Core Decomposition in Multilayer Networks: Theory, Algorithms, and Applications

Multilayer networks are a powerful paradigm to model complex systems, where multiple relations occur between the same entities. Despite the keen interest in a variety of tasks, algorithms, and analyses in this type of network, the problem of extracting dense subgraphs has remained largely unexplored so far. In this work we study the problem of core decomposition of a multilayer network. The multilayer context is much challenging as no total order exists among multilayer cores; rather, they form a lattice whose size is exponential in the number of layers. In this setting we devise three algorithms which differ in the way they visit the core lattice and in their pruning techniques. We then move a step forward and study the problem of extracting the inner-most (also known as maximal) cores, i.e., the cores that are not dominated by any other core in terms of their core index in all the layers. Inner-most cores are typically orders of magnitude less than all the cores. Motivated by this, we devise an algorithm that effectively exploits the maximality property and extracts inner-most cores directly, without first computing a complete decomposition. Finally, we showcase the multilayer core-decomposition tool in a variety of scenarios and problems. We start by considering the problem of densest-subgraph extraction in multilayer networks. We introduce a definition of multilayer densest subgraph that trades-off between high density and number of layers in which the high density holds, and exploit multilayer core decomposition to approximate this problem with quality guarantees. As further applications, we show how to utilize multilayer core decomposition to speed-up the extraction of frequent cross-graph quasi-cliques and to generalize the community-search problem to the multilayer setting

Cores and Other Dense Structures in Complex Networks

Complex networks are a powerful paradigm to model complex systems. Specific network models, e.g., multilayer networks, temporal networks, and signed networks, enrich the standard network representation with additional information to better capture real-world phenomena. Despite the keen interest in a variety of problems, algorithms, and analysis methods for these types of network, the problem of extracting cores and dense structures still has unexplored facets. In this work, we present advancements to the state of the art by the introduction of novel definitions and algorithms for the extraction of dense structures from complex networks, mainly cores. At first, we define core decomposition in multilayer networks together with a series of applications built on top of it, i.e., the extraction of maximal multilayer cores only, densest subgraph in multilayer networks, the speed-up of the extraction of frequent cross-graph quasi-cliques, and the generalization of community search to the multilayer setting. Then, we introduce the concept of core decomposition in temporal networks; also in this case, we are interested in the extraction of maximal temporal cores only. Finally, in the context of discovering polarization in large-scale online data, we study the problem of identifying polarized communities in signed networks. The proposed methodologies are evaluated on a large variety of real-world networks against na\"{\i}ve approaches, non-trivial baselines, and competing methods. In all cases, they show effectiveness, efficiency, and scalability. Moreover, we showcase the usefulness of our definitions in concrete applications and case studies, i.e., the temporal analysis of contact networks, and the identification of polarization in debate networks.Comment: arXiv admin note: text overlap with arXiv:1812.0871

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

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

Frequent Subgraph Mining via Sampling with Rigorous Guarantees

Frequent subgraph mining is a fundamental task in the analysis of collections of graphs that aims at finding all the subgraphs that appear with more than a user-specified frequency in the dataset. While several exact approaches have been proposed to solve the task, it remains computationally challenging on large graph datasets due to the complexity of the subgraph isomorphism problem inherent in the task and the huge number of candidate patterns even for fairly small subgraphs. In this thesis, we study two statistical learning measures of complexity, VC-dimension and Rademacher averages, for subgraphs, and derive efficiently computable bounds for both. We then show how such bounds can be applied to devise efficient sampling-based approaches for rigorously approximating the solutions of the frequent subgraph mining problem, providing sample sizes which are much tighter than what would be obtained by a straightforward application of Chernoff and union bounds. We also show that our bounds can be used for true frequent subgraph mining, which requires to identify subgraphs generated with probability above a given threshold using samples from an unknown generative process. Moreover, we carried out an extensive experimental evaluation of our methods on real datasets, which shows that our bounds lead to efficiently computable and high-quality approximations for both applications.Frequent subgraph mining is a fundamental task in the analysis of collections of graphs that aims at finding all the subgraphs that appear with more than a user-specified frequency in the dataset. While several exact approaches have been proposed to solve the task, it remains computationally challenging on large graph datasets due to the complexity of the subgraph isomorphism problem inherent in the task and the huge number of candidate patterns even for fairly small subgraphs. In this thesis, we study two statistical learning measures of complexity, VC-dimension and Rademacher averages, for subgraphs, and derive efficiently computable bounds for both. We then show how such bounds can be applied to devise efficient sampling-based approaches for rigorously approximating the solutions of the frequent subgraph mining problem, providing sample sizes which are much tighter than what would be obtained by a straightforward application of Chernoff and union bounds. We also show that our bounds can be used for true frequent subgraph mining, which requires to identify subgraphs generated with probability above a given threshold using samples from an unknown generative process. Moreover, we carried out an extensive experimental evaluation of our methods on real datasets, which shows that our bounds lead to efficiently computable and high-quality approximations for both applications

Quantum-Assisted Learning of Hardware-Embedded Probabilistic Graphical Models

Mainstream machine-learning techniques such as deep learning and probabilistic programming rely heavily on sampling from generally intractable probability distributions. There is increasing interest in the potential advantages of using quantum computing technologies as sampling engines to speed up these tasks or to make them more effective. However, some pressing challenges in state-of-the-art quantum annealers have to be overcome before we can assess their actual performance. The sparse connectivity, resulting from the local interaction between quantum bits in physical hardware implementations, is considered the most severe limitation to the quality of constructing powerful generative unsupervised machine-learning models. Here we use embedding techniques to add redundancy to data sets, allowing us to increase the modeling capacity of quantum annealers. We illustrate our findings by training hardware-embedded graphical models on a binarized data set of handwritten digits and two synthetic data sets in experiments with up to 940 quantum bits. Our model can be trained in quantum hardware without full knowledge of the effective parameters specifying the corresponding quantum Gibbs-like distribution; therefore, this approach avoids the need to infer the effective temperature at each iteration, speeding up learning; it also mitigates the effect of noise in the control parameters, making it robust to deviations from the reference Gibbs distribution. Our approach demonstrates the feasibility of using quantum annealers for implementing generative models, and it provides a suitable framework for benchmarking these quantum technologies on machine-learning-related tasks.Comment: 17 pages, 8 figures. Minor further revisions. As published in Phys. Rev.

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