Fractal-to-dispersive Aggregate Transition From Multi-particle And Small-world Connected Structures

Diffusion-limited aggregation(DLA),which was introduced by Witten and Sander in 1981,can describe many fractal phenomena in nature.In this paper,we discuss variants of DLA model to investigated the effect of particles density and network structure on the fractal characteristics and fractal dimension of the aggregation.The main research can be divided into the following two parts:First,we study the particle aggregation model in a square lattice.We simulated multi-seed DLA model,and the results showed that the small clusters growing from different seeds hardly ever cross each other.Different clusters compete for limited resource of free particles and inhibit the growth of other groups in their direction.We also studied the variant of DLA model in which multiple particles walk together.These simultaneous adding particles could form several clusters in network.When the total number of particles remains unchanged,with the increasing of particle addition rate,the pattern changes from fractal to diffuse,and the fractal dimension gradually decreases.When particle addition rate is constant,with the increasing of the total number of particles,the pattern changes from non-fractal to fractal,and the fractal dimension gradually increases.When the lattice size is large enough,the fractal dimension of the pattern is only related to the particle number density,and has no concern with the size of network.Our results show that there is a critical particle number density under different parameters.When the particle number density in the network is less than the critical value,the pattern is non-fractal.When the particle number density reaches the critical value,the pattern become fractal.Second,we study the particle aggregation models in small world networks.The model mimics how proteins are deposited in the brain.We investigated the effects of the complexity of small-world network,the addition rate of free particles and the removal rate of particles,which correspond to brain network complexity,protein production and metabolism,respectively.With the increasing of the number or length of the long side of the small-world network,the fractal dimension of the pattern decreases.The large particle addition rate leads to many small clusters appearing in the network and results diffusive pattern.Metabolic processes offset the effects of excessive free particles though reducing the number of free particles in the network and eliminating some small clusters.Therefore,the patterns have fractal characteristics and larger fractal dimension.Our study shows that the fractal characteristics of patterns are related to the number of small clusters formed in the network.Different clusters are randomly distributed in the network,they do not cross with each other and compete for a finite number of free particles.The more the number of clusters,the less particles each cluster will adsorb.So the pattern is diffused and has a small fractal dimension.Both reducing the rate of adding particles into the network and removing particles lead to the decreasing of the number of clusters.While the increasing of the particles density result the increasing number of free particles that all of the small clusters can attach to.Therefore,all of the three methods can change the pattern from a scatter point set to a fractal,and have a larger fractal dimension.