Age of Gossip in Networks with Multiple Views of a Source

We consider the version age of information (AoI) in a network where a subset of nodes act as sensing nodes, sampling a source that in general can follow a continuous distribution. Any sample of the source constitutes a new version of the information and the version age of the information is defined with respect to the most recent version of the information available for the whole network. We derive a recursive expression for the average version AoI between different subsets of the nodes which can be used to evaluate the average version AoI for any subset of the nodes including any single node. We derive asymptotic behavior of the average AoI on any single node of the network for various topologies including line, ring, and fully connected networks. The prior art result on version age of a network by Yates [ISIT'21] can be interpreted as in our derivation as a network with a single view of the source, e.g., through a Poisson process with rate <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\lambda_{00}$</tex>. Our result indicates that there is no loss in the average version AoI performance by replacing a single view of the source with distributed sensing across multiple nodes by splitting the same rate <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\lambda_{00}$</tex>. Particularly, we show that asymptotically, the average AoI scales with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$O(\log (n))$</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$O(\sqrt{n})$</tex> for fully connected and ring networks, respectively. More interestingly, we show that for the ring network the same <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$O(\sqrt{n})$</tex> asymptotical performance on average AoI is still achieved with distributed sensing if the number of sensing nodes only scales with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$O(\sqrt{n})$</tex> instead of prior known result which requires <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$O(n)$</tex>. Our results indicate that the sensing nodes can be arbitrarily chosen as long as the maximum number of consecutive non-sensing nodes also scales as <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$O(\sqrt{n})$</tex>.