Classes of behavior of small-world networks

We study the statistical properties of a variety of diverse real-world networks. We present evidence of the occurrence of three classes of small-world networks: (a) scale-free networks, characterized by a vertex connectivity distribution that decays as a power law; (b) broad-scale networks, characterized by a connectivity distribution that has a power law regime followed by a sharp cutoff; and (c) single-scale networks, characterized by a connectivity distribution with a fast decaying tail. Moreover, we note for the classes of broad-scale and single-scale networks that there are constraints limiting the addition of new links. Our results suggest that the nature of such constraints may be the controlling factor for the emergence of different classes of networks.

2003

Brain and Cognitive Science Department, MIT, 1999

Small-world architectures may be implicated in a range of phenomena from networks of neurons in the cerebral cortex to social networks and propogation of viruses [1]- . Small-world networks are interpolations of regular and random networks that retain the advantages of both regular and random networks by being highly clustered like regular networks and having small average path length between nodes, like random networks. While most of the recent attention on small-world networks has focussed on the effect of introducing disorder/randomness into a regular network, we show that that the fundamental mechanism behind the small-world phenomenon is not disorder/randomness, but the presence of connections of many different length scales. Consequently, in order to explain the small-world phenomenon, we introduce the concept of multiple scale networks and then state the multiple length scale hypothesis . We show that small-world behavior in randomly rewired networks is a consequence of features common to all multiple scale networks. To support the multiple length scale hypothesis, novel network architectures are introduced that need not be a result of random rewiring of a regular network. In each case it is shown that whenever the network exhibits small-world behavior, it also has connections of diverse length scales. We also show that the distribution of the length scales of the new connections is significantly more important than whether the new connections are long range, medium range or short range.

2001

Page 1. Characteristics of Small World Networks Petter Holme 20th April 2001 References: [1.] DJ Watts and SH Strogatz, Collective Dynamics of 'Small-World' Networks, Nature 393, 440 (1998). [2.] DJ Watts , Small Worlds: The Dynamics of Networks between Order and Randomness, (Princeton University Press, Princeton, 1999), Part 1. [3.] N. Mathias and V. Gopal, Small Worlds: How and Why, Phys. Rev. E 63, 21117 (2001). [4.] M. Gitterman, Small-World Phenomena in Physics: The Ising Model, J. Phys. A 33, 8373 (2000). Page 2.

Systems as diverse as genetic networks or the world wide web are best described as networks with complex topology. A common property of many large networks is that the vertex connectivities follow a scale-free power-law distribution. This feature is found to be a consequence of the two generic mechanisms that networks expand continuously by the addition of new vertices, and new vertices attach preferentially to already well connected sites. A model based on these two ingredients reproduces the observed stationary scale-free distributions, indicating that the development of large networks is governed by robust self-organizing phenomena that go beyond the particulars of the individual systems. * [email protected]

Computing Research Repository, 2001

Complex networks describe a wide range of systems in nature and society. Frequently cited examples include the cell, a network of chemicals linked by chemical reactions, and the Internet, a network of routers and computers connected by physical links. While traditionally these systems have been modeled as random graphs, it is increasingly recognized that the topology and evolution of real networks are governed by robust organizing principles. This article reviews the recent advances in the field of complex networks, focusing on the statistical mechanics of network topology and dynamics. After reviewing the empirical data that motivated the recent interest in networks, the authors discuss the main models and analytical tools, covering random graphs, small-world and scale-free networks, the emerging theory of evolving networks, and the interplay between topology and the network's robustness against failures and attacks.

Statistical physics is the natural framework to model complex networks. In the last twenty years, it has brought novel physical insights on a variety of emergent phenomena, such as self-organisation, scale invariance, mixed distributions and ensemble non-equivalence, which cannot be deduced from the behaviour of the individual constituents. At the same time, thanks to its deep connection with information theory, statistical physics and the principle of maximum entropy have led to the definition of null models reproducing some features of empirical networks, but otherwise as random as possible. We review here the statistical physics approach for complex networks and the null models for the various physical problems, focusing in particular on the analytic frameworks reproducing the local features of the network. We show how these models have been used to detect statistically significant and predictive structural patterns in real-world networks, as well as to reconstruct the network structure in case of incomplete information. We further survey the statistical physics frameworks that reproduce more complex, semi-local network features using Markov chain Monte Carlo sampling, and the models of generalised network structures such as multiplex networks, interacting networks and simplicial complexes. The science of networks has exploded in the Information Age thanks to the unprecedented production and storage of data on basically any human activity. Indeed, a network represents the simplest yet extremely effective way to model a large class of technological, social, economic and biological systems, as a set of entities (nodes) and of interactions (links) among them. These interactions do represent the fundamental degrees of freedom of the network, and can be of different types—undirected or directed, binary or valued (weighted)—depending on the nature of the system and the resolution used to describe it. Notably, most of the networks observed in the real world fall within the domain of complex systems, as they exhibit strong and complicated interaction patterns, and feature collective emergent phenomena that do not follow trivially from the behaviours of the individual entities [1]. For instance, many networks are scale-free [2–6], meaning that the number of links incident to a node (known as the node's degree) is fat-tailed distributed, sometimes following a power-law: most nodes have a few links, but a few nodes (the hubs) have many of them. The same happens for the distribution of the total weight of connections incident to a node (the node's strength) [7, 8]. Similarly, most real-world networks are organised into modules or feature a community structure [9, 10], and they possess high clustering—as nodes tend to create tightly linked groups, but are also small-world [11–13] as the distance (in terms of number of connections) amongst node pairs scales logarithmically with the system size. The observation of these universal features in complex networks has stimulated the development of a unifying mathematical language to model their structure and understand the dynamical processes taking place on them—such as the flow of traffic on the Internet or the spreading of either diseases or information in a population [14–16]. Two different approaches to network modelling can be pursued. The first one consists in identifying one or more microscopic mechanisms driving the formation of the network, and use them to define a dynamic model which can reproduce some of the emergent properties of real systems. The small-world model [11], the preferential attachment model [2], the fitness model [5], the relevance model [17] and many others follow this approach which is akin to kinetic theory. These models can handle only simple microscopic dynamics, and thus while providing good physical insights they need several refinements to give quantitatively accurate predictions. The other possible approach consists in identifying a set of characteristic static properties of real systems, and then building networks having the same properties but otherwise maximally random. This approach is thus akin to statistical mechanics and therefore is based on rigorous probabilistic arguments that can lead to accurate and reliable predictions. The mathematical framework is that of exponential random graphs (ERG), which has been first introduced in the social sciences and statistics [18–26] as a convenient formulation relying on numerical techniques such as Markov chain Monte Carlo algorithms. The interpretation of ERG in physical terms is due to Park and Newman [27], who showed how to derive them from the principle of maximum entropy and the statistical mechanics of Boltzmann and Gibbs. As formulated by Jaynes [28], the variational principle of maximum entropy states that the probability distribution best representing the current state of (knowledge on) a system is the one which maximises the Shannon entropy, subject in principle to any prior information on the system itself. This means making self-consistent inference assum

Artificial Neural Networks— …, 2002

Abstract. Small-World networks are highly clusterized networks with small distances between their nodes. There are some well known biolog-ical networks that present this kind of connectivity. On the other hand, the usual models of Small-World networks make use of ...

Brain Connectivity, 2011

Small-world networks by Watts and Strogatz are a class of networks that are highly clustered, like regular lattices, yet have small characteristic path lengths, like random graphs. These characteristics result in networks with unique properties of regional specialization with efficient information transfer. Social networks are intuitive examples of this organization with cliques or clusters of friends being interconnected, but each person is really only 5-6 people away from anyone else. While this qualitative definition has prevailed in network science theory, in application, the standard quantitative application is to compare path length (a surrogate measure of distributed processing) and clustering (a surrogate measure of regional specialization) to an equivalent random network. It is demonstrated here that comparing network clustering to that of a random network can result in aberrant findings and networks once thought to exhibit small-world properties may not. We propose a new small-world metric, ω (omega), which compares network clustering to an equivalent lattice network and path length to a random network, as Watts and Strogatz originally described. Example networks are presented that would be interpreted as small-world when clustering is compared to a random network but are not smallworld according to ω. These findings have significant implications in network science as small-world networks have unique topological properties, and it is critical to accurately distinguish them from networks without simultaneous high clustering and low path length.

BioEssays, 2005

Recent observations of power-law distributions in the connectivity of complex networks came as a big surprise to researchers steeped in the tradition of random networks. Even more surprising was the discovery that power-law distributions also characterize many biological and social networks. Many attributed a deep significance to this fact, inferring a ''universal architecture'' of complex systems. Closer examination, however, challenges the assumptions that (1) such distributions are special and (2) they signify a common architecture, independent of the system's specifics. The real surprise, if any, is that power-law distributions are easy to generate, and by a variety of mechanisms. The architecture that results is not universal, but particular; it is determined by the actual constraints on the system in question.