Tree-Based Coarsening and Partitioning of Complex Networks

Abstract

Many applications produce massive complex networks whose analysis would benefit from parallel processing. Parallel algorithms, in turn, often require a suitable network partition. For solving optimization tasks such as graph partitioning on large networks, multilevel methods are preferred in practice. Yet, complex networks pose challenges to established multilevel algorithms, in particular to their coarsening phase.

One way to specify a (recursive) coarsening of a graph is to rate its edges and then contract the edges as prioritized by the rating. In this paper we (i) define weights for the edges of a network that express the edges’ importance for connectivity, (ii) compute a minimum weight spanning tree T ^m w.r.t. these weights, and (iii) rate the network edges based on the conductance values of T ^m’s fundamental cuts. To this end, we also (iv) develop the first optimal linear-time algorithm to compute the conductance values of all fundamental cuts of a given spanning tree.

We integrate the new edge rating into a leading multilevel graph partitioner and equip the latter with a new greedy postprocessing for optimizing the maximum communication volume (MCV). Bipartitioning experiments on established benchmark graphs show that both the postprocessing and the new edge rating improve upon the state of the art by more than 10%. In total, with a modest increase in running time, our new approach reduces the MCV of complex network partitions by 20.4%.