RC-Gossip: Dynamic Distributed Averaging

• RC-Gossip is a class of distributed protocols that modify information mixing by adjusting communication rates through non-uniform clock distributions and dynamic recurrence strategies.
• These methods accelerate consensus by targeting stale nodes and optimizing edge-selection probabilities, thereby improving information freshness in networks.
• The approaches balance increased performance in convergence speed and dissemination efficiency with additional complexity in memory, synchrony, and parameter tuning.

Rate-Changing Gossip (RC-Gossip) denotes a class of gossip-based distributed information dissemination and averaging mechanisms in which the effective mixing dynamics are deliberately altered, rather than kept fixed as in classical uniform pairwise gossip. In the narrowest sense, RC-Gossip refers to protocols with non-uniform node activation rates realized by non-uniform Poisson clocks and optimized edge-selection probabilities (Jafarizadeh, 2015). In a broader sense, the term also covers time-varying polynomial gossip recurrences, momentum-augmented accelerated gossip, and freshness-oriented schemes that retarget transmission effort toward stale nodes, thereby changing the effective contraction, propagation, or freshness rate over time (Berthier et al., 2018, Loizou et al., 2018, Hasan et al., 4 Aug 2025).

1. Conceptual scope and defining interpretations

The most explicit introduction of the term appears in “RC-Gossip: Information Freshness in Clustered Networks with Rate-Changing Gossip” (Hasan et al., 4 Aug 2025), where gossip is directed toward nodes that are still stale, so that already-fresh nodes are excluded from further targeting within the current source-version epoch. In that formulation, the rate change is state dependent: as the number of stale nodes shrinks, the per-stale-node update hazard increases.

A narrower interpretation appears in “Optimizing the Gossip Algorithm with Non-Uniform Clock Distribution over Classical & Quantum Networks” (Jafarizadeh, 2015). There, RC-Gossip means that the node activation probabilities themselves are design variables. If node $i$ has Poisson clock rate $\lambda_i$, then the probability that node $i$ initiates the next gossip event is

$P_i=\frac{\lambda_i}{\sum_j \lambda_j},$

so changing $\lambda_i$ changes the communication rate schedule across the network.

A broader interpretation is justified by accelerated consensus papers that do not change who communicates how often, but do change the effective iteration dynamics. In “Accelerated Gossip in Networks of Given Dimension using Jacobi Polynomial Iterations” (Berthier et al., 2018), the coefficients of the distributed second-order recurrence vary with the iteration index $t$. In “Provably Accelerated Randomized Gossip Algorithms” (Loizou et al., 2018), acceleration is achieved through extra node state and momentum-like parameters $\alpha_k,\beta_k,\gamma_k$, with Option 1 explicitly time varying. This suggests that RC-Gossip is best understood as a family of protocols in which the effective rate of information mixing is modulated either by communication scheduling, by time-varying linear coefficients, or by state augmentation.

2. Consensus foundations and the linear-system view

A central foundation for RC-Gossip is the average-consensus problem on a connected undirected graph

$G=(V,E), \qquad V=\{1,2,\dots,n\},$

where node $i$ initially stores a private scalar $c_i\in\mathbb R$, and all nodes seek the average

$\lambda_i$0

With $\lambda_i$1, consensus means convergence to $\lambda_i$2 (Loizou et al., 2018).

The key structural observation is that consensus can be written as a linear system $\lambda_i$3. With the normalized incidence matrix, an edge $\lambda_i$4 contributes the row

$\lambda_i$5

so $\lambda_i$6 is exactly the constraint $\lambda_i$7 (Loizou et al., 2018). This equivalence leads to the identity

$\lambda_i$8

and, correspondingly,

$\lambda_i$9

(Loizou et al., 2018).

The broader unifying formalism is the sketch-and-project framework developed in “Revisiting Randomized Gossip Algorithms” (Loizou et al., 2019). With a random sketch $i$0, weight matrix $i$1, and relaxation parameter $i$2, the generic update is

$i$3

Under exactness and $i$4, the convergence rate is

$i$5

where

$i$6

This linear-system viewpoint is important because it makes several RC-Gossip mechanisms precise: changing the sketch distribution, block size, relaxation, path length, or momentum changes $i$7, hence changes the effective mixing rate (Loizou et al., 2019).

For ordinary randomized pairwise gossip, the update recovered from randomized Kaczmarz is

$i$8

and the standard rate bound is

$i$9

(Loizou et al., 2018).

3. Principal mechanisms of rate change

Several distinct mechanisms implement rate change in the literature.

Mechanism	Representative formula	Representative paper
Non-uniform clock distribution	$P_i=\frac{\lambda_i}{\sum_j \lambda_j},$0	(Jafarizadeh, 2015)
Time-varying polynomial recursion	$P_i=\frac{\lambda_i}{\sum_j \lambda_j},$1	(Berthier et al., 2018)
Momentum/state augmentation	$P_i=\frac{\lambda_i}{\sum_j \lambda_j},$2	(Loizou et al., 2018)
Stale-node targeting	$P_i=\frac{\lambda_i}{\sum_j \lambda_j},$3, $P_i=\frac{\lambda_i}{\sum_j \lambda_j},$4	(Hasan et al., 4 Aug 2025)

In the non-uniform-clock formulation, the expected gossip operator is

$P_i=\frac{\lambda_i}{\sum_j \lambda_j},$5

with effective symmetric edge weights

$P_i=\frac{\lambda_i}{\sum_j \lambda_j},$6

and optimizing convergence means minimizing $P_i=\frac{\lambda_i}{\sum_j \lambda_j},$7 subject to the probability constraints (Jafarizadeh, 2015). The same work shows that uniform clock distribution is generally suboptimal and that the non-uniform optimum is not unique. It also provides a detailed-balance construction from optimal continuous-time consensus weights: $P_i=\frac{\lambda_i}{\sum_j \lambda_j},$8 which yields $P_i=\frac{\lambda_i}{\sum_j \lambda_j},$9 and

$\lambda_i$0

(Jafarizadeh, 2015).

In the Jacobi-polynomial approach, the update is explicitly time varying: $\lambda_i$1 with

$\lambda_i$2

Here the coefficient schedule depends on the iteration index $\lambda_i$3 and on the spectral dimension parameter $\lambda_i$4, not on the spectral gap (Berthier et al., 2018). The method is simultaneously a polynomial filter and a second-order distributed recursion.

In the accelerated Kaczmarz-based gossip construction, each node stores two local states $\lambda_i$5 and $\lambda_i$6, forms

$\lambda_i$7

and when edge $\lambda_i$8 is selected, only nodes $\lambda_i$9 and $t$0 exchange $t$1, while all nodes update their local registers (Loizou et al., 2018). The selected pair performs exact averaging on the $t$2-values,

$t$3

but the overall dynamics are modified by $t$4 and $t$5. Option 1 uses iteration-dependent parameters, while Option 2 uses constants.

A common misconception is that every RC-Gossip method changes communication frequency or edge-activation probabilities. The acceleration literature shows a more nuanced picture: some protocols change the effective rate of convergence through time-varying recurrences or momentum while keeping one uniformly sampled edge per iteration (Loizou et al., 2018, Berthier et al., 2018).

4. Freshness-oriented RC-Gossip in clustered networks

A distinct line of work studies RC-Gossip as a freshness protocol rather than a consensus solver. In the clustered model of (Hasan et al., 4 Aug 2025), a source updates according to a Poisson process of rate $t$6, $t$7 end-nodes are partitioned into $t$8 clusters of size $t$9, each cluster has a clusterhead, and dissemination proceeds through source-to-clusterhead rate $\alpha_k,\beta_k,\gamma_k$0, clusterhead-to-node rate $\alpha_k,\beta_k,\gamma_k$1, and optionally node-to-node gossip rate $\alpha_k,\beta_k,\gamma_k$2.

The performance metric is the Binary Freshness Metric

$\alpha_k,\beta_k,\gamma_k$3

with long-term average

$\alpha_k,\beta_k,\gamma_k$4

A renewal-reward argument yields the exact identity

$\alpha_k,\beta_k,\gamma_k$5

where $\alpha_k,\beta_k,\gamma_k$6 is the probability that node $\alpha_k,\beta_k,\gamma_k$7 receives the current version before the next source self-update (Hasan et al., 4 Aug 2025).

The defining RC mechanism is stale-node targeting. In a non-hierarchical disconnected network, if $\alpha_k,\beta_k,\gamma_k$8 nodes are already fresh, then each of the remaining $\alpha_k,\beta_k,\gamma_k$9 stale nodes is updated directly by the source at rate

$G=(V,E), \qquad V=\{1,2,\dots,n\},$0

In a fully connected network, each stale node receives source updates at rate

$G=(V,E), \qquad V=\{1,2,\dots,n\},$1

and gossip updates at rate

$G=(V,E), \qquad V=\{1,2,\dots,n\},$2

because fresh nodes push only to stale nodes (Hasan et al., 4 Aug 2025). This removes redundant fresh-to-fresh transmissions.

For disconnected networks, the difference between fixed-rate and RC dissemination is explicit: $G=(V,E), \qquad V=\{1,2,\dots,n\},$3 while

$G=(V,E), \qquad V=\{1,2,\dots,n\},$4

The paper shows

$G=(V,E), \qquad V=\{1,2,\dots,n\},$5

for $G=(V,E), \qquad V=\{1,2,\dots,n\},$6 (Hasan et al., 4 Aug 2025).

For fully connected networks with RC at both source and gossip layers,

$G=(V,E), \qquad V=\{1,2,\dots,n\},$7

In clustered networks, the renewal decomposition factors freshness into source-to-clusterhead and clusterhead-to-node stages: $G=(V,E), \qquad V=\{1,2,\dots,n\},$8 This yields closed-form expressions for clustered disconnected and clustered fully connected variants, and the numerical study shows nontrivial optimal cluster sizes (Hasan et al., 4 Aug 2025).

5. Scaling laws under different rates and topologies

RC-Gossip is also studied through age and freshness scaling laws. In “The Age of Gossip in Networks” (Yates, 2021), the discrete freshness variable is version age,

$G=(V,E), \qquad V=\{1,2,\dots,n\},$9

and the main analytical device is subset age

$i$0

For fixed heterogeneous rates $i$1 and $i$2, the stationary recursion is

$i$3

On a symmetric complete graph, the average version age of each node grows as $i$4, whereas the ring topology is numerically reported to scale approximately as $i$5 when $i$6 (Yates, 2021).

With the binary freshness metric, topology and rate interact differently. In “Gossiping with Binary Freshness Metric” (Bastopcu et al., 2021), the key ratio is

$i$7

For a disconnected network,

$i$8

so freshness decays as $i$9. For a ring,

$c_i\in\mathbb R$0

For a fully connected network,

$c_i\in\mathbb R$1

Thus, when peer gossip is sufficiently fast relative to source evolution, the fully connected topology changes the asymptotic exponent itself (Bastopcu et al., 2021).

A different RC mechanism arises from topology switching. In “Age of Gossip With Time-Varying Topologies” (Srivastava et al., 2024), the active graph state is a finite-state CTMC. If one CTMC state is the fully connected graph and the CTMC transition rates are constant in $c_i\in\mathbb R$2, then the version age of a typical node still scales logarithmically with $c_i\in\mathbb R$3, as in an always-fully-connected network (Srivastava et al., 2024). The same work also reports numerically that when the CTMC rates scale with $c_i\in\mathbb R$4, the age scaling can worsen and reflect the switching timescale.

A further caution appears in “The Role of Gossiping for Information Dissemination over Networked Agents” (Bastopcu et al., 2022). There the dissemination metric is average correctness under a majority rule, and the peer gossip rate is a tunable parameter $c_i\in\mathbb R$5. In the high-gossip regime, if the correct information does not already hold a majority after source seeding, gossip can amplify incorrect information; in the low-gossip regime, the paper derives the gossip-gain approximation

$c_i\in\mathbb R$6

This shows that more gossip is not universally better: its value depends on the seeding mechanism, the current fraction of correct nodes, and the source-change rate (Bastopcu et al., 2022).

6. Assumptions, limitations, and adjacent formulations

The RC-Gossip literature spans several modeling regimes, and the differences are substantial. Accelerated randomized gossip via Kaczmarz requires all nodes to update every iteration, synchronized clocks, and knowledge of parameters such as $c_i\in\mathbb R$7; inactive nodes update locally even when only one edge communicates (Loizou et al., 2018). The Jacobi-polynomial iteration likewise assumes a synchronous linear setting with globally scheduled coefficients $c_i\in\mathbb R$8, and its strongest theory is for graphs of known spectral dimension (Berthier et al., 2018). Freshness analyses based on SHS or renewal reward generally assume memoryless Poisson update processes and either fixed rates or explicitly specified state dependence (Yates, 2021, Hasan et al., 4 Aug 2025).

These distinctions matter because the phrase “rate changing” can refer to different objects. It can mean non-uniform activation clocks, as in the Poisson-clock optimization framework (Jafarizadeh, 2015). It can mean a nonstationary second-order filter, as in Jacobi gossip (Berthier et al., 2018). It can mean state-dependent stale-node targeting, as in clustered freshness RC-Gossip (Hasan et al., 4 Aug 2025). It can also arise from scheduled alternation between communication modes, as in periodic global averaging for decentralized optimization, where sparse gossip steps are punctuated by exact global averages every $c_i\in\mathbb R$9 iterations (Chen et al., 2021).

Several adjacent literatures reinforce this broader interpretation. “Optimal Gossip with Direct Addressing” (Haeupler et al., 2014) achieves $\lambda_i$00-round dissemination through a strongly phase-dependent sequence of random recruitment, cluster growth, cluster merging, and final pull-based cleanup. “Gossip in a Smartphone Peer-to-Peer Network” (Newport, 2017) studies a dynamic model with stability factor $\lambda_i$01, where topology may change completely every round when $\lambda_i$02. “Acceleration of Gossip Algorithms through the Euler-Poisson-Darboux Equation” (Berthier et al., 2022) shows that standard local gossip has a heat-equation limit with spatial spread scale $\lambda_i$03, whereas accelerated Jacobi gossip has an Euler–Poisson–Darboux limit with support radius growing linearly in $\lambda_i$04. A plausible implication is that RC-Gossip should not be identified with a single primitive; it is better understood as a family of methods that modify either communication scheduling, state dimension, coefficient schedules, or topology exposure so as to alter effective propagation and convergence rates.

Across these variants, the recurring tradeoff is between stronger effective mixing and stronger implementation assumptions. Non-uniform clocks enlarge the feasible design space but require rate assignment and probability optimization (Jafarizadeh, 2015). Polynomial and momentum accelerations improve spectral dependence but require extra memory and synchrony (Berthier et al., 2018, Loizou et al., 2018). Stale-node targeting improves freshness and avoids redundant transmissions, but it presumes knowledge of which nodes are stale within a source-version epoch (Hasan et al., 4 Aug 2025). Topology switching can preserve favorable age scaling if good states recur often enough, but $\lambda_i$05-dependent switching rates can negate that advantage (Srivastava et al., 2024). These constraints define the present boundaries of RC-Gossip as a research area.