ABSTRACT: Recently, Barabási and Albert [2] suggested modeling complex real-world networks such as the worldwide web as follows:consider a random graph process in which vertices are added to the graph one at a time and joined to a fixed number of earlier vertices, selected with probabilities proportional to their degrees. In [2] and, with Jeong, in [3], Barabási and Albert suggested that after many steps the proportion P�d � of vertices with degree d should obey a power law P�d � α d −γ. They obtained γ = 2�9 ± 0�1 by experiment and gave a simple heuristic argument suggesting that γ = 3. Here we obtain P�d � asymptotically for all d ≤ n 1/15, where n is the number of vertices, proving as a consequence that γ = 3.

Recent studies concerning the Internet connectivity at the AS level have attracted considerable attention. These studies have exclusively relied on the BGP data from Oregon route-views [1] to derive some unexpected and intriguing results. The Oregon route-views data sets reflect AS peering relationships, as reported by BGP, seen from a handful of vantage points in the global Internet. The possibility that these data sets from Oregon route-views may provide only a very sketchy picture of the complete inter-AS connections that exist in the actual Internet has received surprisingly little scrutiny. In this paper, we will use the term "AS peering relationship" to mean that there is "at least one direct router-level connection" between two existing ASs, and that these two ASs agree to exchange traffic by enabling BGP between them. By augmenting the Oregon route-views data sets with BGP summary information from a large number of Internet Looking Glass sites and with routing policy information from Internet Routing Registry (IRR) databases, we find that (1) a significant number of existing AS connections remain hidden from most BGP routing tables, (2) the AS connections to tier-1 ASs are in general more easily observed than those to non tier-1 ASs, and (3) there are at least about 25--50% more AS connections in the Internet than commonly-used BGP-derived AS maps reveal (but only about 2% more ASs). These findings point out the need for an increased awareness of and a more critical attitude toward the applicability and completeness of given data sets at hand when establishing the generality of any particular observations about the Internet.

The popularity of peer-to-peer multimedia file sharing applications such as Gnutella and Napster has created a flurry of recent research activity into peer-to-peer architectures. We believe that the proper evaluation of a peer-to-peer system must take into account the characteristics of the peers that choose to participate in it. Surprisingly, however, few of the peer-to-peer architectures currently being developed are evaluated with respect to such considerations. In this paper, we remedy this situation by performing a detailed measurement study of the two popular peer-to-peer file sharing systems, namely Napster and Gnutella. In particular, our measurement study seeks to characterize the population of end-user hosts that participate in these two systems. This characterization includes the bottleneck bandwidths between these hosts and the Internet at large, IP-level latencies to send packets to these hosts, how often hosts connect and disconnect from the system, how many files hosts share and download, the degree of cooperation between the hosts, and several correlations between these characteristics. Our measurements show that there is significant heterogeneity and lack of cooperation across peers participating in these systems.

Recently there has been much interest in studying large-scale real-world networks and attempting to model their properties using random graphs. Although the study of real-world networks as graphs goes back some time, recent activity perhaps started with the paper of Watts and Strogatz [55] about the ‘smallworld

Abstract. The ‘classical ’ random graph models, in particular G(n, p), are ‘homogeneous’, in the sense that the degrees (for example) tend to be concentrated around a typical value. Many graphs arising in the real world do not have this property, having, for example, power-law degree distributions. Thus there has been a lot of recent interest in defining and studying ‘inhomogeneous ’ random graph models. One of the most studied properties of these new models is their ‘robustness’, or, equivalently, the ‘phase transition ’ as an edge density parameter is varied. For G(n, p), p = c/n, the phase transition at c = 1 has been a central topic in the study of random graphs for well over 40 years. Many of the new inhomogenous models are rather complicated; although there are exceptions, in most cases precise questions such as determining exactly the critical point of the phase transition are approachable only when there is independence between the edges. Fortunately, some models studied have this already, and others can be approximated by models with

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.

Many graphs arising in various information networks exhibit the “power law ” behavior – the number of vertices of degree k is proportional to k −β for some positive β. We show that if β>2.5, the largest eigenvalue of a random power law graph is almost surely (1 + o(1)) √ m where m is the maximum degree. Moreover, the k largest eigenvalues of a random power law graph with exponent β have power law distribution with exponent 2β − 1 if the maximum degree is sufficiently large, where k is a function depending on β,m and d, the average degree. When 2 <β<2.5, the largest eigenvalue is heavily concentrated at cm 3−β for some constant c depending on β and the average degree. This result follows from a more general theorem which shows that the largest eigenvalue of a random graph with a given expected degree sequence is determined by m, the maximum degree, and ˜ d, the weighted average of the squares of the expected degrees. We show that the k-th largest eigenvalue is almost surely (1 + o(1)) √ m k where mk is the k-th largest expected degree provided mk is large enough. These results have implications on the usage of spectral techniques in many areas related to pattern detection and information retrieval. 1

where these ideas were initially discussed and much of the work was done. We are especially grateful to John Padgett, organizer of the States and Markets group at SFI for his support and insights. We have benefited from comments from the audience at seminars

While questions about social cohesion lie at the core of our discipline, definitions are often vague and difficult to operationalize. We link research on social cohesion and social embeddedness by developing a conception of structural cohesion based on network nodeconnectivity. Structural cohesion is defined as the minimum number of actors who, if removed from a group, would disconnect the group. A structural dimension of embeddedness can then be defined through the hierarchical nesting of these cohesive structures. We demonstrate the empirical applicability of our conception of nestedness in two dramatically different substantive settings and discuss additional theoretical implications with reference to a wide array of substantive fields. "...social solidarity is a wholly moral phenomenon which by itself is not amenable to exact observation and especially not to measurement." (Durkheim, (1893 [1984], p.24) "The social structure [of the dyad] rests immediately on the one and on the other of the two, and the secession of either would destroy the whole. ... As soon, however, as there is a sociation of three, a group continues to exist even in case one of the members drops out." (Simmel (1908 [1950], p. 123)

We present two new algorithms for generating uniformly random samples of pages from the World Wide Web, building upon recent work by Henzinger et al. (Henzinger et al. 2000) and Bar-Yossef et al. (Bar-Yossef et al. 2000). Both algorithms are based on a weighted random-walk methodology. The first algorithm (DIRECTED-SAMPLE) operates on arbitrary directed graphs, and so is naturally applicable to the web. We show that, in the limit, this algorithm generates samples that are uniformly random. The second algorithm (UNDIRECTED-SAMPLE) operates on undirected graphs, thus requiring a mechanism for obtaining inbound links to web pages (e.g., access to a search engine). With this additional knowledge of inbound links, the algorithm can arrive at a uniform distribution faster than DIRECTED-SAMPLE, and we derive explicit bounds on the time to convergence. In addition, we evaluate the two algorithms on simulated web data, showing that both yield reliably uniform samples of pages. We also compare our results with those of previous algorithms, and discuss the theoretical relationships among the various proposed methods.