Tree-based Coarsening and Partitioning 1 of Complex Networks 2 Roland Glantz, Henning Meyerhenke, and Christian Schulz 3 Karlsruhe Institute of Technology (KIT), Karlsruhe, Germany 4 Abstract. Many applications produce massive complex networks whose 5 analysis would benefit from parallel processing. Parallel algorithms, in 6 turn, often require a suitable network partition. For solving optimization 7 tasks such as graph partitioning on large networks, multilevel methods 8 are preferred in practice. Yet, complex networks pose challenges to es- 9 tablished multilevel algorithms, in particular to their coarsening phase. 10 One way to specify a (recursive) coarsening of a graph is to rate its edges 11 and then contract the edges as prioritized by the rating. In this paper 12 we (i) define weights for the edges of a network that express the edges’ 13 importance for connectivity, (ii) compute a minimum weight spanning 14 tree T m w. r. t. these weights, and (iii) rate the network edges based 15 on the conductance values of T m ’s fundamental cuts. To this end, we 16 also (iv) develop the first optimal linear-time algorithm to compute the 17 conductance values of all fundamental cuts of a given spanning tree. 18 We integrate the new edge rating into a leading multilevel graph par- 19 titioner and equip the latter with a new greedy postprocessing for op- 20 timizing the maximum communication volume (MCV). Experiments on 21 bipartitioning frequently used benchmark networks show that the post- 22 processing already reduces MCV by 11.3%. Our new edge rating further 23 reduces MCV by 10.3% compared to the previously best rating with the 24 postprocessing in place for both ratings. In total, with a modest increase 25 in running time, our new approach reduces the MCV of complex network 26 partitions by 20.4%. 27 Keywords: Graph coarsening, multilevel graph partitioning, complex net- 28 works, fundamental cuts, spanning trees 29 1 Introduction 30 Complex networks such as social networks or web graphs have become a focus 31 of investigation recently [7]. Such networks are often scale-free, i.e. they have 32 a power-law degree distribution with many low-degree vertices and few high- 33 degree vertices. They also have a small diameter (small-world property), so that 34 the whole network is discovered within a few hops from any vertex. Complex net- 35 works arise in a variety of applications; several of them generate massive data 36 sets. As an example, the social network Facebook currently contains a billion 37 active users (http://newsroom.fb.com/Key-Facts). On this scale many algorith- 38 mic tasks benefit from parallel processing. The efficiency of parallel algorithms 39 on huge networks, in turn, is usually improved by graph partitioning (GP). 40