We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

In applications such as social, energy, transportation, sensor, and neuronal networks, high-dimensional data naturally reside on the vertices of weighted graphs. The emerging field of signal processing on graphs merges algebraic and spectral graph theoretic concepts with computational harmonic analysis to process such signals on graphs. In this tutorial overview, we outline the main challenges of the area, discuss different ways to define graph spectral domains, which are the analogs to the classical frequency domain, and highlight the importance of incorporating the irregular structures of graph data domains when processing signals on graphs. We then review methods to generalize fundamental operations such as filtering, translation, modulation, dilation, and downsampling to the graph setting and survey the localized, multiscale transforms that have been proposed to efficiently extract information from high-dimensional data on graphs. We conclude with a brief discussion of open issues and possible extensions.

The emerging field of signal processing on graphs: Extending high-dimensional data analysis to networks and other irregular domains

This book is an introduction to the modern approach to the theory of Markov chains. The main goal of this approach is to determine the rate of convergence of a Markov chain to the stationary distribution as a function of the size and geometry of the state space. The authors develop the key tools for estimating convergence times, including coupling, strong stationary times, and spectral methods. Whenever possible, probabilistic methods are emphasized. The book includes many examples and provides brief introductions to some central models of statistical mechanics. Also provided are accounts of random walks on networks, including hitting and cover times, and analyses of several methods of shuffling cards. As a prerequisite, the authors assume a modest understanding of probability theory and linear algebra at an undergraduate level. ""Markov Chains and Mixing Times"" is meant to bring the excitement of this active area of research to a wide audience.

Markov Chains and Mixing Times

A large body of work has been devoted to defining and identifying clusters or communities in social and information networks, i.e., in graphs in which the nodes represent underlying social entities and the edges represent some sort of interaction between pairs of nodes. Most such research begins with the premise that a community or a cluster should be thought of as a set of nodes that has more and/or better connections between its members than to the remainder of the network. In this paper, we explore from a novel perspective several questions related to identifying meaningful communities in large social and information networks, and we come to several striking conclusions. Rather than defining a procedure to extract sets of nodes from a graph and then attempting to interpret these sets as "real" communities, we employ approximation algorithms for the graph-partitioning problem to characterize as a function of size the statistical and structural properties of partitions of graphs that could plausibly be i...

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Community Structure in Large Networks: Natural Cluster Sizes and the Absence of Large Well-Defined Clusters

A large body of work has been devoted to defining and identifying clusters or communities in social and information networks. We explore from a novel perspective several questions related to identifying meaningful communities in large social and information networks, and we come to several striking conclusions. We employ approximation algorithms for the graph partitioning problem to characterize as a function of size the statistical and structural properties of partitions of graphs that could plausibly be interpreted as communities. In particular, we define the network community profile plot, which characterizes the "best" possible community--according to the conductance measure--over a wide range of size scales. We study over 100 large real-world social and information networks. Our results suggest a significantly more refined picture of community structure in large networks than has been appreciated previously. In particular, we observe tight communities that are barely connected to the rest of the network at very small size scales; and communities of larger size scales gradually "blend into" the expander-like core of the network and thus become less "community-like." This behavior is not explained, even at a qualitative level, by any of the commonly-used network generation models. Moreover, it is exactly the opposite of what one would expect based on intuition from expander graphs, low-dimensional or manifold-like graphs, and from small social networks that have served as testbeds of community detection algorithms. We have found that a generative graph model, in which new edges are added via an iterative "forest fire" burning process, is able to produce graphs exhibiting a network community profile plot similar to what we observe in our network datasets.

A major consideration we had in writing this survey was to make it accessible to mathematicians as well as to computer scientists, since expander graphs, the protagonists of our story, come up in numerous and often surprising contexts in both fields But, perhaps, we should start with a few words about graphs in general They are, of course, one of the prime objects of study in Discrete Mathematics However, graphs are among the most ubiquitous models of both natural and human-made structures In the natural and social sciences they model relations among species, societies, companies, etc In computer science, they represent networks of communication, data organization, computational devices as well as the flow of computation, and more In mathematics, Cayley graphs are useful in Group Theory Graphs carry a natural metric and are therefore useful in Geometry, and though they are “just” one-dimensional complexes, they are useful in certain parts of Topology, eg Knot Theory In statistical physics, graphs can represent local connections between interacting parts of a system, as well as the dynamics of a physical process on such systems The study of these models calls, then, for the comprehension of the significant structural properties of the relevant graphs But are there nontrivial structural properties which are universally important? Expansion of a graph requires that it is simultaneously sparse and highly connected Expander graphs were first defined by Bassalygo and Pinsker, and their existence first proved by Pinsker in the early ’70s The property of being an expander seems significant in many of these mathematical, computational and physical contexts It is not surprising that expanders are useful in the design and analysis of communication networks What is less obvious is that expanders have surprising utility in other computational settings such as in the theory of error correcting codes and the theory of pseudorandomness In mathematics, we will encounter eg their role in the study of metric embeddings, and in particular in work around the Baum-Connes Conjecture Expansion is closely related to the convergence rates of Markov Chains, and so they play a key role in the study of Monte-Carlo algorithms in statistical mechanics and in a host of practical computational applications The list of such interesting and fruitful connections goes on and on with so many applications we will not even

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S Hoory

Papers

Expander graphs and their applications