A survey of statistical network models

Networks are ubiquitous in science and have become a focal point for discussion in everyday life. Formal statistical models for the analysis of network data have emerged as a major topic of interest in diverse areas of study, and most of these involve a form of graphical representation. Probability models on graphs date back to 1959. Along with empirical studies in social psychology and sociology from the 1960s, these early works generated an active network community and a substantial literature in the 1970s. This effort moved into the statistical literature in the late 1970s and 1980s, and the past decade has seen a burgeoning network literature in statistical physics and computer science. The growth of the World Wide Web and the emergence of online networking communities such as Facebook, MySpace, and LinkedIn, and a host of more specialized professional network communities has intensified interest in the study of networks and network data. Our goal in this review is to provide the reader with an entry point to this burgeoning literature. We begin with an overview of the historical development of statistical network modeling and then we introduce a number of examples that have been studied in the network literature. Our subsequent discussion focuses on a number of prominent static and dynamic network models and their interconnections. We emphasize formal model descriptions, and pay special attention to the interpretation of parameters and their estimation. We end with a description of some open problems and challenges for machine learning and statistics.Comment: 96 pages, 14 figures, 333 reference

Networking - A Statistical Physics Perspective

Efficient networking has a substantial economic and societal impact in a broad range of areas including transportation systems, wired and wireless communications and a range of Internet applications. As transportation and communication networks become increasingly more complex, the ever increasing demand for congestion control, higher traffic capacity, quality of service, robustness and reduced energy consumption require new tools and methods to meet these conflicting requirements. The new methodology should serve for gaining better understanding of the properties of networking systems at the macroscopic level, as well as for the development of new principled optimization and management algorithms at the microscopic level. Methods of statistical physics seem best placed to provide new approaches as they have been developed specifically to deal with non-linear large scale systems. This paper aims at presenting an overview of tools and methods that have been developed within the statistical physics community and that can be readily applied to address the emerging problems in networking. These include diffusion processes, methods from disordered systems and polymer physics, probabilistic inference, which have direct relevance to network routing, file and frequency distribution, the exploration of network structures and vulnerability, and various other practical networking applications.Comment: (Review article) 71 pages, 14 figure

Structure Entropy, Self-Organization, and Power Laws in Urban Street Networks: Evidence for Alexander's Ideas

Easy and intuitive navigability is of central importance in cities. The actual scale-free networking of urban street networks in their topological space, where navigation information is encoded by mapping roads to nodes and junctions to links between nodes, has still no simple explanation. Emphasizing the road-junction hierarchy in a holistic and systematic way leads us to envisage urban street networks as evolving social systems subject to a Boltzmann-mesoscopic entropy conservation. This conservation, which we may interpret in terms of surprisal, ensures the passage from the road-junction hierarchy to a scale-free coherence. To wit, we recover the actual scale-free probability distribution for natural roads in self-organized cities. We obtain this passage by invoking Jaynes's Maximum Entropy principle (statistical physics), while we capitalize on modern ideas of quantification (information physics) and well known results on structuration (lattice theory) to measure the information network entropy. The emerging paradigm, which applies to systems with more intricate hierarchies as actual cities, appears to reflect well the influential ideas on cities of the urbanist Christopher Alexander

A contrasting look at self-organization in the Internet and next-generation communication networks

This article examines contrasting notions of self-organization in the Internet and next-generation communication networks, by reviewing in some detail recent evidence regarding several of the more popular attempts to explain prominent features of Internet structure and behavior as "emergent phenomena." In these examples, what might appear to the nonexpert as "emergent self-organization" in the Internet actually results from well conceived (albeit perhaps ad hoc) design, with explanations that are mathematically rigorous, in agreement with engineering reality, and fully consistent with network measurements. These examples serve as concrete starting points from which networking researchers can assess whether or not explanations involving self-organization are relevant or appropriate in the context of next-generation communication networks, while also highlighting the main differences between approaches to self-organization that are rooted in engineering design vs. those inspired by statistical physics