Application Of Dynamic Random Graph Model In The Study Of Probability Properties Of Random Graph

A complex network is an abstract model for understanding complex systems in the real world.It abstracts the entities in a complex system as nodes and the relationships between entities as connections.Network structure has been one of the popular topics that researchers are keen to explore and can be used to describe real-life complex systems.It has certain stochastic properties that can be represented by graph theory in mathematics.Random graphs are an important class of graphs,which are generated by random processes accompanied by uncertainty.The use of random graphs for modelling networks can uncover key information in the network.However,current theoretical work on modelling the structure of complex networks has focused on static invariant networks,but real-world networks change their structure or properties over time.Therefore,the study of dynamic networks is an interesting and promising topic.Current approaches to the study of dynamic networks mainly divide them into two types: time-series networks and static snapshots.The former has the limitation of ignoring changes in the overall structure of the network and the possible community structures in the network,while the latter has the limitation of ignoring the time-series information generated by changes between networks.Each of these two approaches therefore has its own shortcomings.Based on these issues,this paper combines the two approaches to dynamic networks and considers static random graphs as the limit of dynamic random graphs to study the nature of random graphs and does the following.In this paper,we derive the dynamics of four classical static random graphs starting from the Erd(?)s-Rényi random graph model,the generalised random graph model,the configuration graph model and the dynamics of the random block model.The paper assumes that the number of nodes in the network remains constant,that the appearance and disappearance of edges follow a Markov process in continuous time,and that the rate at which edges appear or disappear is related to the properties of the nodes.The paper calculates the properties of the model at equilibrium,extracting information from the properties of the nodes,the probability of a node pair being connected by an edge,and the dynamic characteristics of edge appearance and disappearance.A maximum likelihood approach is used to infer the properties of the random graph,giving maximum likelihood estimates including the rate of change,the degree of the node and the community structure.A program is also written to perform dynamic simulations and analyse the simulation results to verify the conclusions of this paper.This paper applies the idea of a Markov process,where the limit is the limit of a Marcian chain distribution given a starting point.There are several advantages to viewing a static random graph as the limit of a dynamic random graph:(1)For some more specific models where the distribution is not known,such as the configuration graph model,the traditional approach is to use Monte Carlo methods to generate the graph of the configuration graph model uniformly,but the algorithm is difficult to guarantee randomness.In contrast,this paper uses the idea of Markov chains,where the limit is the limit of the Marcian chain.Markov chains have no requirement for a starting point and can converge well to the desired distribution.This is more concise and efficient,and the algorithm has certain superiority.(2)This method can be used to predict the outcome of community differentiation,where the current situation and evolutionary process are known,and can be used to classify communities using a dynamic random graph model.A better understanding of the community structure existing in the network(3)By considering the static random graph as the limit of the dynamic random graph,the maximum likelihood estimate of the parameters of the static random graph can be obtained,and therefore the dynamic random graph simulation can be used to derive the properties of the static random graph.In summary,this paper is a meaningful exploration in the field of dynamic random graphs.