Double-network Jump Phase Transition And The Scaling Law

This thesis mainly reports a model-analytic, numeric simulation, and some systems empirical investigation on the so-called"double-layered hopping"networks the"hopping-occurrence"phase transition, and the averaged laminar phase length scaling law. The double-layered hopping networks consist of three parts: the upper layer network, the lower layer network, and the hopping edges between the two layers. The hopping-occurrence phase transition is defined as the transition between the situations where the hopping occurs or not occurs in a shortest path between two nodes. The averaged laminar phase length is defined as the ratio between the hopping edge number and the total edge number in a weighted shortest path. The scaling law is defined as the changing dependence of the averaged laminar phase length on the difference of the upper and lower layer averaged edge weights.The averaged laminar phase length scaling law makes us remember the corresponding scaling law of the crisis-induced intermittency in nonlinear science. If compare a weighted shortest path in a layered complex network with a time sequence in a nonlinear dynamic system chaotic attractor, the hopping frequency between layered networks may be thought similar to the hopping frequency between chaotic attractors in a crisis-induced intermittency, which can be transferred to the averaged laminar phase length. It is already proved that the crisis-induced intermittency is a kind of critical phenomena and obeys a universal scaling law, we therefore guess that the layered network hopping-occurrence phase transition is also a kind of critical phenomena, and obeys similar scaling behavior. This guess needs to be explained from the aspects of model-analytic, numeric simulation, and some systems empirical investigation.By using the mean-field approximation idea, we propose a very simplified model of the layered network hopping-occurrence phase transition, so as to analytically deduce that the dependence of the averaged laminar phase length,λ, on the lower layer edge weight, q, follows a linear scaling law, i.e., the scaling exponent equals 1. We also obtain the analytic expression of the critical point, q c.The model simulation considered three situations: both the layers are produced by BA scale-free model; the upper layer is produced by BA scale-free model, but the lower layer is produced by WS small-world model; and the upper layer is produced by BA scale-free model, but the lower layer is produced by ER random model. When the networks are produced, change the lower network edge weightq , and computeλfor the three situations respectively by normalization and accumulation. We found that q andλobey a linear scaling law very well.We empirically investigated the hopping-occurrence phase transition and the averaged laminar phase length scaling law of five systems, which form eight double-layered hopping networks (among them the movie-actor collaboration network form four double-layered hopping networks by dividing the different time periods). The method is that we approximate the possible value range of q in each double-layered hopping network, and imaginarily magnify the range, then extrapolate to find the critical point, q c. Then, we found that the empirical results of all the real world systems approximately obey the predicted q ?λscaling law with the exponent 1.At the end of the thesis we also report an investigation on the combined traffic networks in the Yangtze River delta metropolis circle. In this earlier study we had no conceptions about multiple layered networks and hopping between them yet, only simply overlapped different kinds of nodes and edges on one layered graph. However, this was just the investigation, which stimulated our main ideas reported in this thesis. Thus we still report the study here.