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.

Fig. 2. Clusters survive small external noise in 2d simulations (numbers on the axes denote lattice location), hence the inclusion of a small second temperature does not change the composition of neighbors in the steady state. From ref. G. Ódor, Self-organizing, two-temperature Ising model describing human segregation, arXiv:0710.1496 Int. J. Mod. Phys C 19 (2008) in 393.

Quenched disorder is known to play a relevant role in dynamical processes and phase transitions. Its 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]. 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], in power-grids [83], [86], or in epidemic models [91].

Fig. 4. Weight distribution of the fruit-fly connectome. Right inset: adjacency matrix plot of the fruit-fly connectome. Black/white dots denote connected/unconnected nodes i (vertical), j (horizontal). Left inset: full adjacency matrix down-sampled with a max pooling kernel of size 10 x 10, arXiv:0710.1496

For a recent review see: [93].

Apr 20, 2022.