Locate the Critical Connectivity Threshold for Systemic Cascade Transition

Determine the intermediate connectivity levels at which the stylized interconnection-queue contagion model transitions from progressive failure to systemic collapse, particularly for average degrees between 10 and 20, in order to locate the phase-transition point more precisely.

Background

The paper models interconnection-queue projects as nodes in an Erdős–Rényi cost-sharing network. At average degree k = 10, the model produces moderate cascade amplification, whereas at k = 20 it exhibits an unstable, near-total-collapse regime. This contrast suggests a critical connectivity threshold separating progressive failure from categorical systemic fragility.

The simulations evaluate k = 3, 5, 10, and 20, leaving the intermediate region unresolved. Identifying the threshold more precisely would clarify when cost-sharing interdependencies become capable of sustaining systemic withdrawal cascades and would help assess whether reducing network density could function as a resilience intervention.

References

The transition between k = 10 (stable, 1.38\times net amplification under roughly 10\% shock) and k = 20 (unstable, bimodal collapse) suggests the existence of a critical connectivity threshold. Beyond this threshold, the system transitions from progressive failure, in which shocks produce proportionate additional failures, to a state of categorical fragility. In the latter, virtually any perturbation can trigger systemic collapse. Future research is required to evaluate intermediate connectivity levels (k = 12, 15) to locate the phase transition point more precisely.

— The U.S. Interconnection Queue System: Cascading Vulnerability Analysis and a Resilience Engineering Framework  (2609.10455 - Heidari, 9 Sep 2026) in Section 7.3, “Computational Contagion Model Results”