Extraordinary variability and sharp transitions in a maximally frustrated dynamic network

¹ Department of Physics, Virginia Polytechnic Institute and State University - Blacksburg, VA 24061, USA
² Department of Physics and Astronomy, Iowa State University - Ames, IA 50011, USA

Received: 29 September 2012
Accepted: 3 December 2012

Abstract

Using Monte Carlo and analytic techniques, we study a minimal dynamic network involving two populations of nodes, characterized by different preferred degrees. Reminiscent of introverts and extroverts in a population, one set of nodes, labeled introverts (I), prefers fewer contacts (a lower degree) than the other, labeled extroverts (E). As a starting point, we consider an extreme case, in which an I simply cuts one of its links at random when chosen for updating, while an E adds a link to a random unconnected individual (node). The model has only two control parameters, namely, the number of nodes in each group, N_I and N_E. In the steady state, only the number of crosslinks between the two groups fluctuates, with remarkable properties: Its average (X) remains very close to 0 for all N_I > N_E or near its maximum () if N_I < N_E. At the transition (N_I = N_E), the fraction wanders across a substantial part of [0,1], much like a pure random walk. Mapping this system to an Ising model with spin-flip dynamics and unusual long-range interactions, we note that such fluctuations are far greater than those displayed in either first- or second-order transitions of the latter. 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.