基于节点相关性的网络不动点理论研究

Research on network fixed point theory based on the correlation of nodes

现代复杂的通信网络内部存在着广泛的幂律现象,网络节点之间存在相关特性.根据这种相关特性,提出了网络不动点理论.将Banach不动点理论引入网络模型,证明了网络不动点理论的正确有效性.证明过程是把通信网络看作由路径预测算法产生的似马尔可夫链的路由节点迭代序列形成的网络空间.由节点相关性可知,此空间中的节点序列相对越长就越能折射出搜索的目标所在,预测准确率也会逐步增加,可以更好地进行目标定位、数据挖掘等.通过某种路由准则的算子从源节点最终映射到的目的节点与Banach空间的不动点相对应,即为网络空间的不动点.当网络发展到能为用户提供真正的无处不在的连接时,网络不动点理论的物理特性将非常明显.因为网络规模越大,节点间的群体作用越显著,就越能显现网络不动点理论的物理特性.

Power laws are ubiquitous in the complex communication networks of today. Network nodes are correlated. According to the correlation, network fixed point theory is proposed and analyzed. Banach fixed point theory was used to explain the operation of networks. In this way, the validity of network fixed point theory is proved. The iterative node sequences of Markov-like chains are generated by algorithms of routing. Communication network can be considered as a space formed by the node sequences. Based on the correlation of nodes, the more nodes in the sequence, the more accurately reflected the searching object node. The property makes location finding and data mining more accurate in communication. The object node mapped from the source node by some routing rule corresponds to Banach fixed point. The fixed point in network space is the object node. The physical character of network fixed point theory will be highly evident, when the network can provide ubiquitous connection for users. The reason is that when the network scale becomes greater, the colony action of nodes is more obvious, and network fixed point theory can show its physical character better. It has great significant theoretical and practical meaning for the organic and dynamic characters and congestion analysis of complex traffic communication network.