Fast Gossip-based Rumor Spreading using Small Messages

Abstract: We study gossip algorithms for the fundamental rumor spreading problem, where the goal is to disseminate a rumor from a given source node to all nodes in an arbitrary (and unknown) graph. Gossip algorithms allow each node to call only one neighbor per round and are therefore highly message-efficient, with low per-node communication overhead per round. The state of the art present fast gossip algorithms, however they typically leverage large-sized messages. This undermines the light-weight communication advantage of gossip, since even though only one neighbor is contacted per round, the message size can be linear in $n$, the network size. Hence, a fundamental question is whether one can perform fast gossip using small messages. The main contribution of this paper is to answer the above question in the affirmative and present two gossip algorithms that achieve fast rumor spreading using messages of polylog{n} size. Specifically, we present the following algorithms: 1. An algorithm that runs in $O(c \log n / Φ_c)$ rounds for every $c \geq 1$, and $Φ_c$ is the weak conductance. Our bound in terms of weak conductance is essentially optimal. 2. An algorithm that depends on the network diameter (and is independent of the graph's conductance), which runs in $\tilde{O}(D+\sqrt{n})$ rounds with high probability. Our algorithm can be modified to output a minimum spanning tree (MST) in the same number of rounds, which is essentially round-optimal (even for non-gossip algorithms). Our gossip algorithms use graph sketches [Ahn, Guha, McGregor, SODA 2012] in a novel way to overcome communication bottlenecks and achieve small communication overhead with small message sizes.

• The paper presents a gossip algorithm achieving fast rumor spreading with O(polylog(n)) message sizes, tight with respect to weak conductance.
• It introduces a sparse backbone and cluster decomposition, reducing round complexity to approximately O(D+√n) in various graph settings.
• Graph sketching is employed to enable efficient inter- and intra-cluster communication, facilitating scalable, low-overhead distributed aggregation.

Fast Gossip-Based Rumor Spreading with Small Messages

Motivation and Context

Efficient dissemination of information is a core operation in distributed systems. Rumor spreading via gossip algorithms—where nodes iteratively contact neighbors to propagate messages—is a widely studied primitive due to its simplicity and scalability. The practical utility of gossip lies in message efficiency, achieved by restricting each node to contact a single neighbor per round (the $GOSSIP$ model). However, the strongest prior performance guarantees for fast gossip in general graphs have relied on messages whose size is linear in the network ($n$), undermining the lightweight communication that makes gossip attractive in large-scale systems.

This work directly addresses a fundamental theoretical barrier: Can fast gossip-based rumor spreading be accomplished using only small ($\operatorname{polylog}(n)$) messages? The paper answers affirmatively, presenting two algorithms that, for a broad class of graphs, match (or improve on) the fastest known gossip times while ensuring all messages are small, providing the first such results with tight bounds relative to graph conductance, weak conductance, and diameter.

Model and Problem Statement

The paper formalizes the problem in the $GOSSIP$ or $GOSSIP$-$CONGEST$ model:

• Nodes know only their own ID and the IDs of their neighbors (the KT1 model).
• The underlying topology is unknown to all nodes.
• In each synchronous round, each node can initiate communication with a single neighbor, exchanging a message of $O(\operatorname{polylog}(n))$ bits in each direction.
• The goal is to spread a rumor from a source node $s$ to all nodes, or equivalently, perform broadcast/aggregate functions in this setting.

Prior state-of-the-art for fast non-uniform gossip, particularly the Censor-Hillel & Shachnai algorithm, leveraged large messages to track the set of informed nodes and circumvent communication bottlenecks related to bottleneck cuts (low conductance regions). Uniform gossip (push/pull) with small messages does not suffice in these cases, incurring $\Omega(n \log n)$ round complexity in certain graphs.

Technical Contributions

1. Fast Rumor Spreading Parameterized by Weak Conductance

The first main result gives a $GOSSIP$-$n$0 algorithm for rumor spreading parameterized by weak conductance:

Theorem: There exists a (randomized) algorithm that, for any $n$1, solves rumor spreading in $n$2 rounds with high probability, where $n$3 denotes the weak conductance. All messages are of size $n$4 (2605.14376).

Core Techniques

• Cluster Decomposition and Sunflower Structures: The algorithm exploits the structural property that for any $n$5, the graph can be partitioned into $n$6 "sunflowers" (clusters, each with high internal conductance and few inter-cluster connections).
• Efficient Intra-Cluster Spreading: Within each cluster, modified uniform gossip spreads the rumor efficiently in $n$7 rounds—using only small messages—by leveraging the high conductance of the cluster.
• Graph Sketches for Outgoing Edges: To overcome the main challenge (identifying outgoing edges for cluster merging with limited communication), the algorithm uses graph sketches [Ahn, Guha, McGregor SODA 2012], compressing incidence information to $n$8 bits so roots can efficiently sample outgoing inter-cluster edges.
• Controlled Supercluster Growth: By grouping clusters into superclusters of bounded diameter and leveraging phase-wise diameter doubling (only clusters of appropriately small diameter are permitted to merge, growing exponentially in size each phase), the merging process completes in $n$9 phases, each costing $\operatorname{polylog}(n)$0 rounds.

Figure 1: A $\operatorname{polylog}(n)$1-barbell of $\operatorname{polylog}(n)$2 nodes—demonstrates separation of weak conductance from conductance; $\operatorname{polylog}(n)$3 cliques of size $\operatorname{polylog}(n)$4 connected in a path.

Asymptotic Optimality and Improvements

• No Message Size/Time Tradeoff: Unlike Censor-Hillel & Shachnai, which incur $\operatorname{polylog}(n)$5-factor round overhead and linear message size, this algorithm eliminates the $\operatorname{polylog}(n)$6 overhead and uses only small messages.
• Tight Lower Bounds: The algorithm is existentially (almost) optimal: for the $\operatorname{polylog}(n)$7-barbell graph Figure 1, no algorithm can improve the round bound by more than polylogarithmic factors.
• Exponential Speedup in Bottlenecked Graphs: In low-conductance but large weak-conductance instances (such as the $\operatorname{polylog}(n)$8-barbell), the improvement is exponential—in rounds and total communication—over uniform gossip.

2. Fast Rumor Spreading in General Graphs Parameterized by Diameter

The second principal result presents a $\operatorname{polylog}(n)$9-$GOSSIP$0 algorithm whose round complexity is $GOSSIP$1, matching (up to logs) the lower bounds in the $GOSSIP$2 model and improving over previous $GOSSIP$3-$GOSSIP$4 algorithms (notably, Ghaffari and Kuhn DISC’18, which give only $GOSSIP$5).

Core Techniques

• Sparse Backbone Construction: The algorithm computes a sparse subgraph $GOSSIP$6, with:
  • A small-diameter spanning forest in the high-degree regions.
  • Low-degree ($GOSSIP$7) spanners in the low-degree regions, incurring only $GOSSIP$8 multiplicative stretch.
  • Total time for backbone construction: $GOSSIP$9 rounds.
• Simulating Faster Communication: In low-degree areas, $GOSSIP$0, being $GOSSIP$1-regular, can simulate $GOSSIP$2 rounds ($GOSSIP$3-$GOSSIP$4 blowup only $GOSSIP$5 per round); spanning forest in dense parts ensures long-range communication unfolds in a bounded number of steps.
• Optimized Tradeoff: By carefully choosing sparsification thresholds, the algorithm achieves an additive $GOSSIP$6 term, outperforming prior approaches that did not exploit low-degree spanner structures for locality.

Consequences

• Diameter-Dominated (Sparse) Graphs: For $GOSSIP$7, this runtime is optimal up to polylogarithmic factors, resolving a longstanding open question for the $GOSSIP$8 model.
• MST and Aggregate Computation: The constructed spanning backbone supports leader election, MST construction, and exact aggregate computation in the same round bound, as it is used for efficient convergecast and broadcast.

The Graph Sketching Mechanism

Graph sketching (linear sketches) provides the crucial ingredient for achieving both small message size and fast cluster merging/aggregation:

• Each node encodes its incident edge information into a sketch vector.
• Sketches are linearly aggregatable (nodes sum their own and children's sketches via convergecast).
• The cluster root, having the aggregate, can (using shared randomness) probabilistically sample outgoing edges uniformly at random among those crossing between clusters—without knowing the full set of edges or their endpoints.
• All this is done with $GOSSIP$9 communication per aggregation.

This technique provides a distributed analogue to classical centralized cut-edge sampling and is essential for fast, message-efficient gossip in the presence of graph bottlenecks.

Numerical Results and Bounds

The algorithm achieves the following round and message bounds (all messages $CONGEST$0 bits):

Algorithm	Round bound	Message size per round
This work—weak conductance	$CONGEST$1	$CONGEST$2
[Censor-Hillel & Shachnai]	$CONGEST$3	$CONGEST$4
This work—general diameter ($CONGEST$5)	$CONGEST$6	$CONGEST$7
[Ghaffari & Kuhn DISC’18]	$CONGEST$8	$CONGEST$9
[CHKM SODA’12; Haeupler JACM’15]	$O(\operatorname{polylog}(n))$0	$O(\operatorname{polylog}(n))$1

Key strong/contradictory claims:

• The first non-uniform gossip algorithm to achieve $O(\operatorname{polylog}(n))$2 rumor spreading with small messages, strictly improving both complexity and communication over prior work that used large messages.
• The first gossip algorithm to achieve $O(\operatorname{polylog}(n))$3 rounds in the $O(\operatorname{polylog}(n))$4-$O(\operatorname{polylog}(n))$5 model, closing the gap with $O(\operatorname{polylog}(n))$6 lower bounds and matching non-gossip models up to logs.

Applications and Implications

• Aggregate Function Computation: The structure of the constructed broadcast tree and cluster decomposition enables efficient computation of min, max, sum, and average functions, as aggregation can be routed via the same low-diameter structures.
• MST and Leader Election: The backbone tree enables distributed MST construction and leader election in the same asymptotic round complexity, improving known bounds for these classic problems in lightweight gossip settings.
• Practical distributed systems: These results make lightweight, low-overhead, and scalable rumor spreading practical without incurring the hidden cost of large messages.

Future Directions

The main theoretical open question is to determine whether the $O(\operatorname{polylog}(n))$7 bound for $O(\operatorname{polylog}(n))$8-based rumor spreading (and associated problems) is tight. The authors conjecture that this is indeed optimal up to polylogarithmic factors.

Further directions include:

• Removing or relaxing the limited knowledge assumptions (such as knowledge of $O(\operatorname{polylog}(n))$9, $s$0, and $s$1), or making the algorithms self-stabilizing.
• Extending techniques to dynamic, asynchronous, or non-synchronous models.
• Exploring explicit implementation and engineering of graph sketch primitives for practical protocol stacks in real networks.

Conclusion

This paper establishes that gossip-based rumor spreading—long regarded as efficient only at the expense of large messages or slowed by graph bottlenecks—can be fast and lightweight, achieving essentially optimal round complexity even in poorly connected or bottlenecked graphs. The main innovations are a novel combination of sunflower/cluster decompositions for weak conductance and the application of distributed graph sketching for efficient inter-cluster communication, all within the strict message constraints of the $s$2-$s$3 model. As such, the work bridges the theory/practice gap for lightweight distributed information dissemination and sets the stage for further advances in decentralized network algorithms (2605.14376).