General Epidemic Push-Pull Protocol (GEP3)

• General Epidemic Push-Pull Protocol (GEP3) is an umbrella model for epidemic dissemination using push, pull, or combined push-pull interactions in various network settings.
• It integrates synchronous, asynchronous, and random phone-call models to analyze dissemination time, message complexity, and freshness maintenance.
• GEP3 provides actionable design insights for optimizing protocols across evolving graphs, complete networks, and SIS-type systems using rigorous theoretical bounds.

Searching arXiv for recent and foundational papers relevant to epidemic push–pull protocols and GEP3-style formulations. I’m going to look up arXiv entries on push–pull rumor spreading, epidemic dissemination, and push-pull gossip age analysis to ground the article in the current arXiv record. General Epidemic Push–Pull Protocol (GEP3) denotes an epidemic-style dissemination framework in which networked nodes propagate state by push, pull, or combined push-pull interactions. The literature represented by rumor spreading, epidemic dissemination, and age-of-information analyses suggests that GEP3 is best understood as an umbrella abstraction rather than a single canonical named algorithm: in one line of work, each node chooses neighbors and exchanges a rumor in synchronous rounds on evolving graphs; in another, contacts occur asynchronously according to independent Poisson processes; in a third, push and pull are treated as controllable primitives in generalized random phone-call models and freshness-maintenance systems (Daknama, 2017, Srivastava et al., 2024, Mercier et al., 2017).

1. Protocol semantics and core variants

In the standard synchronous rumor-spreading formulation, one node is informed initially. In Push, every informed node chooses a neighbor uniformly at random and informs it if the chosen neighbor is uninformed. In Pull, each uninformed node chooses a neighbor uniformly at random and becomes informed if that neighbor is informed. In Push&Pull, every node picks a neighbor uniformly at random, and if at least one endpoint knows the rumor at the beginning of the round, then after the round both endpoints are informed (Daknama, 2017).

A broader GEP3 formulation appears in the generalized random phone-call model. At each communication round, each process may call between $0$ and $in$ processes uniformly at random to request a rumor, may call between $0$ and $out$ processes uniformly at random to push a rumor, and has the option not to answer pull requests. Establishing the communication is free, and only messages that contain the rumor are counted. This model is fully connected, synchronous, and address-oblivious, and it generalizes the earlier “polite” model by allowing a process to refuse to push, refuse to answer pull requests, or refuse to request a rumor in a given round (Mercier et al., 2017).

An asynchronous GEP3 formulation is given in the age-of-gossip setting. For each ordered pair of nodes $(i,j)$, there may be a pull Poisson process of rate $\lambda_{ij}^{\text{pull}}$ and a push Poisson process of rate $\lambda_{ij}^{\text{push}}$. A node-to-node contact on edge $(i,j)$ then occurs at combined rate

$$\lambda_{ij}=\lambda_{ji}^{\text{pull}}+\lambda_{ij}^{\text{push}},$$

and the receiver adopts the fresher packet. In that setting, GEP3 is literally an epidemic push-pull protocol with arbitrary link-specific rates (Srivastava et al., 2024).

The literature therefore uses “push,” “pull,” and “push-pull” in two closely related senses. In rumor spreading, the state variable is typically binary—uninformed or informed. In freshness-maintenance systems, the state is versioned and ordered, so the effective rule is not merely “infect if one endpoint is informed,” but “replace stale state by fresher state.” This suggests that GEP3 is most naturally defined by its interaction rule rather than by a single state representation.

2. State variables, network models, and analytical formalisms

A central formalism for synchronous GEP3 on dynamic sparse networks is the homogeneous rumor spreading process. Let $I_t$ be the number of informed nodes at the end of round $in$0, with $in$1. For each $in$2, every uninformed node becomes informed in that round with probability $in$3, depending only on $in$4, and pairwise dependence is controlled by a covariance bound $in$5. The rumor spreading time is

$in$6

often written $in$7. This abstraction is used for evolving Erdős–Rényi graphs $in$8, especially in the sparse regime $in$9 with $0$0 constant (Daknama, 2017).

For Pull in $0$1, conditioned on an uninformed node $0$2 having at least one neighbor, the chosen neighbor is informed with probability $0$3, while an isolated $0$4 cannot become informed. Using

$0$5

the per-node success probability becomes

$0$6

For Push&Pull, if $0$7 denotes “$0$8 is pushed” and $0$9 denotes “$out$0 pulls,” then

$out$1

The overlap term is essential because push and pull are not independent on the same sampled graph (Daknama, 2017).

A second formalism, developed for fully connected networks with fixed fanout $out$2, models one push round as a state-dependent random walk. If $out$3 is the number of newly infected nodes after $out$4 selections, then for $out$5,

$out$6

and $out$7 is the number of newly infected nodes in the round. This representation yields recursive formulas for the distribution and moments of $out$8, fluid limits, diffusion limits, and asymptotic normality of the number of newly infected nodes in a push round (Caglar et al., 2014).

A third formalism treats GEP3 as an SIS-type nonlinear dynamical system on an arbitrary graph $out$9. With pull infection probability $(i,j)$0, push infection capability $(i,j)$1, cure capability $(i,j)$2, and node infection probabilities $(i,j)$3, the update is

$(i,j)$4

For push-only, the sufficient epidemic-threshold condition becomes

$(i,j)$5

where $(i,j)$6 is the largest eigenvalue of the adjacency matrix. The same framework also gives upper and lower bounds on the global mean infection rate and shows that it can be estimated by monitoring a small constant number of nodes, without knowing the parameter values (Xu et al., 2016).

These formalisms are complementary. The homogeneous-process framework yields precise broadcast-time asymptotics on evolving sparse random graphs; the random-walk and fluid-limit approach resolves round-level growth on complete graphs; the nonlinear dynamical-systems model captures persistent push- and pull-based spreading on arbitrary topologies. Together they define the main mathematical vocabulary of GEP3.

3. Dissemination-time asymptotics and efficiency regimes

For evolving Erdős–Rényi graphs $(i,j)$7, the expected rumor spreading times of Push, Pull, and Push&Pull are known explicitly. In this setting,

$(i,j)$8

$(i,j)$9

and

$\lambda_{ij}^{\text{pull}}$0

All three protocols also satisfy exponential concentration bounds of the form

$\lambda_{ij}^{\text{pull}}$1

for suitable constants $\lambda_{ij}^{\text{pull}}$2 (Daknama, 2017).

These formulas separate the spread into an early growth phase and a late shrinking phase. For Pull, the early term $\lambda_{ij}^{\text{pull}}$3 reflects exponential growth with factor roughly $\lambda_{ij}^{\text{pull}}$4, while the late term $\lambda_{ij}^{\text{pull}}$5 comes from the isolation bottleneck. For Push&Pull, the early phase is faster because the effective linear growth factor is $\lambda_{ij}^{\text{pull}}$6, but the gain is reduced by the overlap correction $\lambda_{ij}^{\text{pull}}$7. The analysis therefore shows that push and pull do not add independently on a sparse evolving graph; they “waste effort on the same targets” in the early phase (Daknama, 2017).

In the fully connected generalized random phone-call model, a different asymptotic regime appears. The regular pull algorithm disseminates a rumor to all processes with high probability in

$\lambda_{ij}^{\text{pull}}$8

rounds of communication. If $\lambda_{ij}^{\text{pull}}$9, then the total number of rumor-carrying messages is $\lambda_{ij}^{\text{push}}$0, and when $\lambda_{ij}^{\text{push}}$1 the message count is exactly $\lambda_{ij}^{\text{push}}$2. A regular push-then-pull algorithm achieves $\lambda_{ij}^{\text{push}}$3 rounds and $\lambda_{ij}^{\text{push}}$4 messages of size $\lambda_{ij}^{\text{push}}$5, which the paper states are asymptotically optimal (Mercier et al., 2017).

The same work also records a controversy in the classical literature. Two widely cited lower bounds of Karp et al. are contradicted because the original random phone-call model allowed processes not to share the rumor once communication was established, whereas the proofs implicitly assumed that they always did. The corrected interpretation is that selective non-transmission—especially stopping push once the rumor is old—is precisely what makes $\lambda_{ij}^{\text{push}}$6 message complexity possible in $\lambda_{ij}^{\text{push}}$7 rounds (Mercier et al., 2017).

On complete graphs, round-level growth can also be compared directly for push and pull. With initial susceptible fraction $\lambda_{ij}^{\text{push}}$8 and fanout $\lambda_{ij}^{\text{push}}$9, the limiting expected fraction of newly infected nodes in one push round is

$(i,j)$0

while for pull it is

$(i,j)$1

Because

$(i,j)$2

a pull round infects slightly more nodes on average than a push round when the initial infected fraction is nonzero (Caglar et al., 2014).

Taken together, these results identify the core efficiency regimes of GEP3. Pull dominates the late stage on sparse evolving graphs and can be asymptotically optimal in message complexity on complete graphs. Push is especially useful when the rumor is young. Push&Pull accelerates the early phase, but in sparse dynamic settings its gain is strictly smaller than the naive sum of push and pull because of overlap.

4. Freshness maintenance and age-of-information formulations

GEP3 is not only a dissemination protocol; it is also a freshness-maintenance mechanism. In the age-of-gossip framework, each node $(i,j)$3 stores a version index $(i,j)$4, the source holds $(i,j)$5, and the version age is

$(i,j)$6

For a subset $(i,j)$7, the set version age is

$(i,j)$8

and the limiting average version age is

$(i,j)$9

The stochastic hybrid systems analysis yields the recursive equation

$$\lambda_{ij}=\lambda_{ji}^{\text{pull}}+\lambda_{ij}^{\text{push}},$$0

which expresses the age of a set in terms of its one-node supersets (Srivastava et al., 2024).

This formulation yields a sharp scaling distinction. Prior work had already shown that push-only gossip on a fully connected network has

$$\lambda_{ij}=\lambda_{ji}^{\text{pull}}+\lambda_{ij}^{\text{push}},$$1

The newer analysis shows that pull and push-pull can achieve constant version age, while push-only cannot beat the logarithmic scaling. It further proves the monotonicity property

$$\lambda_{ij}=\lambda_{ji}^{\text{pull}}+\lambda_{ij}^{\text{push}},$$2

for every gossip network $$\lambda_{ij}=\lambda_{ji}^{\text{pull}}+\lambda_{ij}^{\text{push}},$$3 and every set $$\lambda_{ij}=\lambda_{ji}^{\text{pull}}+\lambda_{ij}^{\text{push}},$$4 (Srivastava et al., 2024).

Star networks make the distinction concrete. In one star configuration, pull-only achieves $$\lambda_{ij}=\lambda_{ji}^{\text{pull}}+\lambda_{ij}^{\text{push}},$$5 age while push-only has $$\lambda_{ij}=\lambda_{ji}^{\text{pull}}+\lambda_{ij}^{\text{push}},$$6 age. In another, both push-only and pull-only have $$\lambda_{ij}=\lambda_{ji}^{\text{pull}}+\lambda_{ij}^{\text{push}},$$7 age, but push-pull still achieves $$\lambda_{ij}=\lambda_{ji}^{\text{pull}}+\lambda_{ij}^{\text{push}},$$8 version age. The paper interprets this as a consequence of bidirectional spread and bottleneck relief: allowing both push and pull lets fresh information traverse the network in whichever direction is favorable at a given contact (Srivastava et al., 2024).

For GEP3, this age-theoretic viewpoint changes the performance objective. Completion time to full dissemination becomes only one metric. A protocol may be optimal or near-optimal for broadcast time yet suboptimal for version age, and vice versa. The age results therefore broaden GEP3 from a rumor-spreading primitive into a general framework for maintaining fresh replicated state.

5. Topology, synchrony, and structural bottlenecks

On arbitrary connected graphs, asynchronous and synchronous push&pull satisfy strong universal bounds. In either version, the average spread time is at most linear even if only the pull operation is used, and the guaranteed spread time is within a logarithmic factor of the average spread time, hence $$\lambda_{ij}=\lambda_{ji}^{\text{pull}}+\lambda_{ij}^{\text{push}},$$9. In the asynchronous version, both the average and guaranteed spread times are $I_t$0, and these bounds are best possible up to constant factors (Acan et al., 2014).

The asynchronous formulation associates independent Poisson clocks of rate $I_t$1 with the vertices. When the clock of vertex $I_t$2 rings, it calls a random neighbor $I_t$3; if one knows the rumor and the other does not, the rumor crosses the edge. A key equivalent description is that the communication time on edge $I_t$4 is exponentially distributed with rate $I_t$5, and these edge communication times are mutually independent. This degree-dependent edge rate is a recurring structural theme in GEP3 analyses on heterogeneous graphs (Acan et al., 2014).

A particularly striking topological phenomenon appears in random $I_t$6-trees, which are small-world, highly clustered, power-law graphs with polynomially small conductance, vertex expansion $I_t$7, and constant treewidth. On these graphs, if initially a random node is aware of the rumor, then with probability $I_t$8 after

$I_t$9

rounds the rumor propagates to $in$00 nodes, where $in$01 is any slowly growing function. However, with probability $in$02, the same protocol needs at least

$in$03

rounds to inform all nodes (Mehrabian et al., 2014).

The paper describes this as an “exponential dichotomy” between the time required for informing almost all nodes and all nodes. Its interpretation is that poor global expansion does not preclude rapid dissemination to a giant informed set, but small structural bottlenecks can delay the last fraction of nodes by polynomial time. A closely related upper bound is proved for random $in$04-Apollonian networks: $in$05 rounds suffice to inform $in$06 nodes with high probability, where

$in$07

This reinforces the topological point: high clustering and logarithmic diameter do not by themselves guarantee fast completion to every node (Mehrabian et al., 2014).

For GEP3, these results correct a common oversimplification. Epidemic push-pull is often associated with logarithmic-time spreading on “good” networks, but network goodness is not exhausted by diameter or power-law degree distribution. Synchrony, local degree asymmetries, evolving isolation, and sparse cutsets can all dominate the tail of the completion distribution.

6. Design implications, misconceptions, limitations, and open directions

Several design principles recur across the literature. In evolving $in$08 graphs, increasing $in$09 raises the probability $in$10 that a node is not isolated and therefore speeds the early growth phase, especially for Push&Pull. But the late phase is still constrained by the isolation bottleneck, which yields the unavoidable $in$11 term for Pull and Push&Pull (Daknama, 2017). A plausible implication is that GEP3 tuning should distinguish early amplification from late cleanup rather than treating “faster contacts” as a uniform improvement.

On complete graphs, the recommended operational pattern is to push when the rumor is young and pull when the rumor is old. The regular push-then-pull algorithm is asymptotically optimal when $in$12, provided the push phase is short enough that its communication overhead stays in $in$13. Running the push phase too long recovers the $in$14 message behavior of the classical push-pull algorithms of Karp et al.; stopping earlier yields $in$15 messages instead (Mercier et al., 2017). This suggests that in GEP3, the question is not merely whether push and pull are both available, but when each should be active.

A different class of design implications comes from the arbitrary-network SIS analysis. Because the global mean infection rate can be estimated through localized monitoring of a small constant number of nodes, without knowing the values of the parameters, GEP3 need not rely on global telemetry to track macroscopic state (Xu et al., 2016). In freshness-maintenance settings, the SHS recursion for $in$16 provides an analogous control-theoretic tool: one can identify sets $in$17 with large age and adjust push or pull rates across the cut $in$18 to reduce staleness (Srivastava et al., 2024).

The literature also imposes clear modeling limitations. The evolving-graph rumor-spreading results assume independent per-round graphs, homogeneous nodes, synchronous rounds, perfect transmission once an edge exists and is chosen, and detailed analysis only in the sparse regime $in$19 (Daknama, 2017). The age-of-gossip analysis assumes independent Poisson processes, static topology, perfect communication, and a single-source version process (Srivastava et al., 2024). The nonlinear SIS model assumes discrete-time Markovian infection and cure dynamics on an arbitrary but fixed graph (Xu et al., 2016). These assumptions are analytically productive but leave open the behavior of GEP3 under correlated contacts, non-Poisson timing, multiple concurrent rumors, transmission failures beyond missing edges, and adaptive or topology-aware neighbor selection.

The main conceptual misconception addressed by the literature is that “push-pull” automatically dominates by simple additive combination. In sparse evolving graphs, push and pull overlap substantially in the early phase, reducing the net gain by the term $in$20 (Daknama, 2017). In the random phone-call model, another misconception was that address-oblivious epidemic dissemination with uniformly random peer choice necessarily required $in$21 or $in$22 rumor messages in $in$23 rounds; that conclusion depended on an implicit “always share” assumption that is not optimal once impolite behavior is allowed (Mercier et al., 2017). In the age-of-information setting, a further misconception is that push-only is representative of gossip freshness more generally; the push-pull analysis shows that constant version age is achievable where push alone is only logarithmic or even linear (Srivastava et al., 2024).

In aggregate, GEP3 emerges as a family of epidemic protocols whose behavior is governed by a small set of recurring quantities: per-contact success probabilities, overlap between push and pull, isolation or bottleneck probabilities, graph-dependent spectral or expansion parameters, and the distinction between completion-time and freshness objectives. The arXiv literature does not present a single universal GEP3 theorem, but it does provide a technically coherent theory of when, why, and how push-pull epidemic protocols are fast, message-efficient, fresh, or bottlenecked across a broad range of network and timing models.