Study On The Distance Index Of Three Alternative Networks

With the progress of science and technology and the deepening of human cognition,complex network has attracted more and more researchers'interest.In this paper,we study the structure of three alternative networks and several different distance indexes related to them,such as resistance distance and geodesic distance.Based on the above indexes,we study the Kirchhoff index and weighted average geodesic distance of the network.In the first chapter,we briefly introduce the research status and background of complex networks and the research background of weighted complex networks.We give the concepts of resistance distance and Kirchhoff index,and give the basic knowledge of average geodesic distance.In the second chapter,we know the knowledge of its Kirchhoff index is important to understand its properties for a network.In this chapter,we study three kinds of scalars based on resistance distance:the Kirchhoff index,the multiplicative degree-Kirchhoff index and the additive degree-Kirchhoff index.Firstly,we derive the formula of the resistance distance,and then calculate the formulas of the Kirchhoff index,the additive degree-Kirchhoff index and the multiplicative degree-Kirchhoff index of the quadrilateral substitution network.Then the three Kirchhoff indexes of the quadrilateral substitution network are obtained by using the iterative relation.Finally,the network consistency is obtained by using the relationship between Kirchhoff index and first-order consistency.In the third chapter,we introduce two colored substitution networks based on substitution rules.To calculate the sum of the shortest distances between all nodes,we discuss the following cases:two nodes on adjacent branches and two nodes on non-adjacent branches.The most difficult problem is to calculate the minimum distance sum of all the nodes in the symmetric branch of two non-adjacent positions.After research,we found an effective method.The key to this method is to use the geodesic distance between the two initial nodes in the previous generation.Therefore,we first derive two formulas for the geodesic distance between two initial nodes.Further,we obtain the expression of the sum of the shortest distance between the initial node A_i(or B_i)and other nodes in the substitution network.Finally,using the results of geodesic distance,the sum of the shortest distance of the two colored substitution networks is obtained.So we get the dominant term for the average shortest distance.The results show that with the increase of the order of the colored substitution network,the dominant term of the average shortest distance of the network presents a sublinear dependence on the order of the network.In the fourth chapter,we first study the structural properties of the Sierpinski carpet fractal network Gt constructed by Sierpinski carpet F,including degree distribution and clustering coefficient.Then,the self-similarity measure of weighted vector is analyzed by geodesic distance integral,and the weighted average geodesic distance of Sierpinski carpet fractal F is obtained.Furthermore,the weighted average geodesic distance of Sierpinski carpet fractal network is obtained.