Inspired by empirical studies of networked systems such as the Internet, social networks, and biological networks, researchers have in recent years developed a variety of techniques and models to help us understand or predict the behavior of these systems. Here we review developments in this field, including such concepts as the small-world effect, degree distributions, clustering, network correlations, random graph models, models of network growth and preferential attachment, and dynamical processes taking place on networks.

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.

Recent work on modeling the web graph has dwelt on capturing the degree distributions observed on the web. Pointing out that this represents a heavy reliance on “local” properties of the web graph, we study the distribution of PageRank values on the web. Our measurements suggest that PageRank values on the web follow a power law. We then develop generative models for the web graph that explain this observation and moreover remain faithful to previously studied degree distributions. We analyze these models and compare the analysis to both snapshots from the web and to graphs generated by simulations on the new models. To our knowledge this represents the first modeling of the web that goes beyond fitting degree distributions on the web.

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.

Attacks against Internet routing are increasing in number and severity. Contributing greatly to these attacks is the absence of origin authentication: there is no way to validate claims of address ownership or location. The lack of such services enables not only attacks by malicious entities, but indirectly allow seemingly inconsequential miconfigurations to disrupt large portions of the Internet. This paper considers the semantics, design, and costs of origin authentication in interdomain routing. We formalize the semantics of address delegation and use on the Internet, and develop and characterize broad classes of origin authentication proof systems. We estimate the address delegation graph representing the current use of IPv4 address space using available routing data. This effort reveals that current address delegation is dense and relatively static: as few as 16 entities perform 80% of the delegation on the Internet. We conclude by evaluating the proposed services via traced based simulation. Our simulation shows the enhanced proof systems can significantly reduce resource costs associated with origin authentication.

How does the Web look? How could we tell an abnormal social network from a normal one? These and similar questions are important in many fields where the data can intuitively be cast as a graph; examples range from computer networks to sociology to biology and many more. Indeed, any M : N relation in database terminology can be represented as a graph. A lot of these questions boil down to the following: "How can we generate synthetic but realistic graphs?" To answer this, we must first understand what patterns are common in real-world graphs and can thus be considered a mark of normality/realism. This survey give an overview of the incredible variety of work that has been done on these problems. One of our main contributions is the integration of points of view from physics, mathematics, sociology, and computer science. Further, we briefly describe recent advances on some related and interesting graph problems.

We show that the largest eigenvalues of graphs whose highest degrees are Zipf-like distributed with slope are distributed according to a power law with slope =2. This follows as a direct and almost certain corollary of the degree power law. Our result has implications for the singular value decomposition method in information retrieval.

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

Abstract. There is a large, popular, and growing literature on “scale-free ” networks with the Internet along with metabolic networks representing perhaps the canonical examples. While this has in many ways reinvigorated graph theory, there is unfortunately no consistent, precise definition of scale-free graphs and few rigorous proofs of many of their claimed properties. In fact, it is easily shown that the existing theory has many inherent contradictions and that the most celebrated claims regarding the Internet and biology are verifiably false. In this paper, we introduce a structural metric that allows us to differentiate between all simple, connected graphs having an identical degree sequence, which is of particular interest when that sequence satisfies a power law relationship. We demonstrate that the proposed structural metric yields considerable insight into the claimed properties of SF graphs and provides one possible measure of the extent to which a graph is scale-free. This structural view can be related to previously studied graph properties such as the various notions of self-similarity, likelihood, betweenness and assortativity. Our approach clarifies much of the confusion surrounding the sensational qualitative claims in the current literature, and offers a rigorous and quantitative alternative, while suggesting the potential for a rich and interesting theory. This paper is aimed at readers familiar with the basics of Internet technology and comfortable with a theorem-proof style of exposition, but who may be unfamiliar with the existing literature on scale-free networks. 1.

Abstract—The benefit derived from using multicast is seemingly dependent upon the shape of the distribution tree. In this paper, we attempt to accurately model interdomain multicast trees. We measure a number of key parameters, such as depth, degree frequency, and average degree, for a number of real and synthetic data sets. We find that interdomain multicast trees actually do share a common shape at both the router and autonomous system levels. Furthermore, we develop a characterization of multicast efficiency which reveals that group sizes as small as 20 to 40 receivers offer a 55%–70 % reduction in the total number of links traversed when compared to separately delivered unicast streams. A final contribution of our work consists of a number of data sets, compiled from multicast group membership and path data, that can be used to generate large sample trees, representative of the current multicast infrastructure. Index Terms—Efficiency, modeling, multicast, topology. I.