Using Network Dynamical Influence to Drive Consensus

Abstract

Consensus and decision-making are often analysed in the context of networks, with many studies focusing attention on ranking the nodes of a network depending on their relative importance to information routing. Dynamical influence ranks the nodes with respect to their ability to influence the evolution of the associated network dynamical system. In this study it is shown that dynamical influence not only ranks the nodes, but also provides a naturally optimised distribution of effort to steer a network from one state to another. An example is provided where the “steering” refers to the physical change in velocity of self-propelled agents interacting through a network. Distinct from other works on this subject, this study looks at directed and hence more general graphs. The findings are presented with a theoretical angle, without targeting particular applications or networked systems; however, the framework and results offer parallels with biological flocks and swarms and opportunities for design of technological networks.

Network Communities of Dynamical Influence

“…To validate the claims that the CDI are the communities with the greatest influence over the consensus process, we investigate the optimisation of consensus leadership. The speed of consensus can be captured analytically by assessing the first eigenvalue of the system matrix 26 . Consensus can be driven to a target value by applying a constant input perturbation to a set of vertices that have a directed connection to all other vertices 26 .…”

“…The speed of consensus can be captured analytically by assessing the first eigenvalue of the system matrix 26 . Consensus can be driven to a target value by applying a constant input perturbation to a set of vertices that have a directed connection to all other vertices 26 . Similar perturbation optimisations have been studied extensively in the context of leadership selection for the control of multi-agent and swarm systems [26][27][28][29][30] .…”

“…Consensus can be driven to a target value by applying a constant input perturbation to a set of vertices that have a directed connection to all other vertices 26 . Similar perturbation optimisations have been studied extensively in the context of leadership selection for the control of multi-agent and swarm systems [26][27][28][29][30] . In these cases, the perturbation is often constrained so that only a set number of leaders are chosen with a binary option for perturbation input, i.e.…”

“…There has been an attempt to tackle both problems, but this work was restricted to 1-D community and ring graphs 29 , and an examination of how the proportion of leader vertices affects the convergence rate to consensus in multiplex networks 30 . Of most relevance, to the work herein, is globally bounded input perturbations that can be applied with a variable distribution to any combination of vertices 26 . For such a case, the first left eigenvector of the Laplacian matrix was identified as a sub-optimal resource allocation (equivalent to an input perturbation) for achieving fast convergence to consensus 26 .…”

“…Of most relevance, to the work herein, is globally bounded input perturbations that can be applied with a variable distribution to any combination of vertices 26 . For such a case, the first left eigenvector of the Laplacian matrix was identified as a sub-optimal resource allocation (equivalent to an input perturbation) for achieving fast convergence to consensus 26 . An improvement in this allocation has since been developed for directed k-outdegree graphs, where a near-optimal perturbation vector can be produced by combining first left eigenvectors from manipulated versions of the adjacency matrix 31 .…”