Fully dynamic maintenance of k-connectivity in parallel

©2001 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE.Given a graph G=(V, E) with n vertices and m edges, the k-connectivity of G denotes either the k-edge connectivity or the k-vertex connectivity of G. In this paper, we deal with the fully dynamic maintenance of k-connectivity of G in the parallel setting for k=2, 3. We study the problem of maintaining k-edge/vertex connected components of a graph undergoing repeatedly dynamic updates, such as edge insertions and deletions, and answering the query of whether two vertices are included in the same k-edge/vertex connected component. Our major results are the following: (1) An NC algorithm for the 2-edge connectivity problem is proposed, which runs in O(log n log(m/n)) time using O(n3/4) processors per update and query. (2) It is shown that the biconnectivity problem can be solved in O(log2 n ) time using O(nα(2n, n)/logn) processors per update and O(1) time with a single processor per query or in O(log n logn/m) time using O(nα(2n, n)/log n) processors per update and O(logn) time using O(nα(2n, n)/logn) processors per query, where α(.,.) is the inverse of Ackermann's function. (3) An NC algorithm for the triconnectivity problem is also derived, which takes O(log n logn/m+logn log log n/α(3n, n)) time using O(nα(3n, n)/log n) processors per update and O(1) time with a single processor per query. (4) An NC algorithm for the 3-edge connectivity problem is obtained, which has the same time and processor complexities as the algorithm for the triconnectivity problem. To the best of our knowledge, the proposed algorithms are the first NC algorithms for the problems using O(n) processors in contrast to Ω(m) processors for solving them from scratch. In particular, the proposed NC algorithm for the 2-edge connectivity problem uses only O(n3/4) processors. All the proposed algorithms run on a CRCW PRAMWeifa Liang, Brent, R.P., Hong She

Efficient Truss Maintenance in Evolving Networks

Truss was proposed to study social network data represented by graphs. A k-truss of a graph is a cohesive subgraph, in which each edge is contained in at least k-2 triangles within the subgraph. While truss has been demonstrated as superior to model the close relationship in social networks and efficient algorithms for finding trusses have been extensively studied, very little attention has been paid to truss maintenance. However, most social networks are evolving networks. It may be infeasible to recompute trusses from scratch from time to time in order to find the up-to-date $k$-trusses in the evolving networks. In this paper, we discuss how to maintain trusses in a graph with dynamic updates. We first discuss a set of properties on maintaining trusses, then propose algorithms on maintaining trusses on edge deletions and insertions, finally, we discuss truss index maintenance. We test the proposed techniques on real datasets. The experiment results show the promise of our work

Selecting a suitable Parallel Label-propagation based algorithm for Disjoint Community Detection

Community detection is an essential task in network analysis as it helps identify groups and patterns within a network. High-speed community detection algorithms are necessary to analyze large-scale networks in a reasonable amount of time. Researchers have made significant contributions in the development of high-speed community detection algorithms, particularly in the area of label-propagation based disjoint community detection. These algorithms have been proven to be highly effective in analyzing large-scale networks in a reasonable amount of time. However, it is important to evaluate the performance and accuracy of these existing methods to determine which algorithm is best suited for a particular type of network and specific research problem. In this report, we investigate the RAK, COPRA, and SLPA, three label-propagation-based static community discovery techniques. We pay close attention to each algorithm's minute details as we implement both its single-threaded and multi-threaded OpenMP-based variants, making any necessary adjustments or optimizations and obtaining the right parameter values. The RAK algorithm is found to perform well with a tolerance of 0.05 and OpenMP-based strict RAK with 12 threads was 6.75x faster than the sequential non-strict RAK. The COPRA algorithm works well with a single label for road networks and max labels of 4-16 for other classes of graphs. The SLPA algorithm performs well with increasing memory size, but overall doesn't offer a favourable return on investment. The RAK algorithm is recommended for label-propagation based disjoint community detection.Comment: 11 pages, 1 tabl