Age of Gossip With Time-Varying Topologies

TL;DR

The version age of information metric is used to quantify the freshness of information at the nodes in the gossiping network and shows that, if one of the states of the CTMC is the fully connected graph and the transition rates of the CTMC are constant, then the version age of a typical node in the network scales logarithmically with the number of nodes.

Abstract

We consider a gossiping network, where a source node sends updates to a network of $n$ gossiping nodes. Meanwhile, the connectivity topology of the gossiping network changes over time, among a finite number of connectivity ''states,'' such as the fully connected graph, the ring graph, the grid graph, etc. The transition of the connectivity graph among the possible options is governed by a finite state continuous time Markov chain (CTMC). When the CTMC is in a particular state, the associated graph topology of the gossiping network is in the way indicated by that state. We evaluate the impact of time-varying graph topologies on the freshness of information for nodes in the network. We use the version age of information metric to quantify the freshness of information at the nodes. Using a method similar to the first passage percolation method, we show that, if one of the states of the CTMC is the fully connected graph and the transition rates of the CTMC are constant, then the version age of a typical node in the network scales logarithmically with the number of nodes, as in the case if the network was always fully connected. That is, there is no loss in the age scaling, even if the network topology deviates from full connectivity, in this setting. We perform numerical simulations and analyze more generally how having different topologies and different CTMC rates (that might depend on the number of nodes) affect the average version age scaling of a node in the gossiping network.

Figures (5)

• Figure 1: A description of the time evolution of the connectivity of the gossip network. The CTMC represents the connectivity graph (topology) of the gossip network. In this example, the CTMC has four states, representing the fully connected (FC) topology, the ring topology, the grid topology and the disconnected (DC) topology. The current topology is represented using a red dashed circle. In this example, the network starts in the fully connected topology, moving to the ring topology, the disconnected topology and back to the ring topology. Since the rates of the CTMC are different, the time spent in a state will be random, which is also shown in the figure.
• Figure 2: Top: A sample CTMC containing the fully connected (FC) topology as a state. The other states can be any fixed topology. In this example, the disconnected (DC), ring and grid topologies are present. Bottom: A network where all states are replaced by the DC topology except one FC topology. As we note in Section \ref{['sec: FC']}, the average version age of a node on the top will be less than the average version age of the same node on the bottom.
• Figure 3: Variation of average version age of a node as the number of nodes varies when the CTMC has two states, the ring topology and the grid topology.
• Figure 4: Variation of average version age of a node as the number of nodes varies when the CTMC has two states, the ring topology and the fully connected topology.
• Figure 5: Variation of average version age of a node as the number of nodes varies when the CTMC has four states, the ring topology, the grid topology, the disconnected topology and the fully connected topology.

Theorems & Definitions (4)

• Theorem 1
• Lemma 1
• Lemma 2
• Lemma 3