Boundary value problems in consensus networks

TL;DR

This paper explores how boundary conditions influence consensus networks, revealing that the network's final state is a harmonic function determined solely by boundary nodes, with implications for network robustness and applications.

Contribution

It introduces the analysis of boundary value conditions in consensus networks, showing their impact on convergence and sensitivity to malicious or erroneous nodes.

Findings

Network estimates converge to harmonic functions influenced by boundary nodes.

Consensus networks are highly sensitive to single malicious or erroneous nodes.

Experimental and numerical studies validate the theoretical analysis.

Abstract

This paper studies the effect of boundary value conditions on consensus networks. Consider a network where some nodes keep their estimates constant while other nodes average their estimates with that of their neighbors. We analyze such networks and show that in contrast to standard consensus networks, the network estimate converges to a general harmonic function on the graph. Furthermore, the final value depends only on the value at the boundary nodes. This has important implications in consensus networks -- for example, we show that consensus networks are extremely sensitive to the existence of a single malicious node or consistent errors in a single node. We also discuss applications of this result in social and sensor networks. We investigate the existence of boundary nodes in human social networks via an experimental study involving human subjects. Finally, the paper is concluded with the numerical studies of the boundary value problems in consensus networks.