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23
Visual exploration of complex timevarying graphs

, 2006
"... Many graph drawing and visualization algorithms, such as forcedirected layout and linedot rendering, work very well on relatively small and sparse graphs. However, they often produce extremely tangled results and exhibit impractical running times for highly nonplanar graphs with large edge dens ..."
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Cited by 43 (1 self)
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Many graph drawing and visualization algorithms, such as forcedirected layout and linedot rendering, work very well on relatively small and sparse graphs. However, they often produce extremely tangled results and exhibit impractical running times for highly nonplanar graphs with large edge density. And very few graph layout algorithms support dynamic timevarying graphs; applying them independently to each frame produces distracting temporally incoherent visualizations. We have developed a new visualization technique based on a novel approach to hierarchically structuring dense graphs via stratification. Using this structure, we formulate a hierarchical forcedirected layout algorithm that is both efficient and produces quality graph layouts. The stratification of the graph also allows us to present views of the data that abstract away many small details of its structure. Rather than displaying all edges and nodes at once, resulting in a convoluted rendering, we present an interactive tool that filters edges and nodes using the graph hierarchy and allows users to drill down into the graph for details. Our layout algorithm also accommodates timevarying graphs in a natural way, producing a temporally coherent animation that can be used to analyze and extract trends from dynamic graph data. For example, we demonstrate the use of our method to explore financial correlation data for the U.S. stock market in the period from 1990 to 2005. The user can easily analyze the timevarying correlation graph of the market, uncovering information such as market sector trends, representative stocks for portfolio construction, and the interrelationship of stocks over time.
Visualization of social and other scalefree networks
 IN PROC. OF IEEE INFOVIS
, 2008
"... This paper proposes novel methods for visualizing specifically the large powerlaw graphs that arise in sociology and the sciences. In such cases a large portion of edges can be shown to be less important and removed while preserving component connectedness and other features (e.g. cliques) to more ..."
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Cited by 26 (1 self)
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This paper proposes novel methods for visualizing specifically the large powerlaw graphs that arise in sociology and the sciences. In such cases a large portion of edges can be shown to be less important and removed while preserving component connectedness and other features (e.g. cliques) to more clearly reveal the network’s underlying connection pathways. This simplification approach deterministically filters (instead of clustering) the graph to retain important node and edge semantics, and works both automatically and interactively. The improved graph filtering and layout is combined with a novel computer graphics anisotropic shading of the dense crisscrossing array of edges to yield a full social network and scalefree graph visualization system. Both quantitative analysis and visual results demonstrate the effectiveness of this approach.
Finding Maximal Cliques in Massive Networks by H*graph
"... Maximal clique enumeration (MCE) is a fundamental problem in graph theory and has important applications in many areas such as social network analysis and bioinformatics. The problem is extensively studied; however, the best existing algorithms require memory space linear in the size of the input gr ..."
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Cited by 26 (14 self)
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Maximal clique enumeration (MCE) is a fundamental problem in graph theory and has important applications in many areas such as social network analysis and bioinformatics. The problem is extensively studied; however, the best existing algorithms require memory space linear in the size of the input graph. This has become a serious concern in view of the massive volume of today’s fastgrowing network graphs. Since MCE requires random access to different parts of a large graph, it is difficult to divide the graph into smaller parts and process one part at a time, because either the result may be incorrect and incomplete, or it incurs huge cost on merging the results from different parts. We propose a novel notion, H ∗graph, which defines the core of a network and extends to encompass the neighborhood of the core for MCE computation. We propose the first externalmemory algorithm for MCE (ExtMCE) that uses theH ∗graph to bound the memory usage. We prove both the correctness and completeness of the result computed by ExtMCE. Extensive experiments verify that ExtMCE efficiently processes large networks that cannot be fit in the memory. We also show that the H ∗graph captures important properties of the network; thus, updating the maximal cliques in the H ∗graph retains the most essential information, with a low update cost, when it is infeasible to perform update on the entire network.
Efficient Core Decomposition in Massive Networks
"... Abstract—The kcore of a graph is the largest subgraph in which every vertex is connected to at least k other vertices within the subgraph. Core decomposition finds the kcore of the graph for every possible k. Past studies have shown important applications of core decomposition such as in the study ..."
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Cited by 21 (4 self)
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Abstract—The kcore of a graph is the largest subgraph in which every vertex is connected to at least k other vertices within the subgraph. Core decomposition finds the kcore of the graph for every possible k. Past studies have shown important applications of core decomposition such as in the study of the properties of large networks (e.g., sustainability, connectivity, centrality, etc.), for solving NPhard problems efficiently in real networks (e.g., maximum clique finding, densest subgraph approximation, etc.), and for largescale network fingerprinting and visualization. The kcore is a well accepted concept partly because there exists a simple and efficient algorithm for core decomposition, by recursively removing the lowest degree vertices and their incident edges. However, this algorithm requires random access to the graph and hence assumes the entire graph can be kept in main memory. Nevertheless, realworld networks such as online social networks have become exceedingly large in recent years and still keep growing at a steady rate. In this paper, we propose the first externalmemory algorithm for core decomposition in massive graphs. When the memory is large enough to hold the graph, our algorithm achieves comparable performance as the inmemory algorithm. When the graph is too large to be kept in the memory, our algorithm requires only O(kmax) scans of the graph, where kmax is the largest core number of the graph. We demonstrate the efficiency of our algorithm on real networks with up to 52.9 million vertices and 1.65 billion edges. I.
Finding maximal cliques in massive networks
 ACM Trans. Database Syst
"... Maximal clique enumeration is a fundamental problem in graph theory and has important applications in many areas such as social network analysis and bioinformatics. The problem is extensively studied; however, the best existing algorithms require memory space linear in the size of the input graph. T ..."
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Cited by 13 (5 self)
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Maximal clique enumeration is a fundamental problem in graph theory and has important applications in many areas such as social network analysis and bioinformatics. The problem is extensively studied; however, the best existing algorithms require memory space linear in the size of the input graph. This has become a serious concern in view of the massive volume of today’s fastgrowing networks. We propose a general framework for designing externalmemory algorithms for maximal clique enumeration in large graphs. The general framework enables maximal clique enumeration to be processed recursively in small subgraphs of the input graph, thus allowing inmemory computation of maximal cliques without the costly random disk access. We prove that the set of cliques obtained by the recursive local computation is both correct (i.e., globally maximal) and complete. The subgraph to be processed each time is defined based on a set of base vertices that can be flexibly chosen to achieve different purposes. We discuss the selection of the base vertices to fully utilize the available memory in order to minimize I/O cost in static graphs, and for update maintenance in dynamic graphs. We also apply our framework to design an externalmemory algorithm for maximum clique computation in a large graph.
Fast algorithms for maximal clique enumeration with limited memory
 In Proceedings of the ACM SIGKDD international
, 2012
"... Maximal clique enumeration (MCE) is a longstanding problem in graph theory and has numerous important applications. Though extensively studied, most existing algorithms become impractical when the input graph is too large and is diskresident. We first propose an efficient partitionbased algorith ..."
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Cited by 8 (5 self)
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Maximal clique enumeration (MCE) is a longstanding problem in graph theory and has numerous important applications. Though extensively studied, most existing algorithms become impractical when the input graph is too large and is diskresident. We first propose an efficient partitionbased algorithm for MCE that addresses the problem of processing large graphs with limited memory. We then further reduce the high cost of CPU computation of MCE by a careful nested partition based on a cost model. Finally, we parallelize our algorithm to further reduce the overall running time. We verified the efficiency of our algorithms by experiments in large realworld graphs.
Fast algorithms for the maximum clique problem on massive sparse graphs. arXiv preprint arXiv:1209.5818v2
, 2012
"... Abstract The maximum clique problem is a well known NPHard problem with applications in data mining, network analysis, information retrieval and many other areas related to the World Wide Web. There exist several algorithms for the problem with acceptable runtimes for certain classes of graphs, bu ..."
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Cited by 4 (3 self)
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Abstract The maximum clique problem is a well known NPHard problem with applications in data mining, network analysis, information retrieval and many other areas related to the World Wide Web. There exist several algorithms for the problem with acceptable runtimes for certain classes of graphs, but many of them are infeasible for massive graphs. We present a new exact algorithm that employs novel pruning techniques and is able to find maximum cliques in very large, sparse graphs quickly. Extensive experiments on different kinds of synthetic and realworld graphs show that our new algorithm can be orders of magnitude faster than existing algorithms. We also present a heuristic that runs orders of magnitude faster than the exact algorithm while providing optimal or nearoptimal solutions. We illustrate a simple application of the algorithms in developing methods for detection of overlapping communities in networks.
Statistical Properties of the Foreign Exchange Network at Different Time Scales: Evidence from Detrended CrossCorrelation Coefficient and Minimum Spanning Tree
 ENTROPY
, 2013
"... ..."
Detecting blackholes and volcanoes in directed networks
 In ICDM
, 2010
"... In this paper, we formulate a novel problem for finding blackhole and volcano patterns in a large directed graph. Specifically, a blackhole pattern is a group which is made of a set of nodes in a way such that there are only inlinks to this group from the rest nodes in the graph. In contrast, a volc ..."
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Cited by 3 (0 self)
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In this paper, we formulate a novel problem for finding blackhole and volcano patterns in a large directed graph. Specifically, a blackhole pattern is a group which is made of a set of nodes in a way such that there are only inlinks to this group from the rest nodes in the graph. In contrast, a volcano pattern is a group which only has outlinks to the rest nodes in the graph. Both patterns can be observed in real world. For instance, in a trading network, a blackhole pattern may represent a group of traders who are manipulating the market. In the paper, we first prove that the blackhole mining problem is a dual problem of finding volcanoes. Therefore, we focus on finding the blackhole patterns. Along this line, we design two pruning schemes to guide the blackhole finding process. In the first pruning scheme, we strategically prune the search space based on a set of patternsizeindependent pruning rules and develop an iBlackhole algorithm. The second pruning scheme follows a divideandconquer strategy to further exploit the pruning results from the first pruning scheme. Indeed, a target directed graphs can be divided into several disconnected subgraphs by the first pruning scheme, and thus the blackhole finding can be conducted in each disconnected subgraph rather than in a large graph. Based on these two pruning schemes, we also develop an iBlackholeDC algorithm. Finally, experimental results on realworld data show that the iBlackholeDC algorithm can be several orders of magnitude faster than the iBlackhole algorithm, which has a huge computational advantage over a bruteforce method.
Enumerating Isolated Cliques in Synthetic and Financial Networks
 PROC. 2ND COCOA
"... We do computational studies concerning the enumeration of maximal isolated cliques in graphs. Isolation, as recently introduced, measures the degree of connectedness of the cliques to the rest of the graph. Isolation helps both in getting faster algorithms than for the enumeration of maximal general ..."
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Cited by 2 (2 self)
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We do computational studies concerning the enumeration of maximal isolated cliques in graphs. Isolation, as recently introduced, measures the degree of connectedness of the cliques to the rest of the graph. Isolation helps both in getting faster algorithms than for the enumeration of maximal general cliques and in filtering out cliques with special semantics. We perform experiments with synthetic graphs (in the Gn,m,p model) and financial networks, proposing the enumeration of isolated cliques as a useful instrument in analyzing financial networks.