Theoretical analysis for tree-like networks using random geometry

Among various network topologies, tree-like networks, also known as hierarchical networks are proposed to decrease the overhead of the routing table especially for the situation involving many network nodes. Usually, the routing table size and the routing complexity are the two crucial concerns in designing a large network. Although there have been various algorithms to optimise the routing strategies for the hierarchical networks, hardly exists any work in studying and evaluating the routing table size and the routing complexity rigorously in the statistical sense. In this study, the authors generalise a new mathematical framework by applying the point process in random geometry. The new framework proposed by the authors leads to the explicit statistical measures of the routing table size and the routing complexity, which can be specified as the functions of the hierarchical network parameters including the number of the hierarchical levels and the cluster population for each hierarchical level. After the relationship between the network topology and these two network performance measures (routing complexity and routing table size) is established, a cluster-population optimisation method for hierarchical networks is presented. The simulation results are also provided to demonstrate the advantage of a hierarchical network over the associated conventional network without hierarchy.