Parallel Maximum Clique Algorithms with Applications to Network Analysis and Storage

We propose a fast, parallel maximum clique algorithm for large sparse graphs that is designed to exploit characteristics of social and information networks. The method exhibits a roughly linear runtime scaling over real-world networks ranging from 1000 to 100 million nodes. In a test on a social network with 1.8 billion edges, the algorithm finds the largest clique in about 20 minutes. Our method employs a branch and bound strategy with novel and aggressive pruning techniques. For instance, we use the core number of a vertex in combination with a good heuristic clique finder to efficiently remove the vast majority of the search space. In addition, we parallelize the exploration of the search tree. During the search, processes immediately communicate changes to upper and lower bounds on the size of maximum clique, which occasionally results in a super-linear speedup because vertices with large search spaces can be pruned by other processes. We apply the algorithm to two problems: to compute temporal strong components and to compress graphs.Comment: 11 page

Graph Compression for Adjacency-Matrix Multiplication

19 April 2022 A Correction to this paper has been published: https://doi.org/10.1007/s42979-022-01141-w[Abstract] Computing the product of the (binary) adjacency matrix of a large graph with a real-valued vector is an important operation that lies at the heart of various graph analysis tasks, such as computing PageRank. In this paper, we show that some well-known webgraph and social graph compression formats are computation-friendly, in the sense that they allow boosting the computation. We focus on the compressed representations of (a) Boldi and Vigna and (b) Hernández and Navarro, and show that the product computation can be conducted in time proportional to the compressed graph size. Our experimental results show speedups of at least 2 on graphs that were compressed at least 5 times with respect to the original.We thank Cecilia Hernández for providing us with her software extracting the bicliques, and a helpful description in how to run it. This research has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie [grant agreement No 690941], namely while the first author was visiting the University of Chile, and while the second author was affiliated with the University of Helsinki and visiting the University of A Coruña. The first author was funded by Fundação para a Ciência e a Tecnologia (FCT) [grant number UIDB/50021/2020 and PTDC/CCI-BIO/29676/2017]; the second author was funded by the Academy of Finland [Grant number 268324], Fondecyt [Grant number 1171058] and NSERC [Grant number RGPIN-07185-2020]; the third author was funded by JSPS KAKENHI [grant numbers JP21K17701 and JP21H05847]; the fourth author was funded by AEI and Ministerio de Ciencia e Innovación (PGE and FEDER) [grant number PID2019-105221RB-C41] and Xunta de Galicia (co-funded with FEDER) [Grant numbers ED431C 2021/53 and ED431G 2019/01]; and the fifth author was funded by ANID – Millennium Science Initiative Program – Code ICN17_002Xunta de Galicia; ED431C 2021/53Xunta de Galicia; ED431G 2019/0

Distance Preserving Graph Simplification

Large graphs are difficult to represent, visualize, and understand. In this paper, we introduce "gate graph" - a new approach to perform graph simplification. A gate graph provides a simplified topological view of the original graph. Specifically, we construct a gate graph from a large graph so that for any "non-local" vertex pair (distance higher than some threshold) in the original graph, their shortest-path distance can be recovered by consecutive "local" walks through the gate vertices in the gate graph. We perform a theoretical investigation on the gate-vertex set discovery problem. We characterize its computational complexity and reveal the upper bound of minimum gate-vertex set using VC-dimension theory. We propose an efficient mining algorithm to discover a gate-vertex set with guaranteed logarithmic bound. We further present a fast technique for pruning redundant edges in a gate graph. The detailed experimental results using both real and synthetic graphs demonstrate the effectiveness and efficiency of our approach.Comment: A short version of this paper will be published for ICDM'11, December 201

Finding events in temporal networks: Segmentation meets densest-subgraph discovery

International audienceIn this paper we study the problem of discovering a timeline of events in a temporal network. We model events as dense subgraphs that occur within intervals of network activity. We formulate the event-discovery task as an optimization problem, where we search for a partition of the network timeline into k non-overlapping intervals, such that the intervals span subgraphs with maximum total density. The output is a sequence of dense subgraphs along with corresponding time intervals, capturing the most interesting events during the network lifetime. A naïve solution to our optimization problem has polynomial but prohibitively high running time complexity. We adapt existing recent work on dynamic densest-subgraph discovery and approximate dynamic programming to design a fast approximation algorithm. Next, to ensure richer structure, we adjust the problem formulation to encourage coverage of a larger set of nodes. This problem is NP-hard even for static graphs. However, on static graphs a simple greedy algorithm leads to approximate solution due to submodularity. We extended this greedy approach for the case of temporal networks. However, the approximation guarantee does not hold. Nevertheless, according to the experiments, the algorithm finds good quality solutions

Finding events in temporal networks : segmentation meets densest subgraph discovery

In this paper, we study the problem of discovering a timeline of events in a temporal network. We model events as dense subgraphs that occur within intervals of network activity. We formulate the event discovery task as an optimization problem, where we search for a partition of the network timeline into k non-overlapping intervals, such that the intervals span subgraphs with maximum total density. The output is a sequence of dense subgraphs along with corresponding time intervals, capturing the most interesting events during the network lifetime. A naïve solution to our optimization problem has polynomial but prohibitively high running time. We adapt existing recent work on dynamic densest subgraph discovery and approximate dynamic programming to design a fast approximation algorithm. Next, to ensure richer structure, we adjust the problem formulation to encourage coverage of a larger set of nodes. This problem is NP-hard; however, we show that on static graphs a simple greedy algorithm leads to approximate solution due to submodularity. We extend this greedy approach for temporal networks, but we lose the approximation guarantee in the process. Finally, we demonstrate empirically that our algorithms recover solutions with good quality.Peer reviewe