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.

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The Structure and Function of Complex Networks

A number of recent studies have focused on the statistical properties of networked systems such as social networks and the Worldwide Web. Researchers have concentrated particularly on a few properties that seem to be common to many networks: the small-world property, power-law degree distributions, and network transitivity. In this article, we highlight another property that is found in many networks, the property of community structure, in which network nodes are joined together in tightly knit groups, between which there are only looser connections. We propose a method for detecting such communities, built around the idea of using centrality indices to find community boundaries. We test our method on computer-generated and real-world graphs whose community structure is already known and find that the method detects this known structure with high sensitivity and reliability. We also apply the method to two networks whose community structure is not well known—a collaboration network and a food web—and find that it detects significant and informative community divisions in both cases.

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Community structure in social and biological networks

We propose and study a set of algorithms for discovering community structure in networks-natural divisions of network nodes into densely connected subgroups. Our algorithms all share two definitive features: first, they involve iterative removal of edges from the network to split it into communities, the edges removed being identified using any one of a number of possible "betweenness" measures, and second, these measures are, crucially, recalculated after each removal. We also propose a measure for the strength of the community structure found by our algorithms, which gives us an objective metric for choosing the number of communities into which a network should be divided. We demonstrate that our algorithms are highly effective at discovering community structure in both computer-generated and real-world network data, and show how they can be used to shed light on the sometimes dauntingly complex structure of networked systems.

/pdf/finding-and-evaluating-community-structure-in-networks-1o9f6y8pyr.pdf

Finding and evaluating community structure in networks.

http://www.maths.qmul.ac.uk/%7Elatora/report_06.pdf

Complex networks: Structure and dynamics

The modern science of networks has brought significant advances to our understanding of complex systems. One of the most relevant features of graphs representing real systems is community structure, or clustering, i. e. the organization of vertices in clusters, with many edges joining vertices of the same cluster and comparatively few edges joining vertices of different clusters. Such clusters, or communities, can be considered as fairly independent compartments of a graph, playing a similar role like, e. g., the tissues or the organs in the human body. Detecting communities is of great importance in sociology, biology and computer science, disciplines where systems are often represented as graphs. This problem is very hard and not yet satisfactorily solved, despite the huge effort of a large interdisciplinary community of scientists working on it over the past few years. We will attempt a thorough exposition of the topic, from the definition of the main elements of the problem, to the presentation of most methods developed, with a special focus on techniques designed by statistical physicists, from the discussion of crucial issues like the significance of clustering and how methods should be tested and compared against each other, to the description of applications to real networks.

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Community detection in graphs

A comprehensive introduction to network flows that brings together the classic and the contemporary aspects of the field, and provides an integrative view of theory, algorithms, and applications. presents in-depth, self-contained treatments of shortest path, maximum flow, and minimum cost flow problems, including descriptions of polynomial-time algorithms for these core models. emphasizes powerful algorithmic strategies and analysis tools such as data scaling, geometric improvement arguments, and potential function arguments. provides an easy-to-understand descriptions of several important data structures, including d-heaps, Fibonacci heaps, and dynamic trees. devotes a special chapter to conducting empirical testing of algorithms. features over 150 applications of network flows to a variety of engineering, management, and scientific domains. contains extensive reference notes and illustrations.

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Network Flows: Theory, Algorithms, and Applications

1 Defining Network Flow A flow network is a directed graph G = (V,E) in which each edge (u, v) ∈ E has non-negative capacity c(u, v) ≥ 0. We require that if (u, v) ∈ E, then (v, u) / ∈ E. That is, if an edge exists, then the edge between the same vertices going the reverse direction does not exist. Every flow network has a source s and a sink t, and we assume that for every v ∈ V , there is some path s→ · · · → v → · · · → t. Note that this implies that flow networks are connected. Informally, the intuition behind network flow is to think of the edges as pipes and the weights on the edges as the capacity its corresponding pipe per unit of time. The question we are trying to ask is: how many units of water can we send from the source to the sink per unit of time? Formally, a flow in G is a function f : V × V → R that satisfies the following: • Capacity constraint. For all u, v ∈ V , we require 0 ≤ f(u, v) ≤ c(u, v). Our pipe cannot hold more than is allowed as dictated by its capacity. • Flow conservation. For u ∈ V − {s, t}, we require ∑

Network Flows

http://dspace.mit.edu/bitstream/handle/1721.1/35748/15-053Spring-2002/NR/rdonlyres/Sloan-School-of-Management/15-053Introduction-to-OptimizationSpring2002/1B8B4FF1-07EB-4E68-A420-CD8CE1B515CF/0/s02lec23b.pdf

A survey of very large-scale neighborhood search techniques

Efficient implementations of Dijkstra's shortest path algorithm are investigated. A new data structure, called the radix heap, is proposed for use in this algorithm. On a network with n vertices, m edges, and nonnegative integer arc costs bounded by C, a one-level form of radix heap gives a time bound for Dijkstra's algorithm of O(m + n log C). A two-level form of radix heap gives a bound of O(m + n log C/log log C). A combination of a radix heap and a previously known data structure called a Fibonacci heap gives a bound of O(m + na

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Faster Algorithms for the Shortest Path Problem

In this paper, we study inverse optimization problems defined as follows. LetS denote the set of feasible solutions of an optimization problemP, letc be a specified cost vector, andx0 be a given feasible solution. The solutionx0 may or may not be an optimal solution ofP with respect to the cost vectorc. The inverse optimization problem is to perturb the cost vectorc tod so thatx0 is an optimal solution ofP with respect tod and ||d- c || p is minimum, where ||d- c || p is some selectedLp norm. In this paper, we consider the inverse linear programming problem underL1 norm (where ||d- c || p= ? j?Jw j|d j-c j|, withJ denoting the index set of variablesx jandw jdenoting the weight of the variablej) and underL8 norm (where||d- c || p= max j?J{w j|d j-c j|} ). We prove the following results: (i) If the problemP is a linear programming problem, then its inverse problem under theL1 as well asL8 norm is also a linear programming problem. (ii) If the problemP is a shortest path, assignment or minimum cut problem, then its inverse problem under theL1 norm and unit weights can be solved by solving a problem of the same kind. For the nonunit weight case, the inverse problem reduces to solving a minimum cost flow problem. (iii) If the problemP is a minimum cost flow problem, then its inverse problem under theL1 norm and unit weights reduces to solving a unit-capacity minimum cost flow problem. For the nonunit weight case, the inverse problem reduces to solving a minimum cost flow problem. (iv) If the problemP is a minimum cost flow problem, then its inverse problem under theL8 norm and unit weights reduces to solving a minimum mean cycle problem. For the nonunit weight case, the inverse problem reduces to solving a minimum cost-to-time ratio cycle problem. (v) If the problemP is polynomially solvable for linear cost functions, then inverse versions ofP under theL1 andL8 norms are also polynomially solvable.

Ravindra K. Ahuja

Papers

Network Flows: Theory, Algorithms, and Applications

Network Flows

A survey of very large-scale neighborhood search techniques

Faster Algorithms for the Shortest Path Problem

Inverse Optimization