Moment Analysis of Age-Dependent Gossip Networks

We study a class of gossip networks where a source delivers fresh status updates through networks consisting of a set of gossiping nodes. Contrary to previous works, the source delivers updates subject to an age-dependent point process where the rates are related to the age process at each node. Moreover, our work divides the age-dependent gossip networks into two types, i.e., type-A and type-B, where the former allows only one node sampling from source and the latter allows all. We first prove the necessary Markovity and ergodicity of both age-dependent gossip networks and proceed with deriving the general form of unique stationary distributions. With these premises established, closed-form expressions of the stationary age moments for type-A networks with arbitrary topologies are derived with the help of SHSs, with three specified results given for the line, ring and fully-connected networks. Meanwhile, we prove that no closed-form expressions of stationary age moments can be obtained in type-B networks. An approximated model for arbitrary-connected type-B networks is proposed, where we formulate the stationary moment equations for the approximated SHSs under the guarantee of Lagrange stability. Then, we provide a moment closure method to solve two symmetric cases, i.e., fully-connected and ring networks, approximately and verify the effectiveness of our algorithm by comparing to the simulations.