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.