Time-Delay Dependent Stability Robustness of Small-World Protocols for Fast Distributed Consensus Seeking

In order to achieve a faster consensus seeking over complex networks, recently two main solutions have been proposed; the first one is using the physical communication network as an information flow graph but with optimized weights for all information flow edges, and the second one is introducing a few shortcut (non-local) multi- hop edges to the information flow network without physically adding or changing any edges in the underlying communication network. The most famous and interesting one in later category is small-world networks that can dramatically increase the algebraic connectivity of regular complex networks. We recast the consensus protocol over a small-world information flow network considering time delays and compare its performance and time-delay stability margin with one over an equally weighted information flow network on the same communication network. Our results show the fact that, "the small-world network construction has a negligible effect on the time-delay robustness of the consensus protocol over a initial regular network", is not true. Hence considering this observed fact and also some difficulties in distributed construction of small-world networks, we have to take more care in employing small-world structures to get faster consensus protocols.