his 50th birthday. Our aim is to study the probable structure of a random graph rn N which has n given labelled vertices P, P2,..., Pn and N edges; we suppose_

Each complex network (or class of networks) presents specific topological features which characterize its connectivity and highly influence the dynamics and function of processes executed on the network. The analysis, discrimination, and synthesis of complex networks therefore rely on the use of measurements capable of expressing the most relevant topological features. This article presents a survey of such measurements. It includes general considerations about complex network characterization, a brief review of the principal models, and the presentation of the main existing measurements organized into classes. Special attention is given to relating complex network analysis with the areas of pattern recognition and feature selection, as well as on surveying some concepts and measurements from traditional graph theory which are potentially useful for complex network research. Depending on the network and the analysis task one has in mind, a specific set of features may be chosen. It is hoped that the present survey will help the

Given large, multi-million node graphs (e.g., Facebook, web-crawls, etc.), how do they evolve over time? How are they connected? What are the central nodes and the outliers? In this paper we define the Radius plot of a graph and show how it can answer these questions. However, computing the Radius plot is prohibitively expensive for graphs reaching the planetary scale. There are two major contributions in this paper: (a) We propose HADI (HAdoop DIameter and radii estimator), a carefully designed and fine-tuned algorithm to compute the radii and the diameter of massive graphs, that runs on the top of the Hadoop/MapReduce system, with excellent scale-up on the number of available machines (b) We run HADI on several real world datasets including YahooWeb (6B edges, 1/8 of a Terabyte), one of the largest public graphs ever analyzed. Thanks to HADI, we report fascinating patterns on large networks, like the surprisingly small effective diameter, the multi-modal/bi-modal shape of the Radius plot, and its palindrome motion over time.

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Networked multiagent systems are comprised of many autonomous yet interdependent agents situated in a virtual social network. Two examples of such systems are supply chain networks and sensor networks. A common challenge in many networked multiagent systems is decentralized team formation among the spatially and logically extended agents. Even in cooperative multiagent systems, efficient team formation is made difficult by the limited local information available to the individual agents. We present a model of distributed multiagent team formation in networked multi-agent systems, describe a policy learning framework for joining teams based on local information, and give empirical results on improving team formation performance. In particular, we show that local policy learning from limited information leads to a significant increase in organizational team formation performance compared to a naive heuristic.

This report is meant to summarize the contents of 3 papers by M.E.J Newman, S.H. Strogatz and D. J Watts. To model a network a commonly used method is the random graphs model proposed by Erdős and Rényi between 1959-1961. According to Newman some real-world networks, like social, biological networks and the internet among others, behave differently than the model proposed by Erdős and Rényi. This is mainly because the degree distributions of real-world networks are in most cases different from the Poisson degree distribution which is the basis of the Erdős and Rényi model. Furthermore, networks like the internet are more adequately modeled with directed graphs rather than with simple undirected graphs. The use of probability generating functions will simplify the process of calculating some properties on average of the models. Newman also discusses the issue of modeling networks that show clustering or transitivity. The models he proposes initially do not model clustering efficiently, therefore he makes some adjustments to incorporate it. 1

contained in this document are those of the authors and should not be interpreted as representing the official

Given an undirected, edge-weighted connected graph, the k-cut problem is to partition the vertex set into k non-empty connected components so as to minimize the total weight of edges whose end points are in different components. We present a combinatorial polynomial-time 2-approximation algorithm for the k-cut problem. We use a La-grangean relaxation (also suggested by Barahona [2]) to reduce the problem to the attack problem, for which a polynomial time algorithm was provided by Cunningham [4]. We prove several structural results of the relaxation, and use these results to develop an approximation algorithm. We provide analytical comparisons of our algorithm and lower bound with two others: Saran and Vazirani [10] and Naor and Rabani [8]. We also provide computational results comparing the performance of our algorithm on random graphs with respect to the lower bound provided by the attack problem as well as an alternate 2-approximation algorithm provided by Saran and Vazirani [10].

I semi-empirically study the social networking sites such as LinkedIn. Such sites enable users to maintain contact information of people they know and trust (their first degree connections or friends), and to discover the friends of their friends (their second degree connections), and to access the friends of the friends of their friends (their third degree connections). Connections up to some degree (e.g., third) make up a network. I find the size of such a tree network grows sublinearly with time, even when its owner actively seeks out new friends. Under simplistic assumptions I find that the value of such a network to its owner is three times that of a standard contact list (containing only first degree connection). The total value of a network of N connections up to d degrees of separation to all