Computing Small Temporal Modules in Time Logarithmic in History Length

Binh-Minh Bui-Xuan (1) , Hugo Hourcade (2, 1) , Cédric Miachon (2)

1 APR - Algorithmes, Programmes et Résolution
2 Courtanet

A temporal graph G is a sequence of static graphs indexed by a set of integers representing time instants. Given ∆ an integer, a ∆-module is a set of vertices A having the same neighbourhood outside of A for ∆ consecutive instants. We address specific cases of ∆-module enumeration, when |A| = 2 or when ∆ = ∞. Our main parameter for time complexity analysis is the history length τ = max{t : G t ∈ G not empty } − min{t : G t ∈ G not empty }. Using red-black tree data structure, we give solutions to above enumeration problems in time logarithmic in τ. For the general ∆-module enumeration problem, we give a pre-processing using overlapping properties of minimal ∆modules. Numerical analysis of our implementation on graphs collected from real world data scales up to a history length of 10 8 time instants 1. Keywords graph theory • historical data • modular decomposition • temporal graph Supported by Courtanet-Sorbonne Université convention C19.0665 and ANRT grant 2019.0485.

Domaines: Algorithme et structure de données [cs.DS]

Binh-Minh Bui-Xuan, Hugo Hourcade, Cédric Miachon. Computing Small Temporal Modules in Time Logarithmic in History Length. Social Network Analysis and Mining, 2022, ⟨10.1007/s13278-021-00820-5⟩. ⟨hal-03431380⟩