Networks, brain, sociophysics, population dynamics, long-range interactions

The application of nonequilibrium statistical physics in various other branches of sciences, has been flourished in the past decades. The modeling and tools of physics have opened a new way of analyzing these complex system. Statistical physics can describe population dynamics of animals or plants and can explain extinction or oscillation phenomena observed in nature. Besides the traditional mathematical approaches based on differential equations it can take into account fluctuations observed in predator-prey models.

We have investigated simple predator-prey models [30] and for some values of the control parameters we found that the model exhibits a line of directed percolation like transitions to a single absorbing state. For other values of the control parameters one finds a second line of continuous transitions toward an infinite number of absorbing states, and the corresponding steady-state exponents are mean-field-like. The critical behavior of the special bicritical point, where the two transition lines meet, belongs to a different universality class. A particular strategy for preparing the initial states used for the dynamical Monte Carlo method is devised to correctly describe the physics of the system near the second transition line.





It is worth to mention, that our model is closely related to a model introduced by Drossel and Schwabl to investigate the effect of immunization in an extension of the simple forest-fire model. That three-state model (0 is an empty site, 1 is a tree, and 2 is a burning tree) differs from our model in some details: the growth rate of a tree (s : 0 ->1) is p , independently of the environment, and a tree is ignited (s : 1-> 2) with rate (1-g )Q (n₂ ), (Q is the usual Heaviside function) . This second process models the immunization of trees against fire. The third process (s : 2 -> 0) occurs at rate 1. For nonzero immunity and p>0, Albano showed that a transition toward a single absorbing state is DP like, while for p=0 (at the end point of the DP transition line), the transition belongs to the dynamical percolation universality class, and the absorbing state is not unique.

The problem of human segregation is an important problem of society and politics even in the 21st century. Social sciences have been investigating the reasons and and nature of segregation for a long time. Sociologist have introduced several models, one of them is the Schelling model 2. From physicist point of view that model is a 3-state voter-type non-equilibrium model (groups A,B and empty), with spin-exchange dynamics at zero temperature (T = 0) on a 2-dimensional square lattice. Although the model describes a segregation by a quench without external reasons, unwanted frozen states may also occur. As an application of the knowledge of nonequilibrium models we have also investigated a social related model as a proposal of Prof. D. Stauffer.

A two-temperature Ising-Schelling model is introduced and studied for describing human segregation. The self -organized Ising model with Glauber kinetics simulated by Muller et al. exhibits a phase transition between segregated and mixed phases mimicking the change of tolerance (local temperature) of individuals. The effect of external noise is considered here [50] as a second temperature to the decision of individuals who consider change of accommodation.

Quenched disorder is known to play a relevant role in dynamical processes and phase transitions. Effects on the dynamics of complex networks have hardly been studied. Aimed at filling this gap, we have analyzed the contact process, i.e., the simplest propagation model, with quenched disorder on complex networks. We found Griffiths phases and other rare-region effects, leading generically to anomalously slow (algebraic, logarithmic, . . .) relaxation , on Erdos-Renyi networks. Similar effects are predicted to exist for other topologies with a finite percolation threshold. More surprisingly, we found that Griffiths phases can also emerge in the absence of quenched disorder, as a consequence of topological heterogeneity in networks with finite topological dimension. These results have a broad spectrum of implications for propagation phenomena and other dynamical processes on networks [57],[59],[62],[64],[67],[68],[69],[71],[72],[72],[73]. Even in finite scale-free networks one can also find Griffiths effects and slow dynamics in extended parameter regions [64],[74]. Recently we showed numerical evidence for Griffiths phase in modular networks possessing infinite topological dimensions [82].

More interstingly it turns out that at criticality or in the Griffiths phase bursty behavior of agents can emerge as consequence of quenched network topologies [71]. This has been studied in synthetic brain networks [73], and the possibilty of Griffiths effects in large human connectome networks has also been pointed out [75], [77], [79], [85].
Heterogeneities also play an important role at network synchronization phenomena, [87], which occur in brain models [88], [92], [93], [94], [98], [104], in power-grids [83], [86], or in epidemic models [91].

Dec 9, 2024