Distributed computation of temporal twins in periodic undirected time-varying graphs
Lina Azerouk, Binh-Minh Bui-Xuan, Camille Palisoc, Maria Potop-Butucaru, Massinissa Tighilt
TL;DR
The paper addresses identifying $(\Delta,d)$-twins in $p$-periodic undirected time-varying graphs to support backup routing in dynamic networks. It presents a distributed deterministic algorithm achieving enumeration of all $(\Delta,d)$-twins in $2p$ rounds with message size $O(\delta_{\mathcal{G}}\log n)$, and a randomized hashing-based variant that reduces communication to $O(\log n)$ w.h.p. by sampling neighborhood information. The approach first reduces the static case to detecting $d$-twins via 2-hop neighborhood analysis with inclusion-exclusion bounds, then lifts the technique to the periodic dynamic setting by exploiting repetition over the period and ensuring termination after $2p$ rounds. These results enable resource-efficient twin-based backup strategies and robust path/routing constructs in changing networks, with future work extending to $\epsilon$-modules.
Abstract
Twin nodes in a static network capture the idea of being substitutes for each other for maintaining paths of the same length anywhere in the network. In dynamic networks, we model twin nodes over a time-bounded interval, noted $(Δ,d)$-twins, as follows. A periodic undirected time-varying graph $\mathcal G=(G_t)_{t\in\mathbb N}$ of period $p$ is an infinite sequence of static graphs where $G_t=G_{t+p}$ for every $t\in\mathbb N$. For $Δ$ and $d$ two integers, two distinct nodes $u$ and $v$ in $\mathcal G$ are $(Δ,d)$-twins if, starting at some instant, the outside neighbourhoods of $u$ and $v$ has non-empty intersection and differ by at most $d$ elements for $Δ$ consecutive instants. In particular when $d=0$, $u$ and $v$ can act during the $Δ$ instants as substitutes for each other in order to maintain journeys of the same length in time-varying graph $\mathcal G$. We propose a distributed deterministic algorithm enabling each node to enumerate its $(Δ,d)$-twins in $2p$ rounds, using messages of size $O(δ_\mathcal G\log n)$, where $n$ is the total number of nodes and $δ_\mathcal G$ is the maximum degree of the graphs $G_t$'s. Moreover, using randomized techniques borrowed from distributed hash function sampling, we reduce the message size down to $O(\log n)$ w.h.p.