基于二进制新鲜度度量的八卦_Gossiping with Binary Freshness Metric
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arXiv:2107.14218v1 [cs.IT] 29 Jul 2021
Gossiping with Binary Freshness Metric
Melih Bastopcu Baturalp Buyukates Sennur Ulukus
Departmen t of Electrical and Computer Engineer ing
University of Maryland , College Park, MD 20742
bastopcu@umd.edu baturalp@umd.edu ulukus@umd.edu
Abstract—We consider the binary freshness metric for gossip
networks that consist of a single sou rce and n end-nodes, where
the end-nodes are allowed to share t heir stored versions of the
source information with the other nodes. We develop recursive
equations that characterize binary freshness in arbitrarily con-
nected gossip networks using the stochastic hybrid systems (SHS)
approach. Next, we study binary freshness in several structured
gossip networks, namely disconnected, ring and fully connected
networks. We show that for both disconnected and ring network
topologies, when the number of nodes gets large, the binary
freshness of a node decreases down to 0 as n
−1
, but the freshness
is strictly larger for the ring topology. We also show that for the
fully connected topology, the rate of decrease to 0 is slower, and
it takes the form of n
−ρ
for a ρ smaller than 1, when the update
rates of the source and the end-nodes are sufficiently large.
Finally, we study the binary freshness metric for clustered gossip
networks, where multip le clusters of structured gossip networks
are connected to the source node through designated access nodes,
i.e., clust er h eads. We characterize the binary freshness in such
networks and nu merically study how the optimal cluster si zes
change with respect to the update rates in the system.
I. INTRODUCTION
Timeliness o f the received information has gained signif-
icant attention with the emergence of modern technologies
such as autonomous driving, holographic communications, an d
industrial IoT. To measure the timeliness at the receiver, age
of infor mation (Ao I) h as been proposed as a performance
metric [1]; see also [2] for a recent survey. The traditio nal age
metric increases lin early over time in the absence of a ny update
deliveries and reduces to the age of the most recently received
update upon successful delivery. In its original fo rm, the age
metric does not encapsulate the information change rate at the
source. For example, when there has been no update deliveries,
the age at the receiver g e ts large indicating that the information
at the re ceiver has become stale. On the contrary, the re c eiver
may still have the freshest version of th e infor mation even
though it has not received any updates for a long time, if the
informa tion at the source does not change frequently.
Motivated by this, several variations of the traditional age
metric have been proposed recently to consider information
change rate at the source. In this work, we use the binary
freshness metric to characterize informatio n f reshness at the
receiver nodes. Previously used in [3]–[7], binary freshness
takes the value 1 when the information at the receiver node is
up-to- date whereas it takes the value 0 when the receiver node
has a stale information. This metric is similar to version age
This work was supported by NSF Grants CCF 17-13977 and ECCS 18-
07348.
1
2
3
4
5
6
1source
λ
e
Fig. 1. Gossip network model consisting of a source represented by the blue
node, and the users represented by the green nodes. Here, users form a ring
network. Other network topologies are shown in Fig. 2.
of information [8]–[ 11] in that both metrics remain unc hanged
as long as the information at the sour c e stays the same. While
version age increases by 1 whenever the source generates a
new update, b inary freshne ss immediately becomes 0 a s soon
as the receiver node and the source are out- of-sync. Another
metric that considers the information change statistics a t the
source is age of incorrect information (AoII), which is equal
to 0 if the receiver has fresh information, otherwise, it is larger
than 0 and may incre ase over time [12]–[14]. A special ca se
of AoII is the age of synchronization introduced in [15].
Our aim in this work is to study information timeline ss
in arbitrarily connected and structured gossip networks using
the binary freshness metric. Works that are closely related to
our work are [5]– [7], [9], [10]. The binary freshness metric
has been studied for serially conn ected caching systems in
[5], [6] and f or parallel connected caching systems in [7].
References [5]–[7] maximize information freshness at the end-
nodes in caching systems by using alternating maximization
methods. Different from [5]–[7], in this work, we employ
the stochastic hybrid systems (SHS) approach and develop
a general method that enable us to characterize the binary
freshness in arbitrarily con nected networks. We explore gos-
siping strategies a mong the end-nodes and study freshness in
structured gossip n etworks, i.e., disconnected, ring and fully
connected network topologies. Reference [9] uses the SHS
approa c h to char acterize the version age and finds the scaling
of version age in structured gossip networks. Reference [10]
improves the scaling o f version age by introducing clustering
to gossip networks. Here, we develop the binary freshness
counterpart of the works [9], [10]. As the binary f reshness
metric and encompassing mathematics are different than those
in version ag e, our work is distinct from works [9], [10].
In this work, by using the SHS approach, we first pro-
vide a way to characterize binary freshness in arbitrarily
connected networks. Then, we consider binary freshness in
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