Understanding social network structure and evolution has important implications for many aspects of network and system design including provisioning, bootstrapping trust and reputation systems via social networks, and defenses against Sybil attacks. Several recent results suggest that augmenting the social network structure with user attributes (e.g., location, employer, communities of interest) can provide a more fine-grained understanding of social networks. However, there have been few studies to provide a systematic understanding of these effects at scale. We bridge this gap using a unique dataset collected as the Google+ social network grew over time since its release in late June 2011. We observe novel phenomena with respect to both standard social network metrics and new attribute-related metrics (that we define). We also observe interesting evolutionary patterns as Google+ went from a bootstrap phase to a steady invitation-only stage before a public release. Based on our empirical observations, we develop a new generative model to jointly reproduce the social structure and the node attributes. Using theoretical analysis and empirical evaluations, we show that our model can accurately reproduce the social and attribute structure of real social networks. We also demonstrate that our model provides more accurate predictions for practical application contexts.
Modeling social networks There are two broad classes of models for generating social networks: and . Static models try to reproduce a single static network snapshot @cite_13 @cite_44 @cite_54 @cite_53 . Dynamic models can provide insights on how nodes arrive and create links; these include models such as preferential attachment @cite_84 , copying @cite_39 , nearest neighbor @cite_49 , forest fire @cite_71 . @cite_27 evaluated such models using both network metrics and application benchmarks and showed that the nearest neighbor model outperforms others. The dynamic generative model by Leskovec et al. mimics the nearest neighbor model in a dynamic setting @cite_21 , and thus we use it as our starting point in , However, these models are known to generate networks with power-law degree distributions. Many social networks including Google+, however, exhibit lognormal degree distributions @cite_3 @cite_20 @cite_86 . Our dynamic model extends these prior work to provably generate a lognormal distribution for social outdegree. Our model also provides a more general framework by capturing both social and attribute structure.
@cite_54 @cite_53 @cite_21 @cite_84 @cite_39 @cite_44 @cite_3 @cite_27 @cite_49 @cite_71 @cite_86 @cite_13 @cite_20
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This paper explores two aspects of social network modeling. First, we generalize a successful static model of relationships into a dynamic model that accounts for friendships drifting over time. Second, we show how to make it tractable to learn such models from data, even as the number of entities n gets large. The generalized model associates each entity with a point in p-dimensional Euclidean latent space. The points can move as time progresses but large moves in latent space are improbable. Observed links between entities are more likely if the entities are close in latent space. We show how to make such a model tractable (sub-quadratic in the number of entities) by the use of appropriate kernel functions for similarity in latent space; the use of low dimensional KD-trees; a new efficient dynamic adaptation of multidimensional scaling for a first pass of approximate projection of entities into latent space; and an efficient conjugate gradient update rule for non-linear local optimization in which amortized time per entity during an update is O(log n). We use both synthetic and real-world data on up to 11,000 entities which indicate near-linear scaling in computation time and improved performance over four alternative approaches. We also illustrate the system operating on twelve years of NIPS co-authorship data. In this paper, we study the question of how efficiently a collection of interconnected nodes can perform a global computation in the GOSSIP model of communication. In this model, nodes do not know the global topology of the network, and they may only initiate contact with a single neighbor in each round. This model contrasts with the much less restrictive LOCAL model, where a node may simultaneously communicate with all of its neighbors in a single round. A basic question in this setting is how many rounds of communication are required for the information dissemination problem, in which each node has some piece of information and is required to collect all others. In the LOCAL model, this is quite simple: each node broadcasts all of its information in each round, and the number of rounds required will be equal to the diameter of the underlying communication graph. In the GOSSIP model, each node must independently choose a single neighbor to contact, and the lack of global information makes it difficult to make any sort of principled choice. As such, researchers have focused on the uniform gossip algorithm, in which each node independently selects a neighbor uniformly at random. When the graph is well-connected, this works quite well. In a string of beautiful papers, researchers proved a sequence of successively stronger bounds on the number of rounds required in terms of the conductance φ and graph size n, culminating in a bound of O(φ-1 log n). In this paper, we show that a fairly simple modification of the protocol gives an algorithm that solves the information dissemination problem in at most O(D + polylog (n)) rounds in a network of diameter D, with no dependence on the conductance. This is at most an additive polylogarithmic factor from the trivial lower bound of D, which applies even in the LOCAL model. In fact, we prove that something stronger is true: any algorithm that requires T rounds in the LOCAL model can be simulated in O(T + polylog(n)) rounds in the GOSSIP model. We thus prove that these two models of distributed computation are essentially equivalent. short-lived or high degree peers due to the dynamics of peer participation or the heterogeneity of peer degrees, respectively. This paper presents Respondent-Driven Sampling (RDS) as a promising technique for sampling unstructured P2P overlays. This allows one to accurately estimate the distribution of a desired peer property without capturing the entire overlay structure. RDS is a variant of snowball sampling that has been proposed and used in the social sciences to characterize hidden population in a society [9], [13]. We apply the RDS technique to unstructured P2P network and evaluate its performance over a wide range of static and dynamic graphs as well as a widely deployed P2P system. Throughout our evaluation, we compare and contrast the performance of the RDS technique with another sampling technique, namely Metropolized Random Walk (MRW), that we developed in our earlier work [16]. Our main findings can be summarized as follows: First, RDS outperforms MRW across all scenarios. In particular, RDS exhibits a significantly better performance than MRW when the overlay structure exhibits a combination of highly skewed node degrees and highly skewed (local) clustering coefficients. Second, our simulation and empirical evaluations reveal that both the RDS and MRW techniques can accurately estimate key peer properties over dynamic unstructured overlays. Third, our empirical evaluations suggest that the efficiency of the two sampling techniques in practice is lower than in our simulations involving synthetic graphs. We attribute this to our inability to capture accurate reference snapshots. The rest of the paper is organized as follows: Section II presents an overview of both the RDS and MRW techniques, and sketches our evaluation methodology. We examine both techniques over variety of static and dynamic graphs in Section III and IV, respectively. Section V presents the empirical evaluation of the two sampling techniques over Gnutella network. We study the detailed growth of a social networking site with full temporal information by examining the creation process of each friendship relation that can collectively lead to the macroscopic properties of the network. We first study the reciprocal behavior of users, and find that link requests are quickly responded to and that the distribution of reciprocation intervals decays in an exponential form. The degrees of inviters accepters are slightly negatively correlative with reciprocation time. In addition, the temporal feature of the online community shows that the distributions of intervals of user behaviors, such as sending or accepting link requests, follow a power law with a universal exponent, and peaks emerge for intervals of an integral day. We finally study the preferential selection and linking phenomena of the social networking site and find that, for the former, a linear preference holds for preferential sending and reception, and for the latter, a linear preference also holds for preferential acceptance, creation, and attachment. Based on the linearly preferential linking, we put forward an analyzable network model which can reproduce the degree distribution of the network. The research framework presented in the paper could provide a potential insight into how the micro-motives of users lead to the global structure of online social networks.