Age of $k$-out-of-$n$ Systems on a Gossip Network

We consider information update systems on a gossip network, which consists of a single source and $n$ receiver nodes. The source encrypts the information into $n$ distinct keys with version stamps, sending a unique key to each node. For decoding the information in a $k$-out-of-$n$ system, each receiver node requires at least $k+1$ different keys with the same version, shared over peer-to-peer connections. Each node determines $k$ based on a given function, ensuring that as $k$ increases, the precision of the decoded information also increases. We consider two different schemes: a memory scheme (in which the nodes keep the source's current and previous encrypted messages) and a memoryless scheme (in which the nodes are allowed to only keep the source's current message). We measure the ''timeliness'' of information updates by using the $k$-keys version age of information. Our work focuses on determining closed-form expressions for the time average age of information in a heterogeneous random graph under both with memory and memoryless schemes.