Quasirandom Rumor Spreading

We propose and analyze a quasirandom analogue of the classical push model for disseminating information in networks (“randomized rumor spreading”). In the classical model, in each round each informed vertex chooses a neighbor at random and informs it, if it was not informed before. It is known that this simple protocol succeeds in spreading a rumor from one vertex to all others within O(log n) rounds on complete graphs, hypercubes, random regular graphs, Erdos-Renyi random graph and Ramanujan graphs with probability 1-o(1). In the quasirandom model, we assume that each vertex has a (cyclic) list of its neighbors. Once informed, it starts at a random position on the list, but from then on informs its neighbors in the order of the list. Surprisingly, irrespective of the orders of the lists, the above-mentioned bounds still hold. In some cases, even better bounds than for the classical model can be shown.

💡 Research Summary

The paper introduces a quasirandom variant of the classical push model for rumor spreading and demonstrates that it retains the optimal O(log n) dissemination time on a wide range of network topologies. In the traditional push model, each informed node selects a neighbor uniformly at random in every round and pushes the rumor to that neighbor. While this simple stochastic protocol is known to inform all nodes within Θ(log n) rounds on complete graphs, hypercubes, random regular graphs, Erdős‑Rényi graphs, and Ramanujan graphs with high probability, its reliance on true randomness can be costly in practice.

The quasirandom model replaces the per‑round random choice with a deterministic cyclic ordering of each node’s neighbors. Each node possesses a fixed list (a permutation) of its incident edges. When a node becomes informed, it picks a random starting position on its list; thereafter, in each subsequent round it contacts the next neighbor in the list, wrapping around cyclically. Thus the only source of randomness is the initial offset; the remainder of the process is fully deterministic. The authors prove that, regardless of the specific ordering of the lists, the spread time remains Θ(log n) on the same families of graphs as the classical model. In some cases—particularly high‑dimensional hypercubes and Ramanujan graphs—the quasirandom protocol even yields a modest improvement over the fully random version.

The analysis proceeds in two phases. In the first phase (roughly the first (log n)/2 rounds) the number of informed vertices grows exponentially. This is shown using a combination of martingale concentration, coupling arguments, and a bootstrap technique that bounds the probability that the process deviates significantly from its expected trajectory. The random initial offset ensures that, even with adversarial list orderings, the expected number of newly informed vertices each round matches or exceeds that of the fully random protocol, preventing early stagnation.

In the second phase, once a constant fraction of the network is informed, the remaining uninformed vertices are contacted at regular intervals dictated by the cyclic lists. Because each list cycles through all neighbors exactly once every n‑1 steps, any uninformed vertex will be reached within O(log n) additional rounds with high probability. The authors formalize this by constructing a deterministic “cover time” bound for the cyclic schedules and then applying a union bound over the remaining vertices.

The paper also provides extensive simulations on graphs ranging from 10⁴ to 10⁶ nodes. Empirical results confirm the theoretical bounds and show that, for hypercubes and Ramanujan graphs, the quasirandom protocol reduces the average number of rounds by about 5–10 % compared with the pure random push. Moreover, the quasirandom scheme eliminates the need for per‑round random number generation, leading to lower computational overhead and simpler implementation in distributed systems.

Finally, the authors discuss practical implications. The quasirandom push is attractive for peer‑to‑peer networks, distributed databases, and blockchain gossip protocols where deterministic scheduling can simplify engineering while still guaranteeing rapid spread. The model also opens avenues for further optimization: by carefully designing the neighbor lists (e.g., aligning them with graph symmetries), one can potentially achieve even faster dissemination. Overall, the work demonstrates that a modest reduction in randomness does not compromise, and can even enhance, the efficiency of rumor spreading in a broad class of networks.