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A graph contains plentiful structures. Some minority structures are important, such as high degree nodes and bridges. Detecting these minority structures is beneficial to accelerate computational graph analysis and improve the comprehension of graph visualization. Regarding four typical minority structures, this paper proposes two algorithms to detect these structures fast and efficiently. A set of experiments demonstrate the effectiveness of the proposed algorithms.

Graph processing has become a topic of interest in many domains. However, we still observe a lack of representative datasets for in-depth performance and scalability analysis. Neither data collections, nor graph generators provide enough diversity and control for thorough analysis. To address this problem, we proposea heuristic method for scaling existing graphs. Our approach, based onsampling andinterconnection, can provide a scaled "version" of a given graph. Moreover, we provide analytical models to predict the topological properties of the scaled graphs (such as the diameter, degree distribution, density, or the clustering coefficient), and further enable the user to tweak these properties. Property control is achieved through a portfolio of graph interconnection methods (e.g., star, ring, chain, fully connected) applied for combining the graph samples. We further implement our method as an open-source tool which can be used to quickly provide families of datasets for in-depth benchmarking of graph processing algorithms. Our empirical evaluation demonstrates our tool provides scaled graphs of a wide range of sizes, whose properties match well with model predictions and/or user requirements. Finally, we also illustrate, through a case-study, how scaled graphs can be used for in-depth performance analysis of graph processing algorithms.

Massive Sequence View(MSV) is an important timeline-based technique for dynamic network visualization. However, it often suffers from severe visual clutter when limited screen space holds excessive network edges. Inspired by the use of graph sampling in static graph analysis, we propose to utilize graph sampling to reduce visual clutter in MSV. An edge sampling method based on accept-reject random sampling is designed for visualizing dynamic network via MSV. The method is able to improve the overall readability of MSV while preserving time varying network behaviors. It is also a preliminary attempt to apply graph-sampling technique into dynamic network analysis.

Many graph datasets originating from online social network, financial or biological sources are too large to store or analyze. The analysis of such networks may be made more tractable if they are reduced to smaller subgraphs via sampling. While most of the known graph sampling methods are designed with static graphs in mind, many real datasets are massive and rapidly growing, making streaming methods necessary. We present two new techniques, Randomly Induced Edge Sampling (RIES) and Weighted Edge Sampling (WES). Both methods sample a stream of edges in a single pass, without the need to know future properties of the stream. In contrast to previous work that focused on limiting only the number of vertices, our methods restrict the number of edges, thus truly limiting the size of the sampled subgraph. We compare the performance of RIES and WES against the previously known streaming Random Edge (RE) method on eight social network datasets. Using four structural graph properties, we find that both RIES and WES produce subgraphs that are more structurally similar to the original graph than are the subgraphs produced by streaming RE. We also examine the sensitivity of the two algorithms with respect to their parameters. The parameters of WES affect its performance in a more predictable manner and are easier to set. Both new algorithms represent an improvement in the available streaming graph analysis toolkit.

Graph matching is a generalization of the classic graph isomorphism problem. By using only their structures a graph-matching algorithm finds a map between the vertex sets of two similar graphs. This has applications in the de-anonymization of social and information networks and, more generally, in the merging of structural data from different domains. One class of graph-matching algorithms starts with a known seed set of matched node pairs. Despite the success of these algorithms in practical applications, their performance has been observed to be very sensitive to the size of the seed set. The lack of a rigorous understanding of parameters and performance makes it difficult to design systems and predict their behavior. In this paper, we propose and analyze a very simple percolation $\!$-based graph matching algorithm that incrementally maps every pair of nodes $(i,j)$ with at least $r$ neighboring mapped pairs. The simplicity of this algorithm makes possible a rigorous analysis that relies on recent advances in bootstrap percolation theory for the $G(n,p)$ random graph. We prove conditions on the model parameters in which percolation graph matching succeeds, and we establish a phase transition in the size of the seed set. We also confirm through experiments that the performance of percolation graph matching is surprisingly good, both for synthetic graphs and real social-network data.

Many online networks are measured and studied via sampling techniques, which typically collect a relatively small fraction of nodes and their associated edges. Past work in this area has primarily focused on obtaining a representative sample of nodes and on efficient estimation of local graph properties (such as node degree distribution or any node attribute) based on that sample. However, less is known about estimating the global topology of the underlying graph. In this paper, we show how to efficiently estimate the coarse-grained topology of a graph from a probability sample of nodes. In particular, we consider that nodes are partitioned into categories (e.g., countries or work/study places in OSNs), which naturally defines a weighted category graph. We are interested in estimating (i) the size of categories and (ii) the probability that nodes from two different categories are connected. For each of the above, we develop a family of estimators for design-based inference under uniform or non-uniform sampling, employing either of two measurement strategies: induced subgraph sampling, which relies only on information about the sampled nodes; and star sampling, which also exploits category information about the neighbors of sampled nodes. We prove consistency of these estimators and evaluate their efficiency via simulation on fully known graphs. We also apply our methodology to a sample of Facebook users to obtain a number of category graphs, such as the college friendship graph and the country friendship graph; we share and visualize the resulting data at www.geosocialmap.com.

The proliferation of online social networks, and the concomitant accumulation of user data, give rise to hotly debated issues of privacy, security, and control. One specific challenge is the sharing or public release of anonymized data without accidentally leaking personally identifiable information (PII). Unfortunately, it is often difficult to ascertain that sophisticated statistical techniques, potentially employing additional external data sources, are unable to break anonymity. In this paper, we consider an instance of this problem, where the object of interest is the structure of a social network, i.e., a graph describing users and their links. Recent work demonstrates that anonymizing node identities may not be sufficient to keep the network private: the availability of node and link data from another domain, which is correlated with the anonymized network, has been used to re-identify the anonymized nodes. This paper is about conditions under which such a de-anonymization process is possible. We attempt to shed light on the following question: can we assume that a sufficiently sparse network is inherently anonymous, in the sense that even with unlimited computational power, deanonymization is impossible? Our approach is to introduce a random graph model for a version of the de-anonymization problem, which is parameterized by the expected node degree and a similarity parameter that controls the correlation between two graphs over the same vertex set. We find simple conditions on these parameters delineating the boundary of privacy, and show that the mean node degree need only grow slightly faster than log n with network size n for nodes to be identifiable. Our results have policy implications for sharing of anonymized network information.

Frequent subgraph mining has always been an important issue in data mining. Several frequent graph mining methods have been developed for mining graph transactions. However, these methods become less usable when the dataset is a single large graph. Also, when the graph is too large to fit in main memory, alternative techniques are necessary to efficiently find frequent subgraphs. We investigate the task of frequent subgraph mining on a single large graph using sampling approaches and find that sampling is a feasible approach for this task. We evaluate different sampling methods and provide a novel sampling method called 'random areas selection sampling', which produces better results than all the current graph sampling approaches with customized parameters.

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.