An Output Sensitive Algorithm for Maximal Clique Enumeration in Sparse Graphs

The degeneracy of a graph G is the smallest integer k such that every subgraph of G contains a vertex of degree at most k. Given an n-order k-degenerate graph G, we present an algorithm for enumerating all its maximal cliques. Assuming that c is the number of maximal cliques of G, our algorithm has setup time O(n(k^2+s(k+1))) and enumeration time cO((k+1)f(k+1)) where s(k+1) (resp. f(k+1)) is the preprocessing time (resp. enumeration time) for maximal clique enumeration in a general (k+1)-order graph. This is the first output sensitive algorithm whose enumeration time depends only on the degeneracy of the graph

Shared-Memory Parallel Maximal Clique Enumeration

We present shared-memory parallel methods for Maximal Clique Enumeration (MCE) from a graph. MCE is a fundamental and well-studied graph analytics task, and is a widely used primitive for identifying dense structures in a graph. Due to its computationally intensive nature, parallel methods are imperative for dealing with large graphs. However, surprisingly, there do not yet exist scalable and parallel methods for MCE on a shared-memory parallel machine. In this work, we present efficient shared-memory parallel algorithms for MCE, with the following properties: (1) the parallel algorithms are provably work-efficient relative to a state-of-the-art sequential algorithm (2) the algorithms have a provably small parallel depth, showing that they can scale to a large number of processors, and (3) our implementations on a multicore machine shows a good speedup and scaling behavior with increasing number of cores, and are substantially faster than prior shared-memory parallel algorithms for MCE.Comment: 10 pages, 3 figures, proceedings of the 25th IEEE International Conference on. High Performance Computing, Data, and Analytics (HiPC), 201

Incremental Maintenance of Maximal Cliques in a Dynamic Graph

We consider the maintenance of the set of all maximal cliques in a dynamic graph that is changing through the addition or deletion of edges. We present nearly tight bounds on the magnitude of change in the set of maximal cliques, as well as the first change-sensitive algorithms for clique maintenance, whose runtime is proportional to the magnitude of the change in the set of maximal cliques. We present experimental results showing these algorithms are efficient in practice and are faster than prior work by two to three orders of magnitude.Comment: 18 pages, 8 figure

Sublinear-Space Bounded-Delay Enumeration for Massive Network Analytics: Maximal Cliques

Due to the sheer size of real-world networks, delay and space become quite relevant measures for the cost of enumeration in network analytics. This paper presents efficient algorithms for listing maximum cliques in networks, providing the first sublinear-space bounds with guaranteed delay per enumerated clique, thus comparing favorably with the known literature

Mining maximal cliques from large graphs using MapReduce

Maximal clique enumeration (MCE), a fundamental task in graph analysis, can help identify dense substructures within a graph, and has found applications in graphs arising in biological and chemical networks, and more. While MCE is well studied in the sequential case, a single machine can no longer process large graphs arising in today\u27s applications, and effective ways are needed for processing these in parallel. This work introduces PECO (Parallel Enumeration of Cliques using Ordering); a novel parallel MCE algorithm. Unlike previous works, which require a post-processing step to remove duplicate and non-maximal cliques, PECO enumerates maximal cliques with no duplicates while minimizing work redundancy and eliminating the need for an additional post-processing step. This is achieved by dividing the input graph into smaller overlapping subgraphs, and by inducing a total ordering among the vertices. Then, as a subgraph is processed, the ordering is used in tandem with a sequential MCE algorithm to reduce redundant work while only enumerating a clique if it satisfies a certain condition with respect to the ordering, ensuring that each maximal clique is output exactly once. It is well recognized that in enumerating maximal cliques, the sizes of different subproblems can be non-uniform, and load balancing among the subproblems is a significant issue. Our algorithm uses the above vertex ordering to greatly improve load balancing when compared with straightforward approaches to parallelization. PECO has been designed and implemented for the MapReduce framework, but this technique is applicable to other parallel frameworks as well. Our experiments on a variety of large real world graphs, using several ordering strategies, show that PECO can enumerate cliques in large graphs of well over a million vertices and tens of millions of edges, and that it scales well to at least 64 processors. A comparison of ordering strategies shows that an ordering based on vertex degree performs the best, improving load balance and reducing total work when compared to the other strategies

Efficient maximal cliques enumeration in weakly closed graphs

We show that the algorithm presented in [J. Fox, T. Roughgarden, C. Seshadhri, F. Wei, and N. Wein. Finding cliques in social networks: A new distribution-free model. SIAM journal on computing, 49(2):448-464, 2020.] can be modified to have enumeration time complexity $\alpha\mathcal{O} (npoly(c))$. Here parameter $c$ is the weakly closure of the graph and $\alpha$ its number of maximal cliques. This result improves on their complexity which was not output sensitive and exponential in the closure of the graph