Transmission of packets on a hierarchical network: Statistics and explosive percolation

TL;DR

This paper models packet transmission in hierarchical networks, analyzing statistical properties and percolation transitions, revealing continuous and explosive phase transitions depending on network topology.

Contribution

It provides analytical expressions for occupation numbers and identifies conditions for continuous versus explosive percolation transitions in hierarchical networks.

Findings

Mean occupation numbers derived analytically

Percolation transitions are continuous in 2-D models

Explosive transition observed in V-lattice topology

Abstract

We analyze an idealized model for the transmission or flow of particles, or discrete packets of information, in a weight bearing branching hierarchical 2-D networks, and its variants. The capacities add hierarchically down the clusters. Each node can accommodate a limited number of packets, depending on its capacity and the packets hop from node to node, following the links between the nodes. The statistical properties of this system are given by the Maxwell - Boltzmann distribution. We obtain analytical expressions for the mean occupation numbers as functions of capacity, for different network topologies. The analytical results are shown to be in agreement with the numerical simulations. The traffic flow in these models can be represented by the site percolation problem. It is seen that the percolation transitions in the 2-D model and in its variant lattices are continuous transitions, whereas the transition is found to be explosive (discontinuous) for the V- lattice, the critical case of the 2-D lattice. We discuss the implications of our analysis.