The Maximum Clique Enumeration Problem: Algorithms, Applications and Implementations

Graph-theoretical approaches to biological network analysis have proven to be effective for small networks but are computationally infeasible for comprehensive genome-scale systems-level elucidation of these networks. The difficulty lies in the NP-hard nature of many global systems biology problems that, in practice, translates to exponential (or worse) run times for finding exact optimal solutions. Moreover, these problems, especially those of an enumerative flavor, are often memory-intensive and must share very large sets of data effectively across many processors. For example, the enumeration of maximal cliques - a core component in gene expression networks analysis, cis regulatory motif finding, and the study of quantitative trait loci for high-throughput molecular phenotypes can result in as many as 3^n/3 maximal cliques for a graph with n vertices. Memory requirements to store those cliques reach terabyte scales even on modest-sized genomes. Emerging hardware architectures with ultra-large globally addressable memory such as the SGI Altix and Cray X1 seem to be well suited for addressing these types of data-intensive problems in systems biology. This paper presents a novel framework that provides exact, parallel and scalable solutions to various graph-theoretical approaches to genome-scale elucidation of biological networks. This framework takes advantage of these large-memory architectures by creating globally addressable bitmap memory indices with potentially high compression rates, fast bitwise-logical operations, and reduced search space. Augmented with recent theoretical advancements based on fixed-parameter tractability, this framework produces computationally feasible performance for genome-scale combinatorial problems of systems biology.

Clique is one of the most fundamental models for cohesive subgraph mining in network analysis. Existing clique model mainly focuses on unsigned networks. In real world, however, many applications are modeled as signed networks with positive and negative edges. As the signed networks hold their own properties different from the unsigned networks, the existing clique model is inapplicable for the signed networks. Motivated by this, we propose the balanced clique model that considers the most fundamental and dominant theory, structural balance theory, for signed networks, and study the maximal balanced clique enumeration problem which computes all the maximal balanced cliques in a given signed network. We show that the maximal balanced clique enumeration problem is NP-Hard. A straightforward solution for the maximal balanced clique enumeration problem is to treat the signed network as two unsigned networks and leverage the off-the-shelf techniques for unsigned networks. However, such a solution is inefficient for large signed networks. To address this problem, in this paper, we first propose a new maximal balanced clique enumeration algorithm by exploiting the unique properties of signed networks. Based on the new proposed algorithm, we devise two optimization strategies to further improve the efficiency of the enumeration. We conduct extensive experiments on large real and synthetic datasets. The experimental results demonstrate the efficiency, effectiveness and scalability of our proposed algorithms.

Maximal clique enumeration is a fundamental problem in graph theory and has been extensively studied. However, maximal clique enumeration is time-consuming in large graphs and always returns enormous cliques with large overlaps. Motivated by this, in this paper, we study the diversified top-k clique search problem which is to find top-k cliques that can cover most number of nodes in the graph. Diversified top-k clique search can be widely used in a lot of applications including community search, motif discovery, and anomaly detection in large graphs. A naive solution for diversified top-k clique search is to keep all maximal cliques in memory and then find k of them that cover most nodes in the graph by using the approximate greedy max k-cover algorithm. However, such a solution is impractical when the graph is large. In this paper, instead of keeping all maximal cliques in memory, we devise an algorithm to maintain k candidates in the process of maximal clique enumeration. Our algorithm has limited memory footprint and can achieve a guaranteed approximation ratio. We also introduce a novel light-weight $$\mathsf {PNP}$$PNP-$$\mathsf {Index}$$Index, based on which we design an optimal maximal clique maintenance algorithm. We further explore three optimization strategies to avoid enumerating all maximal cliques and thus largely reduce the computational cost. Besides, for the massive input graph, we develop an I/O efficient algorithm to tackle the problem when the input graph cannot fit in main memory. We conduct extensive performance studies on real graphs and synthetic graphs. One of the real graphs contains 1.02 billion edges. The results demonstrate the high efficiency and effectiveness of our approach.

Here we study a variant of maximal clique enumeration problem by incorporating a minimum size criterion. We describe preprocessing techniques to reduce the graph size. This is of practical interest since enumerating maximal cliques is a computationally hard problem and the execution time increases rapidly with the input size. We discuss basics of an algorithm for enumerating large maximal cliques which exploits the constraint on minimum size of the desired maximal cliques. Social networks are prime examples of large sparse graphs where enumerating large maximal cliques is of interest. We present experimental results on the social network formed by the call detail records of one of the world's largest telecom service providers. Our results show that the preprocessing methods achieve significant reduction in the graph size. We also characterize the execution behaviour of our large maximal clique enumeration algorithm.

As a fundamental task in graph data management, maximal clique enumeration (MCE) has attracted extensive attention from both academic and industrial communities due to its wide range of applications. However, MCE is very challenging as the number of maximal cliques may grow exponentially with the number of vertices. The state-of-the-art methods adopt a recursive paradigm to enumerate maximal cliques exhaustively, suffering from a large amount of redundant computation. In this paper, we propose a novel reduction-based framework for MCE, namely RMCE, that aims to reduce the search space and minimize unnecessary computations. The proposed framework RMCE incorporates three kinds of powerful reduction techniques including global reduction, dynamic reduction, and maximality check reduction. Global and dynamic reduction techniques effectively reduce the size of the input graph and dynamically construct subgraphs during the recursive sub-tasks, respectively. The maximality check reduction minimizes the computation for ensuring maximality by utilizing neighborhood dominance between visited vertices. Extensive experiments on 18 real graphs demonstrate that the existing approaches achieve remarkable speedups powered by the proposed techniques.

The Maximal Clique Enumeration problem (MCE) is a graph problem encountered in many applications such as social network analysis and computational biology. However, this problem is difficult and requires exponential time. Consequently, appropriate solutions must be proposed in the case of massive graph databases. In this paper, we investigate and evaluate an approach that deals with this problem on a compressed version of the graphs. This approach is interesting because compression is a staple of massive data processing. We mainly show, through extensive experimentations, that besides reducing the size of the graphs, this approach enhances the efficiency of existing algorithms.

Characterizing biological pathways at the genome scale is one of the most important and challenging tasks in the post genomic era. To address this challenge, we have developed a computational method to systematically and automatically derive partial biological pathways in yeast using high-throughput biological data, including yeast two hybrid data, protein complexes identified from mass spectroscopy, genetics interactions, and microarray gene expression data in yeast Saccharomyces cerevisiae. The inputs of the method are the upstream starting protein (e.g., a sensor of a signal) and the downstream terminal protein (e.g., a transcriptional factor that induces genes to respond the signal); the output of the method is the protein interaction chain between the two proteins. The high-throughput data are coded into a graph of interaction network, where each node represents a protein. The weight of an edge between two nodes models the closeness of the two represented proteins in the interaction network and it is defined by a rule-based formula according to the high-throughput data and modified by the protein function classification and subcellular localization information. The protein interaction cascade pathway in vivo is predicted as the shortest path identified from the graph of the interaction network using Dijkstra's algorithm. We have also developed a web server of this method (http://compbio.ornl.gov/structure/pathway) for public use. To our knowledge, our method is the first automated method to generally construct partial biological pathways using a suite of high-throughput biological data. This work demonstrates the proof of principle using computational approaches for discoveries of biological pathways with high-throughput data and biological annotation data.

Social network analysis (SNA) is a set of broadly used techniques designed for analyzing structural information contained in interactions. However, current SNA tools have a poor ability to handle large and dynamic social networks. One particular problem of interest is that of maximal clique enumeration used for studying modularity/community. Critical challenges for this problem include limited scalability and poor ability for handling dynamism. In this paper, we design and implement an anytime anywhere approach for maximal clique enumeration problem in SNA. Through a set of experiments on random graphs, we validate and demonstrate the effectiveness and efficiency of our approach.

This thesis delves into critical challenges associated with subgraph analysis in large-scale graphs. The proliferation of data in domains such as social networks, Internet applications, and biological networks necessitates a deep understanding of relationships modeled as graphs. The complexity and scale of these graphs pose challenges for efficient extraction of meaningful information. The thesis focuses on three interconnected research problems: subgraph counting in fully-dynamic graph streams, k-clique listing in large general graphs, and maximal clique enumeration in large general graphs. The first research problem addresses subgraph counting in fully-dynamic graph streams, where graphs evolve dynamically with edge insertions and deletions. Traditional ap- proaches sample edges uniformly, overlooking the varying importance of edges. The thesis introduces a novel weighted sampling framework, WSD, for fully-dynamic graph streams, which samples the edges based on their weights that indicate their importance and reflect their properties. It determines the weights of edges in a data-driven fashion, using a novel method based on reinforcement learning. We conduct extensive experi- ments to verify that our technique can produce estimates with smaller errors while often running faster compared with existing algorithms. The second research problem focuses on k-clique listing in large general graphs, a critical task for applications such as community detection and network analysis. The state-of-the-art algorithms all adopt a branch-and-bound (BB) framework with a vertex-oriented branching strategy (called VBBkC), which forms a sub-branch by expanding a partial k-clique with a vertex. These algorithms have the time complexity of O(km(\delta/2)^(k-2)), where m is the number of edges in the graph and \delta is the degeneracy of the graph. In this thesis, we propose a BB framework with a new edge-oriented branching (called EBBkC), which forms a sub-branch by expanding a partial k-clique with two vertices that connect each other (which correspond to an edge). We explore various edge orderings for EBBkC such that it achieves a time complexity of O(m\delta + km(\tau/2)^(k-2), where \tau is an integer related to the maximum truss number of the graph and we have \tau<\delta. The time complexity of EBBkC is better than that of VBBkC algorithms for k>3 since both O(m\delta) and O(km(\tau/2)^(k-2) are bounded by O(km(\delta/2)^(k-2)). Furthermore, we develop specialized algorithms for sub-branches on dense graphs so that we can early-terminate them and apply the specialized algorithms. We conduct extensive experiments on 19 real graphs, and the results show that our newly developed EBBkC based algorithms with the early termination technique consistently and largely outperform the state-of-the- art (VBBkC based) algorithms. The third research problem centers on maximal clique enumeration (MCE) in large general graphs, crucial for tasks like community detection and biological network analysis. Existing algorithms typically adopt the branch-and-bound framework with the vertex-oriented Bron-Kerbosch (BK) branching strategy, which forms the sub-branches by expanding the partial clique with a vertex. The state-of-the-arts have a time complexity no better than O(n\delta 3^(\delta/3)), where n is the number of vertices in the graph and \delta is the degeneracy of the graph. We introduce a novel approach, HBBMC, a hybrid framework combining vertex-oriented and edge-oriented BK branching, where the latter adopts a new branch-and-bound framework which forms the sub-branches by expanding the partial clique with a edge. This hybrid strategy enables more efficient pruning of sub-branches, resulting in a worst-case time complexity of O(m\delta + m\tau 3^(\tau/3)), where \tau is related to the maximum truss number. This theoretical result is better than that of the state-of-the-arts when \delta > \tau + (3/ln3) \rho, a condition satisfied by many real-world graphs, where \rho is the edge density of the graph defined as \rho = m/n. To further enhance efficiency, we introduce an early termination technique applicable to all branch-and-bound frameworks, which leverages the topological information of the graphs within the branches. This technique accelerates the enumeration process without affecting worst-case time complexity. Extensive experiments on real and synthetic graphs demonstrate the superior performance of our approach. The proposed HBBMC algorithm with the early termination technique consistently outperforms existing vertex-oriented algorithms. In summary, this thesis makes significant contributions by introducing novel algo- rithms, frameworks, and methodologies to address subgraph-related problems. The pro- posed approaches show superior performance across various real-world and synthetic datasets, providing efficient solutions for graph analysis in large-scale, dynamic, and complex scenarios.

The paraclique algorithm provides an effective means for biological data clustering. It satisfies the mathematical quest for density, while fulfilling the pragmatic need for noise abatement on real data. Given a finite, simple, edge-weighted and thresholded graph, the paraclique method first finds a maximum clique, then incorporates additional vertices in a controlled manner, and finally extracts the subgraph thereby defined. When more than one maximum clique is present, however, deciding which to employ is usually left unspecified. In practice, this frequently and quite naturally reduces to using the first maximum clique found. In this paper, maximum clique selection is studied in the context of well-annotated transcriptomic data, with ontological classification used as a proxy for cluster quality. Enrichment p-values are compared using maximum cliques chosen in a variety of ways. The most appealing and intuitive option is almost surely to start with the maximum clique having the highest average edge weight. Although there is of course no guarantee that such a strategy is any better than random choice, results derived from a large collection of experiments indicate that, in general, this approach produces a small but statistically significant improvement in overall cluster quality. Such an improvement, though modest, may be well worth pursuing in light of the time, expense and expertise often required to generate timely, high quality, high throughput biological data.

The identification of disease-associated modules based on protein-protein interaction networks (PPINs) and gene expression data has provided new insights into the mechanistic nature of diverse diseases. However, their identification is hampered by the detection of protein communities within large-scale, whole-genome PPINs. A presented successful strategy detects a PPIN’s community structure based on the maximal clique enumeration problem (MCE), which is a non-deterministic polynomial time-hard problem. This renders the approach computationally challenging for large PPINs implying the need for new strategies. We present ModuleDiscoverer, a novel approach for the identification of regulatory modules from PPINs and gene expression data. Following the MCE-based approach, ModuleDiscoverer uses a randomization heuristic-based approximation of the community structure. Given a PPIN of Rattus norvegicus and public gene expression data, we identify the regulatory module underlying a rodent model of non-alcoholic steatohepatitis (NASH), a severe form of non-alcoholic fatty liver disease (NAFLD). The module is validated using single-nucleotide polymorphism (SNP) data from independent genome-wide association studies and gene enrichment tests. Based on gene enrichment tests, we find that ModuleDiscoverer performs comparably to three existing module-detecting algorithms. However, only our NASH-module is significantly enriched with genes linked to NAFLD-associated SNPs. ModuleDiscoverer is available at http://www.hki-jena.de/index.php/0/2/490 (Others/ModuleDiscoverer).

Structure mining plays an important part in the researches in biology, physics, internet or telecommunications in recently emerging network science. As a main task in this area, the problem of maximal clique enumeration has attracted much interest and been studied in variant avenues in prior works. However, most of these works mainly rely on single chip computational capacity and have been constrained by local optimization. Thus it is an impossible mission for these methods to process terabytes datasets. In this paper, to extract maximal cliques from graphs, we propose a general enumeration process in a distributed manner on cluster system with the help of MapReduce. Graph is firstly split into small subgraphs automatically. Then a novel key-based clique enumeration algorithm is proposed based on subgraphs. We demonstrate that our algorithm has a high parallelism and a prominent performance on extremely huge graphs. Our method is implemented to fully utilize MapReduce execution mechanism and the experiments are soundly discussed as using such a powerful distributed platform. However we not only show the scalability and efficiency of the algorithm but also share some critical experience in using MapReduce computing model.

An algorithm is said to be certifying if it outputs, together with a solution to the problem it solves, a proof that this solution is correct. We explain how state of the art maximum clique, maximum weighted clique, maximal clique enumeration and maximum common (connected) induced subgraph algorithms can be turned into certifying solvers by using pseudo-Boolean models and cutting planes proofs, and demonstrate that this approach can also handle reductions between problems. The generality of our results suggests that this method is ready for widespread adoption in solvers for combinatorial graph problems.

Incremental maximal clique enumeration (IMCE), which maintains maximal cliques in dynamic graphs, is a fundamental problem in graph analysis. A maximal clique has a solid descriptive power of dense structures in graphs. Real-world graph data is often large and dynamic. Studies on IMCE face significant challenges in the efficiency of incremental batch computation and hybrid edge changes. Moreover, with growing graph sizes, new requirements occur on indexing global maximal cliques and obtaining maximal cliques under specific vertex scope constraints. This work presents a new data structure SOMEi to maintain intermediate maximal cliques during construction. SOMEi serves as a space-efficient index to retrieve scope-constrained maximal cliques on the fly. Based on SOMEi, we design a procedure-oriented IMCE algorithm to deal with hybrid edge changes within a unified algorithm framework. In particular, the algorithm is able to process a large batch of edge changes and significantly improve the average processing time of a single edge change through an efficient pruning strategy. Experimental results on real and synthetic graph data demonstrate that the proposed algorithm outperforms all the baselines and achieves good efficiency through pruning.

Efficient enumeration of all maximal cliques in a given graph has many applications in graph theory, data mining and bio informatics. However, the exponentially increasing computation time of this problem confines the scale of the graph. Meanwhile, recent researches show that many networks in our world are complex networks involving massive data. To solve the maximal clique problem in the real-world scenarios, this paper presents a parallel algorithm Peamc (parallel enumeration of all maximal cliques) which exploits several new and effective techniques to enumerate all maximal cliques in a complex network. Furthermore, we provide a performance study on a true-life call graph with up to 2,423,807 vertices and 5,317,183 edges. The experimental results show that Peamc can find all the maximal cliques in a complex network with high efficiency and scalability