SCALE-FREE Property, SCALE-FREE Phenomenon And Their Control In Complex Networks

Complex network is the high abstraction of complex systems. There are plentifulof complex networks in the real world. Since 1998 when the paper about "smallworld" was published in "Nature" and 1999 when the other paper about "scale free"was published in "Science", a great many physical scientists and mathematicians haveshifted their research interests to complex networks. The reason for which is thatsome conclusions about complex networks obtained in the past are wrong, e.g., thedegree distribution of complex networks obeys power law while in the past it isbelieved that it obeys Poisson distribution.In recent years, there appeared large number of achievements in the field ofcomplex networks, e.g., almost every real network is scale-free, the scale-freeproperty of complex networks can affect spreading dynamics and synchronizationdynamics on the networks, several effective algorithms were devised to findcommunity structures in complex networks, several network evolution algorithmswere discovered which can reproduce some properties such as scale-free andsmall-world of networks, and so on. Furthermore, it is found that the research resultsabout complex networks can be applied practically to the complex systems, e.g., thefact that virus spreading on scale-free networks lacks threshold caused traditionalmethod of virus prevention and cure to be changed. Scientists recognize that it isnecessary and important to research further about the complex networks.However, the research on complex networks is still at start-up stage after all. Thespecial research methodology in this field has not formed yet, and the research is farfrom being enough in many aspects of this field, such as the characterization on thetopological structure of complex networks, the characterization on the heterogeneityof complex networks, how the heterogeneity affects dynamics on networks, the globalstability of SIS model on complex networks, how topological structure affects thefinal proportion of infected nodes in SIS model and SIR model, why degreedistribution exponent of most networks locates between 2 and 3, and the controlproblem in complex networks. The problems above have been researched thoroughlyand deeply in this paper, and the following contributions are obtained:(1) The new concepts of connectivity coefficient and attraction coefficient areproposed, and are applied to the Internet AS level network. According to the staticprobability model presented in this paper, following results are verified theoreticallyand empirically: a) connectivity coefficients of subnets obey power law; b) there exists network core in the Internet; c) the degree distribution exponent has a criticalvalue 2.(2) The concepts of Lorenz curve and Gini coefficient in economics areintroduced into complex networks to characterize network heterogeneity. Comparedwith other coefficients such as degree distribution exponent, network structuralentropy and degree distribution entropy, Gini coefficient is found to be a reasonableindex in the characterization of heterogeneity.(3) The degree distribution exponent is studied theoretically. The reason whymost real networks have their degree distribution exponents locate between 2 and 3 isfound. Some special properties of Hub nodes in scale-free networks, such as thenumber of Hub nodes and the maximum degree of Hub nodes, are presented. Therelation between degree distribution exponent and Hub nodes is found, and thequantity definition of Hub nodes is given in this paper first.(4) The empirical study is made on BBS. It is found that the structure of BBSusers network is distinct from that of BBS users network with a given topic in that theformer has its degree distribution exponent less than 2 while the latter greater than 2.It means that in the BBS users network with a given topic, there exist only very fewnumber of Hubs. It is helpful for the tracking of key users.The community finding algorithms are applied succesfully to the topic detectionof BBS forum. The simulation result shows that this new method of topic detection ismore effective.(5) The global stability of SIS model in scale-free networks is analyzedthoroughly. It is proved that if the effective spreading rate is greater than the threshold,then, no matter how small the initial infection proportion is, the final infectionproportion is definite, which is irrelative to the initial infection proportion.The relations between the final infection proportion, the threshold value and thedegree distribution exponent are presented. It is found that, the less the degreedistribution exponent is, the less the threshold value is, and the greater the finalinfection proportion is.(6) The virus breakout problem of SIR model in scale-free networks is analyzedthoroughly. It is proved theoretically that there exists a threshold for effectivespreading rate, above which the virus will break out in the network.The extent to which the network is damaged under virus breakout is found to berelated to the degree distribution exponent. The less the degree distribution exponentis, the more damaged the network will be under virus breakout. (7) The control problem of scale-free networks is studied, and a new model ofevolution and control of Internet is proposed. Through adjustment of some parameters,this model can evolve into many kinds of networks, and can reproduce somedynamical characters of the Internet, e.g., the time when Hubs are produced.