You would expect that how nodes are connected matters enormously for phase transitions. Rewire a lattice — change the long-range connections, the loop structure, the clustering — and the critical behavior should change. Daimari, Borah, and Thongjaomayum test this by studying the random field Ising model on generalized Petersen graphs GP(N,k), where the parameter k controls connectivity while keeping coordination number fixed at z=3.
The generalized Petersen graph consists of an outer loop and an inner loop of N nodes each. Each inner node connects to its k-th neighbors along the inner loop, plus one node on the outer loop. Varying k changes the topology dramatically — short loops, long-range connections, different graph diameters — while every node retains exactly three neighbors.
The result: no topological rearrangement rescues criticality. For z=3 on a random graph, the random field Ising model has no phase transition — this was known from exact solutions. The same absence persists across all generalized Petersen graph geometries, regardless of k. Directed variants show the same behavior. The coordination number alone determines whether a phase transition exists, and topology is irrelevant.
This is a case where the local constraint — how many neighbors each node has — completely overrides global structure. The system cannot “see” its large-scale wiring; it responds only to its immediate crowding. At z ≤ 3, there are simply not enough local connections to sustain collective order against the disordering field, and no amount of clever long-range reconnection changes that arithmetic.
Coordination number overrides connectivity: in disordered systems at low coordination, the system's collective behavior is blind to its own topology.