Online multiplayer games are a popular form of social interaction, used by hundreds of millions of individuals. However, little is known about the social networks within these online games, or how they evolve over time. Understanding human social dynamics within massive online games can shed new light on social interactions in general and inform the development of more engaging systems. Here, we study a novel, large friendship network, inferred from nearly 18 billion social interactions over 44 weeks between 17 million individuals in the popular online game Halo:Reach. This network is one of the largest, most detailed temporal interaction networks studied to date, and provides a novel perspective on the dynamics of online friendship networks, as opposed to mere interaction graphs. Initially, this network exhibits strong structural turnover and decays rapidly from a peak size. In the following period, however, both network size and turnover stabilize, producing a dynamic structural equilibrium. In contrast to other studies, we nd that the Halo friendship network is non-densifying: both the mean degree and the average pairwise distance are stable, suggesting that densication cannot occur when maintaining friendships is costly. Finally, players with greater long-term engagement exhibit stronger local clustering, suggesting a group-level social engagement process. These results demonstrate the utility of online games for studying social networks, shed new light on empirical temporal graph patterns, and clarify the claims of universality of network densication.
Since the underlying processes that produce complex networks are typically not random, a rich body of work related to mathematically modeling various networks whose structure cannot be generated by an Erd o s-R 'enyi random graph model has emerged @cite_4 . One such model is the configuration model, which produces a graph with a predefined degree distribution by randomly assigning edges between vertices according to their degree sequence @cite_24 . Another is the preferential attachment model, which is a generative model that produces heavy tailed degree distributions by assigning edges to vertices according to the notion of the rich get richer". That is, edges are assigned to vertices according to how many they already have. Other approaches that produce specific properties, such as short diameters and communities include the Watts and Strogatz model and stochastic block model respectively @cite_28 @cite_26 .
@cite_24 @cite_28 @cite_4 @cite_26
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The preferential attachment graph G"m(n) is a random graph formed by adding a new vertex at each time step, with m edges which point to vertices selected at random with probability proportional to their degree. Thus at time n there are n vertices and mn edges. This process yields a graph which has been proposed as a simple model of the world wide web [A. Barabasi, R. Albert, Emergence of scaling in random networks, Science 286 (1999) 509-512]. In this paper we show that if m>=2 then whp the cover time of a simple random walk on G"m(n) is asymptotic to 2mm-1nlogn. A detailed understanding of expansion in complex networks can greatly aid in the design and analysis of algorithms for a variety of important network tasks, including routing messages, ranking nodes, and compressing graphs. This has motivated several recent investigations of expansion properties in real-world graphs and also in random models of real-world graphs, like the preferential attachment graph. The results point to a gap between real-world observations and theoretical models. Some real-world graphs are expanders and others are not, but a graph generated by the preferential attachment model is an expander whp. We study a random graph Gn that combines certain aspects of geometric random graphs and preferential attachment graphs. This model yields a graph with power-law degree distribution where the expansion property depends on a tunable parameter of the model. The vertices of Gn are n sequentially generated points x1, x2, ..., xn chosen uniformly at random from the unit sphere in R3. After generating xt, we randomly connect it to m points from those points in x1, x2, ..., xt-1.... We develop an information-theoretic view of the stochastic block model, a popular statistical model for the large-scale structure of complex networks. A graph G from such a model is generated by first assigning vertex labels at random from a finite alphabet, and then connecting vertices with edge probabilities depending on the labels of the endpoints. In the case of the symmetric two-group model, we establish an explicit ‘single-letter’ characterization of the per-vertex mutual information between the vertex labels and the graph, when the graph average degree diverges. The explicit expression of the mutual information is intimately related to estimation-theoretic quantities, and -in particular- reveals a phase transition at the critical point for community detection. Below the critical point the per-vertex mutual information is asymptotically the same as if edges were independent of the vertex labels. Correspondingly, no algorithm can estimate the partition better than random guessing. Conversely, above the threshold, the per-vertex mutual information is strictly smaller than the independent-edges upper bound. In this regime there exists a procedure that estimates the vertex labels better than random guessing. The proliferation of models for networks raises challenging problems of model selection: the data are sparse and globally dependent, and models are typically high-dimensional and have large numbers of latent variables. Together, these issues mean that the usual model-selection criteria do not work properly for networks. We illustrate these challenges, and show one way to resolve them, by considering the key network-analysis problem of dividing a graph into communities or blocks of nodes with homogeneous patterns of links to the rest of the network. The standard tool for undertaking this is the stochastic block model, under which the probability of a link between two nodes is a function solely of the blocks to which they belong. This imposes a homogeneous degree distribution within each block; this can be unrealistic, so degree-corrected block models add a parameter for each node, modulating its overall degree. The choice between ordinary and degree-corrected block models matters because they make very different inferences about communities. We present the first principled and tractable approach to model selection between standard and degree-corrected block models, based on new large-graph asymptotics for the distribution of log-likelihood ratios under the stochastic block model, finding substantial departures from classical results for sparse graphs. We also develop linear-time approximations for log-likelihoods under both the stochastic block model and the degree-corrected model, using belief propagation. Applications to simulated and real networks show excellent agreement with our approximations. Our results thus both solve the practical problem of deciding on degree correction and point to a general approach to model selection in network analysis.