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

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. The scaling behavior of the second order percolation case is studied in detail. We discuss the implications of our analysis.

Transmission of Packets on a Hierarchical Network: Avalanches, Statistics and Explosive Percolation

“…The numerical study by [7] shows that the percolation transition in the V lattice is a discontinuous or explosive one, which can be observed from the Figure. Fig.…”

“…The V lattice exhibits critical behavior for avalanche phenomena [20], site percolation phenomena [7], and capacity distribution and failure rate [6] quite distinct from the behavior seen for the base lattice model, manifesting its criticality. Here, we focus only on the avalanche and percolation properties of the V lattice.…”

“…6 The order parameter S as a function of packet density µ, averaged over 500 realizations, for the free subnetwork in the base lattice network of (100 × 100) sites. [7], with associated second order percolation exponents. Here, again the V lattice shows critical behavior.…”

“…For the free sub-network, one anticipate a transition from the percolating to the non-percolating state when the packet density µ increases, where the packet density is the ratio of total number of packets to total number of sites in the network. In order to analyze the transition, we simulate the stochastic dynamics of packet transmission in the V lattice for different values of the packet density µ [7].…”

“…Earlier studies of transport processes on networks has been carried out for important classes like scale free and random networks [3,4,5]. However, an important class of networks, viz., that of branching hierarchical networks [6,7], has not yet been extensively explored. Networks of the branching, hierarchical type are common in both real and engineering contexts.…”