Order-to-chaos transition in the hardness of random Boolean satisfiability problems.

Transient chaos is a ubiquitous phenomenon characterizing the dynamics of phase-space trajectories evolving towards a steady-state attractor in physical systems as diverse as fluids, chemical reactions, and condensed matter systems. Here we show that transient chaos also appears in the dynamics of certain efficient algorithms searching for solutions of constraint satisfaction problems that include scheduling, circuit design, routing, database problems, and even Sudoku. In particular, we present a study of the emergence of hardness in Boolean satisfiability (k-SAT), a canonical class of constraint satisfaction problems, by using an analog deterministic algorithm based on a system of ordinary differential equations. Problem hardness is defined through the escape rate κ, an invariant measure of transient chaos of the dynamical system corresponding to the analog algorithm, and it expresses the rate at which the trajectory approaches a solution. We show that for a given density of constraints and fixed number of Boolean variables N, the hardness of formulas in random k-SAT ensembles has a wide variation, approximable by a lognormal distribution. We also show that when increasing the density of constraints α, hardness appears through a second-order phase transition at α_{χ} in the random 3-SAT ensemble where dynamical trajectories become transiently chaotic. A similar behavior is found in 4-SAT as well, however, such a transition does not occur for 2-SAT. This behavior also implies a novel type of transient chaos in which the escape rate has an exponential-algebraic dependence on the critical parameter κ∼N^{B|α-α_{χ}|^{1-γ}} with 0<γ<1. We demonstrate that the transition is generated by the appearance of metastable basins in the solution space as the density of constraints α is increased.

Dynamics of conflicts in Wikipedia.

In this work we study the dynamical features of editorial wars in Wikipedia (WP). Based on our previously established algorithm, we build up samples of controversial and peaceful articles and analyze the temporal characteristics of the activity in these samples. On short time scales, we show that there is a clear correspondence between conflict and burstiness of activity patterns, and that memory effects play an important role in controversies. On long time scales, we identify three distinct developmental patterns for the overall behavior of the articles. We are able to distinguish cases eventually leading to consensus from those cases where a compromise is far from achievable. Finally, we analyze discussion networks and conclude that edit wars are mainly fought by few editors only.

Circadian patterns of Wikipedia editorial activity: a demographic analysis.

Wikipedia (WP) as a collaborative, dynamical system of humans is an appropriate subject of social studies. Each single action of the members of this society, i.e., editors, is well recorded and accessible. Using the cumulative data of 34 Wikipedias in different languages, we try to characterize and find the universalities and differences in temporal activity patterns of editors. Based on this data, we estimate the geographical distribution of editors for each WP in the globe. Furthermore we also clarify the differences among different groups of WPs, which originate in the variance of cultural and social features of the communities of editors.