Distributed network size estimation and average degree estimation and control in networks isomorphic to directed graphs

Many properties of interest in graph structures are based on the nodes' average degree (i.e., the average number of edges incident to/from each node). In this work, we present asynchronous distributed algorithms, based on ratio consensus, that can be used to accurately estimate the number of nodes in a multi-component system whose communication topology is described by a directed graph. In addition, we describe an asynchronous distributed algorithm that allows each node to introduce or terminate links in order to reach a target average degree in the network. Such an approach can be useful in many realistic scenarios; for example, for the introduction and removal of renewable energy resources in a power network, while maintaining an average degree that fulfils some structural and dynamical properties and/or optimises some performance indicators of the network. The effectiveness of the proposed algorithms is demonstrated via illustrative examples.