. Moreover, this behavior is exactly the opposite of what one would expect based on experience with and intuition from expander graphs, from graphs that are well-embeddable in a low-dimensional structure, and from small social networks that have served as testbeds of community detection algorithms. We have found, however, that a generative model, in which new edges are added via an iterative “forest fire” burning process, is able to produce graphs exhibiting a network community structure similar to our observations.

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.

The rapid growth of the web has been noted and tracked extensively. Recent studies have however documented the dual phenomenon: web pages have small half lives, and thus the web exhibits rapid death as well. Consequently, page creators are faced with an increasingly burdensome task of keeping links up-to-date, and many are falling behind. In addition to just individual pages, collections of pages or even entire neighborhoods of the web exhibit significant decay, rendering them less e#ective as information resources. Such neighborhoods are identified only by frustrated searchers, seeking a way out of these stale neighborhoods, back to more up-to-date sections of the web; measuring the decay of a page purely on the basis of dead links on the page is too naive to reflect this frustration. In this paper we formalize a strong notion of a decay measure and present algorithms for computing it e#ciently. We explore this measure by presenting a number of validations, and use it to identify interesting artifacts on today's web. We then describe a number of applications of such a measure to search engines, web page maintainers, ontologists, and individual users.

A large body of work has been devoted to defining and identifying clusters or communities in social and information networks, i.e., in graphs in which the nodes represent underlying social entities and the edges represent some sort of interaction between pairs of nodes. Most such research begins with the premise that a community or a cluster should be thought of as a set of nodes that has more and/or better connections between its members than to the remainder of the network. In this paper, we explore from a novel perspective several questions related to identifying meaningful communities in large social and information networks, and we come to several striking conclusions. Rather than defining a procedure to extract sets of nodes from a graph and then attempt to interpret these sets as a “real ” communities, we employ approximation algorithms for the graph partitioning problem to characterize as a function of size the statistical and structural properties of partitions of graphs that could plausibly be interpreted as communities. In particular, we define the network community profile plot, which characterizes the “best ” possible community—according to the conductance measure—over a wide range of size scales. We study over 100 large real-world networks, ranging from traditional and on-line social networks, to technological and information networks and

Graph models for real-world complex networks such as the Internet, the WWW and biological networks are necessary for analytic and simulation-based studies of network protocols, algorithms, engineering and evolution. To date, all available data for such networks suggest heavy tailed statistics, most notably on the degrees of the underlying graphs. A practical way to generate network topologies that meet the observed data is the following degree-driven approach: First predict the degrees of the graph by extrapolation from the available data, and then construct a graph meeting the degree sequence and additional constraints, such as connectivity and randomness. Within the networking community, this is currently accepted as the most successful approach for modeling the inter-domain topology of the Internet.

In this paper we study a random graph with N nodes, where node j has degree Dj and {Dj} N j=1 are i.i.d. with P(Dj ≤ x) = F(x). We assume that 1 − F(x) ≤ cx −τ+1 for some τ> 3 and some constant c> 0. This graph model is a variant of the so-called configuration model, and includes heavy tail degrees with finite variance. The minimal number of edges between two arbitrary connected nodes, also known as the graph distance or the hopcount, is investigated when N → ∞. We prove that the graph distance grows like log ν N, when the base of the logarithm equals ν = E[Dj(Dj − 1)]/E[Dj]> 1. This confirms the heuristic argument of Newman, Strogatz and Watts [35]. In addition, the random fluctuations around this asymptotic mean log ν N are characterized and shown to be uniformly bounded. In particular, we show convergence in distribution of the centered graph distance along exponentially growing subsequences. 1

We study random graphs with an i.i.d. degree sequence of which the tail of the distribution function F is regularly varying with exponent τ ∈ (1, 2). Thus, the degrees have infinite mean. Such random graphs can serve as models for complex networks where degree power laws are observed. The minimal number of edges between two arbitrary nodes, also called the graph distance or the hopcount, in a graph with N nodes is investigated when N → ∞. The paper is part of a sequel of three papers. The other two papers study the case where τ ∈ (2, 3), and τ ∈ (3, ∞), respectively. The main result of this paper is that the graph distance converges for τ ∈ (1, 2) to a limit random variable with probability mass exclusively on the points 2 and 3. We also consider the case where we condition the degrees to be at most N α for some α> 0. For τ −1 < α < (τ −1) −1, the hopcount converges to 3 in probability, while for α> (τ − 1) −1, the hopcount converges to the same limit as for the unconditioned degrees. Our results give convincing asymptotics for the hopcount when the mean degree is infinite, using extreme value theory.

Abstract — Simulations are an important tool in network research. As the selected topology often influences the outcome of the simulation, realistic topologies are needed to produce realistic simulation results. We first discuss the different types of topologies and present our collection of real-world topologies that can be used for simulation. We then define several similarity metrics to compare artificially generated topologies with real world topologies. We use them to find out what the input parameter range of the topology generators of BRITE, TIERS and GT-ITM are to create realistic topologies. These parameters can act as a valuable starting point for researchers that have to generate artificial topologies.

Random graphs with a given degree sequence are a useful model capturing several features absent in the classical Erdős-Rényi model, such as dependent edges and non-binomial degrees. In this paper, we use a characterization due to Erdős and Gallai to develop a sequential algorithm for generating a random labeled graph with a given degree sequence. The algorithm is easy to implement and allows surprisingly efficient sequential importance sampling. Applications are given, including simulating a biological network and estimating the number of graphs with a given degree sequence. 1. Introduction. Random