We describe our work in the collection and analysis of massive data describing the connections between participants to online social networks. Alternative approaches to social network data collection are defined and evaluated in practice, against the popular Facebook Web site. Thanks to our ad-hoc, privacy-compliant crawlers, two large samples, comprising millions of connections, have been collected; the data is anonymous and organized as an undirected graph. We describe a set of tools that we developed to analyze specific properties of such social-network graphs, i.e., among others, degree distribution, centrality measures, scaling laws and distribution of friendship.
First, Leskovec and Faloutsos @cite_28 cover sampling techniques applied to graphs, exploiting several algorithms for efficiently visiting large graphs and avoiding bias of data. Several work covers aspects regarding crawling techniques for mining social networks: @cite_20 provide an exhaustive walk-through about this field.
@cite_28 @cite_20
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While data mining in chemoinformatics studied graph data with dozens of nodes, systems biology and the Internet are now generating graph data with thousands and millions of nodes. Hence data mining faces the algorithmic challenge of coping with this significant increase in graph size: Classic algorithms for data analysis are often too expensive and too slow on large graphs. While one strategy to overcome this problem is to design novel efficient algorithms, the other is to 'reduce' the size of the large graph by sampling. This is the scope of this paper: We will present novel Metropolis algorithms for sampling a 'representative' small subgraph from the original large graph, with 'representative' describing the requirement that the sample shall preserve crucial graph properties of the original graph. In our experiments, we improve over the pioneering work of Leskovec and Faloutsos (KDD 2006), by producing representative subgraph samples that are both smaller and of higher quality than those produced by other methods from the literature. Breadth First Search (BFS) and other graph traversal techniques are widely used for measuring large unknown graphs, such as online social networks. It has been empirically observed that incomplete BFS is biased toward high degree nodes. In contrast to more studied sampling techniques, such as random walks, the bias of BFS has not been characterized to date. In this paper, we quantify the degree bias of BFS sampling. In particular, we calculate the node degree distribution expected to be observed by BFS as a function of the fraction of covered nodes, in a random graph RG(p k ) with a given (and arbitrary) degree distribution p k . Furthermore, we also show that, for RG(p k ), all commonly used graph traversal techniques (BFS, DFS, Forest Fire, and Snowball Sampling) lead to the same bias, and we show how to correct for this bias. To give a broader perspective, we compare this class of exploration techniques to random walks that are well-studied and easier to analyze. Next, we study by simulation the effect of graph properties not captured directly by our model. We find that the bias gets amplified in graphs with strong positive assortativity. Finally, we demonstrate the above results by sampling the Facebook social network, and we provide some practical guidelines for graph sampling in practice. We pose a new and intriguing question motivated by distributed computing regarding random walks on graphs: How long does it take for several independent random walks, starting from the same vertex, to cover an entire graph? We study the cover time - the expected time required to visit every node in a graph at least once - and we show that for a large collection of interesting graphs, running many random walks in parallel yields a speed-up in the cover time that is linear in the number of parallel walks. We demonstrate that an exponential speed-up is sometimes possible, but that some natural graphs allow only a logarithmic speed-up. A problem related to ours (in which the walks start from some probablistic distribution on vertices) was previously studied in the context of space efficient algorithms for undirected s-t-connectivity and our results yield, in certain cases, an improvement upon some of the earlier bounds. Breadth First Search (BFS) is a widely used approach for sampling large graphs. However, it has been empirically observed that BFS sampling is biased toward high-degree nodes, which may strongly affect the measurement results. In this paper, we quantify and correct the degree bias of BFS. First, we consider a random graph RG(pk) with an arbitrary degree distribution pk. For this model, we calculate the node degree distribution expected to be observed by BFS as a function of the fraction f of covered nodes. We also show that, for RG(pk), all commonly used graph traversal techniques (BFS, DFS, Forest Fire, Snowball Sampling, RDS) have exactly the same bias. Next, we propose a practical BFS-bias correction procedure that takes as input a collected BFS sample together with the fraction f. Our correction technique is exact (i.e., leads to unbiased estimation) for RG(pk). Furthermore, it performs well when applied to a broad range of Internet topologies and to two large BFS samples of Facebook and Orkut networks. Various aspects of the theory of random walks on graphs are surveyed. In particular, estimates on the important parameters of access time, commute time, cover time and mixing time are discussed. Connections with the eigenvalues of graphs and with electrical networks, and the use of these connections in the study of random walks is described. We also sketch recent algorithmic applications of random walks, in particular to the problem of sampling. Our goal in this paper is to develop a practical framework for obtaining a uniform sample of users in an online social network (OSN) by crawling its social graph. Such a sample allows to estimate any user property and some topological properties as well. To this end, first, we consider and compare several candidate crawling techniques. Two approaches that can produce approximately uniform samples are the Metropolis-Hasting random walk (MHRW) and a re-weighted random walk (RWRW). Both have pros and cons, which we demonstrate through a comparison to each other as well as to the "ground truth." In contrast, using Breadth-First-Search (BFS) or an unadjusted Random Walk (RW) leads to substantially biased results. Second, and in addition to offline performance assessment, we introduce online formal convergence diagnostics to assess sample quality during the data collection process. We show how these diagnostics can be used to effectively determine when a random walk sample is of adequate size and quality. Third, as a case study, we apply the above methods to Facebook and we collect the first, to the best of our knowledge, representative sample of Facebook users. We make it publicly available and employ it to characterize several key properties of Facebook. This article deals with random walks on arbitrary graphs. We consider the cover time of finite graphs. That is, we study the expected time needed for a random walk on a finite graph to visit every vertex at least once. We establish an upper bound ofO(n2) for the expectation of the cover time for regular (or nearly regular) graphs. We prove a lower bound of Ω(n logn) for the expected cover time for trees. We present examples showing all our bounds to be tight. The analysis of large graphs plays a prominent role in various fields of research and is relevant in many important application areas. Effective visual analysis of graphs requires appropriate visual presentations in combination with respective user interaction facilities and algorithmic graph analysis methods. How to design appropriate graph analysis systems depends on many factors, including the type of graph describing the data, the analytical task at hand and the applicability of graph analysis methods. The most recent surveys of graph visualization and navigation techniques cover techniques that had been introduced until 2000 or concentrate only on graph layouts published until 2002. Recently, new techniques have been developed covering a broader range of graph types, such as timevarying graphs. Also, in accordance with ever growing amounts of graph-structured data becoming available, the inclusion of algorithmic graph analysis and interaction techniques becomes increasingly important. In this State-of-the-Art Report, we survey available techniques for the visual analysis of large graphs. Our review first considers graph visualization techniques according to the type of graphs supported. The visualization techniques form the basis for the presentation of interaction approaches suitable for visual graph exploration. As an important component of visual graph analysis, we discuss various graph algorithmic aspects useful for the different stages of the visual graph analysis process. We also present main open research challenges in this field. Efficient graph search is a central issue in many aspects of AI. In most of existing work there is a distinction between the active “searcher”, which both executes the algorithm and holds the memory, and the passive “searched graph”, over which the searcher has no control at all. Large dynamic networks like the Internet, where the nodes are powerful computers and the links have narrow bandwidth and are heavilydloaded, call for a different paradigm, in which most of the burden of computing and memorizing is moved from the searching agent to the nodes of the network. In this paper we suggest a method for searching an undirected, connected graph using the VertexdAntdWalk method, where an a(ge)nt walks along the edges of a graph G, occasionally leaving “pheromone” traces at nodes, and using those traces to guide its exploration. We show that the ant can cover the graph within time O (nd), where n is the number of vertices and d the diameter of G. The use of traces achieves a tradedoff between random and selfdavoiding walks, as it dictates a lower priority for alreadydvisited neighbors. Further properties of the suggested method are: (a) modularity: a group of searching agents, each applying the same protocol, can cooperate on a mission of covering a graph with minimal explicit communication between thems (b) possible convergence to a limit cycle: a Hamiltonian path in G (if one exists) is a possible limit cycle of the process. The cover time is the expected time it takes a random walk to cover all vertices of a graph. Despite the fact that it can be approximated with arbitrary precision by a simple polynomial time Monte-Carlo algorithm which simulates the random walk, it is not known whether the cover time of a graph can be computed in deterministic polynomial time. In the present paper we establish a deterministic polynomial time algorithm that, for any graph and any starting vertex, approximates the cover time within polylogarithmic factors. More generally, our algorithm approximates the cover time for arbitrary reversible Markov chains. The new aspect of our algorithm is that the starting vertex of the random walk may be arbitrary and is given as part of the input, whereas previous deterministic approximation algorithms for the cover time assume that the walk starts at the worst possible vertex. In passing, we show that the starting vertex can make a difference of up to a multiplicative factor of Θ(n3 2 √log n) in the cover time of an n-vertex graph. Extensive research has been conducted on top of online social networks (OSNs), while little attention has been paid to the data collection process. Due to the large scale of OSNs and their privacy control policies, a partial data set is often used for analysis. The data set analyzed is decided by many factors including the choice of seeds, node selection algorithms, and the sample size. These factors may introduce biases and further contaminate or even skew the results. To evaluate the impact of different factors, this paper examines the OSN graph crawling problem, where the nodes are OSN users and the edges are the links (or relationship) among these users. More specifically, by looking at various factors in the crawling process, the following problems are addressed in this paper:* Efficiency: How fast different crawlers discover nodes links;* Sensitivity: How different OSNs and the number of protected users affect crawlers;* Bias: How major graph properties are skewed.To the best of our knowledge, our simulations on four real world online social graphs provide the first in-depth empirical answers to these questions. With the increasing deployment and use of GPS-enabled devices, massive amounts of GPS data are becoming available. We propose a general framework for the mining of semantically meaningful, significant locations, e.g., shopping malls and restaurants, from such data. We present techniques capable of extracting semantic locations from GPS data. We capture the relationships between locations and between locations and users with a graph. Significance is then assigned to locations using random walks over the graph that propagates significance among the locations. In doing so, mutual reinforcement between location significance and user authority is exploited for determining significance, as are aspects such as the number of visits to a location, the durations of the visits, and the distances users travel to reach locations. Studies using up to 100 million GPS records from a confined spatio-temporal region demonstrate that the proposal is effective and is capable of outperforming baseline methods and an extension of an existing proposal. State-of-the-art techniques for probability sampling of users of online social networks (OSNs) are based on random walks on a single social relation (typically friendship). While powerful, these methods rely on the social graph being fully connected. Furthermore, the mixing time of the sampling process strongly depends on the characteristics of this graph. In this paper, we observe that there often exist other relations between OSN users, such as membership in the same group or participation in the same event. We propose to exploit the graphs these relations induce, by performing a random walk on their union multigraph. We design a computationally efficient way to perform multigraph sampling by randomly selecting the graph on which to walk at each iteration. We demonstrate the benefits of our approach through (i) simulation in synthetic graphs, and (ii) measurements of Last.fm- an Internet website for music with social networking features. More specifically, we show that multigraph sampling can obtain a representative sample and faster convergence, even when the individual graphs fail, i.e., are disconnected or highly clustered. In this work, we investigate the use of online or “crawling” algorithms to sample large social networks in order to determine the most influential or important individuals within the network (by varying definitions of network centrality). We describe a novel sampling technique based on concepts from expander graphs. We empirically evaluate this method in addition to other online sampling strategies on several real-world social networks. We find that, by sampling nodes to maximize the expansion of the sample, we are able to approximate the set of most influential individuals across multiple measures of centrality. Prediction is one of the most attractive aspects in data mining. Link prediction has recently attracted the attention of many researchers as an effective technique to be used in graph based models in general and in particular for social network analysis due to the recent popularity of the field. Link prediction helps to understand associations between nodes in social communities. Existing link prediction-related approaches described in the literature are limited to predict links that are anticipated to exist in the future. To the best of our knowledge, none of the previous works in this area has explored the prediction of links that could disappear in the future. We argue that the latter set of links are important to know about; they are at least equally important as and do complement the positive link prediction process in order to plan better for the future. In this paper, we propose a link prediction model which is capable of predicting both links that might exist and links that may disappear in the future. The model has been successfully applied in two different though very related domains, namely health care and gene expression networks. The former application concentrates on physicians and their interactions while the second application covers genes and their interactions. We have tested our model using different classifiers and the reported results are encouraging. Finally, we compare our approach with the internal links approach and we reached the conclusion that our approach performs very well in both bipartite and non-bipartite graphs. We study the cover time of multiple random walks. Given a graph G of n vertices, assume that k independent random walks start from the same vertex. The parameter of interest is the speed-up defined as the ratio between the cover time of one and the cover time of k random walks. Recently developed several bounds that are based on the quotient between the cover time and maximum hitting times. Their technique gives a speed-up of *** (k ) on many graphs, however, for many graph classes, k has to be bounded by @math . They also conjectured that, for any 1 ≤ k ≤ n , the speed-up is at most @math on any graph. As our main results, we prove the following: We present a new lower bound on the speed-up that depends on the mixing-time. It gives a speed-up of *** (k ) on many graphs, even if k is as large as n . We prove that the speed-up is @math on any graph. Under rather mild conditions, we can also improve this bound to @math , matching exactly the conjecture of We find the correct order of the speed-up for any value of 1 ≤ k ≤ n on hypercubes, random graphs and expanders. For d -dimensional torus graphs (d > 2), our bounds are tight up to a factor of @math . Our findings also reveal a surprisingly sharp dichotomy on several graphs (including d -dim. torus and hypercubes): up to a certain threshold the speed-up is k , while there is no additional speed-up above the threshold. In this work, we study the use of Twitter by House, Senate and gubernatorial candidates during the midterm (2010) elections in the U.S. Our data includes almost 700 candidates and over 690k documents that they produced and cited in the 3.5 years leading to the elections. We utilize graph and text mining techniques to analyze differences between Democrats, Republicans and Tea Party candidates, and suggest a novel use of language modeling for estimating content cohesiveness. Our findings show significant differences in the usage patterns of social media, and suggest conservative candidates used this medium more effectively, conveying a coherent message and maintaining a dense graph of connections. Despite the lack of party leadership, we find Tea Party members display both structural and language-based cohesiveness. Finally, we investigate the relation between network structure, content and election results by creating a proof-of-concept model that predicts candidate victory with an accuracy of 88.0 . With more than 250 million active users, Facebook (FB) is currently one of the most important online social networks. Our goal in this paper is to obtain a representative (unbiased) sample of Facebook users by crawling its social graph. In this quest, we consider and implement several candidate techniques. Two approaches that are found to perform well are the Metropolis-Hasting random walk (MHRW) and a re-weighted random walk (RWRW). Both have pros and cons, which we demonstrate through a comparison to each other as well as to the "ground-truth" (UNI - obtained through true uniform sampling of FB userIDs). In contrast, the traditional Breadth-First-Search (BFS) and Random Walk (RW) perform quite poorly, producing substantially biased results. In addition to offline performance assessment, we introduce online formal convergence diagnostics to assess sample quality during the data collection process. We show how these can be used to effectively determine when a random walk sample is of adequate size and quality for subsequent use (i.e., when it is safe to cease sampling). Using these methods, we collect the first, to the best of our knowledge, unbiased sample of Facebook. Finally, we use one of our representative datasets, collected through MHRW, to characterize several key properties of Facebook. Information visualization and data visualization are often viewed as similar, but distinct domains, and they have drawn an increasingly broad range of interest from diverse sectors of academia and industry. This study systematically analyzes and compares the intellectual landscapes of the two domains between 2000 and 2014. The present study is based on bibliographic records retrieved from the Web of Science. Using a topic search and a citation expansion, we collected two sets of data in each domain. Then, we identified emerging trends and recent developments in information visualization and data visualization, captivated in intellectual landscapes, landmark articles, bursting keywords, and citation trends of the domains. We found out that both domains have computer engineering and applications as their shared grounds. Our study reveals that information visualization and data visualization have scrutinized algorithmic concepts underlying the domains in their early years. Successive literature citing the datasets focuses on applying information and data visualization techniques to biomedical research. Recent thematic trends in the fields reflect that they are also diverging from each other. In data visualization, emerging topics and new developments cover dimensionality reduction and applications of visual techniques to genomics. Information visualization research is scrutinizing cognitive and theoretical aspects. In conclusion, information visualization and data visualization have co-evolved. At the same time, both fields are distinctively developing with their own scientific interests. A recurrent challenge for modern applications is the processing of large graphs. The ability to generate representative samples of smaller size is useful not only to circumvent scalability issues but also, per se, for statistical analysis and other data mining tasks. For such purposes adequate sampling techniques must be devised. We are interested, in this paper, in the uniform random sampling of a connected subgraph from a graph. We require that the sample contains a prescribed number of vertices. The sampled graph is the corresponding induced graph. We devise, present and discuss several algorithms that leverage three different techniques: Rejection Sampling, Random Walk and Markov Chain Monte Carlo. We empirically evaluate and compare the performance of the algorithms. We show that they are effective and efficient but that there is a trade-off, which depends on the density of the graphs and the sample size. We propose one novel algorithm, which we call Neighbour Reservoir Sampling (NRS), that very successfully realizes the trade-off between effectiveness and efficiency.