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.

Models for the processes by which ideas and influence propagate through a social network have been studied in a number of domains, including the diffusion of medical and technological innovations, the sudden and widespread adoption of various strategies in game-theoretic settings, and the effects of “word of mouth ” in the promotion of new products. Recently, motivated by the design of viral marketing strategies, Domingos and Richardson posed a fundamental algorithmic problem for such social network processes: if we can try to convince a subset of individuals to adopt a new product or innovation, and the goal is to trigger a large cascade of further adoptions, which set of individuals should we target? We consider this problem in several of the most widely studied models in social network analysis. The optimization problem of selecting the most influential nodes is NP-hard here, and we provide the first provable approximation guarantees for efficient algorithms. Using an analysis framework based on submodular functions, we show that a natural greedy strategy obtains a solution that is provably within 63 % of optimal for several classes of models; our framework suggests a general approach for reasoning about the performance guarantees of algorithms for these types of influence problems in social networks. We also provide computational experiments on large collaboration networks, showing that in addition to their provable guarantees, our approximation algorithms significantly out-perform nodeselection heuristics based on the well-studied notions of degree centrality and distance centrality from the field of social networks.

3 The research was done while at HP Labs.

How do blogs cite and influence each other? How do such links evolve? Does the popularity of old blog posts drop exponentially with time? These are some of the questions that we address in this work. Blogs (weblogs) have become an important medium of information because of their timely publication, ease of use, and wide availability. In fact, they often make headlines, by discussing and discovering evidence about political events and facts. Often blogs link to one another, creating a publicly available record of how information and influence spreads through an underlying social network. Aggregating links from several blog posts creates a directed graph which we analyze to discover the patterns of information propagation in blogspace, and thereby understand the underlying social network. Here we report some surprising findings of the blog linking and information propagation structure, after we analyzed one of the largest available datasets, with 45, 000 blogs and ≈ 2.2 million blog-postings. Our analysis also sheds light on how rumors, viruses, and ideas propagate over social and computer networks.

Information cascades are phenomena in which individuals adopt a new action or idea due to influence by others. As such a process spreads through an underlying social network, it can result in widespread adoption overall. We consider information cascades in the context of recommendations, and in particular study the patterns of cascading recommendations that arise in large social networks. We investigate a large person-to-person recommendation network, consisting of four million people who made sixteen million recommendations on half a million products. Such a dataset allows us to pose a number of fundamental questions: What kinds of cascades arise frequently in real life? What features distinguish them? We enumerate and count cascade subgraphs on large directed graphs; as one component of this, we develop a novel efficient heuristic based on graph isomorphism testing that scales to large datasets. We discover novel patterns: the distribution of cascade sizes is approximately heavy-tailed; cascades tend to be shallow, but occasional large bursts of propagation can occur. The relative abundance of di#erent cascade subgraphs suggests subtle properties of the underlying social network and recommendation process.

Online social networking sites like MySpace, Facebook, and Flickr have become a popular way to share and disseminate content. Their massive popularity has led to viral marketing techniques that attempt to spread content, products, and ideas on these sites. However, there is little data publicly available on viral propagation in the real world and few studies have characterized how information spreads over current online social networks. In this paper, we collect and analyze large-scale traces of information dissemination in the Flickr social network. Our analysis, based on crawls of the favorite markings of 2.5 million users on 11 million photos, aims at answering three key questions: (a) how widely does information propagate in the social network? (b) how quickly does information propagate? and (c) what is the role of word-of-mouth exchanges between friends in the overall propagation of information in the network? Contrary to viral marketing “intuition, ” we find that (a) even popular photos do not spread widely throughout the network, (b) even popular photos spread slowly through the network, and (c) information exchanged between friends is likely to account for over 50 % of all favoritemarkings, but with a significant delay at each hop.

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

A dynamic network is a special type of network which is comprised of connected transactors which have repeated evolving interaction. Data on large dynamic networks such as telecommunications networks and the Intenlet are pervaive. However, represe~lting dyna~nic networks in a manner that is conducive to efficient large-scale analysis is a challenge. In this paper, we represent dynamic graphs using a data structure introduced by Cortes et. a]. [Q]. We advocate their representation because it accounts for the evolution of relationships between transactors through time, mitigates noise at the local transactor level, and allows for the removal of stale relationships. Our work improves on their heuristic arguments by formalizing the representation with three tunable parameters. In doing this, we develop a generic framework for evaluating and tuning any dynamic graph. We show that the storage saving approximations involved in the representation do not affect predictive performance, and typically improve it. We motivate our approach using a fraud detection example from the telecommunications industry, and demonstrate that we can outperform published results on the fraud detection task. In addition, we present preliminary analysis on web logs and email networks.

We propose an analytically tractable model of loading-dependent cascading failure that captures some of the salient features of large blackouts of electric power transmission systems. This leads to a new application and derivation of the quasibinomial distribution and its generalization to a saturating form with an extended parameter range. The saturating quasibinomial distribution of the number of failed components has a power law region at a critical loading and a significant probability of total failure at higher loadings.

Models of the convergence of opinion in social systems have been the subject of a considerable amount of recent attention in the physics literature. These models divide into two classes, those in which individuals form their beliefs based on the opinions of their neighbors in a social network of personal acquaintances, and those in which, conversely, network connections form between individuals of similar beliefs. While both of these processes can give rise to realistic levels of agreement between acquaintances, practical experience suggests that opinion formation in the real world is not a result of one process or the other, but a combination of the two. Here we present a simple model of this combination, with a single parameter controlling the balance of the two processes. We find that the model undergoes a continuous phase transition as this parameter is varied, from a regime in which opinions are arbitrarily diverse to one in which most individuals hold the same opinion. We characterize the static and dynamical properties of this transition.