Sampling properties of random graphs: the degree distribution

TL;DR

This paper analyzes how different sampling schemes affect the degree distribution in networks, deriving conditions for distribution preservation and applying findings to biological network data.

Contribution

It introduces a necessary and sufficient condition for degree distribution preservation under sampling and explores its implications for various network types.

Findings

Classical random graphs satisfy the distribution preservation condition under random sampling.

Most real-world networks do not meet the condition, leading to altered degree distributions.

Degree-dependent sampling breaks the closure property even in classical random graphs.

Abstract

We discuss two sampling schemes for selecting random subnets from a network: Random sampling and connectivity dependent sampling, and investigate how the degree distribution of a node in the network is affected by the two types of sampling. Here we derive a necessary and sufficient condition that guarantees that the degree distribution of the subnet and the true network belong to the same family of probability distributions. For completely random sampling of nodes we find that this condition is fulfilled by classical random graphs; for the vast majority of networks this condition will, however, not be met. We furthermore discuss the case where the probability of sampling a node depends on the degree of a node and we find that even classical random graphs are no longer closed under this sampling regime. We conclude by relating the results to real {\it E.coli} protein interaction network data.