Lower Bounds on Information Dissemination in Dynamic Networks

We study lower bounds on information dissemination in adversarial dynamic networks. Initially, k pieces of information (henceforth called tokens) are distributed among n nodes. The tokens need to be broadcast to all nodes through a synchronous network in which the topology can change arbitrarily from round to round provided that some connectivity requirements are satisfied. If the network is guaranteed to be connected in every round and each node can broadcast a single token per round to its neighbors, there is a simple token dissemination algorithm that manages to deliver all k tokens to all the nodes in O(nk) rounds. Interestingly, in a recent paper, Dutta et al. proved an almost matching Omega(n + nk/log n) lower bound for deterministic token-forwarding algorithms that are not allowed to combine, split, or change tokens in any way. In the present paper, we extend this bound in different ways. If nodes are allowed to forward b < k tokens instead of only one token in every round, a straight-forward extension of the O(nk) algorithm disseminates all k tokens in time O(nk/b). We show that for any randomized token-forwarding algorithm, Omega(n + nk/(b^2 log n log log n)) rounds are necessary. If nodes can only send a single token per round, but we are guaranteed that the network graph is c-vertex connected in every round, we show a lower bound of Omega(nk/(c log^{3/2} n)), which almost matches the currently best O(nk/c) upper bound. Further, if the network is T-interval connected, a notion that captures connection stability over time, we prove that Omega(n + nk/(T^2 log n)) rounds are needed. The best known upper bound in this case manages to solve the problem in O(n + nk/T) rounds. Finally, we show that even if each node only needs to obtain a delta-fraction of all the tokens for some delta in [0,1], Omega(nk delta^3 log n) are still required.

💡 Research Summary

The paper investigates fundamental lower bounds for the classic token dissemination problem in adversarially dynamic networks. The setting consists of n nodes initially holding k distinct tokens, which must be spread to all nodes through synchronous rounds. The network topology may change arbitrarily each round, but must satisfy certain connectivity constraints. The study focuses on token‑forwarding algorithms that are not allowed to combine, split, or code tokens, reflecting a realistic limitation for many distributed systems.

First, the authors recall the simple O(nk) algorithm that works when the graph is connected in every round and each node can broadcast a single token per round. They also reference the recent deterministic lower bound Ω(n + nk/ log n) by Dutta et al. Their contribution is to extend these bounds under several natural extensions of the model.

Multiple tokens per round (b < k).
If each node may forward up to b tokens per round, a straightforward extension of the O(nk) algorithm yields O(nk/b) rounds. The paper proves that any randomized token‑forwarding algorithm still requires Ω(n + nk/(b² log n log log n)) rounds. The proof uses a potential‑function argument: the total “information potential” (the sum over all nodes of distinct tokens known) can increase by at most O(b² log n log log n) per round under an adversarially chosen connected graph. Hence, to accumulate the required nk potential, at least the stated number of rounds is necessary.

c‑vertex connectivity.
When the network is guaranteed to be c‑vertex‑connected each round, there exist at least c internally disjoint paths that could be used for parallel dissemination. Nevertheless, the authors show a lower bound of Ω(nk/(c log^{3/2} n)). The argument again bounds the per‑round increase of potential by O(c log^{3/2} n), reflecting the fact that even with many disjoint routes, the adversary can arrange the topology to limit the spread of new tokens. This bound almost matches the best known upper bound O(nk/c).

T‑interval connectivity.
The notion of T‑interval connectivity requires that the graph remains connected over any window of T consecutive rounds. The paper proves a lower bound of Ω(n + nk/(T² log n)). The analysis shows that over a T‑round window the potential can increase by at most O(T² log n), so the total time must scale inversely with T² up to logarithmic factors. The best known algorithm solves the problem in O(n + nk/T) rounds, leaving only a log n gap.

Partial dissemination (δ‑fraction).
Finally, the authors consider a relaxed goal: each node only needs to acquire a δ‑fraction of the k tokens (0 ≤ δ ≤ 1). Even in this weaker setting, they establish a lower bound of Ω(nk δ³ log n). The proof adapts the potential method to the reduced target, showing that the adversary can still keep the growth of known tokens limited enough that the same order of rounds is required, scaled by δ³.

Across all results, the central technical tool is a carefully designed potential function that captures the total amount of distinct token knowledge in the system. By constructing adversarial sequences of connected graphs that minimize the increase of this potential while respecting the given connectivity constraints, the authors derive tight lower bounds for a wide range of parameters. The bounds reveal that, in pure token‑forwarding models, logarithmic factors are unavoidable regardless of how many tokens a node can send per round or how robust the connectivity is. Consequently, achieving substantially faster dissemination in dynamic networks likely requires more powerful techniques such as network coding, token combination, or richer communication primitives.