Random hierarchies that facilitate self-organization

Abstract

Since it is widely accepted that self-organization is difficult to achieve using constructive or centrally run algorithms a random hierarchy is proposed that intrinsically facilitates self-organization. The random hierarchy consists of each node in the network independently choosing a rank at random such that a mean 2(\Delta-1)\Delta^{i-1} nodes have rank i, where \Delta is a network wide hierarchy parameter. Each node of rank i chooses the nearest node of rank i-1 as its leader which forms the hierarchy. The mean and variance of the relevant properties is derived, for example it is shown that each leader has a mean \Delta followers. Simulations were used to demonstrate the effectiveness of the proposed hierarchy and a "bare-bones" set of procedures where provided that may be used to implement the hierarchy over a network of autonomous nodes in a robust way.