Node-to-node and node-to-medium synchronization in quorum sensing networks affected by state-dependent noise

A quorum sensing network is a form of communication system where nodes talk to each other through a shared environment or medium. Such networks arise in many applications, such as bacterial quorum sensing, where diffusing signaling molecules are exchanged with the extracellular environment, and in social networks, where decisions might be influenced by social media. In this paper, we analyze node-to-node and node-to-medium synchronization in these quorum sensing networks when nodes are affected by relative-state-dependent noise and the medium has a different dynamics from the nodes. By using stochastic Lyapunov arguments, we give a number of sufficient conditions for the stability of the synchronization manifold and compare the synchronization dynamics induced by common (extrinsic) noise and independent (intrinsic) noise. We also carry out a stochastic phase plane analysis of the dynamics on the synchronization manifold by introducing the notion of a stochastic invariant manifold.