An Extraordinary Transition in a Minimal Adaptive Network of Introverts and Extroverts

We study a minimal adaptive network involving two populations, modeling the behavior of extreme introverts (I) and extroverts (E). When chosen to update, an I simply cuts one of its links at random while an E adds a link to any other yet-to-be-connected individual (node). In the steady state, the active links in the system are obviously only the cross-links between the I's and the E's. With no free parameters other than the numbers of each population (NI, NE), this minimal model displays remarkable properties: Through simulations using O(10)-O(1000) nodes, we find that the typical number of cross-links (X) fluctuates surprisingly close to the minimum or the maximum allowed values, depending on whether NI > NE or otherwise. At the transition point (i.e., NI = NE), the fraction X/(NINE) wanders across a substantial part of the unit interval, much like a pure random walk confined between two walls. Since this system can be mapped to a NINE Ising model with spin flip dynamics, we note that such fluctuations are far greater than those in the standard Ising model (at either first or second order transitions). Thus, we refer to the case here as an “extraordinary transition.” Thanks to the restoration of detailed balance and the existence of a “Hamiltonian,” several qualitative aspects of these remarkable phenomena can be understood analytically