Computing Zigzag Persistence on Graphs in Near-Linear Time

Abstract

Graphs model real-world circumstances in many applications where they may constantly change to capture the dynamic behavior of the phenomena. Topological persistence which provides a set of birth and death pairs for the topological features is one instrument for analyzing such changing graph data. However, standard persistent homology defined over a growing space cannot always capture such a dynamic process unless shrinking with deletions is also allowed. Hence, zigzag persistence which incorporates both insertions and deletions of simplices is more appropriate in such a setting. Unlike standard persistence which admits nearly linear-time algorithms for graphs, such results for the zigzag version improving the general O(m ω ) time complexity are not known, where ω < 2.37286 is the matrix multiplication exponent. In this paper, we propose algorithms for zigzag persistence on graphs which run in near-linear time. Specifically, given a filtration with m additions and deletions on a graph with n vertices and edges, the algorithm for 0-dimension runs in O(m log 2 n + m log m) time and the algorithm for 1-dimension runs in O(m log 4 n) time. The algorithm for 0-dimension draws upon another algorithm designed originally for pairing critical points of Morse functions on 2-manifolds. The algorithm for 1-dimension pairs a negative edge with the earliest positive edge so that a 1-cycle containing both edges resides in all intermediate graphs. Both algorithms achieve the claimed time complexity via dynamic graph data structures proposed by Holm et al. In the end, using Alexander duality, we extend the algorithm for 0-dimension to compute the (p − 1)-dimensional zigzag persistence for R p -embedded complexes in O(m log 2 n + m log m + n log n) time. * This research is supported by NSF grants CCF 1839252 and 2049010. (a) (b) (c) (d) Figure 1: A sequence of graphs with four prominent clusters each colored differently. Black edges connect different clusters and forward (resp. backward) arrows indicate additions (resp. deletions) of vertices and edges. From (a) to (b), two clusters split; from (b) to (c), two clusters merge; from (c) to (d), one cluster disappears.

Temporal network analysis using zigzag persistence

“…Until recently, there has been limited involvement of zigzag persistence in these types of applications since even though in theory the running time should be similar to standard persistence [48], in practice it has not seen the flurry of optimizations available in the regular case [49,50]. However, recent work [51][52][53] promises substantial improvements in the potentially available code, which should further make the tools discussed in this paper more accessible to a wide array of data sets.…”