[Paper Review] Swap connectivity for two graph spaces between simple and pseudo graphs and disconnectivity for triangle constraints

Joel Nishimura|arXiv (Cornell University)|Apr 6, 2017

Markov Chains and Monte Carlo Methods11 references3 citations

TL;DR

This paper investigates the connectivity of double edge-swap Markov chains for two graph spaces between simple and pseudographs: multiloop-graphs (multiple self-loops allowed, no multiedges) and loopy-multigraphs (multiedges and single self-loops allowed). It proves that double edge-swaps can uniformly sample all loopy-multigraphs for any degree sequence, but cannot uniformly sample all multiloop-graphs for certain degree sequences, which are fully characterized. The work completes the classification of swap-based MCMC connectivity across all combinations of self-loops and multiedges in graph spaces.

ABSTRACT

With sufficient time, double edge-swap Markov chain Monte Carlo (MCMC) methods are able to sample uniformly at random from many different and important graph spaces. For instance, for a fixed degree sequence, MCMC methods can sample any graph from: simple graphs; multigraphs (which may have multiedges); and pseudographs (which may have multiedges and/or multiple self-loops). In this note we extend these MCMC methods to `multiloop-graphs', which allow multiple self-loops but not multiedges and `loopy-multigraphs' which allow multiedges and single self-loops. We demonstrate that there are degree sequences on which the standard MCMC methods cannot uniformly sample multiloop-graphs, and exactly characterize which degree sequences can and cannot be so sampled. In contrast, we prove that such MCMC methods can sample all loopy-multigraphs. Taken together with recent work on graphs which allow single self-loops but no multiedges, this work completes the study of the connectivity (irreducibility) of double edge-swap Markov chains for all combinations of allowing self-loops, multiple self-loops and/or multiedges. Looking toward other possible directions to extend edge swap sampling techniques, we produce examples of degree and triangle constraints which have disconnected spaces for all edges swaps on less than or equal to 8 edges.

Motivation & Objective

• To determine whether double edge-swap Markov chains can uniformly sample all graphs in the multiloop-graph and loopy-multigraph spaces for any fixed degree sequence.
• To identify the precise conditions under which double edge-swaps fail to connect the graph of graphs (gog) for multiloop-graphs.
• To complete the classification of swap-based MCMC connectivity across all combinations of self-loops and multiedges in graph spaces between simple and pseudographs.
• To examine the limitations of edge-swap methods when sampling graphs under triangle constraints, including fixed triangle counts and triangle sequences.

Proposed method

• The paper constructs the graph of graphs (gog) for each degree sequence, where nodes represent graphs and edges represent valid double edge-swaps.
• It analyzes the irreducibility (connectivity) of the gog for multiloop-graphs and loopy-multigraphs using combinatorial and structural graph theory.
• It derives exact necessary and sufficient conditions for gog disconnectivity in multiloop-graphs based on degree sequence properties.
• It provides explicit counterexamples with up to 8 edges to demonstrate that double edge-swaps cannot connect all graphs under fixed triangle counts or triangle sequences.
• It compares the behavior of k-edge-swaps (e.g., 3- and 4-edge-swaps) on constrained graph spaces, showing disconnection even for small graphs.
• It uses exhaustive enumeration and isomorphism class analysis to verify disconnection in specific cases, such as degree sequences with 4 triangles or specific triangle sequences.

Experimental results

Research questions

• RQ1 For which degree sequences is the graph of graphs (gog) disconnected under double edge-swaps in the multiloop-graph space?
• RQ2 Can double edge-swaps uniformly sample all loopy-multigraphs for any fixed degree sequence?
• RQ3 Are there degree and triangle constraint combinations for which no sequence of double edge-swaps can connect all graphs in the space, even with small graphs?
• RQ4 How do triangle constraints (fixed number of triangles or per-vertex triangle counts) affect the connectivity of edge-swap MCMC samplers?
• RQ5 To what extent do the disconnection issues in triangle-constrained spaces limit the effectiveness of biased or importance-weighted sampling methods?

Key findings

• The graph of graphs for loopy-multigraphs is connected for all degree sequences, meaning double edge-swap MCMC methods can uniformly sample all such graphs.
• There exist degree sequences for which the graph of graphs for multiloop-graphs is disconnected, and these are fully characterized by the paper.
• The paper provides an exact criterion to determine whether a given degree sequence leads to a disconnected gog in the multiloop-graph space.
• For degree sequences with exactly 4 triangles, such as {3,3,3,3,2,2,2,2}, there are two isomorphism classes that are disconnected under double edge-swaps, requiring at least 8 edge changes to connect.
• Even for small graphs with 8 edges, there are degree and triangle sequence combinations where no double edge-swap sequence can connect all graphs, indicating fundamental limitations in swap-based sampling.
• The disconnectivity issues are more severe and general than previously known problems with self-loops, as the isomorphism classes differ by many edges, challenging both strict and biased sampling methods.