Extraordinary variability and sharp transitions in a maximally frustrated dynamic network

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 (N ≡ NINE) if NI < NE . At the transition (NI = NE), the fraction X/N 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. Introduction. – Though their significance may not be understood at first glance, network structures can often be easily recognized in nature, from microscopic neurons to galactic filaments [1–5]. While these natural phenomena existed for ages and eons, more recently humans started building artificial counterparts in widely distinct arenas, e.g., in social [6, 7], infrastructural [8, 9], economic [10], and political [11] contexts. Their importance for modern societies cannot be understated. Meanwhile, quantitative efforts to characterize and model networks emerged even more recently, involving developments in many branches of science and engineering, including graph theory, statistical physics, neuroscience, computer science, etc. While these efforts led to much progress, many aspects of networks remain to be explored and/or modeled. For example, much of the literature focuses on static characteristics. While many situations may be well-served by a static model of networks (e.g., highways on timescales of days or months), there are many others for which a dynamic network description would be more appropriate. In particular, social contacts are generally in a state of flux, as new friendships or alliances are formed or existing links are severed. Our goal here is to study such evolving networks, to seek steady states (if any) and characterize their statistical properties. Are they like random Erdös-Rényi graphs [12], with Gaussian degree distributions? Are there strong or weak clustering and/or modularity characteristics? To make the network model easier to understand, we use the language of social network, so that nodes and links represent individuals and contacts (between pairs of persons), respectively. We will model the interactions between nodes stochastically and dynamically, i.e., through probabilistic evolution rules for adding/cutting links. In the language of graphs, the degree of each node will typically change, with a set of prescribed rates. Inspired by the facts that an individual tends to prefer a certain number of contacts in social networks, and that as individuals adapt to changing circumstances, the network p-1 ar X iv :1 21 2. 05 76 v1 [ co nd -m at .s ta tm ec h] 3 D ec 2 01 2