Dynamical Small World Network Model And Stability Analysis

As the human history advances continuously, a number of new concepts and areasderived from the scientific development. Such as transportation network, the Internet ofthings, Internetwork, etc. And after more in-depth study of the different fields combinedwith mathematical models, they gave birth to a whole new discipline, the "networkscience". The "network science" had an actual background. It benefited from Euler’sfamous treatise the Seven Bridges of Konigsberg problem firstly. The Seven Bridgesproblem extended the graph theory to explore a series of the regular network’scharacteristics and conclusions. However, most of the networks in real life are not so "regular and perfect". They have more complex features. Between the late1950s and1960s,Erd s and Renyi founded formal random graph theory. While Watts and Strogatz describedthe―Small-World Networks‖at the end of the20th century. In1999Barabási and Albertgave birth to the―scale-free network‖, which was closer to the network in real life, and thatmarked a new era of complex network research.The complex networks dynamics include transmission dynamics of complex networks,complex network synchronization, evolutionary game, control of complex networks,search on complex networks, robustness of complex networks and so on. This paperfocuses on the spread of disease on complex networks. Transmission dynamics is animportant topic of complex networks, which has a strong value. Currently, researches inthis area had many achievements, however, they were all in the exploratory stage, a systemtheory has not yet been formed. For example, previous studies on the spread of the diseaseof complex networks based on the mean field theory, and did not take the impact ofnetwork structure on the spread of disease into account. In this paper, the actual structureof the network will be considered, and further discussion on the spread of the disease ofcomplex networks will be made.This paper firstly reviewed the classification of complex networks and four typicalnetwork model since the concept of the network science was established, including ERrandom network, WS small-world network, BA scale-free networks, and BBV weightedscale-free networks. Then it introduced three classic models of infectious diseases,including SI, SIS, SIR model and their spreading situations on complex networks.The main work of this paper are: the introduction and study of the dynamicalbehavior of the dynamic small-world networks (DSW), the model is as follows: 1. Let the total number of nodes in the network be N, and arranged them in a ring.2. Each node connected to its nearest m nodes.3. Fixed short-range connectivity and time-varying long-range connections.4. Each node in the network is equivalent of a person in the social networks, short-range connection is equivalent to a person closely connected to their friends or colleagues, and long-range connection is equivalent to a person moves to the strange location, contact someone once and then go back to the original position.Based on dynamic small-world network, I found that it showed different spreading situations on the SI model at various long-range connection probability:When the probability is not zero, the number of patients grows exponentially. While the probability is zero, the number of patients increases linearly. For the SIR model, When the long-range connection probability p satisfies the inequality (?), there is locally stable point which is not zero. And when γ,δ are fixed, we can see the bigger I*as the p is bigger.These work lay a foundation to the further research of the properties and applications of complex networks dynamics.