Exploring Evolution Model On Complex Networks

In the real world, there are many complex systems composed by a large number of highly interconnected dynamical units, such as ecological, social and economic systems. Complex networks provide a powerful tool for further investigations of complex systems. In recent years, complex networks have gradually developed into an emerging discipline, which have a wide application in various fields and received wide attention of scholars all over the world. In order to better study the behavior and function of realistic networks, one needs to make a scrutiny into the topological structure of realistic networks. The main task of this thesis is to build up network models by generation mechanism that can simulate evolving behavior of realistic networks, and to find rigorous methods for the statistical properties of networks. The main points and innovations of this thesis are as follows:1. We have proposed a non-growth model, which captures the essential evolution of some real-world migration when the alteration of the total numbers of communities and individuals are ignored. We noticed that non-growth models also can generate the power-law distribution with the appropriate preference, and in the gravity law, the num-ber of individuals that move between locations is proportional to some power of the population of the source and destination. Following the idea of Johnson et al. and using a continuous-time approach, we get the exact solution of the stationary size distribution, and all the results presented here were checked by numerical simulations, providing a perfect match with the theoretical curves. This model builds a nontrivial bridge between the microscopic migration and the macroscopic distribution. It is expected to shed light on the population and asset distributions in social and economical activities.2. We have proposed a rank-based deactivation model, which base on two mech-anisms:deactivation mechanism and rank mechanism. For the deactivation mechanism, nodes of networks are not always able to keep active. For example in the citation net-works, the life of scientists research is limited. The rank mechanism originates from the idea that the absolute importance of an object is often difficult or impossible for strangers to measure in social networks. Instead, it is quite common to have a clear knowledge about the relative values of two object, i.e., who is more popular or richer between two individuals. In this model, each node of the network can be in two different states:active or inactive. A new node added to the network is always in the active state at first. We adopt the age as a criterion of ranking and provide a comparable analysis of two deactivation models that generalize previous research. In model A, the older active node possesses the higher rank, which implies that old nodes have high possibility to gain new edges. Statistically old nodes have a long time to acquire edges, being responsible for the high-k part of degree distribution, and based on the cumulative advantage the in-degree distribution is a power law. On the contrary, for model B, the rank of the new added node always takes the largest value, which implies that newcomers have high possibility to gain new edges. Hence, old nodes no longer have the cumulative advantage and large degree. Here the mechanism is responsible for a polynomial in-degree distribution. Both distributions are truncated by an exponential. In the limit αâ†'∞, we obtain a structured exponential network for both models. These results are consistent with what have been empirically observed in many real-world networks.3. We have proposed a deactivation model with reactivation mechanism. In this model, each node of the network can be in two different states:active and dormancy. A new node added to the network is always in the active state at first, and only active nodes receive edges. As the evolution of time, new nodes continually add to the network, and the activity of node will continue to reduce until the complete loss. So active nodes may become dormant gradually. However dormant nodes may be awaked and receive edges from subsequent node again. We studied random reactivation, random deacti-vation model and random reactivation, preferential deactivation model, respectively. The results of theoretical analysis were checked by numerical simulations. In many realistic networks, reactivation is universal. For example, in citation networks, there is a universal phenomenon call "delayed recognition", that is, papers did not seem to achieve any sort of recognition until some years after their original publication. To examine theoretical prediction, we used random reactivation, preferential deactivation model to simulate citation networks. By comparing the degree distribution with empirical citation networks, we noticed a good agreement. So this model provides a new way to understand citation networks with age.