Universality and scaling in complex networks from periods of Chinese history.

Critical physical systems with large numbers of molecules can show universal and scaling behaviors. It is of interest to know whether human societies with large numbers of people can show the same behaviors. Here, we use network theory to analyze Chinese history in periods 209 BCE-23 CE and 515-618 CE) related to the Western Han-Xin Dynasty and the late Northern Wei-Sui Dynasty, respectively. Two persons are connected when they appear in the same historical event. We find that the historical networks from two periods separated about 500 years have interesting universal and scaling behaviors, and they are small-world networks; their average cluster coefficients as a function of degree are similar to the network of movie stars. In the historical networks, the persons with larger degrees prefer to connect with persons with a small degree; however, in the network of movie stars, the persons with larger degrees prefer to connect with persons with large degrees. We also find an interesting similar mechanism for the decline or collapse of historical Chinese dynasties. The collapses of the Xin dynasty (9-23 CE) and the Sui dynasty (581-618 CE) were initiated from their arrogant attitude toward neighboring states.

Dependence of extreme events on spatial location.

To model the dependence of extreme events on locations, we consider extreme events of Brownian particles in a potential. We find that barring the exception of very large potentials and/or very small regions, in general, the probability of extreme events increases with the potential. Our approach is general and can be useful for studying several complex systems.

Physical mechanism for biopolymers to aggregate and maintain in non-equilibrium states.

Many human or animal diseases are related to aggregation of proteins. A viable biological organism should maintain in non-equilibrium states. How protein aggregate and why biological organisms can maintain in non-equilibrium states are not well understood. As a first step to understand such complex systems problems, we consider simple model systems containing polymer chains and solvent particles. The strength of the spring to connect two neighboring monomers in a polymer chain is controlled by a parameter s with s → ∞ for rigid-bond. The strengths of bending and torsion angle dependent interactions are controlled by a parameter s A with s A → -∞ corresponding to no bending and torsion angle dependent interactions. We find that for very small s A , polymer chains tend to aggregate spontaneously and the trend is independent of the strength of spring. For strong springs, the speed distribution of monomers in the parallel (along the direction of the spring to connect two neighboring monomers) and perpendicular directions have different effective temperatures and such systems are in non-equilibrium states.

Stochastic dynamical model for stock-stock correlations.

We propose a model of coupled random walks for stock-stock correlations. The walks in the model are coupled via a mechanism that the displacement (price change) of each walk (stock) is activated by the price gradients over some underlying network. We assume that the network has two underlying structures, describing the correlations among the stocks of the whole market and among those within individual groups, respectively, each with a coupling parameter controlling the degree of correlation. The model provides the interpretation of the features displayed in the distribution of the eigenvalues for the correlation matrix of real market on the level of time sequences. We verify that such modeling indeed gives good fitting for the market data of US stocks.