摘要
Internet作为一个典型的复杂网络实例,对其宏观拓扑结构的特征分析及建模研究是目前研究的热点问题,受到学术界广泛关注。近年来人们在该领域的研究取得了长足的进展,尤其是在自治系统层面(autonomous system, AS-level)。但是由于路由级Internet拓扑结构的复杂性,人们至今也未能对其有个清楚的认识,在这个看似混沌的网络之中还蕴涵着一些不为人知的规律有待进行深入的挖掘。对Internet路由级宏观拓扑结构的层次分析以及建模研究,可以帮助人们更好地理解Internet基础架构的特点,这对下-代Internet的设计,及与Internet有关的协议体系的研究都将产生巨大的促进作用,因此具有重要的意义。
本文首先分析了Internet路由级拓扑的k-核结构,通过对CAIDA2007年5月各监测点的实测数据进行横向分析,发现各监测点的k-核结构与整体相似,说明采样偏见问题对Internet的k-核结构影响不大；继续对CAIDA riesling监测点在2006年12个月的采样数据进行纵向分析,结果表明Internet的k-核结构受时间演化的影响很小,只有网络拓扑核数的大小在变,而各核中节点分布的规律是不变的。在此基础上,定性定量地分析了节点的度值与核数的关系,实验证明,这两者之间并没有明显的联系。
其次,本文对Internet路由级拓扑结构的层次特征进行了详细的分析,以k-核解析作为主要分析手段,通过对CAIDA实测数据的分析,表明Internet路由级拓扑具有明显的层次结构：从最高核开始,节点由高核到低核,由内层至外层分布,在最内层,节点集中在有限的几个网段地址上,慢慢向外层发散,网络分布越来越广,节点度分布的幂率性体现越来越明显,至最外层,节点数达到最多,分布的网段地址也达到最广,幂率性体现最明显。
再次,在k-核解析的基础上研究了Internet路由级拓扑的分形特征。通过对CAIDA实测数据各核内的度分布、度相关性以及聚集性等主要特征量的分析,发现随着网络由外到内的逐层剪切,不仅度分布,而且聚类性质以及相关结构也被保留了下来。这意味着一种对逐渐趋于网络中心的区域的全局的自相似性,以及根据网络中心性定义的Internet的任意区域都有和整个网络相同的性质的一种结构。进一步对各核拓扑图分别进行谱密度-特征值分布分析以及无符号拉普拉斯谱(SLS)分布分析,发现各核分析结果表现出高度的一致性,从另一方面证明了Internet拓扑结构具有自相似性,从而说明了Internet路由级拓扑具有分形特征。
接下来,对Internet路由级拓扑的可视化进行了研究。首先提出了一个基于k-核解析的算法：基于节点的核数由内核至外核逐层演化的算法。并以CAIDA Riesling监测点在2007年5月份的Internet路由级拓扑测量数据为例,描绘了Internet的可视化结果。从可视化结果中可以看出,该算法在描绘Internet拓扑的层次性演化上是比较突出的,但是这一算法没有考虑最高核有几个连通成分的情况。因此,接下来,对这一算法进行了改进,并以CAIDA2007年5月份的路由级拓扑数据为例,描绘了Internet的可视化结果,结果表明该算法可以适应最高核中含有多个连通成分的网络,演化的结果突出了Internet路由级拓扑的某些特征,如聚类特性以及社团特性。通过对两种算法的时间复杂度的分析,表明这两种算法运行时间较短,适合巨大网络的可视化。
最后,通过对各层中节点的分布分析以及节点的高层连接分布分析,提出了一个基于k-核解析的Internet路由级拓扑的层次模型并实现了该模型。对模型的分析表明,该模型在大多数性质上都能重现Internet,并且具有和Internet相接近的拓扑核数。实验结果表明,与传统的经典模型相比,本文模型在满足幂率分布的前提下,充分体现了Internet路由级拓扑的层次性。
Being a typical instance of complex network, the research and modeling on Internet topology has become a hot topic at present and attracted more and more attention of academia. In recent years, researches in this field have made considerable progress, especially in the autonomous system level (AS-level) Internet topology. However, because of the complexity of Internet router-level topology, people have not got a clear understanding on it up to now. There are still some laws in this seemingly chaotic network that we have not known, and need us to further excavate. The research and modeling on the hierarchy of Internet router-level topology can help people better understand the characteristics of Internet basic architecture, which will bring great effect on the design of the next generation of Internet and the research on the protocol systems related with Internet.
Firstly, the k-core decomposition of Internet router-level topology was analyzed in the paper. By analyzing the measuring data of CAIDA monitors in May 2007, it has been found that the k-core structures of the topologies measured by the various monitors are similar with that of the whole topology, which indicates that sample bias has a little effect on the k-core structure of Internet. The further analysis on the measuring data of Riesling monitor from Jau 2006 to Dec 2006 shows that the the k-core structure of Internet changes little with the evolution of time. As time went by, only the size of the topology coreness changes, while the distribution law of the nodes in each k-core remains unchanged. After that, we analyzed the relationship between the node degree and node coreness on qualitation and quantitation. The results show that there is no evident relation between them.
Secondly, a detailed analysis of the hierarchy characteristics of Internet router-level topology was given. Taking k-core decomposition as main method, by analyzing the actural data measured by CAIDA, it is indicated that Internet router-level topology has evident hierarchical structure:from inner hierarchy to outer hierarchy, the nodes distribute from the highest coreness to the lowest. In the innermost hierarchy, the nodes distribute on only a few network addresses. From inner to outer, the distribution of the network addresses becomes more and more expanded, and the frequency-degree power law is increasingly finer. In the outmost hierarchy, the distribution of the network addresses is the widest and the power law is the finest.
Thirdly, the fractal features of Internet router-level topology was studied based on k-core decomposition. By analyzing the main measurement quantities such as degree distribution, degree correlations and clustering coefficient of each k-core of the measuring data of CAIDA, it is shown that not only the degree distribution but also the clustering and correlations structures of Internet topology are essentially preserved as the more and more external parts of the network are pruned. This hints to a sort of global self-similarity for regions of increasingly approaching to the centrality of the network, and to a structure in which each region of the Internet as defined in terms of network centrality has the same properties with the whole network. The further analysis on the distribution of the spectral density with eigenvalues and signless Laplacian spectra(SLS) of each k-core shows that the results of them are highly coherence proving in another aspect that Internet topology has the property of self-similarity, which describes that the Internet topology has fractal characteristics.
In the following chapter, the visualization on Internet router-level topology was studied. An algorithm based on k-core decomposition which evolved from inner core to outer core based on node coreness was proposed firstly, then the visualization results with the Internet topology measuring data of Riesling monitor of CAIDA in May 2007 was given. It can be seen from the visualization results that the algorithm is very outstanding on describing the hierarchy evolvement of the Internet topology, but this algorithm was not able to distinguish two or more disconnected components in the innermost core. So it was improved in the next content and the visualization results with the Internet topology measuring data of CAIDA in May 2007 was given. The visualization results show that the improved algorithm can present networks in which the highest k-core are composed by several connected components. Some characteristics, for example, clusterring and community, of Internet router-level topology can be seen from the evolvement results. By analyzing the complexity of the two algorithm, it can be seen that their running time is short and suit to the visualization of huge network.
Finally, by analyzing the distribution of the nodes in each shell and the distribution of the links to the higher shells, a hierarchical model of Internet router-level topology based on k-core decomposition was proposed and realized. The analysis shows that this model can reproduce on Internet on most properties and has got topology coreness close to that of Internet. The experiments show that compared with the previous classical models, the model in the paper can fully reflects the hierarchy characteristics of Internet topology while preserving the power-law distribution of node degree.
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