Differential equivalence yields network centrality

One of the most distinctive features of collective adaptive systems (CAS) is the presence of many individuals which interact with each other and with the environment, giving rise to a system-level behaviour that cannot be analyzed by studying the single agents in isolation. The interaction structure among the individuals of CAS is often captured by networks where nodes denote individuals and edges interactions. Understanding the interplay between the network topology and the CAS dynamics calls for tools from network theory in order, for instance, to identify the most important nodes of a network. Centrality measures address this task by assigning an importance measure to each node, a possible example being the famous PageRank algorithm of Google. In this paper we investigate the relationship between centrality measures and model reduction techniques, such as lumpability of Markov chains, which seek to reduce a model into a smaller one that can be processed more efficiently, while preserving information of interest. In particular, we focus on the relation between network centrality and backward differential equivalence, a generalization of lumpability to general dynamical systems. We show that any two backward differential equivalent nodes enjoy identical centrality measures. By efficiently obtaining substantial reductions of real-world networks from biochemistry, social sciences and computer engineering, we demonstrate the applicability of the result.