11 The Age of Gossip in Networks
Roy D. YatesWINLAB, Department of Electrical and Computer EngineeringRutgers University [email protected]
Abstract —A source node updates its status as a point processand also forwards its updates to a network of observer nodes.Within the network of observers, these updates are forwardedas point processes from node to node. Each node wishes itsknowledge of the source to be as timely as possible. In thisnetwork, timeliness is measured by a discrete form of age ofinformation: each status change at the source is referred to as aversion and the age at a node is how many versions out of dateis its most recent update from the source. This work introducesa method for evaluating the average version age at each nodein the network when nodes forward updates using a memorylessgossip protocol. This method is then demonstrated by versionage analysis for a collection of simple networks. For gossip on acomplete graph with symmetric updating rates, it is shown thateach node has average age that grows as the logarithm of thenetwork size.
I. I
NTRODUCTION
Gossip is a popular mechanism to convey status informationin a distributed systems and networks. The efficacy of gossipmechanisms for distributed computation [1], [2] and messagedissemination [3] is well known. While it is also known thatgossip mechanisms can be inefficient relative to more complexor application-specific algorithms, it is recognized that gossipremains an attractive option in settings when protocols needto be simple or the network topology or connectivity is time-varying [4]. For example, gossip protocols could be a goodchoice for low latency vehicular safety messaging. And yet,while vehicular message exchange was the early motivationfor age of information (AoI) research [5], [6], there has beenlittle (if any) effort to examine AoI for gossip protocols.In this work, we begin to re-examine gossip from an age-of-information (AoI) perspective [7], [8]. Specifically, a sourcewishes to share its status update messages with a networkof n nodes. These nodes, which can be viewed monitors ofthe source, employ gossip to randomly forward these updatemessages amongst themselves in order that all nodes havetimely knowledge of the state of the source.This work extends AoI analysis in a class of status samplingnetworks, a networking paradigm that is consistent with gossipmodels in that short messages, representing samples of anode’s status update process, are delivered as point processesto neighbor nodes. This “zero service time” model may beuseful in a high speed network in which updates representsmall amounts of protocol information (requiring negligibletime for transmission) that are given priority over data traffic.This model has also been widely used in the age analysis This work was supported by NSF award CCF-1717041. of enegry harvesting updaters [9]–[15] where updating ratesare constrained by energy rather than bandwidth. While thetransmission of a single update may be negligible, the updaterates are limited so that protocol information in the aggregatedoes not consume an excessive fraction of network resources.Prior work on status sampling networks [16], [17] developedtools for analyzing age in line networks in which each node i only received updates from node i − . The key advance of thiswork is the development of an average age analysis methodfor monitors that receive updates via multiple network paths.II. S YSTEM M ODEL AND S UMMARY OF R ESULTS
Status updates of a source node are shared via a networkwith a set of nodes N = { , , . . . , n } . Motivated by sen-sor networks in which accurate clocks may be unavailable,timeliness at each node is measured by update versions . Thesource node maintains the current (fresh) version of its statusand thus node always has age X ( t ) = 0 . Starting at time t = 0 , status updates at node occur as a rate λ Poissonprocess N ( t ) . That is, at time t > , the most recent updateat node is version N ( t ) . If the current update at node i is version N i ( t ) , then the age at node i , as measured inversions, is X i ( t ) = N ( t ) − N i ( t ) . An example of versionage sample paths is depicted in Figure 1. In particular, if node has an update at time t , the age at each node i becomes X (cid:48) i ( t ) = X i ( t ) + 1 . On the other hand, X i ( t ) = 0 when node i has observed the current update version of node . In thissense, the version AoI metric is similar to the age of incorrectinformation [18] and age of synchronization [19] metrics. Forall three metrics, the age at a node is zero as long as that nodehas the current status of the source.In this work, we develop a method for evaluating thelimiting average age lim t →∞ E[ X i ( t )] , which we refer to asthe version AoI at node i . Building on prior work [17], [20],this paper employs the methodology of the stochastic hybridsystem (SHS) to analyze the convergence of the expected age.Specifically, we assume the nodes forward updates usinggossip. Node i sends its most recent update to node j as arate λ ij Poisson process. If node i sends its update to node j at time t , the age at node j becomes X (cid:48) j ( t ) = min[ X i ( t ) , X j ( t )] . (1)Implicit in (1) is that updates are version-stamped so thatnode j can adopt fresher updates from node i but ignore olderupdates.The SHS approach is to develop a set of ordinary differ-ential equations for E[ X i ( t )] that enables the evaluation of a r X i v : . [ c s . I T ] F e b the limiting age lim t →∞ E[ X i ( t )] . As we see in (1), thiswill require the characterization of age variables such as X { i,j } ( t ) ≡ min( X i ( t ) , X j ( t )) . More generally, for arbitrarysubsets S ⊆ N , the analysis will need to track the age X S ( t ) ≡ min j ∈ S X j ( t ) (2a)and its expected value v S ( t ) ≡ E[ X S ( t )] . (2b)We can interpret X S ( t ) as the status age of an observer ofupdates arriving at any node in S and we may refer to X S ( t ) as the (version) age of subset S .The main result of the paper is the development of asystem of linear equations for the calculation of the limitingaverage age ¯ v S = lim t →∞ E[ X S ( t )] . To describe this systemof equations, define the update rate of node i into set S as λ i ( S ) ≡ (cid:40)(cid:80) j ∈ S λ ij i (cid:54)∈ S, i ∈ S, (3)and the set of updating neighbors of S as N ( S ) ≡ { i ∈ N : λ i ( S ) > } . (4)With this notation, we state our main result. Theorem 1:
The expected status age v S ( t ) = E[ X S ( t )] ofan observer of node set S converges to ¯ v S = lim t →∞ v S ( t ) satisfying ¯ v S = λ + (cid:80) i ∈ N ( S ) λ i ( S )¯ v S ∪{ i } λ ( S ) + (cid:80) i ∈ N ( S ) λ i ( S ) . (5)Proof of this claim is deferred to Section V-B.In Section IV, we demonstrate the use of Theorem 1 firstfor the n = 3 node network in Figure 2 and second for the n node symmetric gossip network on a complete graph, asdepicted in Figure 3 for n = 6 nodes. In the complete graph, λ ij = λ/ ( n − for all node pairs i, j ∈ N . This correspondsto each node i ∈ N randomly sending its current updates toeach of the other n − nodes as a rate λ/ ( n − Poissonprocess. In addition, the source sends symmetrically to eachnode j ∈ N with Poisson rate λ j = λ/n . By exploiting thesymmetry of the complete graph, Theorem 1 shows that theaverage age at a node grows as log n . Theorem 2:
For the symmetric complete gossip networkwith the source sending updates to each node i ∈ N at rate λ/n , the average version age of each node i is λ λ (cid:34) n − n n − (cid:88) k =1 k + 1 n (cid:35) ≤ lim t →∞ E[ X i ( t )] ≤ λ λ n (cid:88) k =1 k . (6)Hence, as the network size n grows, the average age ateach node only grows logarithmically in n . Although thecommunication models are different in various small ways,this average result is analogous to [3, Theorem 3.1] in whichthe (cid:15) -dissemination time, i.e. the time until the probability asource message has not reached all nodes is less than (cid:15) , isshown to grow as O (log n ) . version age tt , t , t , t , t , t , t , t , t , t , X ( t ) X ( t ) Fig. 1. Fresh updates from a source pass through the network as pointprocesses; t i,jn marks the n th update sent on link ( i, j ) . Node gets updatesfrom the source node . Node gets updates from both the source and alsofrom node . Age is measured in versions, X i ( t ) records how many versionsout-of-date the update at node i is relative to the source. III. R
ELATED W ORK
AoI analysis of updating systems started with the analysesof status age in single-source single-server first-come first-served (FCFS) queues [7], the M/M/1 last-come first-served(LCFS) queue with preemption in service [21], and the M/M/1FCFS system with multiple sources [22]. Here we discuss AoIcontributions relating to networks carrying the updates of asingle source, as in this work. A more extensive overview ofAoI research can be found in [8].To evaluate AoI for a single source sending updates througha network cloud [23] or through an M/M/ m server [24]–[26],out-of-order packet delivery was the key analytical challenge.The first evaluation of the average AoI over multihop networkroutes [27] employed a discrete-time version of the statussampling network also employed in [16], [17]. These worksobtained simple AoI results because the updates followeda single path to a destination monitor. This avoided thecomplexity of multiple paths and the consequent accountingfor repeated and out-of-order update message deliveries.When multiple sources employ wireless networks subjectto interference constraints, AoI has been analyzed under avariety of link scheduling strategies [28]–[36]. Age boundswere developed from graph connectivity properties [37] wheneach node needs to update every other node. For DSRC-based vehicular networks, update piggybacking strategies weredeveloped and evaluated [6].When update transmission times over network links areexponentially distributed, sample path arguments were used[38]–[40] to show that a preemptive Last-Generated, First-Served (LGFS) policy results in smaller age processes at allnodes of the network than any other causal policy. Note thatthe status sampling network model in this work can also beviewed as a network of preemptive LGFS server; see [17]for details. With that equivalence, [40] and this work canbe viewed as complementary in that [40] proves the age-optimality of LGFS policies and this work provides analytictools for the evaluation of those policies.IV. A PPLICATIONS OF T HEOREM n . We start with S = { n } and generatean equation for ¯ v { n } in terms of the variables ¯ v { i,n } for nodes i such that λ i,n > . For each such node i , the next step isto apply (5) recursively with S = { i, n } . This generates an λ λ , λ , λ , λ , λ , Fig. 2. Updates generated at node are forwarded to nodes , , and . equation for each ¯ v { i,n } in terms of variables ¯ v i,j,n for eachnode j that sends updates to one or both nodes in { i, n } .In general, at stage k , we construct equations for ¯ v S forsets S with size | S | = k in terms of variables ¯ v S (cid:48) suchthat each S (cid:48) has size | S (cid:48) | = k + 1 . In the worst case, thisprocedure terminates at stage k = n when S = N . For afully connected graph, this procedure generates equations forall n − non-empty subsets of N . On the other hand, whenthe network graph is sparse, substantially fewer equations maybe generated.In the next three sections, we demonstrate Theorem 1with three examples; a three-node toy network with arbitraryrates, version age analysis of the n -node symmetric completegraph that provides the proof of Theorem 2, and an n -nodesymmetric ring network. A. Toy example of Theorem 1
Here we demonstrate Theorem 1 by solving for the averageversion age ¯ v { } at node for the network shown in Figure 2.The recursive application of (5) with S = { } , S = { , } , S = { , } and S = { , , } yields ¯ v { } = λ , + λ , ¯ v { , } + λ , ¯ v { , } λ , + λ , , (7a) ¯ v { , } = λ , + λ , ¯ v { , , } λ , + λ , , (7b) ¯ v { , } = λ , + ( λ , + λ , )¯ v { , , } λ , + λ , + λ , , (7c) ¯ v { , , } = λ , λ , + λ , . (7d)We note that (7d) is an example of the general result that ¯ v N = λ /λ ( N ) . For this network, it follows from (7) that ¯ v { } = λ , λ , + λ , (cid:20) λ , λ , + λ , (cid:18) λ , λ , + λ , (cid:19) + λ , λ , + λ , + λ , (cid:18) λ , + λ , λ , + λ , (cid:19)(cid:21) . (8)The solution (8) is complicated because it includes a varietyof special cases. For example, when λ , → ∞ , ¯ v { , , } and ¯ v { , } are unchanged but ¯ v { } → ¯ v { , } because nodes and become equivalent to a single node with update rates λ , from node and λ , from node . On the other hand, when λ , → , ¯ v { , , } is unchanged while ¯ v { } → λ , λ , + ¯ v { , } , ¯ v { , } → λ , + λ , ¯ v { , , } λ , + λ , . (9)In this case, the solution for ¯ v { } reflects the path diversityoffered by the two paths from the source to node . λ Fig. 3. Updates generated at node at rate λ are forwarded to nodes in N = { , . . . , } which form a complete graph. Node sends updates toeach node i ∈ N at rate λ/ . Each node i ∈ N send updates to every othernode j at rate λ/ . B. Proof of Theorem 2
We now use Theorem 1 to find the average of a node forthe n -node complete graph, as depicted for n = 6 in Figure 3.Here the symmetry of the complete graph is essential to deriveTheorem 2. In the absence of symmetry, the recursion ofTheorem 1 would generate equations for all n − nontrivialsubsets of N .Let S j denote an arbitrary j -node subset of the completegraph. By symmetry, the age processes X S j ( t ) for all subsets S j are statistically identical. Hence we define ˜ v j = ¯ v S j .Moreover, each subset S j has | N ( S j ) | = n − j neighbor nodes i that send updates to S j at rate λ i ( S j ) = jλ/ ( n − . Foreach such neighbor i , S j ∪ { i } is a j + 1 node subset S j +1 .Also, because the source symmetrically updates all nodes in N , each subset S j receives updates from the source node atrate λ ( S j ) = jλ/n . Thus Theorem 1 yields ˜ v j = λ + | N ( S j ) | λ i ( S j )˜ v j +1 λ ( S j ) + N ( S j ) λ i ( S j ) = λ + j ( n − j ) λn − ˜ v j +1 jλn + j ( n − j ) λn − . (10)For j = n , S j = S n is the set of all nodes. With allnodes in the observer set, the neighbor set N ( N ) is empty, λ ( N ) = λ , and Theorem 1 yields ˜ v n = ¯ v N = λ /λ . Withthis initial condition, (10) enables iterative computation of ˜ v n − , ˜ v n − , . . . until we reach ˜ v , the average age of a singlenode. However to complete the proof, let j = n − k , implying ˜ v n − k = λ ( n − k ) λ + kn − ˜ v n − k +11 n + kn − . (11)With the definition ˆ v k ≡ ˜ v n − k +1 , (11) becomes ˆ v k +1 = λ ( n − k ) λ + kn − ˆ v k n + kn − ≤ λ ( n − k ) λ + kn ˆ v k n + kn . (12)The upper bound in (12) holds iff ˆ v k ≤ nλ / ( n − k ) λ . Since ˆ v = ˜ v n = λ /λ this requirement holds at k = 1 and can beshown by induction to hold for all k . Defining y k = k ˆ v k /n ,it follows from (12) that y k +1 ≤ λ ( n − k ) λ + y k . (13)Since y = λ / ( nλ ) , it follows from (13) that y n ≤ λ λ n − (cid:88) k =0 n − k = λ λ n (cid:88) k =1 k . (14) Nodes n a g e ApproximationAge on Ring
Fig. 4. Average version age of a node on the symmetric n node ring with λ /λ = 1 . The dashed blue line . √ n is an empirical approximation, butnot an upper bound; the age will exceed the approximation for n > . Since y n = ˆ v n = ˜ v , this completes the proof of theTheorem 2 upper bound. For the lower bound, the equalityin (12) implies ˆ v k +1 ≥ n − k − (cid:20) λ ( n − k ) λ + kn − v k (cid:21) . (15)Defining ˆ y k ≡ k ˆ v k / ( n − , (15) implies ˆ y k +1 ≥ λ ( n − k ) λ + ˆ y k . (16)It follows from (16) that ˆ v n = ( n − y n /n satisfies the lowerbound of Theorem 2. C. Age on a Symmetric Ring
In the ring network, the source sends updates to each nodeat rate λ/n while each node i sends updates to each of itsneighbor nodes i + 1 and i − at rates λ/ at rate λ/ . Thusthe network resembles the complete graph of Figure 3, exceptthe interior transitions are deleted.For the ring graph, let R j = { i, i + 1 , . . . , i + j − } denote an arbitrary contiguous j -node subset of the ring. Bysymmetry, the age processes X R j ( t ) for all subsets R j arestatistically identical. Hence we define ˜ v j = ¯ v R j . Moreover,for j < n − , each subset R j has | N ( R j ) | = 2 neighbor nodes k ∈ { i − , j } that send updates to R j at rate λ k ( R j ) = λ/ .For each neighbor k , R j ∪ { k } is a j + 1 node subset R j +1 .In addition, for a subset R n − , there is a single neighbor k who sends updates at rate λ k ( R n − ) = λ (at rate λ/ toneighbor nodes k + 1 and k − that are the head and tailof R n − . Thus | N ( R j ) | λ k ( R j ) = λ for j ∈ { , . . . , n − } .Also, because the source symmetrically updates all nodes in N , each subset R j receives updates from the source node atrate λ ( R j ) = jλ/n . Thus Theorem 1 yields ˜ v j = λ + | N ( R j ) | λ k ( R j )˜ v j +1 λ ( R j ) + N ( R j ) λ k ( R j ) = λ + λ ˜ v j +1 jλn + λ . (17)For j = n , R j = R n is the set of all nodes. With allnodes in the observer set, the neighbor set N ( N ) is empty, λ ( N ) = λ , and Theorem 1 yields ˜ v n = ¯ v N = λ/λ , as itdoes for all graphs. With this initial condition, (17) enablesiterative computation of ˜ v n − , ˜ v n − , . . . until we reach ˜ v , theaverage age of an individual node. As of this writing, thedownward iteration (17) has not yet yielded a simple bound We assume node indexing modulo the n node ring, i.e., node n + 1 refersto node and node − refers to node n . for ˜ v However, as the numerical evaluation is nearly trivial,an age plot is presented in Figure 4 for λ /λ = 1 . From thefigure, it is empirically observed that ∆ ring ( n ) ≈ . √ n .This numerical evidence may seem surprising since O ( n log n ) dissemination time has been reported for the ringgraph [3]. However, to enable age comparisons with thecomplete graph, the ring model in this work sends its freshupdates randomly to the ring. By contrast, the ring graphmodel in [3] assumes the source is a node on the ring andthus the dissemination time to all nodes must be Ω( n ) . Ifsource updates were passed only to a single node on the ring,the average age would indeed grow as O ( n ) .V. S TOCHASTIC H YBRID S YSTEMS FOR A O I A
NALYSIS
In this section we use a stochastic hybrid system (SHS)model to derive Theorem 1. While there are many SHSvariations [41], this work follows [17], [20], which employ themodel and notation in [42]. In general, the SHS is describedby a discrete state q ( t ) ∈ Q = { , , . . . , q max } that evolves asa point process, a continuous component X ( t ) ∈ R n describedby a stochastic differential equation in each state q ∈ Q , and aset L of transition/reset maps that correspond to both changesin the discrete state and jumps in the continuous state. A. Version AoI for gossip networks as an SHS
In this work, the operation of the gossip network is mem-oryless; each node i sends its current update to node j asa Poisson process of rate λ ij . Hence, the SHS discrete statespace is the trivial set Q = { } . Furthermore, because ageis measured in versions, the normally continuous age state X ( t ) in fact becomes discrete in the version gossip network.That is, X ( t ) changes only when there is a transition thatcorresponds to an update being forwarded. In the absence ofsuch a transition, the stochastic differential equation of theSHS is trivially ˙ X ( t ) = .The remaining component of the SHS model is the set L of discrete transition/reset maps. In the gossip network, L corresponds to the set of directed edges ( i, j ) over whichnode i updates node j . However, because of the special roleof node as the source, there are three kinds of transitions.First, ( i, j ) = (0 , corresponds to the source node generatinga new version so that the version age at all other nodes k increases by one. The second type of transition is given by (0 , j ) , corresponding to the source node sending the currentversion to node j , reducing the age at node j to zero. In thethird type, a gossiping node i forwards its current update tonode j ; node j accepts the update if it is a fresher than itsexisting version. To summarize, the set of transitions is L = { (0 , } ∪ { (0 , j ) : j ∈ N } ∪ { ( i, j ) : i, j ∈ N } , (18)transition ( i, j ) occurs at rate λ i,j , and in that transition theage vector becomes φ i,j ( X ) = (cid:2) X (cid:48) · · · X (cid:48) n (cid:3) such that X (cid:48) k = X k + 1 i = 0 , j = 0 , k ∈ N , i = 0 , k = j ∈ N , min( X i , X j ) i ∈ N , k = j ∈ N ,X k otherwise . (19a) Because of the generality and power of the SHS model,complete characterization of the X ( t ) process is often impos-sible. The approach in [42] is to define test functions ψ ( q, X , t ) whose expected values E[ ψ ( q ( t ) , X ( t ) , t )] are performancemeasures of interest that can be evaluated as functions of time;see [42], [43], and the survey [41] for additional background.Since the simplified SHS for the gossip network is timeinvariant and has a trivial discrete state, it is sufficient toemploy the time invariant test functions ψ S ( X ) = X S . Thesetest functions yield the processes ψ S ( X ( t )) = X S ( t ) , (20)which have expected values E (cid:2) ψ S ( X ( t )) (cid:3) = E (cid:2) X S ( t ) (cid:3) ≡ v S ( t ) . (21)The objective here is to use the SHS framework to derive asystem of differential equations for the v S ( t ) . To do so, theSHS mapping ψ → Lψ known as the extended generator isapplied to every test function ψ ( X ) . The extended generator Lψ is simply the function whose expected value is theexpected rate of change of the test function ψ . Specifically, atest function ψ ( X ( t )) has an extended generator ( Lψ )( X ( t )) that satisfies Dynkin’s formula d E[ ψ ( X ( t ))] dt = E[( Lψ )( X ( t ))] . (22)For each test function ψ ( X ) , (22) yields a differential equationfor E[ ψ ( X ( t ))] .From [42, Theorem 1], it follows from the trivial discretestate, the trivial stochastic differential equation ˙ X ( t ) = ,and the time invariance of ψ S ( X ) in (20) that the extendedgenerator of a piecewise linear SHS is given by ( Lψ S )( X ) = (cid:88) ( i,j ) ∈L λ ij [ ψ S ( φ i,j ( X )) − ψ S ( X )] . (23) B. Proof of Theorem 1
In (23), it follows from (2a), (19), and (20) that the effecton the test function of transition ( i, j ) is ψ S ( φ i,j ( X )) = X (cid:48) S = min k ∈ S X (cid:48) k . (24)Evaluation of (24) depends on the transition type ( i, j ) , asgiven in (19). In transition (0 , , the source node has a versionupdate and each node k ∈ S becomes one more version outof date. This implies X (cid:48) k = X k + 1 for all k ∈ N and thus X (cid:48) S = min k ∈ S X (cid:48) k = X S + 1 . (25)For other transitions ( i, j ) , only the age X j at node j ischanged. Thus if j (cid:54)∈ S , then X S = min k ∈ S X k is unchanged.However, if j ∈ S , then X (cid:48) S = min k ∈ S X (cid:48) k = min(min( X i , X j ) , min k ∈ S \{ j } X k )= min k ∈ S ∪{ i } X k = X S ∪{ i } . (26)In addition to the common ( i, j ) transition in which i ∈ N isa gossiping neighbor of j ∈ S , we note that (26) incorporatessome special cases. If i = 0 , then X (cid:48) S = X S ∪{ } = 0 since X = 0 . On the other hand, if i ∈ S , then S ∪ { i } = S and X (cid:48) S = X S . That is, an update sent by a node in S cannotreduce the age X S .Based on the three types of transitions, namely (0 , , (0 , j ) ,and ( i, j ) , we conclude that ( Lψ S )( X ) = λ ( X S + 1 − X S ) + (cid:88) j ∈ S λ j [0 − X S ]+ (cid:88) i> i (cid:54)∈ S (cid:88) j ∈ S λ ij (cid:2) X S ∪{ i } − X S (cid:3) . (27)We note that X , X S , and X S ∪{ i } in (27) refer to the ageprocesses X ( t ) , X S ( t ) and X S ∪{ i } ( t ) . With this in mind,we take the expectation of (27). On the left side of (27), E[( Lψ S )( X ( t ))] = ˙ v S ( t ) by Dynkin’s formula (22). On theright side, E[ X S ( t )] = v S ( t ) and E (cid:2) X S ∪{ i } ( t ) (cid:3) = v S ∪{ i } ( t ) for all i . These substitutions yield ˙ v S ( t ) = λ − v S ( t ) (cid:88) j ∈ S λ j + (cid:88) i> i (cid:54)∈ S (cid:88) j ∈ S λ ij [ v S ∪{ i } ( t ) − v S ( t )] . Employing the definitions (3) and (4) of the update rate λ i ( S ) of node i into S , and the neighbor set N ( S ) , we obtain ˙ v S ( t ) = λ − v S ( t ) (cid:104) λ ( S )+ (cid:88) i ∈ N ( S ) λ i ( S ) (cid:105) + (cid:88) i ∈ N ( S ) λ i ( S ) v S ∪{ i } ( t ) . By setting the derivatives ˙ v S ( t ) = 0 , we obtain a linearequation for the time average age ¯ v S = lim t →∞ v S ( t ) in termsof the necessary ¯ v S ∪{ i } . This yields (5).VI. C ONCLUSION
This work has introduced AoI analysis tools for gossipalgorithms on network graphs. In Theorem 1 we developed aset of linear equations for the computation of average versionage at any node in a gossip network described by an arbitrarygraph. While the general solution has exponential complexityin the number of nodes, we believe this unavoidably reflectsthe multiplicity of paths from the source to a node. When thismethod is applied to the n node complete graph, it was shownusing symmetry properties that the average version age at eachnode grows as log n . This promising result suggests that gossipnetworks may indeed be suitable for low latency measurementdissemination, particularly in sensor network settings.As age analysis for gossip networks is new, considerablework remains. Since this work has examined only the simplestnetwork graphs, age analysis over more complex graphs isneeded. Age analysis of gossip for energy harvesting sensorswould also be another obvious area of interest. While this workemploys the version age metric, we expect to see analogousresults for the traditional sawtooth age metric that tracks theevolution of time. We also believe it may be possible to derivedistributional properties of the age in a gossip network byextending the moment generating function (MGF) approachto age analysis in [17]. R EFERENCES[1] S. Boyd, A. Ghosh, B. Prabhakar, and D. Shah, “Analysis and optimiza-tion of randomized gossip algorithms,” in , vol. 5.IEEE, 2004, pp. 5310–5315.[2] ——, “Randomized gossip algorithms,”
IEEE transactions on informa-tion theory , vol. 52, no. 6, pp. 2508–2530, 2006.[3] D. Shah,
Gossip algorithms . Now Publishers Inc, 2009.[4] K. Birman, “The promise, and limitations, of gossip protocols,”
ACMSIGOPS Operating Systems Review , vol. 41, no. 5, pp. 8–13, 2007.[5] S. Kaul, M. Gruteser, V. Rai, and J. Kenney, “Minimizing age ofinformation in vehicular networks,” in
IEEE Conference on Sensor, Meshand Ad Hoc Communications and Networks (SECON) , 2011.[6] S. K. Kaul, R. D. Yates, and M. Gruteser, “On piggybacking invehicular networks,” in
IEEE Global Telecommunications Conference,GLOBECOM 2011 , Dec. 2011.[7] S. Kaul, R. Yates, and M. Gruteser, “Real-time status: How often shouldone update?” in
Proc. IEEE INFOCOM , March 2012, pp. 2731–2735.[8] R. D. Yates, Y. Sun, D. R. Brown III, S. K. Kaul, E. Modiano, andS. Ulukus, “Age of information: An introduction and survey,” arXivpreprint arXiv:2007.08564 , 2020.[9] X. Wu, J. Yang, and J. Wu, “Optimal status update for age of informationminimization with an energy harvesting source,”
IEEE Trans. GreenCommun. Netw. , vol. 2, no. 1, pp. 193–204, March 2018.[10] S. Feng and J. Yang, “Age of information minimization for an energyharvesting source with updating erasures: Without and with feedback,”available Online: arXiv:1808.05141.[11] A. Arafa, J. Yang, S. Ulukus, and H. V. Poor, “Age-minimal transmissionfor energy harvesting sensors with finite batteries: Online policies,”
IEEETrans. Inf. Theory , vol. 66, no. 1, pp. 534–556, January 2020.[12] ——, “Online timely status updates with erasures for energy harvestingsensors,” in
Proc. Allerton , October 2018.[13] ——, “Using erasure feedback for online timely updating with an energyharvesting sensor,” in
Proc. IEEE ISIT , July 2019.[14] B. T. Bacinoglu, E. T. Ceran, and E. Uysal-Biyikoglu, “Age of infor-mation under energy replenishment constraints,” in
Proc. ITA , February2015.[15] W. Liu, X. Zhou, S. Durrani, H. Mehrpouyan, and S. D. Blostein,“Energy harvesting wireless sensor networks: Delay analysis consid-ering energy costs of sensing and transmission,”
IEEE Trans. WirelessCommun. , vol. 15, no. 7, pp. 4635–4650, July 2016.[16] R. D. Yates, “Age of information in a network of preemptive servers,”in
IEEE Conference on Computer Communications (INFOCOM) Work-shops , Apr. 2018, pp. 118–123, arXiv preprint arXiv:1803.07993.[17] ——, “The age of information in networks: Moments, distributions, andsampling,”
IEEE Transactions on Information Theory , vol. 66, no. 9, pp.5712–5728, 2020.[18] A. Maatouk, S. Kriouile, M. Assaad, and A. Ephremides, “The age ofincorrect information: A new performance metric for status updates,”
IEEE/ACM Transactions on Networking , vol. 28, no. 5, pp. 2215–2228,2020.[19] J. Zhong, R. Yates, and E. Soljanin, “Two freshness metrics for localcache refresh,” in
Proc. IEEE Int’l. Symp. Info. Theory (ISIT) , Jun. 2018,pp. 1924–1928.[20] R. D. Yates and S. K. Kaul, “The age of information: Real-time statusupdating by multiple sources,”
IEEE Trans. Info. Theory , vol. 65, no. 3,pp. 1807–1827, March 2019.[21] S. Kaul, R. Yates, and M. Gruteser, “Status updates through queues,” in
Conf. on Information Sciences and Systems (CISS) , Mar. 2012.[22] R. Yates and S. Kaul, “Real-time status updating: Multiple sources,” in
Proc. IEEE Int’l. Symp. Info. Theory (ISIT) , Jul. 2012.[23] C. Kam, S. Kompella, and A. Ephremides, “Age of information underrandom updates,” in
Proc. IEEE Int’l. Symp. Info. Theory (ISIT) , 2013,pp. 66–70.[24] ——, “Effect of message transmission diversity on status age,” in
Proc.IEEE Int’l. Symp. Info. Theory (ISIT) , June 2014, pp. 2411–2415.[25] C. Kam, S. Kompella, G. D. Nguyen, and A. Ephremides, “Effect ofmessage transmission path diversity on status age,”
IEEE Trans. Info.Theory , vol. 62, no. 3, pp. 1360–1374, Mar. 2016.[26] R. D. Yates, “Status updates through networks of parallel servers,” in
Proc. IEEE Int’l. Symp. Info. Theory (ISIT) , Jun. 2018, pp. 2281–2285.[27] R. Talak, S. Karaman, and E. Modiano, “Minimizing age-of-informationin multi-hop wireless networks,” in , Oct 2017, pp. 486–493. [28] Q. He, D. Yuan, and A. Ephremides, “Optimal link scheduling for ageminimization in wireless systems,”
IEEE Trans. Info. Theory , vol. 64,no. 7, pp. 5381–5394, July 2018.[29] N. Lu, B. Ji, and B. Li, “Age-based scheduling: Improving data freshnessfor wireless real-time traffic,” in
Proceedings of the Eighteenth ACMInternational Symposium on Mobile Ad Hoc Networking and Computing ,ser. Mobihoc ’18. New York, NY, USA: ACM, 2018, pp. 191–200.[Online]. Available: http://doi.acm.org/10.1145/3209582.3209602[30] R. Talak, S. Karaman, and E. Modiano, “Distributed scheduling algo-rithms for optimizing information freshness in wireless networks,” in
IEEE 19th International Workshop on Signal Processing Advances inWireless Communications (SPAWC) , 2018, pp. 1–5.[31] ——, “Optimizing age of information in wireless networks with perfectchannel state information,” in , 2018, pp. 1–8.[32] R. Talak, I. Kadota, S. Karaman, and E. Modiano, “Scheduling policiesfor age minimization in wireless networks with unknown channel state,”in
IEEE International Symposium on Information Theory (ISIT) , 2018,pp. 2564–2568.[33] A. Maatouk, M. Assaad, and A. Ephremides, “The age of updates ina simple relay network,” in , 2018, pp. 1–5.[34] H. H. Yang, A. Arafa, T. Q. S. Quek, and H. V. Poor, “Locallyadaptive scheduling policy for optimizing information freshness inwireless networks,” in , 2019, pp. 1–6.[35] B. Buyukates, A. Soysal, and S. Ulukus, “Age of information in multihopmulticast networks,”
Journal of Communications and Networks , vol. 21,no. 3, pp. 256–267, 2019.[36] S. Leng and A. Yener, “Age of information minimization for anenergy harvesting cognitive radio,”
IEEE Transactions on CognitiveCommunications and Networking , vol. 5, no. 2, pp. 427–439, 2019.[37] S. Farazi, A. G. Klein, and D. R. Brown, “Fundamental bounds on theage of information in multi-hop global status update networks,”
Journalof Communications and Networks , vol. 21, no. 3, pp. 268–279, 2019.[38] A. M. Bedewy, Y. Sun, and N. B. Shroff, “Optimizing data freshness,throughput, and delay in multi-server information-update systems,” in
Proc. IEEE Int’l. Symp. Info. Theory (ISIT) , 2016, pp. 2569–2574.[39] ——, “Age-optimal information updates in multihop networks,” in
Proc.IEEE Int’l. Symp. Info. Theory (ISIT) , June 2017, pp. 576–580.[40] A. M. Bedewy, Y. Sun, and N. B. Shroff, “The age of information inmultihop networks,”
IEEE/ACM Transactions on Networking , vol. 27,no. 3, pp. 1248–1257, 2019.[41] A. R. Teel, A. Subbaraman, and A. Sferlazza, “Stability analysis forstochastic hybrid systems: A survey,”
Automatica , vol. 50, no. 10, pp.2435–2456, 2014.[42] J. Hespanha, “Modelling and analysis of stochastic hybrid systems,”
IEE Proceedings-Control Theory and Applications ∼∼