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.
  We present a detailed study of network evolution by analyzing four large online social networks with full temporal information about node and edge arrivals. For the first time at such a large scale, we study individual node arrival and edge creation processes that collectively lead to macroscopic properties of networks. Using a methodology based on the maximum-likelihood principle, we investigate a wide variety of network formation strategies, and show that edge locality plays a critical role in evolution of networks. Our findings supplement earlier network models based on the inherently non-local preferential attachment. Based on our observations, we develop a complete model of network evolution, where nodes arrive at a prespecified rate and select their lifetimes. Each node then independently initiates edges according to a "gap" process, selecting a destination for each edge according to a simple triangle-closing model free of any parameters. We show analytically that the combination of the gap distribution with the node lifetime leads to a power law out-degree distribution that accurately reflects the true network in all four cases. Finally, we give model parameter settings that allow automatic evolution and generation of realistic synthetic networks of arbitrary scale. Systems as diverse as genetic networks or the World Wide Web are best described as networks with complex topology. A common property of many large networks is that the vertex connectivities follow a scale-free power-law distribution. This feature was found to be a consequence of two generic mechanisms: (i) networks expand continuously by the addition of new vertices, and (ii) new vertices attach preferentially to sites that are already well connected. A model based on these two ingredients reproduces the observed stationary scale-free distributions, which indicates that the development of large networks is governed by robust self-organizing phenomena that go beyond the particulars of the individual systems. The pages and hyperlinks of the World-Wide Web may be viewed as nodes and edges in a directed graph. This graph is a fascinating object of study: it has several hundred million nodes today, over a billion links, and appears to grow exponentially with time. There are many reasons -- mathematical, sociological, and commercial -- for studying the evolution of this graph. In this paper we begin by describing two algorithms that operate on the Web graph, addressing problems from Web search and automatic community discovery. We then report a number of measurements and properties of this graph that manifested themselves as we ran these algorithms on the Web. Finally, we observe that traditional random graph models do not explain these observations, and we propose a new family of random graph models. These models point to a rich new sub-field of the study of random graphs, and raise questions about the analysis of graph algorithms on the Web.  We analyze the social network emerging from the user comment activity on the website Slashdot. The network presents common features of traditional social networks such as a giant component, small average path length and high clustering, but differs from them showing moderate reciprocity and neutral assortativity by degree. Using Kolmogorov-Smirnov statistical tests, we show that the degree distributions are better explained by log-normal instead of power-law distributions. We also study the structure of discussion threads using an intuitive radial tree representation. Threads show strong heterogeneity and self-similarity throughout the different nesting levels of a conversation. We use these results to propose a simple measure to evaluate the degree of controversy provoked by a post. Access to realistic, complex graph datasets is critical to research on social networking systems and applications. Simulations on graph data provide critical evaluation of new systems and applications ranging from community detection to spam filtering and social web search. Due to the high time and resource costs of gathering real graph datasets through direct measurements, researchers are anonymizing and sharing a small number of valuable datasets with the community. However, performing experiments using shared real datasets faces three key disadvantages: concerns that graphs can be de-anonymized to reveal private information, increasing costs of distributing large datasets, and that a small number of available social graphs limits the statistical confidence in the results. The use of measurement-calibrated graph models is an attractive alternative to sharing datasets. Researchers can "fit" a graph model to a real social graph, extract a set of model parameters, and use them to generate multiple synthetic graphs statistically similar to the original graph. While numerous graph models have been proposed, it is unclear if they can produce synthetic graphs that accurately match the properties of the original graphs. In this paper, we explore the feasibility of measurement-calibrated synthetic graphs using six popular graph models and a variety of real social graphs gathered from the Facebook social network ranging from 30,000 to 3 million edges. We find that two models consistently produce synthetic graphs with common graph metric values similar to those of the original graphs. However, only one produces high fidelity results in our application-level benchmarks. While this shows that graph models can produce realistic synthetic graphs, it also highlights the fact that current graph metrics remain incomplete, and some applications expose graph properties that do not map to existing metrics. The linear preferential attachment hypothesis has been shown to be quite successful in explaining the existence of networks with power-law degree distributions. It is then quite important to determine if this mechanism is the consequence of a general principle based on local rules. In this work it is claimed that an effective linear preferential attachment is the natural outcome of growing network models based on local rules. It is also shown that the local models offer an explanation for other properties like the clustering hierarchy and degree correlations recently observed in complex networks. These conclusions are based on both analytical and numerical results for different local rules, including some models already proposed in the literature. How do real graphs evolve over time? What are "normal" growth patterns in social, technological, and information networks? Many studies have discovered patterns in static graphs, identifying properties in a single snapshot of a large network, or in a very small number of snapshots; these include heavy tails for in- and out-degree distributions, communities, small-world phenomena, and others. However, given the lack of information about network evolution over long periods, it has been hard to convert these findings into statements about trends over time.Here we study a wide range of real graphs, and we observe some surprising phenomena. First, most of these graphs densify over time, with the number of edges growing super-linearly in the number of nodes. Second, the average distance between nodes often shrinks over time, in contrast to the conventional wisdom that such distance parameters should increase slowly as a function of the number of nodes (like O(log n) or O(log(log n)).Existing graph generation models do not exhibit these types of behavior, even at a qualitative level. We provide a new graph generator, based on a "forest fire" spreading process, that has a simple, intuitive justification, requires very few parameters (like the "flammability" of nodes), and produces graphs exhibiting the full range of properties observed both in prior work and in the present study.   