Bron-Kerbosch Algorithm Research Articles

Faster maximal clique enumeration in large real-world link streams

Link streams offer a good model for representing interactions over time. They consist of links $(b,e,u,v)$, where $u$ and $v$ are vertices interacting during the whole time interval $[b,e]$. In this paper, we deal with the problem of enumerating maximal cliques in link streams. A clique is a pair $(C,[t_0,t_1])$, where $C$ is a set of vertices that all interact pairwise during the full interval $[t_0,t_1]$. It is maximal when neither its set of vertices nor its time interval can be increased. Some main works solving this problem are based on the famous Bron-Kerbosch algorithm for enumerating maximal cliques in graphs. We take this idea as a starting point to propose a new algorithm which matches the cliques of the instantaneous graphs formed by links existing at a given time $t$ to the maximal cliques of the link stream. We prove its correctness and compute its complexity, which is better than the state-of-the art ones in many cases of interest. We also study the output-sensitive complexity, which is close to the output size, thereby showing that our algorithm is efficient. To confirm this, we perform experiments on link streams used in the state of the art, and on massive link streams, up to 100 million links. In all cases our algorithm is faster, mostly by a factor of at least 10 and up to a factor of $10^4$. Moreover, it scales to massive link streams for which the existing algorithms are not able to provide the solution.

Distributed memory implementation of Bron-Kerbosch algorithm

Distributed memory implementation of Bron-Kerbosch algorithm

FireProt 2.0: web-based platform for the fully automated design of thermostable proteins.

Thermostable proteins find their use in numerous biomedical and biotechnological applications. However, the computational design of stable proteins often results in single-point mutations with a limited effect on protein stability. However, the construction of stable multiple-point mutants can prove difficult due to the possibility of antagonistic effects between individual mutations. FireProt protocol enables the automated computational design of highly stable multiple-point mutants. FireProt 2.0 builds on top of the previously published FireProt web, retaining the original functionality and expanding it with several new stabilization strategies. FireProt 2.0 integrates the AlphaFold database and the homology modeling for structure prediction, enabling calculations starting from a sequence. Multiple-point designs are constructed using the Bron-Kerbosch algorithm minimizing the antagonistic effect between the individual mutations. Users can newly limit the FireProt calculation to a set of user-defined mutations, run a saturation mutagenesis of the whole protein or select rigidifying mutations based on B-factors. Evolution-based back-to-consensus strategy is complemented by ancestral sequence reconstruction. FireProt 2.0 is significantly faster and a reworked graphical user interface broadens the tool's availability even to users with older hardware. FireProt 2.0 is freely available at http://loschmidt.chemi.muni.cz/fireprotweb.

A Diagnosability-Integrated Design Approach Based on Graph Theory

In order to take into account the influence of both system structure and diagnosis algorithm in the diagnosability design of the system, a diagnosability-integrated design method based on graph theory was proposed in this paper. Firstly, based on the diagnosability evaluation results, the difficulty of fault diagnosis was qualitatively analyzed using the K-means method, and the diagnosis plot of measurement point was drawn based on the analysis results. Secondly, the Bron–Kerbosch algorithm was used to extract the maximal cliques from the diagnosis plot of measurement point and determine the set of maximal cliques that can diagnose faults in the system based on the hypergraph edge coverage theorem. Finally, a cascade classifier was set on the maximal clique set to classify and identify faults in the system, and the performance of the diagnosis scheme was evaluated using the posterior probabilities of the classifier outputs combined with the Shannon entropy. At the same time, the method incorporated a measurement point update mechanism, which can decide whether to add additional measurement point according to the evaluation results of Shannon entropy to ensure better diagnosis effect. The results of simulation experiments showed that the fault diagnosis scheme designed by the method of this paper improved the correct rate of diagnosis results by 3.25 percentage points compared with other diagnosis schemes due to the simultaneous consideration of the structure of the system and the diagnosis method, and the diagnosis results of this paper were relatively stable in repeated experiments, which proved the practicality and effectiveness of the method of this paper.

An asynchronous P system with the Bron-Kerbosch algorithm for solving the maximum clique

An asynchronous P system with the Bron-Kerbosch algorithm for solving the maximum clique

Identifying protein function and functional links based on large-scale co-occurrence patterns.

ObjectiveThe vast majority of known proteins have not been experimentally tested even at the level of measuring their expression, and the function of many proteins remains unknown. In order to decipher protein function and examine functional associations, we developed "Cliquely", a software tool based on the exploration of co-occurrence patterns.Computational modelUsing a set of more than 23 million proteins divided into 404,947 orthologous clusters, we explored the co-occurrence graph of 4,742 fully sequenced genomes from the three domains of life. Edge weights in this graph represent co-occurrence probabilities. We use the Bron–Kerbosch algorithm to detect maximal cliques in this graph, fully-connected subgraphs that represent meaningful biological networks from different functional categories.Main resultsWe demonstrate that Cliquely can successfully identify known networks from various pathways, including nitrogen fixation, glycolysis, methanogenesis, mevalonate and ribosome proteins. Identifying the virulence-associated type III secretion system (T3SS) network, Cliquely also added 13 previously uncharacterized novel proteins to the T3SS network, demonstrating the strength of this approach. Cliquely is freely available and open source. Users can employ the tool to explore co-occurrence networks using a protein of interest and a customizable level of stringency, either for the entire dataset or for a one of the three domains—Archaea, Bacteria, or Eukarya.

On the overall and delay complexity of the CLIQUES and Bron-Kerbosch algorithms

We revisit the maximal clique enumeration algorithm cliques by Tomita et al. that appeared in Theoretical Computer Science in 2006. It is known to work in O(3n/3)-time in the worst-case for an n-vertex graph. This is worst-case optimal with respect to the input size, but there is little knowledge about its performance with respect to the output. In this paper, we extend the time-complexity analysis with respect to the maximum size and the number of maximal cliques, and to its delay, solving issues that were left as open problems since the original paper. In particular, we prove that cliques has Ω(3n/6) delay and that, even if we allow to change the pivoting strategy, a variant having polynomial delay cannot be designed unless P=NP. These same results apply to the related Bron-Kerbosch algorithm. On the positive side, we show that the complexity of cliques and Bron-Kerbosch is amortized polynomial on graphs with logarithmic clique number. As these algorithms are widely used and regarded as fast “in practice”, we are interested in observing their practical behavior: we run an evaluation of cliques and three Bron-Kerbosch variants on over 130 real-world and synthetic graphs, observing how the clique number almost always satisfies our logarithmic constraint, and that their performance seems far from its theoretical worst-case behavior in terms of both total time and delay.1

Accelerating the Bron-Kerbosch Algorithm for Maximal Clique Enumeration Using GPUs

Maximal clique enumeration (MCE) is a classic problem in graph theory to identify all complete subgraphs in a graph. In prior MCE work, the Bron-Kerbosch algorithm is one of the most popular solutions, and there are several improved algorithms proposed on CPU platforms. However, while few studies have focused on the related issue of parallel implementation, recently, there have been numerous explorations of the acceleration of general purpose applications using a graphics processing unit (GPU) to reduce the computing power consumption. In this article, we develop a GPU-based Bron-Kerbosch algorithm that efficiently solves the MCE problem in parallel by optimizing the process of subproblem decomposition and computing resource usage. To speed up the computations, we use coalesced memory accesses and warp reductions to increase bandwidth and reduce memory latency. Our experimental results show that the proposed algorithm can fully exploit the resources of GPU architectures, allowing for the vast acceleration of operations to solve the MCE problem.

Robust ENF Estimation Based on Harmonic Enhancement and Maximum Weight Clique

We present a framework for robust electric network frequency (ENF) extraction from real-world audio recordings, featuring multi-tone ENF harmonic enhancement and graph-based optimal harmonic selection. Specifically, We first extend the recently developed single-tone ENF signal enhancement method to the multi-tone scenario and propose a harmonic robust filtering algorithm (HRFA). It can respectively enhance each harmonic component without cross-component interference, thus further alleviating the effects of unwanted noise and audio content on the much weaker ENF signal. In addition, considering the fact that some harmonic components could be severely corrupted even after enhancement, disturbing rather than facilitating ENF estimation, we propose a graph-based harmonic selection algorithm (GHSA), which finds the optimal combination of harmonic components for more accurate ENF estimation. Noticeably, the harmonic selection problem is equivalently formulated as a maximum weight clique (MWC) problem in graph theory, and the Bron-Kerbosch algorithm (BKA) is adopted in the GHSA. With the enhanced and optimally selected harmonic components, both the existing maximum likelihood estimator (MLE) and weighted MLE (WMLE) are incorporated to yield the final ENF estimation results. The proposed framework is extensively evaluated using both synthetic signals and our ENF-WHU dataset consisting of $130$ real-world audio recordings, demonstrating substantially improved capability of extracting the ENF from realistically noisy observations over the existing single- and multi-tone competitors. This work further improves the applicability of the ENF as a forensic criterion in real-world situations.

NOVEL MACHINE LEARNING ALGORITHMS FOR CENTRALITY AND CLIQUES DETECTION IN YOUTUBE SOCIAL NETWORKS

The goal of this research project is to analyze the dynamics of social networks using machine learning techniques to locate maximal cliques and to find clusters for the purpose of identifying a target demographic. Unsupervised machine learning techniques are designed and implemented in this project to analyze a dataset from YouTube to discover communities in the social network and find central nodes. Different clustering algorithms are implemented and applied to the YouTube dataset. The well-known Bron-Kerbosch algorithm is used effectively in this research to find maximal cliques. The results obtained from this research could be used for advertising purposes and for building smart recommendation systems. All algorithms were implemented using Python programming language. The experimental results show that we were able to successfully find central nodes through clique-centrality and degree centrality. By utilizing clique detection algorithms, the research shown how machine learning algorithms can detect close knit groups within a larger network.

Improved Overlapping Community Detection in Networks based on Maximal Cliques Enumeration

Abstract Social Network Analysis has received great interest in recent years in several areas, including communities detection. Many studies have been carried out in this sense. The Bron Kerbosch Algorithm (BK) is one of the most widely known and most efficient algorithm to detect overlapping communities based on the maximal clique notion. It is a linear and fast algorithm, but its major disadvantage lies in “Lost nodes”, which are isolated nodes not belonging to any communities. In addition, this method represents a difficulty in extracting cliques. This difficulty is due to the strict definition of the clique. Since, to get better detection, this definition requires finding nodes which are all linked in the network. In this paper, we introduce DOCNA a new algorithm for detecting overlapping communities in networks based on maximal cliques. DOCNA adopts an improved version of BK Algorithm. In fact, we add an assignment phase which allows each “Lost node” to be a member of one or more detected communities. Moreover, our algorithm uses a special Click_Graph construction process to detect communities. The experimental results, on synthetic and real Networks with different sizes and overlapping rates, illustrate the effectiveness of our approach in detecting dynamic overlapping community structures.

On-Demand Receiver-Centric Channel Allocation via Constrained VCG Auction for Spatial Spectrum Reuse

Spatial spectrum reuse significantly enhances spectrum utilization but requires delicate design to avoid cochannel interference. Instead of focusing solely on spectrum efficiency, we consider maximizing social welfare via channel allocation. Specifically, we propose an on-demand receiver-centric mechanism with a new factor, the supply and demand relationship. The optimal channel allocation that maximizes social welfare can be achieved by the constrained Vickrey–Clarke–Groves (VCG) auction for secondary users (SUs), but with high complexity. To simplify the constrained VCG auction, we group the SUs into maximal independent groups (MIGs) using a modified Bron–Kerbosch algorithm. We prove that truthful bidding is the optimal strategy for the SUs even though they do not participate in the VCG auction for MIGs directly. Therefore, the MIGs are bidding truthfully and social welfare is maximized. We also prove that the optimal channel allocation is not unique and thus a weakly-dominant strategy for the primary user, and the VCG style pricing based on truthful bidding can be implemented by using a decision tree repeatedly. Furthermore, we approximate and simplify the optimal channel allocation with a greedy algorithm, Dijkstra's algorithm, and batch allocation. In our simulation, we compare the proposed methods and demonstrate that on-demand channel allocation increases social welfare.

SimUrb – software for identifying similar municipalities by comparing Urban indices using a graph algorithm

The paper describes the SimUrb tool, which was used to calculate the similarity between municipalities (or other territorial units) according to several characteristics (attributes). The SimUrb tool allows similar groups between ordered sequences to be found. The tool was designed to work with hundreds of records. In order to find similar groups, the tool employs graph theory, in which similar groups are represented as cliques on a simple graph. The Bron-Kerbosch algorithm was used to search for them. The degree of similarity was determined from a metric based on the Euclidean distance between strings. Finding all non-trivial cliques would have been difficult or impossible for such data to be feasible in this study. Therefore, the SimUrb tool was used to find similar groups as the largest disjoint cliques in the respective graph. A case study is introduced in the second part of the paper to illustrate SimUrb’s functionality. The results of standard grouping methods (two methods offered by ArcGIS software) and groups defined under official planning documents were compared to the results from the SimUrb software. We concluded that SimUrb can be used in many applications where the user needs to define groups of objects with the same degree of similarity.

Structural Interpretation of Sparse Fault Data Using Graph Theory and Geological Rules

Structural uncertainty exists when associating sparse fault interpretations made from two-dimensional seismic lines or limited outcrop observations. Here, a graph formalism is proposed that describes the problem of associating spatial fault evidence. A combinatorial analysis, relying on this formalism, shows that the number of association scenarios is given by the Bell number, and increases exponentially with the number of pieces of evidence. As a result, the complete exploration of uncertainties is computationally highly challenging. The available prior geological knowledge is expressed by numerical rules to reduce the number of scenarios, and the graph formalism makes structural interpretation easier to reproduce than manual interpretation. The Bron–Kerbosch algorithm, which finds maximal cliques in undirected graphs, is used to detect major possible structures. This framework opens the way to a numerically assisted exploration of uncertainties during structural interpretation.

Listing all maximal cliques in large graphs on vertex-centric model

Maximal Clique Enumeration (MCE), which consists to enumerate all maximal complete subgraphs in a given graph, is a fundamental problem in graph theory, and it is used in several applications. It is one of Karp’s 21 NP-complete problems. In the literature, this problem has been widely studied. One of the most notable, efficient, successful and extensively used solutions is the Bron–Kerbosch (BK) algorithm. The latter is a sequential algorithm which is able to enumerate all maximal cliques in an undirected graph without duplication. Furthermore, it is used to solve large problems such as maximum clique problem, community detection and graph clustering. However, for large graphs, sequential algorithms are slow and do not scale well. Thus, processing efficiently this kind of graphs needs to develop distributed algorithms under parallel and distributed platforms or large graph mining frameworks. In this setting, we propose new efficient distributed algorithms for maximal clique enumerating based on the vertex-centric model. These algorithms use the BK algorithm principle to deal with the MCE problem on large cluster. The proposed algorithms are implemented in Giraph and evaluated by using real-world graphs and computer-generated benchmark networks. Our experiments on a Hadoop cluster show that the proposed algorithms can effectively process a variety of large real-world and computer-generated graphs and scale well with increasing the dataset size and the number of nodes in the cluster. Furthermore, the proposed algorithms are provably work-efficient compared with other algorithms including BK algorithm.

MAPSkew: Metaheuristic Approaches for Partitioning Skew in MapReduce

MapReduce is a parallel computing model in which a large dataset is split into smaller parts and executed on multiple machines. Due to its simplicity, MapReduce has been widely used in various applications domains. MapReduce can significantly reduce the processing time of a large amount of data by dividing the dataset into smaller parts and processing them in parallel in multiple machines. However, when data are not uniformly distributed, we have the so called partitioning skew, where the allocation of tasks to machines becomes unbalanced, either by the distribution function splitting the dataset unevenly or because a part of the data is more complex and requires greater computational effort. To solve this problem, we propose an approach based on metaheuristics. For evaluating purposes, three metaheuristics were implemented: Simulated Annealing, Local Beam Search and Stochastic Beam Search. Our experimental evaluation, using a MapReduce implementation of the Bron-Kerbosch Clique Algorithm, shows that the proposed method can find good partitionings while better balancing data among machines.

Two-stage coalition formation and radio resource allocation with Nash bargaining solution for inband underlaid D2D communications in 5G networks

For D2D communications in the inband underlaid 5G networks, to improve the utilization of cellular uplink spectrum, a two-stage coalition formation and resource allocation scheme is proposed to allocate the radio resources based on the numbers of resource blocks (RBs) requested by D2D pairs. Any set of D2D pairs whose pairwise distances longer than a threshold value are organized into a Stage-1 coalition by the Bron-Kerbosch algorithm. Otherwise, they are organized into Stage-2 coalitions based on the total number of RBs they requested. Since D2D pairs in a Stage-1 coalition are far away from each other, the co-channel interference can be ignored. To improve the RB utilization, RBs that can be used by D2D pairs in the same Stage-1 coalition are first allocated to that Stage-1 coalition. Then, based on the total number of RBs requested by D2D pairs in a Stage-2 coalition, RBs are allocated to Stage-2 coalitions. Since RBs allocated to a Stage-2 coalition cannot be shared by D2D pairs in that Stage-2 coalition, the Nash bargaining solution (NBS) is used to further allocate RBs to D2D pairs in that Stage-2 coalition. Simulation results show that, the proposed two-stage coalition formation and resource allocation scheme allocates more number of equivalent RBs to D2D pairs than the number of RBs allocated by the greedy algorithm. Specifically, based on the simulation results, the total number of equivalent RBs allocated by the proposed scheme can be up to 325.52% of that allocated by the greedy algorithm.

Adapting the Bron–Kerbosch algorithm for enumerating maximal cliques in temporal graphs

Dynamics of interactions play an increasingly important role in the analysis of complex networks. A modeling framework to capture this is temporal graphs which consist of a set of vertices (entities in the network) and a set of time-stamped binary interactions between the vertices. We focus on enumerating Δ-cliques, an extension of the concept of cliques to temporal graphs: for a given time period Δ, a Δ-clique in a temporal graph is a set of vertices and a time interval such that all vertices interact with each other at least after every Δ time steps within the time interval. Viard, Latapy, and Magnien (ASONAM 2015, TCS 2016) proposed a greedy algorithm for enumerating all maximal Δ-cliques in temporal graphs. In contrast to this approach, we adapt the Bron–Kerbosch algorithm—an efficient, recursive backtracking algorithm which enumerates all maximal cliques in static graphs—to the temporal setting. We obtain encouraging results both in theory (concerning worst-case running time analysis based on the parameter “Δ-slice degeneracy” of the underlying graph) as well as in practice with experiments on real-world data. The latter culminates in an improvement for most interesting Δ-values concerning running time in comparison with the algorithm of Viard, Latapy, and Magnien.

Conceptualization with Incremental Bron-Kerbosch Algorithm in Big Data Architecture

Conceptualization with Incremental Bron-Kerbosch Algorithm in Big Data Architecture

Refined pivot selection for maximal clique enumeration in graphs

This article re-examines the pivoting Bron–Kerbosch algorithm for identifying all maximal cliques within a graph. A curious result is presented: there exist pivot candidates which may be selected with no penalty to the worst-case running time, even if these vertices do not satisfy the established conditions for the known complexity bound. Hence, such pivots may be selected eagerly. A refined pivot selection process is described, and the preservation of the established worst case bound is proved. Additionally, empirical evidence shows that the revised algorithm is notably faster than Bron–Kerbosch with Tomita-style pivot selection.