Semantic Freshness Optimal Sampling and Transmission for Gossiping Receivers

Semantic Freshness Optimal Sampling and Transmission for Gossiping Receivers

arXiv:2608.31140v1 [cs.IT] 31 Aug 2026

Irtiza Hasan Ahmed Arafa Department of Electrical and Computer Engineering University of North Carolina at Charlotte, NC 28223 ihasan@charlotte.edu aarafa@charlotte.edu Abstract—We study the optimal joint sampling and transmission policy for a transmitter communicating with two gossiping receivers that share information with each other, with the objective of tracking a source under the Version Age of Information (VAoI) metric. The transmitter can observe source-version changes, but it has to pay a sampling cost to get the current source information content. Similarly, it can communicate with a receiver by paying a transmission cost. Gossiping enables local information exchange and is able to reduce costly direct transmissions. With imperfect communication links, we formulate an infinite-horizon average-cost Markov Decision Process (MDP) to jointly minimize receiver VAoI, sampling cost, and transmission cost. Using Relative Value Iteration (RVI), we evaluate the optimal policy and establish several properties of its structure. We prove that sampling has a threshold structure in the transmitter VAoI. Among direct transmissions, it is optimal to serve the older receiver. We further characterize the transmit or idle decision through the receiver VAoI difference. Our analysis shows that link reliability and receiver VAoI imbalance have a significant effect on the optimal policy structure. Numerical results verify the structural properties and demonstrate the performance gains of the optimal policy over multiple baselines.

I. I NTRODUCTION Information freshness is considered of critical importance in status-update systems such as sensor networks [1], autonomous vehicles [2], and Internet-of-Things (IoT) networks [3] and wireless edge systems [4], [5], where timely information is needed for control, estimation, and decisionmaking [6]. Traditionally, information freshness is quantified using Age of Information (AoI) [7], which has become a central metric for tracking timeliness in recent years [8]. However, AoI measures freshness purely through elapsed time and may therefore penalize time staleness even when the underlying source content has not changed and information is fresh. This motivates freshness metrics suitable for semantic communication and semantic freshness [9], [10]. Version Age of Information (VAoI) is one such semantic freshness metric, introduced in [11], that tracks the destination’s lag from the source in terms of generated versions, thus quantifying semantic freshness [12]. Many networked status-update systems contain local receiver communication links. These networks do not require every update in the network to pass through the central This work was supported by the U.S. National Science Foundation under Grant ECCS 21-46099.

𝐑𝐗# 𝒑𝟏 𝒑𝒗$ Source

TX

𝒑𝒗#

𝒑𝟐 𝐑𝐗 $

Fig. 1. Transmitter samples the information source (dashed lines and a sampler) and updates two receivers via direct links (solid lines) and receivers may gossip (dashed lines). Symbols above the lines indicate successful communication probabilities.

transmitter. We refer to this decentralized information exchange as gossiping. Existing works have studied the effects of network topology, cluster formation, mobility patterns, energy constraints, and other relevant properties in gossip networks [13]–[19]. The structure of optimal freshness policies has been studied extensively in non-gossip network settings [20]–[32]. The advantage of such structural characterizations is that they provide analytical insight which reduces policy-search complexity and paves the way for physics-informed reinforcementlearning methods [33]–[39]. Policy optimization for freshness in gossip networks has recently begun to receive attention [40]–[44]. In this work, we study a two-receiver VAoI gossip network, for structural characterization of the joint sampling and transmission policy. The transmitter can observe source-version changes, but it has to sample the source at a sampling cost to get the information content. The transmitter can transmit to receivers by paying a transmission cost. Gossiping can reduce dependence on costly direct transmissions as receivers spread fresh versions among themselves. The transmitter may idle and allow receiver gossip to reduce VAoI without incurring a direct transmission cost. However, gossip can only circulate information available at the receivers. Thus, the transmitter needs to make a decision whether to transmit or sample/idle to allow gossiping. We formulate this as an infinite-horizon average-cost MDP [45]. We show that the optimal sampling decision has a threshold structure in the transmitter VAoI and that, among direct transmissions, it is optimal to serve the older receiver. We further characterize the transmit-versus-idle decision through the receiver VAoI difference and show that its threshold direction depends on the relative reliabilities of gossip and direct transmission.

II. S YSTEM M ODEL AND MDP F ORMULATION We consider a time-slotted system indexed by t ∈ {0, 1, 2, . . .}, consisting of one information source, one transmitter (TX), and two receivers, denoted by RX1 and RX2 . Let X(t) denote the source information content in slot t. The source remains unchanged from one slot to the next with probability 1 − α and changes with probability α, independently across slots. TX observes source-version changes but does not obtain the new source content unless it samples it. An RX may either receive communication directly from TX, or through gossiping from the other RX. We assume that RXs acknowledge reception of successful communication, regardless of its source, by sending ACK/NACK feedback to TX, thereby allowing it to track the RXs’ VAoIs [43]. Let Ns (t) denote the source version index at the beginning of slot t, and let Nx (t) denote the freshest version index available at node x (TX, RX1 or RX2 ). The version age of information (VAoI) for node x is defined as vx (t) ≜ Ns (t) − Nx (t).

(1)

Thus, vx (t) = 0 means that node x has the source version at slot t, while vx (t) > 0 means that it lags behind the source by vx (t) versions, which is its VAoI. We define the system state as
s(t) = vtx (t), v1 (t), v2 (t) , (2)

Let s̃ denote the next state. For u = 1, a successful transmission makes RX1 adopt TX’s version, while a failed transmission leaves the receiver versions unchanged: ( (ṽtx , ṽtx , ṽ2 ), w.p. p1 , s̃ = (9) (ṽtx , ṽ1 , ṽ2 ), w.p. 1 − p1 . The transition for u = 2 follows symmetrically. Under idling, u = 0, the receivers gossip using their currently available versions. Hence,  (ṽtx , ṽ, ṽ), w.p. pv1 pv2 ,    (ṽ , ṽ , ṽ), w.p. p (1 − p ), tx 1 v1 v2 s̃ = (10)  (ṽ , ṽ, ṽ ), w.p. (1 − p )p tx 2 v1 v2 ,    (ṽtx , ṽ1 , ṽ2 ), w.p. (1 − pv1 )(1 − pv2 ). For u = 3, TX samples the source and obtains the current source version, resetting its VAoI to zero. In this slot as well, gossiping occurs as in (10). The sampling transition is obtained by replacing the first coordinate ṽtx in (10) with 0. The one-step cost is the sum receiver VAoI plus the transmitter action cost:

i ∈ {1, 2}.

(3)

We truncate all VAoIs at Vmax . At each slot, TX selects an action u from the set A = {0, 1, 2, 3},

(4)

where u = 0 denotes idling, u = 1 and u = 2 denote transmitting to RX1 and RX2 , respectively, and u = 3 denotes sampling the source. A direct transmission to RXi succeeds with probability pi . We assume that direct transmission and RX gossip share the same communication resources. Hence, gossip occurs only in slots without a direct transmission. That is, only during idling or sampling slots, RXs gossip with each other, independently. A gossip communication from RX1 (resp. RX2 ) to RX2 (resp. RX1 ) succeeds with probability pv1 (resp. pv2 ). Let δ ∈ {0, 1} denote the source evolution in a slot, with P(δ = 1) = α,

P(δ = 0) = 1 − α.

(5)

We define the VAoIs after source evolution as ṽtx = (vtx + δ) ∧ Vmax , ṽi = (vi + δ) ∧ Vmax , ṽ = ṽ1 ∧ ṽ2 .

  u = 0, 0, C(u) = Ctx , u ∈ {1, 2},   Cs , u = 3.

(12)

Here, Ctx and Cs denote the direct transmission and sampling costs, respectively. A stationary policy π : S → A maps each state to an action. Its long-term average cost starting from s(0) = s is # "T −1 1 π X π c(s(t), π(s(t))) s(0) = s . ρ (s) = lim sup E T →∞ T t=0 (13) The optimal average cost is ρ⋆ = minπ ρπ (s) and is independent of the initial state since the finite-state MDP is communicating [46]. We use Relative Value Iteration (RVI) [47] to obtain ρ⋆ and the relative value function U (s) satisfying the Bellman optimality equation [48]: ( ) X ⋆ ρ + U (s) = min c(s, u) + P (s̄|s, u)U (s̄) . (14) u∈A

s̄∈S

Define Q(s, u) ≜ c(s, u) +

(6) i = 1, 2,

(11)

where

where vtx (t) is TX’s VAoI and vi (t) is that of RXi , i ∈ {1, 2}. All RX versions satisfy 0 ≤ vtx (t) ≤ vi (t),

c(s, u) = v1 + v2 + C(u),

X

P (s̄|s, u)U (s̄).

(15)

s̄∈S

(7)

Then our goal is to characterize an optimal stationary policy

(8)

π ⋆ (s) ∈ arg min Q(s, u). u∈A

(16)

III. P ROPERTIES OF THE R ELATIVE VALUE F UNCTION AND S TRUCTURE OF THE O PTIMAL P OLICY We first establish two monotonicity properties that will be used to characterize the optimal policy. For states s = ′ (vtx , v1 , v2 ) and s′ = (vtx , v1′ , v2′ ), let s ⪯ s′ denote componentwise ordering. For action u, let Fu,ω,δ (s) denote the next state mapping for a realization ω of the channel outcomes and source evolution δ ∈ {0, 1}. Proofs are omitted due to space constraints. Lemma 1 (Order preservation). For any u ∈ A, δ ∈ {0, 1}, and realization ω of the channel outcomes under action u, the next state is order-preserving: s ⪯ s′

=⇒

Fu,ω,δ (s) ⪯ Fu,ω,δ (s′ ).

(17)

Lemma 2 (Monotonicity). The relative value function U (s) is coordinate-wise nondecreasing: s ⪯ s′

=⇒

U (s) ≤ U (s′ ).

(18)

We now characterize the optimal sampling decision as a function of the transmitter VAoI vtx . Theorem 1 (Sampling threshold). Fix the receiver VAoIs (v1 , v2 ). If sampling is optimal at some feasible vtx = k, then it remains optimal for every feasible vtx ≥ k. Therefore, the optimal sampling decision has a threshold structure in vtx . Proof. Fix (v1 , v2 ) and consider the feasible states sk = (k, v1 , v2 ),

k = 0, 1, . . . , min{v1 , v2 }.

(19)

(20)

which is independent of k. Moreover, sampling resets the transmitter VAoI to zero. Hence, for every source realization δ ∈ {0, 1} and every realization ω of the gossip outcomes, F3,ω,δ (si ) = F3,ω,δ (sj ),

∀i, j.

(21)

Moreover, the probability of each realization (δ, ω) is the same from si and sj . Thus, every possible next state occurs with the same probability under the two initial states, and hence E[U (S̄) | si , 3] = E[U (S̄) | sj , 3],

(22)

and therefore Q(sk , 3) is independent of k. Now consider any non-sampling action u ∈ {0, 1, 2}. For i ≤ j, si ⪯ sj ,

c(si , u) = c(sj , u).

(23)

By Lemma 1, for every source and channel realization, Fu,ω,δ (si ) ⪯ Fu,ω,δ (sj ).

Q(si , u) ≤

min u∈{0,1,2}

Q(sj , u),

(24)

∆(k) ≜ Q(sk , 3) −

min u∈{0,1,2}

Q(sk , u).

(25)

(26)

(27)

Since Q(sk , 3) is constant in k, whereas the minimum nonsampling Q-value is nondecreasing in k, ∆(k) is nonincreasing. Hence, if ∆(k) ≤ 0 at some feasible k, then ∆(j) ≤ 0 for every feasible j ≥ k, and sampling remains optimal. If sampling is optimal for at least one feasible value of k, the corresponding threshold is τ (v1 , v2 ) ≜ min{k : ∆(k) ≤ 0},

(28)

which completes the proof. Theorem 1 separates the sampling decision from the remaining control problem. The remaining decision is whether to transmit the transmitter’s version to one of the receivers or idle and rely on gossip. To characterize this decision, we assume symmetric links, p1 = p2 = p, and pv1 = pv2 = pv . Under this symmetry, the relative value function is invariant to exchanging the receiver coordinates. Lemma 3 (Symmetry). The relative value function satisfies ∀(vtx , v1 , v2 ) ∈ S. (29)

The result follows from the fact that swapping the two symmetric receivers with their corresponding transmission actions does not change the stage cost or any success probability. Lemma 4 (Older-receiver transmission). For any feasible state s = (vtx , v1 , v2 ) ∈ S, v1 ≥ v2

=⇒

Q(s, 1) ≤ Q(s, 2).

(30)

Thus, among the two direct transmissions, it is optimal to serve the receiver with the larger VAoI. Proof. The two transmission actions have the same immediate cost and the same transition upon transmission failure. Upon a successful transmission, action u = 1 yields (ṽtx , ṽtx , ṽ2 ),

(31)

whereas action u = 2 yields (ṽtx , ṽ1 , ṽtx ).

(32)

By Lemma 3, the latter state has the same relative value as (33)

Since v1 ≥ v2 implies ṽ1 ≥ ṽ2 , Lemma 2 gives U (ṽtx , ṽtx , ṽ2 ) ≤ U (ṽtx , ṽtx , ṽ1 ),

u ∈ {0, 1, 2}.

i ≤ j.

Define the sampling gap as

(ṽtx , ṽtx , ṽ1 ).

Lemma 2 then implies Q(si , u) ≤ Q(sj , u),

min u∈{0,1,2}

U (vtx , v1 , v2 ) = U (vtx , v2 , v1 ),

Thus, si and sj differ only in their transmitter VAoIs i and j, respectively. For the sampling action u = 3, c(sk , 3) = v1 + v2 + Cs ,

Consequently,

and therefore Q(s, 1) ≤ Q(s, 2).

(34)

Hence, in the half-space v1 ≥ v2 , the only non-sampling comparison needed is between transmitting to RX1 and idling. Define (35)

Transmission to RX1 is preferred when ∇(k, v1 , v2 ) ≤ 0, whereas idling is preferred when ∇(k, v1 , v2 ) > 0. Let k̃ = (k + δ) ∧ Vmax .

Idle Tx RX1 Tx RX2 Sample

(36)

20.0 17.5 15.0

vtx

∇(k, v1 , v2 ) ≜ Q((k, v1 , v2 ), 1) − Q((k, v1 , v2 ), 0).

α = 0.2, p = 0.4, pv = 0.8, Ctx = 20.0, Cs = 10.0

12.5 10.0 7.5

Expanding the two Q-values gives h X ∇(k, v1 ,v2 ) = Ctx + P(δ) p U (k̃, k̃, ṽ2 )

5.0 2.5 0.0 20.0

δ∈{0,1}

i − pv U (k̃, ṽ2 , ṽ2 ) + (pv − p) U (k̃, ṽ1 , ṽ2 ) .

(37)

To characterize the effect of receiver imbalance, fix the transmitter VAoI k and the smaller receiver VAoI m, and parameterize the state as sd = (k, m + d, m),

d = v1 − v2 ≥ 0.

(38)

Defining g(d) ≜ ∇(k, m + d, m),

(39)

the expression in (37) becomes h X g(d) = Ctx + P(δ) p U (k̃, k̃, m̃) − pv U (k̃, m̃, m̃) δ∈{0,1}

i ^ + (pv − p) U (k̃, m + d, m̃) ,

(40)

^ where m̃ = (m + δ) ∧ Vmax and m + d = (m + d + δ) ∧ Vmax . Theorem 2 (VAoI-difference threshold). For fixed transmitter VAoI k and smaller receiver VAoI m, g(d) is monotone in the receiver VAoI difference d. Consequently, the transmit-versusidle decision has a threshold structure in d. Proof. For d2 > d1 , subtracting (40) evaluated at the two values gives h X ^ g(d2 ) − g(d1 ) = (pv − p) P(δ) U (k̃, m + d2 , m̃) δ∈{0,1}

i ^ − U (k̃, m + d1 , m̃) . (41) Since d2 > d1 implies ^ ^ m + d2 ≥ m + d1 ,

(42)

Lemma 2 shows that the quantity in brackets is nonnegative. If pv > p, then g(d) is nondecreasing in d. Since transmission is preferred when g(d) ≤ 0, the optimal policy can switch at most once from transmission to idling as d increases. If pv < p, then g(d) is nonincreasing, and the policy can switch at most once from idling to transmission. Finally, if pv = p, the d-dependent term in (40) vanishes, so the transmit-versus-idle decision is independent of the receiver VAoI difference.

20.0 17.5 15.0 12.5 10.0 17.5 15.0 7.5 12.5 v2 10.0 5.0 7.5 2.5 5.0 v1 2.5 0.0 0.0

Fig. 2. Optimal policy obtained using RVI.

A useful case occurs when k = m, i.e., when the transmitter and the fresher receiver have the same version. In this case, k̃ = m̃, and (40) reduces to h X ^ g(d) = Ctx + (pv − p) P(δ) U (m̃, m + d, m̃) δ∈{0,1}

i − U (m̃, m̃, m̃) .

(43)

By Lemma 2, the quantity in brackets is nonnegative. Therefore, if pv ≥ p and Ctx > 0, then g(d) > 0 for every d. Hence, idling is strictly preferred to direct transmission. Thus, when the transmitter and the fresher receiver hold the same version and gossip is at least as reliable as the direct link, direct transmission is strictly suboptimal. IV. N UMERICAL A NALYSIS AND P OLICY V ISUALIZATION We evaluate the optimal joint sampling and transmission policy obtained using RVI and compare it with five baselines: Sample and Transmit, Sample and Transmit with Threshold, Round Robin, Round Robin with Threshold, and Random. Unless varied, the parameters are specified in the corresponding figure. Fig. 2 visualizes the optimal policy over (vtx , v1 , v2 ), while Fig. 3 shows fixed-vtx slices. When vtx = 0, direct transmission and idling occupy most of the state space since the transmitter already has the current source version. The transmission regions are symmetric about v1 = v2 , and the older receiver is selected whenever direct transmission is optimal, consistent with Lemma 4. As vtx increases, the sampling region expands and eventually dominates the feasible state space, illustrating the sampling-threshold structure in Theorem 1. Fig. 4 shows that increasing the direct-link reliability p reduces the long-term average cost, where the optimal policy adapts its transmission decisions to the link quality. Similarly, Fig. 5 shows that increasing gossip reliability pv improves performance by making receiver-to-receiver information dissemination more effective. These observations are consistent with Theorem 2, which shows that the relative

α = 0.2, p = 0.4, pv = 0.8, Ctx = 5.0, Cs = 10.0 vtx = 5 vtx = 10

vtx = 0

20

vtx = 15 Idle

v1

15

Tx RX1 10

Tx RX2

5

Sample

0 0

5

10

v2

15

20 0

5

10

v2

15

20 0

5

10

v2

15

20 0

5

10

v2

15

20

Fig. 3. Fixed-vtx two-dimensional slices of the optimal policy in Fig. 2 over (v1 , v2 ) for different values of vtx .

Fig. 4. Performance versus direct-transmission reliability.

Fig. 5. Performance versus gossip reliability.

reliabilities of gossip and direct transmission determine the structure of the transmit-versus-idle decision. Fig. 6 shows that increasing the transmission cost Ctx makes direct transmission less attractive, increasing reliance on gossip. Finally, Fig. 7 shows that, as Cs increases sampling becomes less attractive, causing the optimal policy to delay source acquisition until the freshness benefit justifies the sampling cost. The optimal RVI policy consistently achieves the lowest long-term average cost among the considered policies across the parameter sweeps. V. C ONCLUSION We studied joint sampling, transmission, and gossiping for semantic freshness under VAoI and formulated the problem as an infinite-horizon average-cost MDP. We showed that the

Fig. 6. Performance versus transmission cost.

Fig. 7. Performance versus sampling cost.

sampling decision has a threshold structure in the transmitter VAoI and that, among direct transmissions, it is optimal to serve the older receiver. We further characterized the remaining decision through the receiver VAoI imbalance. In particular, increasing receiver imbalance can shift the optimal decision from transmission to idling when pv > p, from idling to transmission when pv < p, while it has no effect when pv = p. Finally, when the transmitter and the fresher receiver hold the same version and gossip is at least as reliable as the direct link, direct transmission is strictly suboptimal. Numerical results demonstrate the resulting performance gain of jointly adapting all three decisions. Future work includes extensions to larger and general gossip networks.

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