^{1}School of Electronics Engineering and Computer Science, Peking University, Beijing, China^{2}Key Laboratory of High Confidence Software Technologies, Peking University, Beijing, China^{3}China Northwest Center of Financial Research, Lanzhou University of Finance and Economics, Lanzhou, China^{4}School of Information Engineering, Lanzhou University of Finance and Economics, Lanzhou, China^{5}Key Laboratory of E-Business Technology and Application, Lanzhou, China^{6}College of Mathematics and Statistics, Northwest Normal University, Lanzhou, China

Controlling the trapping process is one of the important themes in the study of random walk in real complex systems. We studied two types of random walks that are different from the traditional random walk on a directed weighted network. The first type of random walk is the weighted random walk controlled by the weight *θ*, and the other is the delayed-weighted random walk affected by both delay probability *p* and weight *θ*. Furthermore, we derived analytically the average trapping time (ATT) measuring the efficiency of the two types of trapping processes; the result shows that the ATT grows sub-linearly, linearly, and super-linearly with the network order when the weight satisfies *θ* of the directed network can be adjusted by direction, the delay parameter *p* only changes the pre-factor of the ATT, and the weight *θ* modifies both the pre-factor and scaling of the ATT.

## 1 Introduction

In addition to the topological structure and characteristic of complex networks, the dynamic process in the network is also worth exploring because of its wide range of applications. Random walk, as a fundamental tool to describe the dynamic process of network, has been widely used to evaluate dynamical properties of different complex networks, involving practical problems such as page searching [1], epidemic spreading [2], target searching [3, 4], and energy transporting [5, 6]. As a main sub-topic in the research of random walks, the trapping problem refers to a special random walk in which a trap is placed at a given trap. A related basic quantity is the mean first passage time (MFPT), which represents the expected time required for the walker to reach the trap from the node *i* for the first time. The average trapping time (ATT) is the average of MFPTs over all starting nodes except the trap; the ATT is used as an index to measure the trapping efficiency of the trapping process [7, 8].

In order to improve the trapping efficiency, it is highly desirable to seek an effective method to adjust the trapping process in complex systems, such as treelike fractals [9] and one-dimensional systems. Researchers have studied the dynamic process of several types of networks without weight and direction, such as the Sierpinski network [10, 11], Koch network [12], and fractal lattice [13]. In view of the fact that many research studies on random walks mainly focus on undirected and unweighted networks, Zhang et al. [14] discussed two types of random walks for a class of weighted and undirected networks. Dai et al. [15–18] introduced several weighted random walks on the complex network. In fact, many real networks contain the relationships between individuals with different weights in different directions. For example, the degree of understanding between individuals in social networks, the traffic flow between two places on the transportation network, and the number of vaccines put in the epidemic prevention process are all related to the direction and weight of real networks [19, 20].

There are delays existing in some complex systems; this is the result of transportation or chemical reactions that require action time. The random walks that reveal the effects of delay have been proposed and studied through a mathematical framework [21]. On the other hand, a large amount of work on the trap problem of undirected networks cannot fully describe the real complex networks. The difference between the out-degree and in-degree of the node is considered in a directed network, and the weight of network is combined; we have explored the ATT of a class of directed weighted networks, and the random walk in the directed weighted network can be made more efficient by adjusting the weight factor and delay parameter, thus achieving the purpose of adjusting the random walk process.

The main contents of this study are divided into the following sections. First, we introduced some basic concepts related to random walks. In section 3, we designed a class of directed weighted edge-iteration networks, in which each pair of nodes is connected by two edges in opposite directions, and the weight of edge is controlled by the weight parameter *θ*; the delay phenomenon is measured by the delay parameter *p*. In section 4, we derived the exact analytical expression of the ATT for a given target node and found that the ATT of this network grows sub-linearly, linearly, and super-linearly, the three ways when parameter *θ* goes from 0 to infinity. Also, the delay parameter *p* not only changes the pre-factor of the ATT but also the weight *θ* can modify the pre-factor and scaling of the ATT.

## 2 Preliminaries

We focused on two types of biased random walks with a given target node. The directed weighted network constructed in this study is abbreviated as *t* = 0 in *t* are labeled *N*_{t−1} + 1, *N*_{t−1} + 2, … , *N*_{t} in order. We set the target node on the node labeled 1. The key role in the random walk process is the transition matrix *P*_{t}, whose entries *p*_{ij}(*t*) represent the transition probability of walking from the node *i* to node *j* in each time step; they satisfy *i*.

The mean first passage time for a walker moving from node *i* to the target node in the network

Furthermore, Eq. 1 can be transformed into the following matrix form;

where *N*_{t} − 1)-dimension column vector, and *N*_{t} − 1)-dimension column vector whose entries are equal to 1; the transition probability matrix

*N*_{t} − 1) × (*N*_{t} − 1)-order matrix whose diagonal entries are 1, and other entries are 0.

The ATT is defined as the mean of *T*⟩_{t}; we described the ATT by the following equation:

where *τ*_{ij} is essentially the *ij*-th entry of matrix (*I* − *P*)^{−1} combining Eq.3.

The aforementioned equation shows that the calculation of the ATT can be simplified as summing all the entries of the matrix (*I* − *P*)^{−1}. It is worth noting that the network order *N*_{t} increases exponentially with *t* when *t* → +*∞*; this calculation for a large *t* is very time-consuming. However, we can use Eq. 4 to verify the analytical solution of the ATT.

## 3 Design of a DWEI-Network

Many problems can be abstracted as graphs for research [22–24]. In view of the differences between the out-degree and in-degree of node in the real network, we proposed a network operator called the directed weighted edge-iteration transformation (DWEI-transformation) to construct our network by an iterative method. We set *w*_{i→j} as the weight of an edge from node *i* to node *j* in our network; it satisfies

Directed weighted edge-iteration transformation (DWEI-transformation). For an edge *e* = *AB* with two end nodes *A* and *B*, the two weights of this edge are *w*_{A→B} and *w*_{B→A}. A new edge with end nodes *C* and *D* is added; then, two new edges are connected between *A* and *C*, and *B* and *D*, respectively; the weights of new edges are distributed according to the following rules.

(a) *w*_{A→C} = *θw*_{A→B} and *w*_{C→A} = *w*_{B→A};

(b) *w*_{C→D} = *θw*_{C→A} and *w*_{D→C} = *w*_{A→C};

(c) *w*_{B→D} = *θw*_{B→A} and *w*_{D→B} = *w*_{A→B}.

Figure 1 is a schematic diagram of the DWEI-transformation.

Let *t* time steps; we generated a DWEI-network model, according to Algorithm 1. For the initial time step, the network *w*_{1→2} = *w*_{2→1} = 1. When *t* ≥ 1, the DWEI-transformation is performed on every edge that exists in *t* = 0, 1, 2, 3.

**Algorithm 1. **DWEI-network construction algorithm.

The out-strength of node i at time step t is defined as the sum of all weights of the edges starting from node i in −˗G(t), denoted as s + i (t) = PNt j=1 wi˗j (t). Meanwhile, the in-strength of node i represents the sum of the weights of all edges whose terminating nodes are i, expressed as s − i (t) = PNt j=1 wj˗i(t). The growth pattern determined by the node of network allows us to calculate the total number of nodes in −˗G(t) is Nt = 2 3 (4t + 2). Let ki(t) be the degree of a node i in network −˗G(t), the relationship between the degree of node i is ki(t) = 2ki(t − 1) in two consecutive time steps t − 1 and t. We can also get s + i (t) = (1 + Θ) t−ti+1 according to the definition of s+i(t), ti represents the time step when node i joined the network.

## 4 ATT for Weighted Walks

Different from the method of calculating the ATT by eigenvalues [25, 26], we derived a relationship governing the evolution for *t*, according to the construction process of DWEI networks. Let *Ω*_{t} denote the set of all nodes in *t* in *t* − 1 and before are called old nodes, denoted as *Ω*_{t−1}.

**Lemma 1. ***For the weighted walks of the DWEI network* *, let θ* > 0 *be its weight factor, and the mean first passage time of any node i* ∈ *Ω*_{t} *satisfies the recursive relation*

**Proof. **: The mean first-passage time between any two nodes in the DWEI network can be divided into the following three categories, and their simplified representation is as follows. Let *X* be the MFPT starting from the node *i* to any of its *k*_{i}(*t* − 1) old neighbors, which are directly connected to node *i* at the time step *t* − 1. *Y* represents the MFPT from any of *k*_{i}(*t* − 1) new neighbors of node *i* to one of its *k*_{i}(*t* − 1) old neighbors, and *Z* is the MFPT for starting from others new neighbors other than the aforementioned new nodes of node *i* to one of its *k*_{i}(*t* − 1) old neighbors. Thus, considering two consecutive time steps, we found that *X*, *Y*, *Z* satisfy the following equations:

We get a solution *X* = 1 + 2*θ* by solving the aforementioned formula; it means that from the time step *t* to the next time *t* + 1, the trapping time for an arbitrary node *i* increases 1 + 2*θ* times the previous moment, that is, *τ* ≤ *t*; the sum of the MFPT of all nodes at time step *τ* is denoted as

and the sum of the MFPT of all new nodes at a time step *τ* is formulated as

Then, the average trapping time ⟨*T*⟩_{t} of the DWEI network

**Theorem 2. ***For the weighted random walks, N*_{t} *is the DWEI network order, let θ* > 0 *be the weight factor and t be the time step, then*

*(1) For* *, the ATT of the DWEI network is*

*The dominating term of* ⟨*T*⟩_{t} *is* ⟨*T*⟩_{t} ∼ *N*_{t} *for t* → *∞.*

*(2) When* *, the ATT of the DWEI network is*

*the relationship between* ⟨*T*⟩_{t} *and the network order N*_{t} *satisfy* *for t* → *∞.*

**Proof. **It can be seen that the problem for evaluating ⟨*T*⟩_{t} is reduced to determining

Obviously, once we calculated the result of *t*, and the other is an old node added before the time step *t*. There are two new nodes in the next time step for each edge *e* = *uv*; we labeled them as *x*_{1} and *x*_{2}, respectively. If the MFPTs of the four nodes *u*, *v*, *x*_{1}, *x*_{2} at a certain moment are expressed as *T*(*u*), *T*(*v*), *T* (*x*_{1}), *T* (*x*_{2}), the relationships between the MFPTs of new nodes *x*_{1}, *x*_{2} and its neighbors can be derived as

Combining the two formulas in Eq. 14, we can get

For the DWEI network corresponding to the first few time steps, since the number of nodes in the network is small, we can directly calculate the value of *i* ≤ *t*. For example, when *t* = 1, the way to obtain the value of *θ*. The weights on its edges to node 1 and node 4 are 1 and 0, respectively; then, the second equation in Eq.16 holds, the equations related to node 3 and 4 can be obtained similarly.

Among the four nodes, the two nodes labeled 1 and 2 are called old nodes, and 3 and 4 are called new nodes; thus, the sum of MFPTs of all new nodes is equal to

Therefore, the sum of MFPTs of all nodes in the network

Considering Eq. 15 and summing this equation over all new nodes in

Similarly, the following equation about

Then, subtracting the product of Eq.19 multiplied by 2 (1 + 2*θ*) from Eq. 20, we have

After merging

Considering a value

1) For

We can get

2) When

With the help of a condition

Furthermore, substituting Eq.27 into Eq. 10, we get the average of all MFPTs in the network

Actually, the DWEI network can be regarded as a binary network for a special case *θ* = 1; based on this situation, the value of ⟨*T*⟩_{t} we obtained is consistent with the result in the literature [27]. Next, we showed how to express ⟨*T*⟩_{t} in terms of the network order *N*_{t}; Substituting

as *t* → *∞*. According to Eq. 29, we found that the weight *θ* can not only modify the pre-factor of ⟨*T*⟩_{t} but also the scaling log_{4} (1 + 2*θ*) of ⟨*T*⟩_{t}. In addition, we can obtain the following conclusions in weighted random walks by analyzing Eqs 25, 29, when *T*⟩_{t} grows linearly with *N*_{t}; however, ⟨*T*⟩_{t} grows sub-linearly and super-linearly with the network order *N*_{t} for

**FIGURE 4**. Solids indicate the numerical results, and the hollows indicate the results of our analytical results.

## 5 ATT for Delayed Weighted Walks

In this section, a delay probability 0 ≤ *p* ≤ 1 is introduced to explore its impact on the ATT of our network in the weighted random walk with delay [28]. The delayed weighted random walk is defined as follows; for delay probability 0 ≤ *p* ≤ 1, the walker in network *p* and 1 − *p*, respectively, which is the transition probability *p*_{ij} for the walker jumping from node *i* to another node *j* and is formulated as

The two special cases of delayed weighted random walks are as follows:

1) When *p* = 0, delayed weighted random walks reduce to regular weighted random walks discussed in the previous section because the walker moves from node *i* to any neighboring node *j* with transition probability

2) When *p* = 1, if node *i* is created before the time step *t*, then the walker jumping from the current node *i* to any neighboring node *j* in *i* to any neighbor node *j* on

**Lemma 3. ***For the delayed weighted random walks of the DWEI network* *; let θ* > 0 *be a weight factor and* 0 ≤ *p* ≤ 1 *represents a delay probability,* *is the MFPT corresponding to the regular weighted random walks; the relationship between MFPT* *of delayed weighted random walks and* *for node i is*

**Proof. **: Let *X*, *Y*, and *Z* is the same as section 4, and they satisfy the following three equations in the delayed weighted random walks:

The solved *X* involves the delay probability *p* and the weight factor *θ*; its expression is

Therefore, we have deduced the relationship between

**Theorem 4. ***For the delayed weighted random walks, let θ* > 0 *and t* ≥ 0 *be the weight factor and time step, N*_{t} *is the DWEI-network order; then, the exact dependence of ATT on the network order N*_{t} *is as follows:*

*(1) For* *,*

*2) When* *,*

*for t* → *∞.*

**Proof. **: We defined ⟨*F*⟩_{t} as the ATT for delayed weighted random walks in the network *F*⟩_{t}, two quantities *τ* ≤ *t*. Extending Eq.10 to the delayed weighted random walks, a formula about ⟨*F*⟩_{t} is as follows:

The sum of MFPT of all nodes in the network is composed of the sum of MFPT of old nodes and new nodes; it can be formulated as

Similar to the calculation of Eqs 13, 17, it shows that our calculation focuses on solving

We can directly get *t* + 1:

Multiplying Eq.38 with 2 (1 + 2*θ*) and subtracting the result from Eq. 39, we derived the recursive connection between

where *t* is

1) When the weight factor satisfies

In this case, we can calculate the sum of MFPT of all nodes by combining Eqs 24, 37, 42:

Substituting Eq.43 into Eq. 36, we can get the expression of ⟨*F*⟩_{t}. Furthermore, we deduce that the trend of ⟨*F*⟩_{t} increases with *N*_{t} for a large network:

The aforementioned formula shows that ⟨*F*⟩_{t} increases linearly with *N*_{t} when *A*, *B*, and *C*, respectively, which are all controlled by the weight factor *θ* and have no dependencies on the time step *t*.

Among them, three variables *A*, *B*, *C* equal to

Next, we gave the average of MFPT for all nodes except the trap node; substituting Eq.45 into Eq. 36, we obtain

Considering *F*⟩_{t} in Eq.47 can be represented in terms of the network order *N*_{t} in the following form:

For a large *t*, we have the following expression for the dominating term of ⟨*F*⟩_{t}:

Therefore, Eq.49 shows that ⟨*F*⟩_{t} grows sub-linearly and super-linearly with the network size, when *F*⟩_{t} grows linearly with the network size from Eq.44. The coefficient *p* and the weight factor *θ*. However, the exponent log_{4} (1 + 2*θ*) of *N*_{t} is only controlled by the weight factor *θ* and has no dependencies on the delay probability *p*. Also, we have known that the delay parameter *p* can only change the coefficient of the ATT; it has less influence on the trapping efficiency.

## 6 Conclusion

Based on the undirected unweighted network, we considered the weight and the out-degree and in-degree of nodes in complex systems. We proposed a network operator and a class of directed weighted edge-iteration networks and derived the closed form solution of the average trapping time of two different random walks with a given trap node; one of the random walks is based on the weight *θ* alone acting on the transition probability, and the other is the weight factor *θ* and delay parameter *p* impacting on the transition probability at the same time. The solution shows that the directed weighted construction has a significant effect on the trapping efficiency, and the leading scale of the ATT can be sub-linearly, linearly, and super-linearly with the network size when

## Data Availability Statement

The original contributions presented in the study are included in the article/Supplementary Material; further inquiries can be directed to the corresponding author.

## Author Contributions

JS provided this topic and wrote the manuscript. MM and MZ discussed and drew all figures. BY modified and guided the manuscript. All authors contributed to the manuscript and approved the submitted version.

## Funding

This research was supported by the National Key Research and Development Plan under the Grant No. 2019YFA0706401 and the National Natural Science Foundation of China under Grants 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).

## Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

## Publisher’s Note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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Keywords: random walk, directed network, weight, mean first passage time, trapping efficiency

Citation: Su J, Ma M, Zhang M and Yao B (2022) Adjusting the Trapping Process of a Directed Weighted Edge-Iteration Network. *Front. Phys.* 10:822712. doi: 10.3389/fphy.2022.822712

Received: 26 November 2021; Accepted: 06 June 2022;

Published: 14 July 2022.

Edited by:

Nuno A. M. Araújo, University of Lisbon, PortugalReviewed by:

Yilun Shang, Northumbria University, United KingdomEnqiang Zhu, Guangzhou University, China

Copyright © 2022 Su, Ma, Zhang and Yao. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Mingjun Zhang, zhangmjlz@163.com