Abstract
In this article, we design a family of scalefree networks and study its random target access time and weighted spanning trees through the eigenvalues of transition weight matrix. First, we build a type of fractal network with a weight factor
1 Introduction
In the research related to complex networks, in addition to the topological properties, it is also necessary to describe their dynamic processes. Random walk is one of the powerful research tools that can describe various dynamical behaviors in complex systems, such as remote sensing image segmentation [1], epidemic spreading [2], community detecting [3], cell sampling [4], just to name but a few. Random walks have been studied for many decades on both regular lattices and networks with a variety of structures [5]. For undirected weighted networks, the weight matrices of several classes of local random walk dynamics have been extensively studied, including normal random walks, biased random walks and preferential navigation, random walks in the context of digital image processing and maximum entropy random walks, and nonlocal random walks, including applications in the context of human mobility [6].
In the research methods of random walks, the random target access time [7], mean first passage time [8], mixing time, and other indicators [9] can be used to measure the efficiency of propagation [10,11]. The random target access time is a manifestation of the global characteristics of the network; it refers to the average of mean first passage time from one node to another node over all target nodes according to the stationary state; it is formulated as the sum of reciprocals of every nonzero eigenvalue of the normalized Laplacian matrix for the researched network. For example, some related studies on biased random walks of nonfractal and fractal structure [12].
The eigenvalues and eigenvectors of the transition matrix play a very vital role [13], as they are closely related to determining the aforementioned measures. There has been some research related to the eigenvalues of transition matrices on common networks in this work, such as Sierpinski gaskets and extended Tfractals, polymer networks, and it is worth emphasizing that these studies are based on networks without weight [14,15,16]. However, the weight of the network has important research significance in biological neural network, railway and air transportation [17,18]; therefore, finding the relationship between random walks and weight factor on different weighted networks is meaningful. Zhang et al. [19] studied the spectra of a weighted scalefree network and proved that random target access time
The main research content of this article is arranged as follows. In the next section, we propose a graphic operation called the weighted edge multidivision operation and give the generation algorithm about a family of weighted networks. In Section 3, we find all eigenvalues of the transition weight matrix and determine their multiplicities. Then, we deduce all the eigenvalues of the normalized Laplacian matrix of our networks, and then derive the closedform expressions for random target access time and enumerate the weighted spanning trees in Section 4. In Section 5, we summarize the work of this article.
2 Designweighted network models
Many phenomena can be reduced to graphs for research [22,23,24], so we are about to show the corresponding model of a family of weighted scalefree networks. But, before that, we define a graphic operation called the weighted edge multidivision operation and obtain our weighted networks through the recursion of this graphic operation.
Weighted edge multidivision operation: For a given edge
With the preparation of the weighted edge multidivision operation, the weighted scalefree networks are constructed in an iterative way as depicted. Let
The iterative growth of the network allows us to precisely analyze its topological characteristics, such as the order and the size, the sum of weights of all edges, and so on. At each time step
The network size
Let symbol
The networks studied in this article display some representative topological characteristics observed in different reallife systems. They obey a significant powerlaw degree distribution with exponent
Let
then by considering the initial condition
For an edge
where
3 Eigenvalues and their multiplicities of transition weight matrix
In this section, we determine all eigenvalues and multiplicities of transition weight matrix in our networks, according to the relationship between the eigenvalues of the transition weight matrix of two consecutive generations.
3.1 Relationship between eigenvalues
Let
which shows that matrix
For
Let
Given two old nodes
For each element
We know that
For an old node
Comparing equations (13) and (8) corresponding to any old node
furthermore, the solution of
The above result shows that for an eigenvalue
3.2 Multiplicities of eigenvalues
On the basis of the eigenvalues obtained above, we can determine their corresponding multiplicity for matrix
For small networks, we can calculate their eigenvalues and corresponding multiplicities directly, for instance, the set of eigenvalues of
We can observe the fact that in addition to the eigenvalues
In general, we usually use
For the set of all nodes in
where the fact that
Because the matrix
where the first
from which we obtain
In addition, we have
Therefore, we obtain
The total number of eigenvalue 0 and its descendants in
For eigenvalues
Apparently, eigenvalues
which proves that we have found all the eigenvalues, and the corresponding multiplicities of the matrix
4 Applications of eigenvalues
The eigenvalues of the normalized Laplacian matrix of
4.1 Random target access time
We set
The quantity
Equation (27) implies that the random target access time
The normalized Laplacian matrix of
Theorem 1
For
when
Proof
According to equation (15) and
We suppose that
It is easy to obtain the following result due to equation (29):
In order to facilitate calculations, we divide the set
Hence, we still need to evaluate
which indicates that
Combining the above results in equations (30) and (32), the relationship between
Furthermore, using the initial condition
Finally, for large weighted fractal networks, that is
Equation (35) implies that in the large
4.2 Weighted counting of spanning trees
The spanning tree of a connect network
Different from using the electrical networks theory to obtain the closedform formula for the spanning tree enumeration [30,31] and calculating the weighted spanning trees through the special structure of the networks [32], in this subsection, we count
Theorem 2
For
Proof
We first need to determine the three terms in equation (36). For the sum term in the denominator, we have
Consider the two product terms in the numerator of equation (36), let
Based on the results of the eigenvalues obtained above, the following equation holds:
Then, multiplying equations (38) by (39) results in
For the simple case of
Inserting the results of two equations (37) and (41) into equation (36), we can obtain the following expression of
The result of equation (42) is consistent with the result of direct enumeration, which verifies that our calculation for the eigenvalues of the transition weight matrix of the weighted network
5 Conclusion
There are many documents that have verified that the weights of some networks have a serious impact on the random target access time. Unlike the existing weighted networks, we have found a family of weighted networks whose random target access time is not controlled by its weight factor, and these networks have been proven to exhibit the remarkable scalefree properties observed in various reallife complex systems. We have listed all eigenvalues for the transition weight matrix of

Funding information: This research was supported by the National Key Research and Development Plan under Grant No. 2019YFA0706401 and the National Natural Science Foundation of China under Grant Nos. 61872166 and 61662066, the Technological Innovation Guidance Program of Gansu Province: Soft Science Special Project (21CX1ZA285), and the Northwest China Financial Research Center Project of Lanzhou University of Finance and Economics (JYYZ201905).

Author contributions: B.Y. and J.S. created and conceptualized the idea. J.S. wrote the original draft. X.W. and M.Z. reviewed and edited the draft. All authors have accepted responsibility for the entire content of this manuscript and approved its submission. The authors applied the SDC approach for the sequence of authors.

Conflict of interest: The authors state no conflict of interest.

Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during this study.
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