Strong connectivity of the JCM-constrained state space under k-switches

Determine whether the state space Z of simple vertex-colored graphs that preserve a fixed degree sequence and a fixed Joint Color Matrix (JCM) is strongly connected under k-switch operations that simultaneously swap k edges while preserving the degree sequence and the JCM; and address the computational challenges associated with employing such k-switches for sampling from Z.

Background

The paper shows that for simple vertex-colored graphs with more than two colors, the state space defined by all graphs sharing the same degree sequence and Joint Color Matrix (JCM) is not strongly connected under JCM-preserving double edge swaps (JDESs). A specific counterexample demonstrates that no sequence of JDESs can connect certain states.

In that counterexample, the authors note that a k-switch operation (simultaneous swapping of k edges) can connect the two graphs, suggesting a possible path to restoring strong connectivity. However, the general question of whether k-switches suffice to strongly connect the entire JCM-constrained state space remains unresolved, and practical algorithmic issues for using k-switches in sampling are not yet addressed.

References

However, this fact alone does not imply that the state space is strongly con- nected by k-switches. Investigating whether the state space is in- deed strongly connected under k-switches, and addressing the as- sociated computational challenges are open questions for future research.

— Polaris: Sampling from the Multigraph Configuration Model with Prescribed Color Assortativity  (2409.01363 - Preti et al., 2024) in Appendix A.2: State Space is Disconnected for Simple Graphs with More Than 2 Colors