[Paper Review] Smaller Universes for Uniform Sampling of 0,1-matrices with fixed row and column sums

Annabell Berger, Corrie Jacobien Carstens|arXiv (Cornell University)|Mar 7, 2018

Complex Network Analysis Techniques2 references3 citations

TL;DR

This paper proposes a projected Markov chain approach for uniform sampling of 0,1-matrices with fixed row and column sums by ignoring node labels, effectively sampling isomorphism classes of graphs. It proves that the projected chain mixes at least as fast as the original, often significantly faster due to a smaller state space and better initial distribution, and introduces a preprocessing step that enables efficient sampling of labeled graphs without hexagonal moves, improving practical performance for both switch and Curveball chains.

ABSTRACT

An important problem arising in the study of complex networks, for instance in community detection and motif finding, is the sampling of graphs with fixed degree sequence. The equivalent problem of generating random 0,1 matrices with fixed row and column sums is frequently used as a quantitative tool in ecology. It has however proven very challenging to design sampling algorithms that are both fast and unbiased. This article focusses on Markov chain approaches for sampling, where a close-to-random graph is produced by applying a large number N of small changes to a given graph. Examples are the switch chain and Curveball chain, which are both commonly used by practitioners as they are easy to implement and known to sample unbiased when N is large enough. Within theoretical research, much effort has gone into proving bounds on N. However, existing theoretical bounds are impractically large for most applications while experiments suggest that much fewer steps are needed to obtain a good sample. The contribution of this article is twofold. Firstly it is a step towards better understanding of the discrepancy between experimental observations and theoretically proven bounds. In particular, we argue that while existing Markov chain algorithms run on the set of all labelled graphs with a given degree sequence, node labels are unimportant in practice and are usually ignored in determining experimental bounds. We prove that ignoring node labels corresponds to projecting a Markov chain onto equivalence classes of isomorphic graphs and that the resulting projected Markov chain converges to its stationary distribution at least as fast as the original Markov chain. Often convergence is much faster, as we show in examples, explaining part of the difference between theory and experiments...

Motivation & Objective

• To explain the discrepancy between theoretical mixing time bounds and practical performance in Markov chain sampling of 0,1-matrices with fixed margins.
• To formalize the effect of ignoring node labels in sampling by modeling it as a projection of the Markov chain onto isomorphism classes of graphs.
• To demonstrate that projected chains mix at least as fast as original chains, with significant speedups in many cases due to reduced state space and improved initialization.
• To develop a preprocessing step that, when combined with projected chains, enables fast, unbiased sampling of labeled graphs without requiring hexagonal moves in switch or Curveball chains.
• To show that the resulting combined method yields ergodic chains for all directed graphs, removing the need for complex move types in practice.

Proposed method

• Project the original Markov chain on labeled graphs onto equivalence classes of isomorphic graphs, reducing the state space to isomorphism types.
• Prove that the mixing time of the projected chain is bounded above by that of the original chain, using spectral properties and eigenvalue inclusion.
• Introduce a linear-time preprocessing step that randomly permutes node labels to achieve uniform sampling of labeled graphs after projection.
• Apply the projected chain to isomorphism classes and then use the preprocessing step to recover a uniformly random labeled graph.
• Use spectral gap analysis to quantify mixing time improvements, showing a spectral gap of $ \frac{2}{n} $ for the projected switch chain on certain graph families.
• Demonstrate that for specific degree sequences, the projected chain state space reduces from exponential to constant size, e.g., from $ \binom{2l}{l} $ to 1 for $ l $-regular secondary nodes.

Experimental results

Research questions

• RQ1 Does projecting a Markov chain onto isomorphism classes of graphs preserve or improve mixing time compared to the original labeled chain?
• RQ2 To what extent does ignoring node labels explain the faster convergence observed in practice compared to theoretical bounds?
• RQ3 Can a preprocessing step be used to efficiently sample labeled graphs after projecting the chain, without requiring complex moves like hexagonal moves?
• RQ4 What is the impact of state space reduction on the spectral gap and mixing time in projected chains?
• RQ5 Are there degree sequences for which the projected chain yields a constant-sized state space, enabling near-instant sampling?

Key findings

• The projected Markov chain converges at least as fast as the original chain, with mixing time bounded above by that of the original, due to eigenvalue inclusion and reduced state space.
• For the family of graphs with two primary nodes of degree $ n-1 $ and $ n $ secondary nodes of degree 1, the projected switch chain has a spectral gap of $ \frac{2}{n} $, yielding a mixing time of $ O(n \log(n/\epsilon)) $.
• In the case of two primary nodes of degree $ l $ and $ 2l $ secondary nodes of degree 1, the projected chain reduces the state space from $ \binom{2l}{l} $ to a single state, enabling immediate uniform sampling after preprocessing.
• The combination of the projected chain with a linear-time preprocessing step produces an ergodic chain for all directed graphs, eliminating the need for hexagonal moves in the switch and Curveball chains.
• The speed-up from projection is most pronounced when the original state space has high symmetry, as the reduction in state space size and improved initial distribution lead to faster convergence.
• Theoretical bounds on mixing time for the original chain are often impractically large, but the projected chain provides a practical path to faster, unbiased sampling by focusing on topological structure rather than labels.