在反馈节点集上求布尔网络的不动点及其应用

发布时间：2018-06-17 20:17

本文选题：布尔网络 + 不动点　；参考：《河北师范大学》2017年硕士论文

【摘要】：布尔网络是自然和人造的非线性动态网络的一种紧凑模型,近年来,对于有关布尔网络的研究受到了人们的广泛关注.布尔网络是一个有向图,但是与一般图的的区别是:布尔网络中节点的状态变量取值只能是“0”或“1”,关于布尔网络的相关计算都是在二元布尔代数上进行的.需要注意的是这里的“0”和“1”不再是单纯的数字,而是符号.形式逻辑中命题的“真”和“假”,开关网络中出现的有电与无电,高压与低压,导通与截止等,都可以视为二元布尔代数中两个元素“1”和“0”的现实模型.由于现实生活中的网络模型并不总是强连通的,当这些网络不特殊的时候,我们希望用更少的条件来找到它们的不动点.本文主要研究了一类布尔网络,首先根据有向图中节点的等价关系,把布尔网络分成了若干个极大强连通子网,并为子网进行编号;基于子网的反馈节点集,给出了为每个子网构造对应新子网的算法,这些新子网也合成了新网络;由此,证明了原布尔网络的不动点与新布尔网络的不动点相同.然后,通过新增加的反馈节点集中节点的树形函数,给出了求解布尔网络不动点的一个充分必要条件,即:在新构造网络的基础上,通过移动网络中的弧,使其变为非循环网络,按照节点的拓扑排序,给出了为新增加的反馈节点集中节点构造树形函数的算法;在此基础上,利用状态变量和节点的树形函数所满足的方程,给出了确定不动点的必要条件以及条件加强下的求解不动点的充分条件.从而用更少的方程和更少的条件给出了求解布尔网络不动点的充要条件.
[Abstract]:Boolean network is a compact model of natural and artificial nonlinear dynamic networks. In recent years, the research on Boolean networks has been paid more and more attention. Boolean network is a directed graph, but the difference between Boolean network and general graph is that the state variable of node in Boolean network can only be "0" or "1", and the calculation of Boolean network is carried out on binary Boolean algebra. It is important to note that the "0" and "1" here are no longer simple numbers, but symbols. The "truth" and "falsehood" of propositions in formal logic, the existence of electricity and no electricity in switching networks, high voltage and low voltage, conduction and cutoff can all be regarded as the realistic models of two elements "1" and "0" in binary Boolean algebra. Because network models in real life are not always strongly connected, when these networks are not special, we hope to find their fixed points with fewer conditions. In this paper, we mainly study a class of Boolean networks. Firstly, according to the equivalent relations of nodes in directed graphs, Boolean networks are divided into several maximal strongly connected subnets and numbered for subnets. An algorithm for constructing new subnets for each subnet is presented, and the new subnets are also synthesized, and it is proved that the fixed points of the original Boolean networks are the same as the fixed points of the new Boolean networks. Then, a necessary and sufficient condition for solving fixed points of Boolean network is given by using the tree function of the nodes in the new feedback node set, that is, on the basis of the new construction of the network, through the arc in the mobile network, a necessary and sufficient condition is given for solving the fixed point of the Boolean network. The algorithm of constructing the tree function for the newly added nodes with feedback nodes is given according to the topological sort of nodes, on the basis of which, the equations satisfied by the state variables and the tree functions of nodes are used. The necessary conditions for determining fixed points and the sufficient conditions for solving fixed points under strengthened conditions are given. The necessary and sufficient conditions for solving fixed points of Boolean networks are given by using fewer equations and less conditions.
【学位授予单位】：河北师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O157.5

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