Causality, Influence, and Computation in Possibly Disconnected Dynamic Networks

In this work, we study the propagation of influence and computation in dynamic distributed systems. We focus on broadcasting models under a worst-case dynamicity assumption which have received much attention recently. We drop for the first time in worst-case dynamic networks the common instantaneous connectivity assumption and require a minimal temporal connectivity. Our temporal connectivity constraint only requires that another causal influence occurs within every time-window of some given length. We establish that there are dynamic graphs with always disconnected instances with equivalent temporal connectivity to those with always connected instances. We present a termination criterion and also establish the computational equivalence with instantaneous connectivity networks. We then consider another model of dynamic networks in which each node has an underlying communication neighborhood and the requirement is that each node covers its local neighborhood within any time-window of some given length. We discuss several properties and provide a protocol for counting, that is for determining the number of nodes in the network.

💡 Research Summary

This paper investigates information propagation and distributed computation in worst‑case dynamic networks without assuming any instantaneous connectivity. Instead, the authors introduce a minimal temporal connectivity requirement: in every time window of a given length at least one new causal influence must occur. They formalize this notion through two metrics – outgoing influence time (oit) and incoming influence time (iit). The oit of a node is the smallest number k such that, starting from any round t, the node’s state influences at least one previously uninfluenced node within k rounds; iit is defined symmetrically for receiving influence.

The authors first ask whether unit oit (oit = 1) necessarily implies that each instantaneous graph is connected. By constructing explicit dynamic graphs they show the answer is negative. A simple construction uses a ring of 2ℓ nodes with alternating perfect matchings, yielding always‑disconnected snapshots yet achieving oit = 1. To improve edge‑periodicity, they employ Soifer’s geometric edge‑coloring of a complete graph: for an even‑sized node set they assign n − 1 colors to the edges and schedule each color in a distinct round. This yields a dynamic graph where every edge reappears only after at least n − 1 rounds, the snapshots are always disconnected, and nevertheless every node influences a new node in each round, i.e., unit oit. Hence, a dynamic network can guarantee the fastest possible spread of information without ever being momentarily connected.

Next, the paper derives a termination criterion based on the known oit value. If a node knows that the network’s oit is k, then after it has already been influenced ℓ times, the next new influence must occur within at most k·ℓ·(ℓ + 1)/2 rounds. This bound allows any algorithm to detect global termination without additional synchronisation. The authors also show that the incoming influence time can be as large as O(k·n) even when oit = k, demonstrating an asymmetry between spreading and being spread to.

The authors prove that any algorithm designed for 1‑interval‑connected graphs (the classic model where every time window of length T contains a static spanning connected subgraph) can be adapted to work on graphs with arbitrary oit = k by scaling the time parameters by a factor of k. Consequently, fundamental tasks such as flooding, routing, and all‑to‑all token dissemination retain their asymptotic complexities under the weaker temporal connectivity assumption.

A second model, called “local communication windows,” assumes a fixed underlying communication graph G₀. The adversary may delete edges arbitrarily, but must guarantee that in every window of length τ each node communicates with all of its static neighbors at least once. This captures realistic scenarios where devices periodically encounter each other (e.g., mobile phones, robots). The authors prove basic properties of this model, including bounds on dynamic diameter, and present a counting protocol that determines the exact network size n. The protocol works by each node aggregating counts received from its neighbors; when a node observes no change for τ consecutive rounds it can safely output n. The algorithm terminates in O(n·τ) rounds, uses O(log n) bits per message, and requires only constant local memory.

Overall, the paper makes several key contributions: (1) it introduces the minimal temporal connectivity notion (oit/iit) and shows that unit oit does not imply instantaneous connectivity; (2) it provides a constructive method (geometric edge‑coloring) to achieve optimal influence spread on always‑disconnected dynamic graphs; (3) it derives a practical termination condition based solely on oit; (4) it establishes computational equivalence between the new model and the classic T‑interval‑connected model; and (5) it defines the local communication windows model and supplies a concrete counting algorithm. The work bridges a gap between theoretical worst‑case dynamic network models and practical, highly dynamic environments, and opens several avenues for future research, such as heterogeneous oit values, probabilistic adversaries, and experimental validation on real mobile platforms.