Eigenvalues of transition weight matrix for a family of weighted networks

: In this article, we design a family of scale - free 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 r and a parameter m . Then, we obtain all the eigenvalues of its transition weight matrix by revealing the recursive relationship between eigenvalues in every two consec - utive time steps and obtain the multiplicities corresponding to these eigenvalues. Furthermore, we provide a closed - form expression of the random target access time for the network studied. The obtained results show that the random target access is not a ﬀ ected by the weight; it is only a ﬀ ected by parameters m and t . Finally, we also enumerate the weighted spanning trees of the studied networks through the obtained eigenvalues.


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 non-local 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 non-fractal 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 T-fractals, 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 scale-free network and proved that random target access time H t is closely related to a weight factor r. Dai et al. [20]  Zou et al. [21] proved the for > r 1 of weighted network with two hub nodes. These studies show that weight has a great influence on random target access time of the networks.
The main research content of this article is arranged as follows. In the next section, we propose a graphic operation called the weighted edge multi-division 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 closed-form 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.

Design-weighted 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 scale-free networks. But, before that, we define a graphic operation called the weighted edge multi-division operation and obtain our weighted networks through the recursion of this graphic operation.
Weighted edge multi-division operation: For a given edge ′ ii with two end-nodes i and ′ i with weight w, replace the original edge ′ ii with m roads ′ i j i q q q ( = … q m 1, 2, ); the end-nodes i q and ′ i q of these roads coincide with i and ′ i , respectively. The weight of the new edges ij q and ′ j i q is r times the original edge ′ ii , and the diagram of a weighted edge multi-division operation is shown in Figure 1.
With the preparation of the weighted edge multi-division operation, the weighted scale-free networks are constructed in an iterative way as depicted. Let ( ) G m r t , , ( ≥ m 2, > r 0, ≥ t 0) denote the weighted scale-free networks after t time steps. Initially, = t 0, ( ) G m r , , 0 is an edge with unit weight connecting two nodes. For ≥ t 1, ( ) G m r t , , can be obtained from ( ) − G m r t , , 1 by performing weighted edge multi-division operation on each existing edge in , , becomes ( ) − rw t 1 according to the above graphic operation. 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.
The network size E t is the number of total edges in ( ) G m r t , , , then , , that was generated at time step t i , then we can obtain a recursive formula for the degree of node as ( ) The networks studied in this article display some representative topological characteristics observed in different real-life systems. They obey a significant power-law degree distribution with exponent = + / γ m 2 ln2 ln , and have a fractal dimension , both of which show that the networks are fractal and scalefree [25,26].
Let Q t represent the total weight of all edges in ( ) G m r t , , , by construction, we have then by considering the initial condition = Q 1 0 , we can solve For an edge = e ij connecting two nodes i and j in ( ) G m r t , , , ( ) w t ij denote the weight of edge ij, let ( ) s t i be the strength of node i in time step t, which represents the sum of the weights of all adjacent edges of node i, it can be formulated as where ( ) N i is the set of all neighbors of node i in the network ( ) G m r t , , .

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.

Relationship between eigenvalues
Let W t be the weighted adjacency matrix of network ( ) G m r t , , and its entries ( ) if nodes i and j are connected by an edge with weight ( ) w t ij in ( ) G m r t , , , or ( ) = W i j , 0 t if there is no edge between node i and node j. Another important matrix T t involved in describing weighted random walks is called transition weight matrix, which is defined as , which represents the corresponding transition probability for a particle going from starting node i to ending node j. Since T t is asymmetric, we introduce a real and symmetric matrix P t to assist our research, which shows that matrix P t is similar to matrix T t , and they have the same eigenvalue set. In addition, the entry in the ith row and jth column of P t is essentially ( ) ( ) according to equation (6). Furthermore, we set ϕ as an eigenvector of matrix P t associated with eigenvalue λ, then the eigenvector corresponding to eigenvalue λ in T t can be written as − S ϕ 1 2 . Therefore, we only need to find all the eigenvalues of matrix P t , and the eigenvalues of the another matrix T t can be obtained through transformation. Furthermore, we introduce the normalized Laplacian matrix In the following, we use the decimation approach [27] to enumerate all eigenvalues of normalized Laplacian matrix in our network. For where ϕ i is the element corresponding to node i. In order to distinguish between the newly generated nodes at time step t and the existing old nodes (they were generated before time step t), we let * ϕ represent a vector with − N t 1 dimension that is obtained from ϕ by restricting its components to the old nodes at time step t, with the set of all old nodes in ( ) ϕ as an eigenvector of matrix − P t 1 , it associated with the eigenvalue ( ) − λ t 1 i , similar to equation (7), we can obtain , , , we also have the following equation according to equation (7), Given two old nodes , , , and the weight of the connecting edge between these two nodes in . According to the way the network grows in two consecutive time steps, let … j j j , , , m corresponding to a new node, for every new neighbor j q , we can We know that , and substituting equation (11) into equation (10), equation (10) becomes For an old node ( ) ∈ − i V t 1 , we can further simplify equation (12) to obtain Comparing equations (13) and (8) corresponding to any old node i, we obtain the following equation: furthermore, the solution of ( ) The above result shows that for an eigenvalue ( of the matrix P t through the recursive relationship as shown in equation (15). Therefore, if all the eigenvalues of − P t 1 are known, then we can calculate all the eigenvalues of P t . If there are also eigenvalues ( ) λ t i that cannot be derived from equation (15), then they are zero eigenvalues.

Multiplicities of eigenvalues
On the basis of the eigenvalues obtained above, we can determine their corresponding multiplicity for matrix P t . For convenience, let ( ( )) D λ t t i mul be the degeneracy of eigenvalue ( ) λ t i for matrix P t , because − P t 1 is a real and symmetrical matrix; hence, every eigenvalue ( For small networks, we can calculate their eigenvalues and corresponding multiplicities directly, for instance, the set of eigenvalues of P 0 is { } −1, 1 .  (15), except for eigenvalue −1 and 1, because −1 can only produce eigenvalue 0, and 1 of − P t 1 produce −1 and 1 in the next generation. Therefore, we need to individually consider the multiplicity of eigenvalue 0, as well as the multiplicities of its descendants.
In general, we usually use ( ) r M to represent the rank of a matrix M, and the multiplicity of the zero eigenvalues for matrix P t is expressed as For the set of all nodes in ( ) G m r t , , , let α be the set of all nodes in ( ) − G m r t , , 1 , and β denote the set of nodes newly added at time step t. Now, we identify that the rank of P t , P t can be written in a block form due to the topology of the network, where the fact that P α α 1 zero matrix are substituted into equation (17).
Because the matrix P β α , is the transpose of the matrix P α β , , then we only analyze the rank of the matrix P α β , , P α β , , which is an , and consider = P P β α α β T , , , then, In addition, we have

(23)
For eigenvalues −1 and 1, we found that the eigenvalue −1 in − P t 1 produces eigenvalue 0 according to equation (15); hence, for eigenvalue −1, we have included the number of its descendants in the number of eigenvalue 0, which has been calculated in equation (23). Next, considering the multiplicity of eigenvalues 1 and −1 in ( ) Apparently, eigenvalues −1 and 1 in P t are generated by the eigenvalue 1 in − P t 1 . Adding the number of all the eigenvalues obtained above, we obtain the following result: mul mul (25) which proves that we have found all the eigenvalues, and the corresponding multiplicities of the matrix P t are determined. In addition, then we also obtain all the eigenvalues and their multiplicities of transition weight matrix T t by simple transformation.

Applications of eigenvalues
The eigenvalues of the normalized Laplacian matrix of ( ) G m r t , , can be determined from the eigenvalues obtained in Section 3.2, and it can be used to calculate some quantities for the weighted scale-free networks, such as the random target access time for random walks, and also to enumerate weighted spanning trees [27]. where

Random target access time
T . The random target access time H t for random walks on ( ) G m r t , , is defined as the expected time needed by a particle from a node i to another target node j, chosen randomly from all nodes according to the steady-state distribution [9], then it can be formulated as The quantity H t does not change due to different starting nodes, so it can be reexpressed as Equation (27) implies that the random target access time H t can be looked upon as the average trapping time of a particular trapping problem; it contains much valuable information about trapping in network ( ) G m r t , , . The random target access time in network ( ) G m r t , , can be obtained by the sum of the reciprocal of 1 minus each eigenvalue of T t , and the eigenvalue 1 is not included here [7].
The normalized Laplacian matrix of ( ) G m r t , , is defined as when → ∞ t , and the relationship between H t and the network order Proof. According to equation (15) and ( ) ( ) = − σ t λ t 1 i i , we can easily obtain the following recursive relation commanding the eigenvalues of two normalized Laplacian matrices − L t 1 and L t , We suppose that Ω t contains all eigenvalues of matrix L t at time step t, then equation (29) means that each eigenvalue ( It is easy to obtain the following result due to equation (29): ; ∈ 0 Ω 1 generates eigenvalue ∈ 0, 2 Ω 2 ; ∈ 2 Ω 1 generates eigenvalue ∈ 1 Ω 2 ; and ∈ 1 Ω 1 generates two eigenvalues + , and these eigenvalues generated by equation (29) from Hence, we still need to evaluate in order to determine H t . According to Vieta's formulas, . In addition, we have which indicates that Combining the above results in equations (30) and (32), the relationship between H t and − H t 1 can be obtained as follows: Equation (35) implies that in the large t limit, the random target access time increases as a linear function of the network order; H t is only affected by the topological parameter m; and it has no relationship with the weight factor r. In Figure 3, we give the schematic diagram to reveal the relationships between H t and two parameters t and m. In more detail, when the value of m is determined, H t increases monotonically with t for ≥ t 0 as shown in Figure 4. Similarly, for a given t, H t increases monotonically with m for ≥ m 2 as shown in Figure 5. □

Weighted counting of spanning trees
The spanning tree of a connect network ( ) G m r t , , is a subgraph of it, that is, a tree T , and includes all the nodes of ( ) G m r t , , [29]. For a weighted network ( ) G m r t , , , let ( ( )) G m r t Λ , , be its spanning trees set, and ( ) = ∏ ∈ w T w e T e , which is defined to be the product of weight of all edges in T , where w e is the weight of edge e. Let ( ( )) N G m r t , , st w denote the weight counting of spanning trees of weighted fractal network ( ) G m r t , , , that is, . Different from using the electrical networks theory to obtain the closed-form 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 ( ( )) N G m r t , , st w by using the eigenvalues of normalized Laplacian matrix L t of ( ) G m r t , , , that is,  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 Φ t represent the product ( ) and Θ t represent the product term ( ) = σ t Π i N i 2 g , respectively. According to the calculation formula of the strength of the node, the quantity Φ t obeys the following recursive relations:  Based on the results of the eigenvalues obtained above, the following equation holds: Then, multiplying equations (38) by (39) results in 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 ( ) G m r t , , is correct. □

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 scale-free properties observed in various real-life complex systems. We have listed all eigenvalues for the transition weight matrix of ( ) G m r t , , by giving the explicit recursive expression governing the eigenvalues of networks of two consecutive generations, it means that two eigenvalues of P t can be derived from the ( ) − λ t 1 i of − P t 1 . On this basis, we have harvested all the eigenvalues and their corresponding multiplicities for transition weight matrix T t of the network ( ) G m r t , , and prove that H t is only controlled by the parameter m. Finally, we also use the obtained eigenvalues of the normalized Laplacian matrix to further enumerate the weighted spanning tree of the network. In the future, we will explore the influence of the weight on the efficiency of random walks in the network with other properties besides scale-free. 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.