Performance Analysis of Consensus Algorithms Over Prism Networks Using Laplacian Spectra

Abstract

Prism networks belong to generalized petersen graph topologies with planar and polyhedral properties. These networks are extensively used to study the complex networks in the context of computer science, biological networks, brain networks, and social networks. In this letter, the performance analysis of consensus algorithms over prism networks is investigated in terms of convergence time, network coherence, and communication delay. Specifically, in this letter, we first derive the explicit expressions for eigenvalues of Laplacian matrix over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${m}$ </tex-math></inline-formula> -dimensional prism networks using spectral graph theory. Subsequently, the study of consensus metrics such as convergence time, maximum communication time delay, first order network coherence, and second order network coherence is performed. Our results indicate that the effect of noise and communication time-delay on the consensus dynamics in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${m}$ </tex-math></inline-formula> -dimensional prism networks is minimal. The obtained results illustrate that the scale-free topology of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${m}$ </tex-math></inline-formula> -dimensional prism networks along with loopy structure is responsible for strong robustness with respect to consensus dynamics in prism networks.