Duncan J. Watts

Collective dynamics of ‘small-world’ networks

Networks of coupled dynamical systems have been used to model biological oscillators, Josephson junction arrays,, excitable media, neural networks, spatial games, genetic control networks and many other self-organizing systems. Ordinarily, the connection topology is assumed to be either completely regular or completely random. But many biological, technological and social networks lie somewhere between these two extremes. Here we explore simple models of networks that can be tuned through this middle ground: regular networks ‘rewired’ to introduce increasing amounts of disorder. We find that these systems can be highly clustered, like regular lattices, yet have small characteristic path lengths, like random graphs. We call them ‘small-world’ networks, by analogy with the small-world phenomenon, (popularly known as six degrees of separation). The neural network of the worm Caenorhabditis elegans, the power grid of the western United States, and the collaboration graph of film actors are shown to be small-world networks. Models of dynamical systems with small-world coupling display enhanced signal-propagation speed, computational power, and synchronizability. In particular, infectious diseases spread more easily in small-world networks than in regular lattices.

Random graphs with arbitrary degree distributions and their applications

Recent work on the structure of social networks and the internet has focused attention on graphs with distributions of vertex degree that are significantly different from the Poisson degree distributions that have been widely studied in the past. In this paper we develop in detail the theory of random graphs with arbitrary degree distributions. In addition to simple undirected, unipartite graphs, we examine the properties of directed and bipartite graphs. Among other results, we derive exact expressions for the position of the phase transition at which a giant component first forms, the mean component size, the size of the giant component if there is one, the mean number of vertices a certain distance away from a randomly chosen vertex, and the average vertex-vertex distance within a graph. We apply our theory to some real-world graphs, including the world-wide web and collaboration graphs of scientists and Fortune 1000 company directors. We demonstrate that in some cases random graphs with appropriate distributions of vertex degree predict with surprising accuracy the behavior of the real world, while in others there is a measurable discrepancy between theory and reality, perhaps indicating the presence of additional social structure in the network that is not captured by the random graph.

A simple model of global cascades on random networks

The origin of large but rare cascades that are triggered by small initial shocks is a phenomenon that manifests itself as diversely as cultural fads, collective action, the diffusion of norms and innovations, and cascading failures in infrastructure and organizational networks. This paper presents a possible explanation of this phenomenon in terms of a sparse, random network of interacting agents whose decisions are determined by the actions of their neighbors according to a simple threshold rule. Two regimes are identified in which the network is susceptible to very large cascades—herein called global cascades—that occur very rarely. When cascade propagation is limited by the connectivity of the network, a power law distribution of cascade sizes is observed, analogous to the cluster size distribution in standard percolation theory and avalanches in self-organized criticality. But when the network is highly connected, cascade propagation is limited instead by the local stability of the nodes themselves, and the size distribution of cascades is bimodal, implying a more extreme kind of instability that is correspondingly harder to anticipate. In the first regime, where the distribution of network neighbors is highly skewed, it is found that the most connected nodes are far more likely than average nodes to trigger cascades, but not in the second regime. Finally, it is shown that heterogeneity plays an ambiguous role in determining a systems stability: increasingly heterogeneous thresholds make the system more vulnerable to global cascades; but an increasingly heterogeneous degree distribution makes it less vulnerable.

Network Robustness and Fragility: Percolation on Random Graphs

Recent work on the Internet, social networks, and the power grid has addressed the resilience of these networks to either random or targeted deletion of network nodes or links. Such deletions include, for example, the failure of Internet routers or power transmission lines. Percolation models on random graphs provide a simple representation of this process but have typically been limited to graphs with Poisson degree distribution at their vertices. Such graphs are quite unlike real-world networks, which often possess power-law or other highly skewed degree distributions. In this paper we study percolation on graphs with completely general degree distribution, giving exact solutions for a variety of cases, including site percolation, bond percolation, and models in which occupation probabilities depend on vertex degree. We discuss the application of our theory to the understanding of network resilience.

Influentials, Networks, and Public Opinion Formation

A central idea in marketing and diffusion research is that influentials—a minority of individuals who influence an exceptional number of their peers—are important to the formation of public opinion. Here we examine this idea, which we call the “influentials hypothesis,” using a series of computer simulations of interpersonal influence processes. Under most conditions that we consider, we find that large cascades of influence are driven not by influentials but by a critical mass of easily influenced individuals. Although our results do not exclude the possibility that influentials can be important, they suggest that the influentials hypothesis requires more careful specification and testing than it has received.

Networks, dynamics, and the small-world phenomenon

The small‐world phenomenon formalized in this article as the coincidence of high local clustering and short global separation, is shown to be a general feature of sparse, decentralized networks that are neither completely ordered nor completely random. Networks of this kind have received little attention, yet they appear to be widespread in the social and natural sciences, as is indicated here by three distinct examples. Furthermore, small admixtures of randomness to an otherwise ordered network can have a dramatic impact on its dynamical, as well as structural, properties‐a feature illustrated by a simple model of disease transmission.

Scaling and percolation in the small-world network model.

In this paper we study the small-world network model of Watts and Strogatz, which mimics some aspects of the structure of networks of social interactions. We argue that there is one nontrivial length-scale in the model, analogous to the correlation length in other systems, which is well-defined in the limit of infinite system size and which diverges continuously as the randomness in the network tends to zero, giving a normal critical point in this limit. This length-scale governs the crossover from large- to small-world behavior in the model, as well as the number of vertices in a neighborhood of given radius on the network. We derive the value of the single critical exponent controlling behavior in the critical region and the finite size scaling form for the average vertex-vertex distance on the network, and, using series expansion and Padé approximants, find an approximate analytic form for the scaling function. We calculate the effective dimension of small-world graphs and show that this dimension varies as a function of the length-scale on which it is measured, in a manner reminiscent of multifractals. We also study the problem of site percolation on small-world networks as a simple model of disease propagation, and derive an approximate expression for the percolation probability at which a giant component of connected vertices first forms (in epidemiological terms, the point at which an epidemic occurs). The typical cluster radius satisfies the expected finite size scaling form with a cluster size exponent close to that for a random graph. All our analytic results are confirmed by extensive numerical simulations of the model.

Predicting consumer behavior with Web search

Recent work has demonstrated that Web search volume can “predict the present,” meaning that it can be used to accurately track outcomes such as unemployment levels, auto and home sales, and disease prevalence in near real time. Here we show that what consumers are searching for online can also predict their collective future behavior days or even weeks in advance. Specifically we use search query volume to forecast the opening weekend box-office revenue for feature films, first-month sales of video games, and the rank of songs on the Billboard Hot 100 chart, finding in all cases that search counts are highly predictive of future outcomes. We also find that search counts generally boost the performance of baseline models fit on other publicly available data, where the boost varies from modest to dramatic, depending on the application in question. Finally, we reexamine previous work on tracking flu trends and show that, perhaps surprisingly, the utility of search data relative to a simple autoregressive model is modest. We conclude that in the absence of other data sources, or where small improvements in predictive performance are material, search queries provide a useful guide to the near future.

Origins of Homophily in an Evolving Social Network

The authors investigate the origins of homophily in a large university community, using network data in which interactions, attributes, and affiliations are all recorded over time. The analysis indicates that highly similar pairs do show greater than average propensity to form new ties; however, it also finds that tie formation is heavily biased by triadic closure and focal closure, which effectively constrain the opportunities among which individuals may select. In the case of triadic closure, moreover, selection to “friend of a friend” status is determined by an analogous combination of individual preference and structural proximity. The authors conclude that the dynamic interplay of choice homophily and induced homophily, compounded over many “generations” of biased selection of similar individuals to structurally proximate positions, can amplify even a modest preference for similar others, via a cumulative advantage–like process, to produce striking patterns of observed homophily.

Universal Behavior in a Generalized Model of Contagion

Models of contagion arise broadly in both the biological and the social sciences, with applications ranging from the transmission of infectious diseases to the spread of cultural fads. In this Letter, we introduce a general model of contagion which, by explicitly incorporating memory of past exposures to, for example, an infectious agent, rumor, or new product, includes the main features of existing contagion models and interpolates between them. We obtain exact solutions for a simple version of the model, finding that under general conditions only three classes of collective dynamics exist. Furthermore, we find that, for a given length of memory, the class into which a particular system falls is determined by only two parameters. Our model suggests novel measures for assessing the susceptibility of a population to large contagion events, and also a possible strategy for inhibiting or facilitating them.