Relation between tree-approximation failure and transition-order change

Determine whether the failure of the tree approximation at large σ and the change from a first-order to a continuous transition in the multiplex of two one-dimensional long-range networks have the same underlying origin.

Background

The tree approximation predicts a first-order transition for every value of σ and provides a reasonably accurate estimate of the critical coupling for σ up to approximately 0.5. The simulations, however, indicate that the actual multiplex transition is first order for σ≤0.35 and continuous for σ≥0.4, while the approximation becomes increasingly inaccurate at larger σ.

The paper identifies two distinct discrepancies: the approximation does not reproduce the continuous transition, and its critical-value estimate deteriorates as long-range links become less dominant. It explicitly leaves unresolved whether these phenomena arise from a common mechanism.

References

Whether the failure of the tree approximation at large σ and the change in the order of the transition have the same origin is an open question.

— Percolation on interdependent one-dimensional long-range networks  (2610.10097 - Sharabi et al., 7 Oct 2026) in Section Discussion, paragraph beginning “We note that the tree approximation predicts”