广义生灭过程与随机分枝树演化
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摘要
近十五年来,复杂网络的研究发展迅猛,但复杂网络演化的动态性及复杂性使其理论研究非常困难,有效的理论方法并不多。随机图过程作为研究复杂网络演化的主要理论工具之一,受到了众多学者的青睐。迄今,随机图过程构建的主要思路是基于离散时间马氏演化的,即在已知随机图过程当前状态的条件下,每隔一单位时间加(删)边、加(删)点来实现图过程的构建。本文突破随机图过程构建的传统思路,基于生物无性繁衍机理,用节点的出生率函数和死亡率函数刻划随机图过程的图拓扑演化特征,建立了基于生灭过程的随机分枝树演化。本文用连续时间实值随机过程刻划随机图过程的某类图拓扑演化特征来构建图过程,开创了随机图过程研究的新思路,为随机图研究领域提供了连续时间马氏或非马氏随机图过程范例,丰富了随机图过程的研究内容,提供了更为客观的生物繁衍或网络传染模型(传统的生物繁衍模型或传染模型——分枝过程和生灭过程,主要研究种群规模和种群最终灭绝概率等问题,而忽视了生物网络内部结构和个体之间关系的研究)。本文用马氏过程、图论和微分方程等理论工具,主要给出了随机分枝树的存在性定理、图拓扑结构、节点年龄结构及辈分结构等解析分析。详细内容按照章节介绍如下：
第一章概述了本文研究背景,简述了本文主要研究内容及组织安排、主要创新和研究意义。
第二章研究了广义生灭过程,讨论了后代数的生成函数有关的微分方程并得到方程的解,给出了后代数的分布。
第三章研究了马氏生灭分枝树演化(基于线性生灭过程的随机分枝树演化)。(一)给出了马氏生灭分枝树的构建即存在性定理的证明；(二)给出了分枝树在不同时刻的图拓扑结构特征量解析分析：度(虚出度、拟出度、出度及入度),实节点数和虚节点数,孤立节点数,度(出度、虚出度等)为不同值的实节点数,实连通分支的个数,以各代子节点为根节点的连通分支的个数,单个连通分支的节点数；(三)用初等而简洁的方法重新得到了线性生灭过程的灭绝概率；(四)给出了分枝树在不同时刻的年龄结构和辈分结构特征量解析分析：处在不同年龄段的实节点数,适龄生的节点数(包括活着的和死亡的节点)和实节点数,超龄生的节点数和实节点数,在不同年龄段死亡的节点数,各代的子节点数,各代节点在不同年龄段的子节点数,各代节点出度(或虚出度)取不同值的节点数；(五)讨论了节点生产年龄的分布：首生年龄和末生年龄的分布,生产年龄顺序统计量的分布；(六)讨论了马氏生灭分枝树演化的拓广,其中重点给出了出生率与年龄段有关的随机分枝树演化的构建即存在性定理的证明。
第四章将马氏生灭分枝树拓广到一胎多子的随机分枝树演化(在一胎多子的随机分枝树演化中,假定节点每次分娩的产子数是一随机变量),给出了一胎多子的随机分枝树存在性定理的证明,研究了不同时刻分枝树中活着的节点数及死亡的节点数、连通分支的个数、任一节点在活着的条件下在不同年龄时的子节点数及临死前的子节点数。
第五章给出本文研究展望。在马氏生灭分枝树演化中,子节点的到达计数过程是时齐泊松过程,另外还可考虑子节点的到达计数过程是一般泊松过程或更新过程或其他点过程(比如具有平均发生率或发生强度的点过程)等情形。
In the last15years, the study of complex networks is developing very fast. Due toits dynamics and complexity, it is hard to do the theoretic research and we have not manyefective theory methods. As the major theory method of studying network evolution,the random graph process draws the favour of many scholars. So far, the main methodof constructing random graph process is based on the the discrete time and Markov evo-lution. Given the the current state of graph process, during a unit time one link (orone node) is added (or deleted) to realize the graph process constructing. This arti-cle, breaking through the traditional way of constructing random graph process, basedon the biological mechanism of asexual reproduction, constructs the random branchingtree evolution based on birth death process, using node birth-rate function and death-ratefunction to figure the topology evolution characteristics of random graph. This paper,viareal values stochastic process with continuous time characterize certain graph topolo-gy evolution characteristics to build graph process, pioneers new ideas about the studyof random graphs, and provides continuous time Markov random graph or non-Markovrandom graph process paradigm, and enriches the research contents of the random graphprocess, provides a more objective biological reproduction or epidemic model(The tra-ditional biological reproduction or epidemic model(branching process and birth-deathprocess) mainly study the problem such as population size and extinction probability,but ignored the internal structure of biological networks and relationships between indi-viduals). This paper using theory tools such as Markov process, graph theory and thetheory of diferential equations, study the existence theorem of random branching tree,make analytical analysis for such as graph topology structure, age structure and the re-productive structure etc.. Detailed content as follows:
Chapter one briefly described the research background,and Outlines the main re-search contents, the organization arrangement, main innovation and research signifi-cance.
Chapter two study general birth-death process. The generating function of thenumber of ofspring is mainly studied and the distribution of the number of ofspring isgot.
Chapter3research the evolution of Markov birth-death branching tree (randombranching tree evolution based on the birth-death process).(1) the evolution of Markovbirth-death branching tree is constructed, namely the existence theorem of graph processis show;(2) make analytical analysis for some characteristic variables of graph topologystructure such as: degree(out-degree, in-degree etc.), the number of real nodes or imagi-nary nodes or Isolated nodes,the number of real nodes with diferent degree(out-degree,imaginary out-degree etc.), the number of real connected components, the number ofreal connected components which.s root node is any generation node,the number of n-odes in one connected component;(3) get the extinction probability of linear birth-death process using the elementary and concise method;(4) make analytical analysis for somecharacteristic variables of age structure and reproductive structure such as: the numberof real nodes at diferent age, the number of nodes born at appropriate age, the numberof real nodes born at appropriate age, the number of nodes born at over-age, the numberof real nodes born at over-age, the number of nodes dead at diferent age, the numberof nodes in any generation, the number of nodes at diferent age in any generation, thenumber of nodes with diferent degree in any generation;(5) study the distribution of pro-ductive age such as:first-born age and last-born age, productive age order statistic;(6)discuss the possible extension of the Markov birth death branching tree and show theexistence of the random branching tree evolution with age-dependent birth rate.
In chapter four, the Markov birth-death branching trees evolution is extended tothe branching trees model with nodes being multiparous(In the branching trees modelwith nodes being multiparous, it is assumed that the number of son-nodes born at eachdelivery is a random variable). The existence of the model is shown, and it is studiedthat the number of alive nodes, the number of dead nodes, the number of connectedcomponents, the number of son-nodes of any node at any age or at dying moment.
In chapter five, this paper research prospect is presented. In the Markov birth-deathbranching trees model, the arrival process of the child-nodes is homogeneous Poissonprocess. So It can be extended as that the arrival process of the child-nodes is a renewalprocess or other point process.
第一章概述了本文研究背景,简述了本文主要研究内容及组织安排、主要创新和研究意义。
第二章研究了广义生灭过程,讨论了后代数的生成函数有关的微分方程并得到方程的解,给出了后代数的分布。
第三章研究了马氏生灭分枝树演化(基于线性生灭过程的随机分枝树演化)。(一)给出了马氏生灭分枝树的构建即存在性定理的证明；(二)给出了分枝树在不同时刻的图拓扑结构特征量解析分析：度(虚出度、拟出度、出度及入度),实节点数和虚节点数,孤立节点数,度(出度、虚出度等)为不同值的实节点数,实连通分支的个数,以各代子节点为根节点的连通分支的个数,单个连通分支的节点数；(三)用初等而简洁的方法重新得到了线性生灭过程的灭绝概率；(四)给出了分枝树在不同时刻的年龄结构和辈分结构特征量解析分析：处在不同年龄段的实节点数,适龄生的节点数(包括活着的和死亡的节点)和实节点数,超龄生的节点数和实节点数,在不同年龄段死亡的节点数,各代的子节点数,各代节点在不同年龄段的子节点数,各代节点出度(或虚出度)取不同值的节点数；(五)讨论了节点生产年龄的分布：首生年龄和末生年龄的分布,生产年龄顺序统计量的分布；(六)讨论了马氏生灭分枝树演化的拓广,其中重点给出了出生率与年龄段有关的随机分枝树演化的构建即存在性定理的证明。
第四章将马氏生灭分枝树拓广到一胎多子的随机分枝树演化(在一胎多子的随机分枝树演化中,假定节点每次分娩的产子数是一随机变量),给出了一胎多子的随机分枝树存在性定理的证明,研究了不同时刻分枝树中活着的节点数及死亡的节点数、连通分支的个数、任一节点在活着的条件下在不同年龄时的子节点数及临死前的子节点数。
第五章给出本文研究展望。在马氏生灭分枝树演化中,子节点的到达计数过程是时齐泊松过程,另外还可考虑子节点的到达计数过程是一般泊松过程或更新过程或其他点过程(比如具有平均发生率或发生强度的点过程)等情形。
In the last15years, the study of complex networks is developing very fast. Due toits dynamics and complexity, it is hard to do the theoretic research and we have not manyefective theory methods. As the major theory method of studying network evolution,the random graph process draws the favour of many scholars. So far, the main methodof constructing random graph process is based on the the discrete time and Markov evo-lution. Given the the current state of graph process, during a unit time one link (orone node) is added (or deleted) to realize the graph process constructing. This arti-cle, breaking through the traditional way of constructing random graph process, basedon the biological mechanism of asexual reproduction, constructs the random branchingtree evolution based on birth death process, using node birth-rate function and death-ratefunction to figure the topology evolution characteristics of random graph. This paper,viareal values stochastic process with continuous time characterize certain graph topolo-gy evolution characteristics to build graph process, pioneers new ideas about the studyof random graphs, and provides continuous time Markov random graph or non-Markovrandom graph process paradigm, and enriches the research contents of the random graphprocess, provides a more objective biological reproduction or epidemic model(The tra-ditional biological reproduction or epidemic model(branching process and birth-deathprocess) mainly study the problem such as population size and extinction probability,but ignored the internal structure of biological networks and relationships between indi-viduals). This paper using theory tools such as Markov process, graph theory and thetheory of diferential equations, study the existence theorem of random branching tree,make analytical analysis for such as graph topology structure, age structure and the re-productive structure etc.. Detailed content as follows:
Chapter one briefly described the research background,and Outlines the main re-search contents, the organization arrangement, main innovation and research signifi-cance.
Chapter two study general birth-death process. The generating function of thenumber of ofspring is mainly studied and the distribution of the number of ofspring isgot.
Chapter3research the evolution of Markov birth-death branching tree (randombranching tree evolution based on the birth-death process).(1) the evolution of Markovbirth-death branching tree is constructed, namely the existence theorem of graph processis show;(2) make analytical analysis for some characteristic variables of graph topologystructure such as: degree(out-degree, in-degree etc.), the number of real nodes or imagi-nary nodes or Isolated nodes,the number of real nodes with diferent degree(out-degree,imaginary out-degree etc.), the number of real connected components, the number ofreal connected components which.s root node is any generation node,the number of n-odes in one connected component;(3) get the extinction probability of linear birth-death process using the elementary and concise method;(4) make analytical analysis for somecharacteristic variables of age structure and reproductive structure such as: the numberof real nodes at diferent age, the number of nodes born at appropriate age, the numberof real nodes born at appropriate age, the number of nodes born at over-age, the numberof real nodes born at over-age, the number of nodes dead at diferent age, the numberof nodes in any generation, the number of nodes at diferent age in any generation, thenumber of nodes with diferent degree in any generation;(5) study the distribution of pro-ductive age such as:first-born age and last-born age, productive age order statistic;(6)discuss the possible extension of the Markov birth death branching tree and show theexistence of the random branching tree evolution with age-dependent birth rate.
In chapter four, the Markov birth-death branching trees evolution is extended tothe branching trees model with nodes being multiparous(In the branching trees modelwith nodes being multiparous, it is assumed that the number of son-nodes born at eachdelivery is a random variable). The existence of the model is shown, and it is studiedthat the number of alive nodes, the number of dead nodes, the number of connectedcomponents, the number of son-nodes of any node at any age or at dying moment.
In chapter five, this paper research prospect is presented. In the Markov birth-deathbranching trees model, the arrival process of the child-nodes is homogeneous Poissonprocess. So It can be extended as that the arrival process of the child-nodes is a renewalprocess or other point process.
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