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.

We review the recent fast progress in statistical physics of evolving networks. Interest has focused mainly on the structural properties of random complex networks in communications, biology, social sciences and economics. A number of giant artificial networks of such a kind came into existence recently. This opens a wide field for the study of their topology, evolution, and complex processes occurring in them. Such networks possess a rich set of scaling properties. A number of them are scale-free and show striking resilience against random breakdowns. In spite of large sizes of these networks, the distances between most their vertices are short — a feature known as the “smallworld” effect. We discuss how growing networks self-organize into scale-free structures and the role of the mechanism of preferential linking. We consider the topological and structural properties of evolving networks, and percolation in these networks. We present a number of models demonstrating the main features of evolving networks and discuss current approaches for their simulation and analytical study. Applications of the general results to particular networks in Nature are discussed. We demonstrate the generic connections of the network growth processes with the general problems

The application of kernel-based learning algorithms has, so far, largely been confined to realvalued data and a few special data types, such as strings. In this paper we propose a general method of constructing natural families of kernels over discrete structures, based on the matrix exponentiation idea. In particular, we focus on generating kernels on graphs, for which we propose a special class of exponential kernels called diffusion kernels, which are based on the heat equation and can be regarded as the discretization of the familiar Gaussian kernel of Euclidean space.

Abstract. In recent years, the gossip-based communication model in large-scale distributed systems has become a general paradigm with important applications which include information dissemination, aggregation, overlay topology management and synchronization. At the heart of all of these protocols lies a fundamental distributed abstraction: the peer sampling service. In short, the aim of this service is to provide every node with peers to exchange information with. Analytical studies reveal a high reliability and efficiency of gossip-based protocols, under the (often implicit) assumption that the peers to send gossip messages to are selected uniformly at random from the set of all nodes. In practice—instead of requiring all nodes to know all the peer nodes so that a random sample could be drawn—a scalable and efficient way to implement the peer sampling service is by constructing and maintaining dynamic unstructured overlays through gossiping membership information itself. This paper presents a generic framework to implement reliable and efficient peer sampling services. The framework generalizes existing approaches and makes it easy to introduce new ones. We use this framework to explore and compare several implementations of our abstract scheme. Through extensive experimental analysis, we show that all of them lead to different peer sampling services none of which is uniformly random. This clearly renders traditional theoretical approaches invalid, when the underlying peer sampling service is based on a gossip-based scheme. Our observations also help explain important differences between design choices of peer sampling algorithms, and how these impact the reliability of the corresponding service. 1

Unstructured overlays form an important class of peer-to-peer networks, notably when content-based searching is at stake. The construction of these overlays, which is essentially a membership management issue, is crucial. Ideally, the resulting overlays should have low diameter and be resilient to massive node failures, which are both characteristic properties of random graphs. In addition, they should be able to deal with a high node churn (i.e., expect high-frequency membership changes). Inexpensive membership management while retaining random-graph properties is therefore important. In this paper, we describe a novel gossip-based membership management protocol that meets these requirements. Our protocol is shown to construct graphs that have low diameter, low clustering, highly symmetric node degrees, and that are highly resilient to massive node failures. Moreover, we show that the protocol is highly reactive to restoring randomness when a large number of nodes fail. KEY WORDS: Membership management; peer-to-peer; epidemic/gossiping protocols; unstructured overlays; random graphs.

Previous work in social network analysis (SNA) has modeled the existence of links from one entity to another, but not the language content or topics on those links. We present the Author-Recipient-Topic (ART) model for social network analysis, which learns topic distributions based on the direction-sensitive messages sent between entities. The model builds on Latent Dirichlet Allocation (LDA) and the Author-Topic (AT) model, adding the key attribute that distribution over topics is conditioned distinctly on both the sender and recipient—steering the discovery of topics according to the relationships between people. We give results on both the Enron email corpus and a researcher’s email archive, providing evidence not only that clearly relevant topics are discovered, but that the ART model better predicts people’s roles. 1 Introduction and Related Work Social network analysis (SNA) is the study of mathematical models for interactions among people, organizations and groups. With the recent availability of large datasets of human

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.

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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

In a recent paper, Faloutsos et al. [1] found that the inter Autonomous System (AS) topology exhibits a power-law vertex degree distribution. This result was quite unexpected in the networking community and stirred significant interest in exploring the possible causes of this phenomenon. The work of Barabasi and Albert [2] and its application to network topology generation in the work of Medina et al. [3] have explored a promising class of models that yield strict power-law vertex degree distributions. In this paper, we re-examine the BGP measurements that form the basis for the results reported in [1]. We find that by their very nature (i.e., being strictly BGP-based), the data provides a very incomplete picture of Internet connectivity at the AS level. The AS connectivity maps constructed from this data (the original maps) typically miss 20--50% or even more of the physical links in AS maps constructed using additional sources (the extended maps). Subsequently, we find that while the vertex degree distributions resulting from the extended maps are heavy-tailed, they deviate significantly from a strict power law. Finally, we show that available historical data does not support the connectivity-based dynamics assumed in [2]. Together, our results suggest that the Internet topology at the AS level may well have developed over time following a very different set of growth processes than those proposed in [2].