Descents and nodal load in scale-free networks

TL;DR

This paper investigates the relationship between node descents in BFS trees and load distribution in scale-free networks, revealing that load peaks are sharply defined and challenge previous power-law assumptions.

Contribution

It introduces a semi-analytical approach linking BFS descents to node load and uncovers that load distribution consists of distinct peaks rather than a continuous power-law.

Findings

Load distribution shows sharply delineated peaks.

Contradicts previous power-law conjecture for load distribution.

Provides a new interpretation of load based on BFS descents.

Abstract

The load of a node in a network is the total traffic going through it when every node pair sustains a uniform bidirectional traffic between them on shortest paths. We show that nodal load can be expressed in terms of the more elementary notion of a node's descents in breadth-first-search (BFS or shortest-path) trees, and study both the descent and nodal-load distributions in the case of scale-free networks. Our treatment is both semi-analytical (combining a generating-function formalism with simulation-derived BFS branching probabilities) and computational for the descent distribution; it is exclusively computational in the case of the load distribution. Our main result is that the load distribution, even though it can be disguised as a power-law through subtle (but inappropriate) binning of the raw data, is in fact a succession of sharply delineated probability peaks, each of which can be clearly interpreted as a function of the underlying BFS descents. This find is in stark contrast with previously held belief, based on which a power law of exponent -2.2 was conjectured to be valid regardless of the exponent of the power-law distribution of node degrees.