Structural Sign Herdability in Temporally Switching Networks with Fixed Topology

TL;DR

This paper demonstrates that temporally switching networks with fixed topology can achieve complete herdability under conditions where static networks cannot, highlighting the advantages of temporal dynamics.

Contribution

It reveals that fixed-topology temporal networks can attain herdability with fewer snapshots than static networks, introducing a new mechanism to overcome classical herdability obstructions.

Findings

Temporal switching networks achieve complete herdability even with signed or layer dilations.

Two snapshots suffice for herdability when all snapshots share the same topology.

Temporal dynamics provide a structural advantage over static networks.

Abstract

This paper investigates structural herdability in a special class of temporally switching networks with fixed topology. We show that when the underlying digraph remains unchanged across all snapshots, the network attains complete SS herdability even in the presence of signed or layer dilations, a condition not applicable to static networks. This reveals a fundamental structural advantage of temporal dynamics and highlights a novel mechanism through which switching can overcome classical obstructions to herdability. To validate these conclusions, we utilize a more relaxed form of sign matching within each snapshot of the temporal network. Furthermore, we show that when all snapshots share the same underlying topology, the temporally switching network achieves $\mathcal{SS}$ herdability within just two snapshots, which is fewer than the number required for structural controllability. Several examples are included to demonstrate these results.