We describe some new exactly solvable models of the structure of social networks, based on random graphs with arbitrary degree distributions. We give models both for simple unipartite networks, such as acquaintance networks, and bipartite networks, such as affiliation networks. We compare the predictions of our models to data for a number of real-world social networks and find that in some cases the models are in remarkable agreement with the data, while in others the agreement is poorer, perhaps indicating the presence of additional social structure in the network that is not captured by the random graph.

Given a large, real graph, how can we generate a synthetic graph that matches its properties, i.e., it has similar degree distribution, similar (small) diameter, similar spectrum, etc? We propose to use “Kronecker graphs”, which naturally obey all of the above properties, and we present KronFit, a fast and scalable algorithm for fitting the Kronecker graph generation model to real networks. A naive approach to fitting would take super-exponential time. In contrast, Kron-Fit takes linear time, by exploiting the structure of Kronecker product and by using sampling. Experiments on large real and synthetic graphs show that KronFit indeed mimics very well the patterns found in the target graphs. Once fitted, the model parameters and the resulting synthetic graphs can be used for anonymization, extrapolations, and graph summarization. 1.

In this paper we analyze multiple wireless LAN (WLAN) traces from university and corporate campuses. In particular, we consider an important event between mobile nodes in wireless networks – encounters. We seek to understand encounter patterns in the mobile network from a holistic view by a graph analysis approach. Such an analysis sheds light on the diverse, non-homogeneous nature of users in the given environments in terms of their encounter events with other nodes. Furthermore, we evaluate the feasibility of forming an infrastructure-less network to reach most of the nodes utilizing time-varying inter-node connectivity through encounters, and the robustness of such an ad hoc communication network. Our analysis shows that while the encounter events are “sparse ” (i.e., any given node does not encounter with many other nodes), the connectivity of the whole network is well-maintained, and a Small World pattern of nodal encounter emerges for the observation periods longer than one day. More interestingly, the encounter events collectively form a robust communication network, in which store-carry-forward message dissemination can be mostly successful with at least 20 % of non-cooperative nodes or removal of short-lived (up to minutes) encounter events.

Realistic workloads are essential in evaluating middleware for computational grids. One important component is the raw grid itself: a network topology graph annotated with the hardware and software available on each node and link. This paper defines our requirements for grid generation and presents GridG, our extensible generator. We describe GridG in two steps: topology generation and annotation. For topology generation, we have both model and mechanism. We extend Tiers, an existing tool from the networking community, to produce graphs that obey recently discovered power laws of Internet topology. We also contribute to network topology theory by illustrating a contradiction between two laws and proposing a new version of one of them. For annotation, GridG captures intra- and inter-host correlations between attributes using conditional probability rules. We construct a set of rules, including one based on empirical evidence of OS concentration in subnets, that produce sensible host annotations.

Two important recent trends in military and civilian communications have been the increasing tendency to base operations around an internal network, and the increasing threats to communications infrastructure. This combination of factors makes it important to study the robustness of network topologies. We use graph-theoretic concepts of connectivity to do this, and argue that node connectivity is the most useful such measure. We examine the relationship between node connectivity and network symmetry, and describe two conditions which robust networks should satisfy. To assist with the process of designing robust networks, we have developed a powerful network design and analysis tool called CAVALIER, which we briefly describe.

Many complex networks have a small-world topology characterized by dense local clustering or cliquishness of connections between neighboring nodes yet a short path length between any (distant) pair of nodes due to the existence of relatively few long-range connections. This is an attractive model for the organization of brain anatomical and functional networks because a small-world topology can support both segregated/specialized and distributed/integrated information processing. Moreover, small-world networks are economical, tending to minimize wiring costs while supporting high dynamical complexity. The authors introduce some of the key mathematical concepts in graph theory required for small-world analysis and review how these methods have been applied to quantification of cortical connectivity matrices derived from anatomical tract-tracing studies in the macaque monkey and the cat. The evolution of small-world networks is discussed in terms of a selection pressure to deliver cost-effective information-processing systems. The authors illustrate how these techniques and concepts are increasingly being applied to the analysis of human brain functional networks derived from electroencephalography/magnetoencephalography and fMRI experiments. Finally, the authors consider the relevance of small-world models for understanding the emergence of complex behaviors and the resilience of brain systems to pathological attack by disease or aberrant development. They conclude that small-world models provide a powerful and versatile approach to understanding the structure and function of human brain

Abstract — In this work we study WLAN traces from five different sources and focus on investigation of encounter patterns between users. We find that typical wireless LAN users encounter with a small portion of the whole population (no more than 60 % in all traces, and on average between 1.88 % to 6.70%). Total encounters of MNs follow BiPareto distribution. These few encounters are sufficient to build a connected relationship network, which is a Small World graph. We further investigate the potential of node-to-node information diffusion, and find that the richness of encounter pattern provides a reliable platform on which information diffusion without infrastructure is feasible and robust. I.

Many complex systems can be described in terms of networks of interacting units. Recent studies have shown that a wide class of both natural and artificial nets display a surprisingly widespread feature: the presence of highly heterogeneous distributions of links, providing an extraordinary source of robustness against perturbations. Although most theories concerning the origin of these topologies use growing graphs, here we show that a simple optimization process can also account for the observed regularities displayed by most complex nets. Using an evolutionary algorithm involving minimization of link density and average distance, four major types of networks are encountered: (a) sparse exponential-like networks, (b) sparse scale-free networks, (c) star networks and (d) highly dense networks, apparently defining three major phases. These constraints provide a new explanation for scaling of exponent about -3.

What does a ‘normal ’ computer (or social) network look like? How can we spot ‘abnormal ’ sub-networks in the Internet, or web graph? The answer to such questions is vital for outlier detection (terrorist networks, or illegal money-laundering rings), forecasting, and simulations (“how will a computer virus spread?”). The heart of the problem is finding the properties of real graphs that seem to persist over multiple disciplines. We list such patterns and “laws”, including the “min-cut plots ” discovered by us. This is the first part of our NetMine package: given any large graph, it provides visual feedback about these patterns; any significant deviations from the expected patterns can thus be immediately flagged by the user as abnormalities in the graph. The second part of NetMine is the A-plots tool for visualizing the adjacency matrix of the graph in innovative new ways, again to find outliers. Third, NetMine contains the R-MAT (Recursive MATrix) graph generator, which can successfully model many of the patterns found in real-world graphs and quickly generate realistic graphs, capturing the essence of each graph in only a few parameters. We present results on multiple, large real graphs, where we show the effectiveness of our approach.

Abstract. We introduce a stochastic model for complex networks possessing three qualitative features: power-law degree distributions, local clustering, and slowly growing diameter. The model is mathematically natural, permits a wide variety of explicit calculations, has the desired three qualitative features, and fits the complete range of degree scaling exponents and clustering parameters. 1.