Parallel Local Algorithms for Core, Truss, and Nucleus Decompositions

Finding the dense regions of a graph and relations among them is a fundamental task in network analysis. Nucleus decomposition is a principled framework of algorithms that generalizes the k-core and k-truss decompositions. It can leverage the higher-order structures to locate the dense subgraphs with hierarchical relations. Computation of the nucleus decomposition is performed in multiple steps, known as the peeling process, and it requires global information about the graph at any time. This prevents the scalable parallelization of the computation. Also, it is not possible to compute approximate and fast results by the peeling process, because it does not produce the densest regions until the algorithm is complete. In a previous work, Lu et al. proposed to iteratively compute the h-indices of vertex degrees to obtain the core numbers and prove that the convergence is obtained after a finite number of iterations. In this work, we generalize the iterative h-index computation for any nucleus decomposition and prove convergence bounds. We present a framework of local algorithms to obtain the exact and approximate nucleus decompositions. Our algorithms are pleasingly parallel and can provide approximations to explore time and quality trade-offs. Our shared-memory implementation verifies the efficiency, scalability, and effectiveness of our algorithms on real-world networks. In particular, using 24 threads, we obtain up to 4.04x and 7.98x speedups for k-truss and (3, 4) nucleus decompositions.