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 $(\Delta,d)$-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 $\Delta$ and $d$ two integers, two distinct nodes $u$ and $v$ in $\mathcal G$ are $(\Delta,d)$-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 $\Delta$ consecutive instants. In particular when $d=0$, $u$ and $v$ can act during the $\Delta$ 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 $(\Delta,d)$-twins in $2p$ rounds, using messages of size $O(\delta_\mathcal G\log n)$, where $n$ is the total number of nodes and $\delta_\mathcal G$ 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.

翻译：静态网络中的孪生节点捕捉了在网络中任意位置维持相同长度路径时彼此可替代的思想。在动态网络中，我们将时间有界区间内的孪生节点建模为$(\Delta,d)$-孪生节点，定义如下：周期为$p$的周期性无向时变图$\mathcal G=(G_t)_{t\in\mathbb N}$是一个无限静态图序列，其中对于所有$t\in\mathbb N$，有$G_t=G_{t+p}$。对于整数$\Delta$和$d$，$\mathcal G$中两个不同节点$u$和$v$称为$(\Delta,d)$-孪生节点，若从某个时刻开始，$u$和$v$的外部邻域交集非空，且在连续$\Delta$个时刻内其差异最多为$d$个元素。特别地，当$d=0$时，$u$和$v$可在$\Delta$个时刻内作为彼此的替代，以维持时变图$\mathcal G$中相同长度的旅程。我们提出一种分布式确定性算法，使每个节点能在$2p$轮内枚举其$(\Delta,d)$-孪生节点，使用大小为$O(\delta_\mathcal G\log n)$的消息，其中$n$为节点总数，$\delta_\mathcal G$为所有图$G_t$的最大度。此外，通过采用源自分布式哈希函数采样的随机化技术，我们以高概率将消息大小降低至$O(\log n)$。