The Critical State Of A Network With Degree-dependent Properties

Self-organized criticality phenomenon generally exists in nature. Some of the systems can be described as networks and the self-organization of the systems is widely studied as an important dynamical behavior. We study the effect of degree correlation on network criticality in this paper. During the study we notice that some problems exist in generating assortative networks. We do some study about it since it is important to generate assortative networks in the study of complex networks. In this paper, we mainly study the effect of two methods generating assortative network on network topology and dynamics, and apply the reconnected networks to the study of complex network criticality. This paper can be divided into three parts and the specific study is as follows:First we study the maximum disaaortativity by directed rewiring a disassortative network. The result shows that the disassortativity can be increased to a certain extent by directed rewiring, and the enhancement is larger in networks with smaller size, larger connection density and higher homogeneity. However the network cannot reach total disassortative by directed rewiring. We also study the effectiveness of directed rewiring by comparing it with two sets of data of real Internets. We obtain that the networks generated by directed rewiring procedure cannot reach the same degree correlation as the real networks. The degree distribution of real networks diverges from the model at the largest degree or the smallest degree, which provides a heuristic explanation for the special degree correlation of real networks. These nodes should not be ignored since few nodes bring significant impact.Then we study the effect of random rewiring on ER random networks and BA scale-free networks. Here we use Pearson correlation coefficient r, degree correlation coefficient p and Assortativity A to characterize the degree correlation of networks. The result shows that random rewiring has no consequences for degree correlation in ER networks. In BA networks, by contrast, random rewiring causes higher disassortativity than in the original BA network. And the enhancement of disassortativity is much obvious in network with larger network size, smaller connection density and lower heterogeneity. In the three measures of the degree correlation coefficients, Pearson correlation coefficient r and degree correlation coefficient p measure the degree correlation considering all links in network which is effective to characterize the degree correlation coefficient while Assortativity A just considering links between degree peers and is not for the measure in our study. We compare the effect of random rewiring and directed rewiring. Although the two rewiring both can generate disassortative network, much topology difference are between the two rewired networks which will directly affect the dynamic behavior of networks.Finally we study the criticality of complex network with degree correlation. We first verify the conclusion in reference and find that the equation of largest eigenvalue is not appropriate to networks with degree correlation. Then we change the method and find that the point of instability of inactive state occurs with largest eigenvalue λ= 1.0 which is same with previous result.To verify the result above we introduce four parameters to characterize the criticality of network which are the distribution of avalanche, relation of stimuli and response, dynamic range and susceptibility. The results show that the distributions of avalanche in BA network and disassortative network obey the power-law distribution with the largest eigenvalue unequal to 1.0 while the distribution dosen’t obey th power-law distribution in assortative network indicating the unconspicuous of criticality. In the networks, the slops of stumulus-response curves are 1/2 with the largest eigenvalue unequal to 1.0 and the same result shows in the peak of dynamic range. For the susceptibility of networks, the peaks in ER network and disassortative network are nearly the same while smaller in BA network and minimum in assortative network. We can furthermore find that all of the peaks of networks occur with largest eigenvalue unequal to 1.0. The deviation in disassortative network is minimum while in assortative network is just the opposite.In conclusion, the point of instability of inactive state still occurs with A= 1.0 while the criticality is not at A= 1.0 which implys that the critical point is not agree with the instability point in degree correlation networks.