Complex networks describe a wide range of systems in nature and society, much quoted examples including the cell, a network of chemicals linked by chemical reactions, or the Internet, a network of routers and computers connected by physical links. While traditionally these systems were modeled as random graphs, it is increasingly recognized that the topology and evolution of real

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

Coupled biological and chemical systems, neural networks, social interacting species, the Internet and the World Wide Web, are only a few examples of systems composed by a large number of highly interconnected dynamical units. The first approach to capture the global properties of such systems is to model them as graphs whose nodes represent the dynamical units, and whose links stand for the interactions between them. On the one hand, scientists have to cope with structural issues, such as characterizing the topology of a complex wiring architecture, revealing the unifying principles that are at the basis of real networks, and developing models to mimic the growth of a network and reproduce its structural properties. On the other hand, many relevant questions arise when studying complex networks ’ dynamics, such as learning how a large ensemble of dynamical systems that interact through a complex wiring topology can behave collectively. We review the major concepts and results recently achieved in the study of the structure and dynamics of complex networks, and summarize the relevant applications of these ideas in many different disciplines,

I survey the recent literature on the formation of networks. I provide definitions of network games, a number of examples of models from the literature, and discuss some of what is known about the (in)compatibility of overall societal welfare with individual incentives to form and sever links.

As computer networks increase in size, become more heterogeneous and span greater geographic distances, applications must be designed to cope with the very large scale, poor reliability, and often, with the extreme dynamism of the underlying network. Aggregation is a key functional building block for such applications: it refers to a set of functions that provide components of a distributed system access to global information including network size, average load, average uptime, location and description of hotspots, and so on. Local access to global information is often very useful, if not indispensable for building applications that are robust and adaptive. For example, in an industrial control application, some aggregate value reaching a threshold may trigger the execution of certain actions; a distributed storage system will want to know the total available free space; load-balancing protocols may benefit from knowing the target average load so as to minimize the load they transfer. We propose a gossip-based protocol for computing aggregate values over network components in a fully decentralized fashion. The class of aggregate functions we can compute is very broad and includes many useful special cases such as counting, averages, sums, products, and extremal values. The protocol is suitable for extremely large and highly dynamic systems due to its proactive structure—all nodes receive the aggregate value continuously, thus being able to track

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.

We present statistical analyses of the large-scale structure of three types of semantic networks: word associations, WordNet, and Roget's thesaurus. We show that they have a small-world structure, characterized by sparse connectivity, short average path-lengths between words, and strong local clustering. In addition, the distributions of the number of connections follow power laws that indicate a scale-free pattern of connectivity, with most nodes having relatively few connections joined together through a small number of hubs with many connections. These regularities have also been found in certain other complex natural networks, such as the world wide web, but they are not consistent with many conventional models of semantic organization, based on inheritance hierarchies, arbitrarily structured networks, or high-dimensional vector spaces. We propose that these structures reflect the mechanisms by which semantic networks grow. We describe a simple model for semantic growth, in which each new word or concept is connected to an existing network by differentiating the connectivity pattern of an existing node. This model generates appropriate small-world statistics and power-law connectivity distributions, and also suggests one possible mechanistic basis for the effects of learning history variables (age-ofacquisition, usage frequency) on behavioral performance in semantic processing tasks.

reviewers, especially the reviewer that provided a remarkable 15, single spaced pages of superb commentary. We particularly wish to thank Mark Newman for his advice and help in developing and interpreting the bipartite-affiliation network statistics. We also wish to give very special thanks to the Santa Fe Institute for creating a rich collaborative environment wherein these ideas first emerged and to John Padgett, the organizer of the States and Markets group at the Santa Fe

The proliferation of web-based social networks has lead to new innovations in social networking, particularly by allowing users to describe their relationships beyond a basic connection. In this dissertation, I look specifically at trust in web-based social networks, how it can be computed, and how it can be used in applications. I begin with a definition of trust and a description of several properties that affect how it is used in algorithms. This is complemented by a survey of web-based social networks to gain an understanding of their scope, the types of relationship information available, and the current state of trust. The computational problem of trust is to determine how much one person in the network should trust another person to whom they are not connected. I present two sets of algorithms for calculating these trust inferences: one for networks with binary trust ratings, and one for continuous ratings. For each rating scheme, the algorithms are built upon the defined notions of trust. Each is then analyzed theoretically and with respect to simulated and actual trust networks to determine how accurately they calculate the opinions of people in the system. I show that in both rating schemes the algorithms

Complex Contagion and the Weakness of Long Ties The strength of weak ties is that they tend to be long – they connect socially distant locations. Recent research on “small worlds ” shows that remarkably few long ties are needed to give large and highly clustered populations the “degrees of separation ” of a random network, in which information can rapidly diffuse. We test whether this effect of long ties generalizes from simple to complex contagions – those in which the credibility of information or the willingness to adopt an innovation requires independent confirmation from multiple sources. Using Watts and Strogatz’s original small world model, we demonstrate that long ties may not only fail to speed up complex contagions, they can even preclude diffusion entirely. Results suggest that the spread of collective actions, social movements, and risky innovations benefit not from ties that are long but from bridges that are wide enough to transmit strong social reinforcement. Balance theory shows how wide bridges might also form in evolving networks, but this turns out to have surprisingly little effect on the propagation of complex contagions. We find that