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
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, NI and NE. 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 NI > NE or near its maximum () if NI < NE. At the transition (NI = NE), 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.
PACS: 64.60.aq – Networks / 89.75.Fb – Structures and organization in complex systems / 05.70.Fh – Phase transitions: general studies
© EPLA, 2012