A stopping rule for randomly sampling bipartite networks with fixed degree sequences

TL;DR

A stopping rule is proposed that focuses on the distance between sampled networks and the observed network, and stops performing trades when this distribution stabilizes, and demonstrates that it is practical for use in empirical applications.

Abstract

Statistical analysis of bipartite networks frequently requires randomly sampling from the set of all bipartite networks with the same degree sequence as an observed network. Trade algorithms offer an efficient way to generate samples of bipartite networks by incrementally `trading' the positions of some of their edges. However, it is difficult to know how many such trades are required to ensure that the sample is random. I propose a stopping rule that focuses on the distance between sampled networks and the observed network, and stops performing trades when this distribution stabilizes. Analyses demonstrate that, for over 650 different degree sequences, using this stopping rule ensures a random sample with a high probability, and that it is practical for use in empirical applications.

Figures (4)

• Figure 1: (A) The set $\mathcal{B}$ of all bipartite networks with top-degree sequence {1,2,1} and bottom-degree sequence {1,2,1}; (B) Example of bipartite network randomization using a trade algorithm.
• Figure 2: Distribution of networks in a sample drawn using a trade algorithm, following the given number of trades, with a $\chi^2$ test of uniform distribution.
• Figure 3: Distribution of distance between a starting matrix and networks in a sample ($D$) drawn using a trade algorithm, following the given number of trades, with an Anderson-Darling (AD) test of equality with an earlier distribution.
• Figure 4: Distribution of sampled networks' distance from observed (A) Darwin's Finches, (B) Southern Women, and (C) US Senate networks following trades