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-attribute networks There has been relatively little work on generating , though a few recent work jointly generating both social structure and node attributes can be viewed as models; the most relevant work is from @cite_57 and Kim and Leskovec @cite_2 . @cite_57 focus on dynamic attributes; their model generates undirected networks with power-law distribution for social degree and non-lognormal distribution for attribute degree (see Figure ). Kim and Leskovec model the social and attribute structure simultaneously @cite_2 . Here, both the social degree of attribute nodes and attribute degrees of social nodes follow binomial distribution, which differs from empirically observed . Our model can generate that we confirm through both analysis and simulations to be consistent with real .
@cite_57 @cite_2
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The effects of social influence and homophily suggest that both network structure and node-attribute information should inform the tasks of link prediction and node-attribute inference. Recently, [2010a, 2010b] proposed an attribute-augmented social network model, which we call Social-Attribute Network (SAN), to integrate network structure and node attributes to perform both link prediction and attribute inference. They focused on generalizing the random walk with a restart algorithm to the SAN framework and showed improved performance. In this article, we extend the SAN framework with several leading supervised and unsupervised link-prediction algorithms and demonstrate performance improvement for each algorithm on both link prediction and attribute inference. Moreover, we make the novel observation that attribute inference can help inform link prediction, that is, link-prediction accuracy is further improved by first inferring missing attributes. We comprehensively evaluate these algorithms and compare them with other existing algorithms using a novel, large-scale Googlep dataset, which we make publicly available (rlhttp: www.cs.berkeley.edu ∼stevgong gplus.html). One of the most influential recent results in network analysis is that many natural networks exhibit a power-law or log-normal degree distribution. This has inspired numerous generative models that match this property. However, more recent work has shown that while these generative models do have the right degree distribution, they are not good models for real-life networks due to their differences on other important metrics like conductance. We believe this is, in part, because many of these real-world networks have very different joint degree distributions, that is, the probability that a randomly selected edge will be between nodes of degree k and l. Assortativity is a sufficient statistic of the joint degree distribution, and it has been previously noted that social networks tend to be assortative, while biological and technological networks tend to be disassortative. We suggest understanding the relationship between network structure and the joint degree distribution of graphs is an interesting avenue of further research. An important tool for such studies are algorithms that can generate random instances of graphs with the same joint degree distribution. This is the main topic of this article, and we study the problem from both a theoretical and practical perspective. We provide an algorithm for constructing simple graphs from a given joint degree distribution, and a Monte Carlo Markov chain method for sampling them. We also show that the state space of simple graphs with a fixed degree distribution is connected via endpoint switches. We empirically evaluate the mixing time of this Markov chain by using experiments based on the autocorrelation of each edge. These experiments show that our Markov chain mixes quickly on these real graphs, allowing for utilization of our techniques in practice. Large scale real-world network data such as social and information networks are ubiquitous. The study of such networks seeks to find patterns and explain their emergence through tractable models. In most networks, and especially in social networks, nodes have a rich set of attributes associated with them. We present the Multiplicative Attribute Graphs (MAG) model, which naturally captures the interactions between the network structure and the node attributes. We consider a model where each node has a vector of categorical latent attributes associated with it. The probability of an edge between a pair of nodes depends on the product of individual attribute-attribute similarities. The model yields itself to mathematical analysis. We derive thresholds for the connectivity and the emergence of the giant connected component, and show that the model gives rise to networks with a constant diameter. We also show that MAG model can produce networks with either log-normal or power-law degree distributions. The emergence of order in natural systems is a constant source of inspiration for both physical and biological sciences. While the spatial order characterizing for example the crystals has been the basis of many advances in contemporary physics, most complex systems in nature do not offer such high degree of order. Many of these systems form complex networks whose nodes are the elements of the system and edges represent the interactions between them. Traditionally complex networks have been described by the random graph theory founded in 1959 by Paul Erdohs and Alfred Renyi. One of the defining features of random graphs is that they are statistically homogeneous, and their degree distribution (characterizing the spread in the number of edges starting from a node) is a Poisson distribution. In contrast, recent empirical studies, including the work of our group, indicate that the topology of real networks is much richer than that of random graphs. In particular, the degree distribution of real networks is a power-law, indicating a heterogeneous topology in which the majority of the nodes have a small degree, but there is a significant fraction of highly connected nodes that play an important role in the connectivity of the network. The scale-free topology of real networks has very important consequences on their functioning. For example, we have discovered that scale-free networks are extremely resilient to the random disruption of their nodes. On the other hand, the selective removal of the nodes with highest degree induces a rapid breakdown of the network to isolated subparts that cannot communicate with each other. The non-trivial scaling of the degree distribution of real networks is also an indication of their assembly and evolution. Indeed, our modeling studies have shown us that there are general principles governing the evolution of networks. Most networks start from a small seed and grow by the addition of new nodes which attach to the nodes already in the system. This process obeys preferential attachment: the new nodes are more likely to connect to nodes with already high degree. We have proposed a simple model based on these two principles wich was able to reproduce the power-law degree distribution of real networks. Perhaps even more importantly, this model paved the way to a new paradigm of network modeling, trying to capture the evolution of networks, not just their static topology.