The Global Mean First-Passage Time for Degree-Dependent Random Walks in a Class of Fractal Scale-Free Trees

Abstract

Fractal scale-free structures are widely observed across a range of natural and synthetic systems, such as biological networks, internet architectures, and social networks, providing broad applications in the management of complex systems and the facilitation of dynamic processes. The global mean first-passage time (GMFPT) for random walks on the underlying networks has attracted significant attention as it serves as an important quantitative indicator that can be used in many different fields, such as reaction kinetics, network transport, random search, pathway elaboration, etc. In this study, we first introduce two degree-dependent random walk strategies where the transition probability is depended on the degree of neighbors. Then, we evaluate analytically the GMFPT of two degree-dependent random walk strategies on fractal scale-free tree structures by exploring the connection between first-passage times in degree-dependent random walk strategies and biased random walks on the weighted network. The exact results of the GMFPT for the two degree-dependent random walk strategies are presented and are compared with the GMFPT of the classical unbiased random walk strategy. Our work not only presents a way to evaluate the GMFPT for degree-dependent biased random walk strategies on general networks but also offers valuable insights to enrich the controlling of chemical reactions, network transport, random search, and pathway elaboration.

1. Introduction

Numerous natural and artificial systems, including biological networks, social networks, and internet architectures [1,2,3], display fractal and scale-free properties. The fractal scale-free trees model, which combines the theoretical models originally introduced by Barabási and Albert [4] and Benoit Mandelbrot [5], is a relatively simple fundamental dynamics network, with extensive applications in network science, epidemiology, ecology, and computer science [6,7,8,9]. The advantages of robustness and scalability, etc., in handling complex systems and dynamic processes with the fractal scale-free trees structures make the fractal scale-free trees model very popular, and many different fractal scale-free trees models have been proposed [7,9,10]. The structural and dynamic properties of these models have also been thoroughly explored; examples of these models include $(u,v)$-trees [1], the spanning scale-free trees of some real-world networks [9], fractal and non-fractal scale-free trees [11], etc.

Among the range of dynamic properties, the expected time it takes the walker to first reach the given target, also known as the mean first-passage time (MFPT), for random walks on underlying networks stands out as especially crucial [12,13]. Numerous other properties can be measured or understood through the lens of the MFPT [14,15,16]. Let ${\mathcal{T}}_{x\to y}$ denote the MFPT to the target y from starting location x. One can define the GMFPT, $T_{global}$, by

$$\begin{array}{c}\hfill {\mathcal{T}}_{global}={\displaystyle \frac{1}{\mathcal{V}(\mathcal{V}-1)}}\sum _{x=1}^{\mathcal{V}}\sum _{x\ne y}{\mathcal{T}}_{x\to y},\end{array}$$

(1)

where $\mathcal{V}$ is used to represent the total node population in the underlying networks.

The GMFPT represents the average duration required for a walker, initiating from a randomly chosen site, to reach a target site that is also selected at random from a uniform distribution for the first time [17]. This is a crucial indicator that can be used in many different fields, such as reaction kinetics [18,19], protein interactions and cellular search [20,21], network transport and information dissemination [22,23,24], random search [25,26], pathway elaboration [27], etc. Studies on the GMFPT are helpful for the control of chemical kinetics, network transport, random search, and pathway elaboration [18,22,28,29]. Over the past decades, considerable efforts have been dedicated to unraveling the GMFPT for classical unbiased random walks across diverse networks, including scale-free Koch networks [30], various tree-like networks [31,32,33,34], and pseudofractal scale-free web [35,36]. These studies have revealed how network topology influences the GMFPT. It is also well known that random walk strategies have a significant impact on the GMFPT. There are also many works that are dedicated to enriching random walk strategies and controlling the transport (search) efficiency. Related random walk strategies include maximal entropy random walks [37,38], true reinforced random walks [39,40], weights-dependent biased random walks [41,42,43,44,45,46,47], and degree-dependent biased random walks [48,49].

For degree-dependent biased random walks, the probability of transitioning from the current node to an adjacent node is related to the degree of that adjacent node, specifically the degrees of the nearest and second-nearest neighbors [48]. Note that the walker determines which neighboring node to move to next by making a weighted choice based on the degree of each neighboring node under this strategy; this is particularly meaningful for controlling information dissemination and search processes. The degree-dependent random strategy is an optimization approach leveraging network topology information, widely used in network science and other fields.

In this work, we study the GMFPT for two types of degree-dependent random walk strategies on a class of fractal scale-free trees [11] to enrich the strategies to control transport efficiency. First, we design two degree-dependent random walk strategies in which the transition probability depends on the degree of neighbors. Then, we evaluate analytically the GMFPT of two degree-dependent random walk strategies on a class of fractal scale-free trees by exploring the connection between the first-passage time for degree-dependent random walk strategies and biased random walks on the weighted network. Accurate results of the GMFPT of two degree-dependent random walk strategies are presented and are compared with the GMFPT for the classical unbiased random walk strategy. Our work not only provides a method to evaluate the GMFPT for degree-dependent biased random walk strategies on general networks but also offers valuable insights for enhancing the control of chemical kinetics, network transport, random search, and pathway optimization.

2. Preliminaries

In this section, we present the structure of the fractal scale-free trees we examined and describe the degree-dependent biased random walk strategies and the method we used to calculate the global mean first-passage time (GMFPT) for degree-dependent biased random walk strategies.

2.1. Fractal Scale-Free Trees

Fractal scale-free trees (FSFTs) are deterministic trees that can be constructed recursively and controlled by an integer parameter q ($q\ge 1$). Let $\mathcal{N}\left(g\right)$ represent the fractal scale-free trees following the g-th iteration, i.e., the FSFTs with generation g. Their construction proceeds as follows [11,50]:

• For $g=0$, $\mathcal{N}\left(0\right)$ is an edge whose two endpoints are labeled as A and B.
• For $g\ge 1$, $\mathcal{N}\left(g\right)$ can be obtained from $\mathcal{N}(g-1)$ as follows: replacing each edge of $\mathcal{N}(g-1)$ by a path of 2 links, with both endpoints of the path remaining as the endpoints of the original edge and adding $2q$ new nodes to each path, one half is linked to one endpoint of the path and the other half is linked to another endpoint of the path.

In this paper, the newly added node with initial degree 2, positioned in the middle of the path, is designated as an internal node; the newly added nodes with initial degree 1 are referred to as external nodes. In other words, $\mathcal{N}\left(g\right)$ is obtained from $\mathcal{N}(g-1)$ by replacing each edge of $\mathcal{N}(g-1)$ with a structure, as shown on the arrow’s right side in Figure 1. The first three generations of the FSFTs with $q=2$ are shown in Figure 2.

Figure 1. The construction process for the FSFTs: each edge in $\mathcal{N}(g-1)$ is substituted by the object on the right-hand side of the arrow, where the red dots represent original nodes, the blue dots represent newly added internal nodes, and the green squares represent newly added external nodes.

Figure 2. Growth process of the first three generations for the particular FSFTs with $q=2$.

As shown in Figure 3, the FSFTs can also be constructed in another way. The FSFTs of generation $g+1$, denoted as $\mathcal{N}(g+1)$, are composed of $2q+2$ copies of $\mathcal{N}\left(g\right)$, referred to as subgraphs, which are labeled by ${\mathcal{N}}^{\left(1\right)}\left(g\right)$, ${\mathcal{N}}^{\left(2\right)}\left(g\right)$, ${\mathcal{N}}^{\left(3\right)}\left(g\right)$, …, ${\mathcal{N}}^{(2q+2)}\left(g\right)$, and the $2q+2$ are subgraphs are connected with each other via their hub nodes (i.e., nodes with the highest degree in these subgraphs). It is known that the FSFTs exhibit a self-similar structure with fractal dimension $d_f=ln(2q+2)/ln\left(2\right)$, the degree distribution of FSFTs follows power law $P\left(k\right)\sim k^{-\gamma }$, with $\gamma =1+{\displaystyle \frac{ln(2q+1)}{ln(q+1)}}$ [11]. Therefore, the FSFTs are known as fractal scale-free trees.

Figure 3. Alternative construction method of FSFTs, which shows the FSFTs have self-similarity characteristics. The FSFTs of generation $g+1$, denoted as $\mathcal{N}(g+1)$, are composed of $2q+2$ subgraphs, which are isomorphic to $\mathcal{N}\left(g\right)$, labeled by ${\mathcal{N}}^{\left(1\right)}\left(g\right)$, ${\mathcal{N}}^{\left(2\right)}\left(g\right)$, ${\mathcal{N}}^{\left(3\right)}\left(g\right)$, …, ${\mathcal{N}}^{(2q+2)}\left(g\right)$, and are interconnected via their hub nodes.

Through its construction, it is possible to determine the total count of edges, referred to as ${\mathcal{E}}_g$, of $\mathcal{N}\left(g\right)$ as

$${\mathcal{E}}_g={(2q+1)}^g,$$

(2)

and the total count of nodes, ${\mathcal{V}}_g$, satisfies

$${\mathcal{V}}_g=1+{\mathcal{E}}_g=1+{(2q+1)}^g.$$

(3)

2.2. Two Degree-Dependent Random Walk Strategies

Note that the random walk strategy is determined entirely by its transition probability matrix. Here, we introduce the transition probability matrix for the two random walk strategies. For the first strategy, the probability that the walker from its current node x to any adjacent node y is proportional to the node degree of node y. For the second strategy, the probability that the walker from its current node x to any adjacent node y is inversely proportional to the node degree of node y. Let $P_{xy}$ represent the transition probability from node x to node y, and $v\left(x\right)$, $d_x$ represent the set for all neighbors of node x, the degree of node x, respectively. Then, for the first random walk strategy,

$$P_{xy}=\left\{\begin{array}{cc}{\displaystyle \frac{d_y}{{\displaystyle \sum _{z\in v\left(x\right)}}{d}_z}},& y\in v\left(x\right),\\ 0,& otherwise,\end{array}\right.$$

(4)

and for the second random walk strategy,

$$P_{xy}=\left\{\begin{array}{cc}{\displaystyle \frac{{d_y}^{-1}}{{\displaystyle \sum _{z\in v\left(x\right)}}{d_z}^{-1}}},& y\in v\left(x\right),\\ 0,& otherwise.\end{array}\right.$$

(5)

For simplicity, we refer to the two random walk strategies in this paper as strategy 1 and strategy 2, respectively.

Note that the transition probabilities for both random walk strategies are dominated entirely by the degree of the neighbors of the current position x. Therefore, we call them degree-dependent random walk strategies.

2.3. Method to Evaluate the GMFPT for the Degree-Dependent Random Walk Strategies

Here, we introduce a method for calculating the GMFPT for the two degree-dependent random walk strategies on general networks. The method relies on the correlation between the MFPT on the weighted network and the effective electrical resistance within the equivalent resistance network.

First, we review the main lemma that will be used to calculate the GMFPT. For any weighted network, with $w_{xy}$ being the weight for its edge $(x,y)$, and for random walk on any weighted network with transition probability,

$$P_{xy}=\left\{\begin{array}{cc}{\displaystyle \frac{w_{xy}}{{\displaystyle \sum _{y\in v\left(x\right)}}{w}_{xy}}},& y\in v\left(x\right),\\ 0,& otherwise,\end{array}\right.$$

(6)

the GMFPT for random walks across the weighted network can be expressed as [42,51]

$$\begin{array}{ccc}\hfill {\mathcal{T}}_{global}& =& {\displaystyle \frac{\mathcal{F}{\mathcal{R}}_{sum}}{\mathcal{V}(\mathcal{V}-1)}},\hfill \end{array}$$

(7)

where $\mathcal{V}$ is the total count of nodes of the weighted network,

$$\mathcal{F}={\sum }_{(x,y)}{w}_{xy}$$

(8)

is the total of the weights assigned to every edge within the weighted network, and

$${\mathcal{R}}_{sum}={\displaystyle \frac{1}{2}}\sum _{x\ne y}{\mathcal{R}}_{xy}$$

(9)

totals the effective resistance between any pair of nodes within the equivalent resistance network with each edge $(x,y)$ and is assigned a resistance $r_{xy}={\displaystyle \frac{1}{w_{xy}}}$, and ${\mathcal{R}}_{xy}$ is the effective resistance between node x and y within the equivalent resistance network.

Then, we build the connection between the degree-dependent random walk and the biased random walk on the weighted network as depicted in Equation (6).

For the degree-dependent biased random walk strategy 1, let $w_{xy}=\left(d_x{d}_y\right)$; one can recover the transition probability as shown in Equation (4) from Equation (6). Therefore, the degree-dependent biased random walk strategy 1 is equivalent to a biased random walk on the corresponding weighted network, where each edge $(x,y)$ is assigned a weight $w_{xy}=\left(d_x{d}_y\right)$.

For strategy 2, let $w_{xy}={\left(d_x{d}_y\right)}^{-1}$; one can also recover transition probability as shown in Equation (5) from Equation (6). Therefore, the degree-dependent biased random walk strategy 2 is equivalent to a biased random walk on the corresponding weighted network, where each edge $(x,y)$ is assigned a weight $w_{xy}={\left(d_x{d}_y\right)}^{-1}$.

Thus, both two degree-dependent random walks can be viewed as biased random walks on the corresponding weighted networks, and we can calculate the GMFPT for the degree-dependent random walk on FSFTs by using Equation (7).

3. GMFPT for Degree-Dependent Biased Random Walk Strategy 1 on the FSFTs

Here, we calculate the GMFPT, ${\mathcal{T}}_{global}$, for degree-dependent biased random walk strategy 1 on FSFTs. As discussed in Section 2.3, the degree-dependent biased random walk strategy 1 is equivalent to biased random walk on the corresponding weighted network, where each edge $(x,y)$ is assigned a weight $w_{xy}=\left(d_x{d}_y\right)$, and the GMFPT can be derived from Equation (7).

For clarity, let ${\mathcal{N}}_w\left(g\right)$ denote the weighted FSFTs of generation g, where each edge $(x,y)$ is assigned a weight $w_{xy}=\left(d_x{d}_y\right)$ and ${\mathcal{T}}_{global}^1\left(g\right)$ represent the GMFPT for strategy 1 on the FSFTs with generation g, and let ${\mathcal{R}}_{sum}\left(g\right)$, ${\mathcal{F}}_w\left(g\right)$ represent ${\mathcal{R}}_{sum}$, $\mathcal{F}$ for the weighted FSFTs, ${\mathcal{N}}_w\left(g\right)$, where each edge $(x,y)$ is assigned a weight $w_{xy}=\left(d_x{d}_y\right)$, respectively.

In the following, we will first derive ${\mathcal{R}}_{sum}\left(g\right)$ and ${\mathcal{F}}_w\left(g\right)$ . Then, ${\mathcal{T}}_{global}^1\left(g\right)$ can be obtained from Equation (7).

3.1. Derivation of ${\mathcal{R}}_{sum}\left(g\right)$ for the Weighted FSFTs ${\mathcal{N}}_w\left(g\right)$

Here, we consider the weighted FSFTs, where each edge $(x,y)$ and is assigned a weight, $w_{xy}=\left(d_x{d}_y\right)$, and calculate ${\mathcal{R}}_{sum}\left(g\right)$, which totals the effective resistance between any pair of nodes within the equivalent resistance network with each edge $(x,y)$ is assigned a resistance $r_{xy}={\displaystyle \frac{1}{w_{xy}}}$ and defined in Equation (9). Because the equivalent resistance network for ${\mathcal{R}}_{sum}\left(g\right)$ is a weighted tree, the effective resistance between any two nodes is just the weighted path length between the two nodes. Thus, ${\mathcal{R}}_{sum}\left(g\right)$ can be derived by exploring the self-similar structure of the network.

Similar to FSFTs (see Figure 3), the weighted FSFTs, ${\mathcal{N}}_w\left(g\right)$, also exhibits a self-similar character. As shown in Figure 4, ${\mathcal{N}}_w(g+1)$ consists of $2q+2$ subunits, labeled by ${\mathcal{N}}_w^{\left(1\right)}\left(g\right)$, ${\mathcal{N}}_w^{\left(2\right)}\left(g\right)$, …, ${\mathcal{N}}_w^{(2q+2)}\left(g\right)$, which have the same structure as ${\mathcal{N}}_w\left(g\right)$ and are connected with each other via their hubs. For any $k=1,2,\cdots ,2q+2$, all edges of ${\mathcal{N}}_w^{\left(k\right)}\left(g\right)$, except edges that are incident on the hubs (i.e., A, B, C), have the same weight as the weight for the corresponding edge of ${\mathcal{N}}_w\left(g\right)$. For any $k=1,2,\cdots ,2q+2$, let ${\mathcal{R}}_{sum}^{\left(k\right)}\left(g\right)$ denote the sum of effective resistances for all node pairs of ${\mathcal{N}}_w^{\left(k\right)}\left(g\right)$ and ${\mathcal{D}}_R^{\left(k\right)}\left(g\right)$ represent the difference between ${\mathcal{R}}_{sum}^{\left(k\right)}\left(g\right)$ and ${\mathcal{R}}_{sum}\left(g\right)$.

Figure 4. Construction of the weighted FSFTs. ${\mathcal{N}}_w(g+1)$ consists of $2q+2$ subunits, labeled by ${\mathcal{N}}_w^{\left(1\right)}\left(g\right)$, ${\mathcal{N}}_w^{\left(2\right)}\left(g\right)$, ${\mathcal{N}}_w^{\left(3\right)}\left(g\right)$, …, ${\mathcal{N}}_2^{(2q+2)}\left(g\right)$, which have the same structure as ${\mathcal{N}}_w\left(g\right)$ and are interconnected via their hub nodes, A, B, and C. All edges of ${\mathcal{N}}_w^{\left(k\right)}\left(g\right)$, except edges that are incident on the hubs (i.e., A, B, C), have the same weight as the weight for the corresponding edge of ${\mathcal{N}}_w\left(g\right)$.

For $g=1$, according to strategy 1, we can get that the effective resistance of each edge, ${\mathcal{R}}_{AC}={\mathcal{R}}_{BC}={\displaystyle \frac{1}{2(q+1)}}$, and the effective resistance of the remaining edges is ${\displaystyle \frac{1}{(q+1)}}$; then, we can calculate the effective resistance between two hub nodes, A and B, as

$${\mathcal{R}}_{AB}={\mathcal{R}}_{AC}+{\mathcal{R}}_{BC}={\displaystyle \frac{1}{(q+1)}}.$$

Thus, it is not hard to calculate the ${\mathcal{R}}_{sum}\left(1\right)=5q+2$, and for any $g\ge 1$, ${\mathcal{R}}_{sum}(g+1)$ is satisfied.

$$\begin{array}{ccc}\hfill {\mathcal{R}}_{sum}(g+1)& =& {\displaystyle \frac{1}{2}}\sum _{x,y\in {\mathcal{N}}_w(g+1)}{\mathcal{R}}_{xy}\hfill \\ & =& {\displaystyle \frac{1}{2}}\sum _{k=1}^{2q+2}\sum _{x,y\in {\mathcal{N}}_w^k\left(g\right)}{\mathcal{R}}_{xy}+{\displaystyle \frac{1}{2}}\sum _{k_1\ne k_2}\sum _{{\displaystyle \genfrac{}{}{0pt}{}{x\in {\mathcal{N}}_w^{k_1}\hfill }{y\in {\mathcal{N}}_w^{k_2}\hfill }}}{\mathcal{R}}_{xy}\hfill \\ & =& {\displaystyle \frac{1}{2}}\sum _{x,y\in {\mathcal{N}}_w^{2q+1}\left(g\right)}{\mathcal{R}}_{xy}+{\displaystyle \frac{1}{2}}\sum _{x,y\in {\mathcal{N}}_w^{2q+2}\left(g\right)}{\mathcal{R}}_{xy}\hfill \\ & +& {\displaystyle \frac{1}{2}}\sum _{k=1}^{2q}\sum _{x,y\in {\mathcal{N}}_w^k\left(g\right)}{\mathcal{R}}_{xy}+{\mathcal{L}}_g\hfill \\ & =& 2q{\mathcal{R}}_{sum}^{\left(1\right)}\left(g\right)+2{\mathcal{R}}_{sum}^{(2q+1)}\left(g\right)+{\mathcal{L}}_g\hfill \\ & =& 2q[{\mathcal{R}}_{sum}\left(g\right)+{\mathcal{D}}_R^{\left(1\right)}\left(g\right)]+2[{\mathcal{R}}_{sum}\left(g\right)+{\mathcal{D}}_R^{(2q+1)}\left(g\right)]+{\mathcal{L}}_g\hfill \\ & =& (2q+2){\mathcal{R}}_{sum}\left(g\right)+2q{\mathcal{D}}_R^{\left(1\right)}\left(g\right)+2{\mathcal{D}}_R^{(2q+1)}\left(g\right)+{\mathcal{L}}_g,\hfill \end{array}$$

(10)

where

$${\mathcal{L}}_g={\displaystyle \frac{1}{2}}\sum _{k_1\ne k_2}\sum _{{\displaystyle \genfrac{}{}{0pt}{}{x\in {\mathcal{N}}_w^{k_1}\hfill }{y\in {\mathcal{N}}_w^{k_2}\hfill }}}{\mathcal{R}}_{xy}$$

is the total resistance sum across all node pairs x and y, where x and y are not in the same subunits as ${\mathcal{N}}_w(g+1)$ and

$$\begin{array}{ccc}\hfill {\mathcal{D}}_R^{\left(1\right)}\left(g\right)& =& -{\displaystyle \frac{q(6q^3+7q^2+3q){(2q+2)}^{2g}}{2(8q^3+18q^2+13q+3){(q+1)}^{g+1}}}\hfill \\ & +& {\displaystyle \frac{(q^2-q){(2q+2)}^g}{2{(q+1)}^2}}+{\displaystyle \frac{q^2{(2q+2)}^g}{2(2q^2+3q+1){(q+1)}^{g+1}}}\hfill \\ & -& {\displaystyle \frac{q^3+3q^2}{2(4q+3){(q+1)}^2}}-{\displaystyle \frac{q^2{(2v+2)}^g}{{(q+1)}^2}},\hfill \end{array}$$

(11)

$$\begin{array}{ccc}\hfill {\mathcal{D}}_R^{(2q+1)}\left(g\right)& =& -{\displaystyle \frac{q(6q^3+7q^2+3q){(2q+2)}^{2g}}{2(8q^3+18q^2+13q+3){(q+1)}^{g+1}}}\hfill \\ & +& {\displaystyle \frac{(q^2-q){(2q+2)}^g}{2{(q+1)}^2}}+{\displaystyle \frac{q^2{(2q+2)}^g}{2(2q^2+3q+1){(q+1)}^{g+1}}}\hfill \\ & -& {\displaystyle \frac{q^3+3q^2}{2(4q+3){(q+1)}^2}}-{\displaystyle \frac{q^2{(2q+2)}^g}{{(q+1)}^2}}-{\displaystyle \frac{q{(2q+2)}^g}{2(q+1)}}\hfill \\ & -& {\displaystyle \frac{(6q^3+7q^2+3q){(2q+2)}^{2g}}{4(8q^3+18q^2+13q+3){(q+1)}^g}}-{\displaystyle \frac{q^3+3q^2}{4(4q+3)(q+1)}}\hfill \\ & +& {\displaystyle \frac{(q-1){(2q+2)}^g}{2(q+1)}}+{\displaystyle \frac{q{(2q+2)}^g}{4(2q^2+3q+1){(q+1)}^g}}.\hfill \end{array}$$

(12)

The detailed derivation of ${\mathcal{D}}_R^{\left(1\right)}\left(g\right)$ and ${\mathcal{D}}_R^{\left(0\right)}\left(g\right)$ is presented in Appendix A.

Using Equation (10) recursively, we can obtain

$$\begin{array}{ccc}\hfill {\mathcal{R}}_{sum}\left(g\right)& =& (2q+2){\mathcal{R}}_{sum}(g-1)+2q{\mathcal{D}}_R^{\left(1\right)}(g-1)+2{\mathcal{D}}_R^{\left(0\right)}(g-1)+{\mathcal{L}}_{g-1}\hfill \\ & =& {(2q+2)}^2{\mathcal{R}}_{sum}(g-2)+2q(2q+2){\mathcal{D}}_R^{\left(1\right)}(g-2)\hfill \\ & +& 2(2q+2){\mathcal{D}}_R^{\left(0\right)}(g-1)+(2q+2){\mathcal{L}}_{g-2}\hfill \\ & +& 2q{\mathcal{D}}_R^{\left(1\right)}(g-1)+2{\mathcal{D}}_R^{\left(0\right)}(g-1)+{\mathcal{L}}_{g-1}\hfill \\ & =& {(2q+2)}^{g-1}{\mathcal{R}}_{sum}\left(1\right)+2q{\sum }_{k=1}^{g-1}{(2q+2)}^{g-1-k}{\mathcal{D}}_R^{\left(1\right)}\left(k\right)\hfill \\ & +& 2{\sum }_{k=1}^{g-1}{(2q+2)}^{g-1-k}{\mathcal{D}}_R^{\left(0\right)}\left(k\right)+{\sum }_{k=1}^{g-1}{(2q+2)}^{g-1-k}{\mathcal{L}}_k.\hfill \end{array}$$

(13)

For any integer $k\ge 1$, as derived in Appendix B, we find

$$\begin{array}{ccc}\hfill {\mathcal{L}}_k& =& {\displaystyle \frac{(5q^2+5q){(2q+2)}^{2k}}{(4q+2){(q+1)}^k}}+{\displaystyle \frac{(6q^2+6q+1)2^k{(2q+2)}^{2k}}{8q+4}}\hfill \\ & +& {\displaystyle \frac{{(2q+1)}^2(4q^2+7q+1){(2q+2)}^{2k}}{4(q+1)(2q^2+4q+1)}}+{\displaystyle \frac{(2q^2+q){(2q+2)}^k}{2q+2}}\hfill \\ & +& {\displaystyle \frac{(1+3q-q^2){(2q+2)}^k}{(8q^3+20q^2+12q+2){(q+1)}^k}}.\hfill \end{array}$$

(14)

Replacing ${\mathcal{D}}_R^{\left(1\right)}\left(g\right)$, ${\mathcal{D}}_R^{(2q+1)}\left(g\right)$, and ${\mathcal{L}}_k$ from Equations (11), (12), and (14), respectively, in Equation (26), we get

$$\begin{array}{ccc}\hfill {\mathcal{R}}_{sum}\left(g\right)& =& {\displaystyle \frac{q(q+3)(8q^2+11q+4){(2q+2)}^{2g}}{4(2q+1)(4q+3){(q+1)}^{g+2}}}\hfill \\ & +& {\displaystyle \frac{(6q^2+6q+1)2^g{(2q+2)}^{2g}}{8\left(q+1\right)\left(2q+2\right)\left(4q+3\right)}}\hfill \\ & -& {\displaystyle \frac{(4q^4+9q^3+8q^2+5q+1){(2q+2)}^g}{4q(2q+1)(2q^2+4q+1){(q+1)}^{g+1}}}\hfill \\ & +& {\displaystyle \frac{(4q^2+7q+1)(2q+1){(2q+2)}^{2g}}{8{\left(q+1\right)}^2(2q^2+4q+1)}}+{\displaystyle \frac{q\left(q+3\right)}{8q^2+14q+6}}\hfill \\ & +& {\displaystyle \frac{(2q+1)(5q^3+5q^2-4g{q}^2+5q-3gq+3){(2q+2)}^g}{4q(4q+3){(q+1)}^2}}.\hfill \end{array}$$

(15)

3.2. Derivation of ${\mathcal{F}}_w\left(g\right)$ for the Weighted FSFTs ${\mathcal{N}}_w\left(g\right)$

Here, we calculate ${\mathcal{F}}_w\left(g\right)$, which totals weight for all edges of ${\mathcal{N}}_w\left(g\right)$. Note that ${\mathcal{N}}_w(g+1)$ is composed of $2q+2$ subunits, labeled by ${\mathcal{N}}_w^{\left(1\right)}\left(g\right)$, ${\mathcal{N}}_w^{\left(2\right)}\left(g\right)$, ${\mathcal{N}}_w^{\left(3\right)}\left(g\right)$, …, ${\mathcal{N}}_w^{(2q+2)}\left(g\right)$, which have the same structure as ${\mathcal{N}}_w\left(g\right)$, and for any $k=1,2,\cdots ,2q+2$, all edges of ${\mathcal{N}}_w^{\left(k\right)}\left(g\right)$, except edges that are incident on the hubs (i.e., A, B, C), have the same weight as the the weight for the corresponding edge of ${\mathcal{N}}_w\left(g\right)$.

Let ${\mathcal{D}}_F^{\left(k\right)}\left(g\right),(k=1,2,\cdots 2q+2)$ denote the differences between ${\mathcal{F}}_w\left(g\right)$ and the total weight sum of all edges of subunit ${\mathcal{N}}_w^{\left(k\right)}\left(g\right)$. Analogous to the derivation process of ${\mathcal{R}}_{sum}\left(g\right)$, for $g=1$, we can get the weight of each edge, ${\mathcal{F}}_{AC}\left(1\right)={\mathcal{F}}_{BC}\left(1\right)=2(q+1)$, and the weight of the remaining edges is $q+1$; therefore, it is easy to calculate ${\mathcal{F}}_w\left(1\right)=4(q+1)(q+2)$, and for $g\ge 1$, ${\mathcal{F}}_w(g+1)$ satisfies

$$\begin{array}{c}\hfill {\mathcal{F}}_w(g+1)=(2q+2){\mathcal{F}}_w\left(g\right)+2q{\mathcal{D}}_F^{\left(1\right)}\left(g\right)+2{\mathcal{D}}_F^{(2q+1)}\left(g\right),\end{array}$$

(16)

where

$$\begin{array}{c}\hfill {\mathcal{D}}_F^{\left(1\right)}\left(g\right)=2q(q+2){(q+1)}^{2g-1},\end{array}$$

(17)

$$\begin{array}{c}\hfill {\mathcal{D}}_F^{(2q+1)}\left(g\right)=2(q+2){(q+1)}^{2g}.\end{array}$$

(18)

Replacing ${\mathcal{D}}_F^{\left(1\right)}\left(g\right)$ and ${\mathcal{D}}_F^{(2q+1)}\left(g\right)$ from Equations (17) and (18), respectively, in Equation (16), we get

$$\begin{array}{c}\hfill {\mathcal{F}}_w\left(g\right)={\displaystyle \frac{4(q+2)(q^2+q+1){(q+1)}^{2g}}{(q-1){(q+1)}^2}}-{\displaystyle \frac{2{(q+2)}^2{(2q+2)}^g}{q^2-1}}.\end{array}$$

(19)

3.3. GMFPT for Degree-Dependent Biased Random Walk Strategy 1 on the FSFTs

Replacing ${\mathcal{R}}_{sum}\left(g\right)$ and ${\mathcal{F}}_w\left(g\right)$ from Equations (28) and (19) in Equation (7), we obtain the exact result for the GMFPT, referred to as ${\mathcal{T}}_{global}^1\left(g\right)$.

The results show, for large networks, i.e., $g\to \infty$,

$$\begin{array}{ccc}\hfill {\mathcal{T}}_{global}^1\left(g\right)& \approx & 2^g{(q+1)}^{2g}\times {\displaystyle \frac{\left(q+2\right)\left(q^2+q+1\right)\left(6q^2+6q+1\right)}{2\left(q-1\right){\left(q+1\right)}^3\left(2q+1\right)\left(4q+3\right)}}\hfill \\ & -& 2^{2g}{(q+1)}^g\times {\displaystyle \frac{{(q+2)}^2(6q^2+6q+1)}{4(q-1)(2q+1)(4q+3){(q+1)}^2}}\hfill \\ & +& {(q+1)}^{2g}\times {\displaystyle \frac{\left(q+2\right)\left(2q+1\right)\left(q^2+q+1\right)\left(4q^2+7q+1\right)}{2\left(q-1\right){\left(q+1\right)}^4\left(2q^2+4q+1\right)}}\hfill \\ & -& {2^g(q+1)}^g\times {\displaystyle \frac{{\left(q+2\right)}^2\left(2q+1\right)\left(4q^2+7q+1\right)}{4\left(q-1\right){\left(q+1\right)}^3\left(2q^2+4q+1\right)}}\hfill \\ & +& {(q+1)}^g\times {\displaystyle \frac{q(q+2)(q+3)(q^2+q+1)(8q^2+11q+4)}{(q-1)(2q+1)(4q+3){(q+1)}^4}}\hfill \\ & -& 2^g\times {\displaystyle \frac{q(q+3){(q+2)}^2(8q^2+11q+4)}{2(q-1)(2q+1)(4q+3){(q+1)}^3}},\hfill \end{array}$$

(20)

Therefore, as $g\to \infty$,

$$\begin{array}{c}\hfill {\mathcal{T}}_{global}^1\left(g\right)\sim 2^g{(q+1)}^{2g}.\end{array}$$

(21)

Note that ${\mathcal{V}}_g={(2q+2)}^g+1$. We can rewrite Equation (21) as

$$\begin{array}{c}\hfill {\mathcal{T}}_{global}^1\left(g\right)\sim {{\mathcal{V}}_g}^{1+{\displaystyle \frac{ln(q+1)}{ln(2q+2)}}}.\end{array}$$

(22)

with the exponent greater than 1 and related to the factor of the node degree $(q+1)$. Thus, for strategy 1, the GMFPT grows super-linearly with the number of network nodes.

4. GMFPT for Degree-Dependent Biased Random Walk Strategy 2 on the FSFTs

Let ${\mathcal{T}}_{global}^2\left(g\right)$ represent the GMFPT for degree-dependent biased random walk strategy 2 on the FSFTs with generation g. As discussed in Section 2.3, the degree-dependent biased random walk strategy 2 is equivalent to a biased random walk on the corresponding weighted network, where each edge $(x,y)$ is assigned a weight $w_{xy}={\left(d_x{d}_y\right)}^{-1}$. In this section, we will calculate ${\mathcal{T}}_{global}^2\left(g\right)$ by using Equation (7).

Let ${\mathcal{N}}_{w2}\left(g\right)$ denote the weighted FSFTs of generation g, where each edge $(x,y)$ is assigned a weight $w_{xy}={\left(d_x{d}_y\right)}^{-1}$, and let ${\overline{\mathcal{R}}}_{sum}\left(g\right)$ and $\overline{\mathcal{F}}\left(g\right)$ represent ${\mathcal{R}}_{sum}$, $\mathcal{F}$ for the weighted FSTs, ${\mathcal{N}}_{w2}\left(g\right)$. Similar to the derivation of ${\mathcal{T}}_{global}^1\left(g\right)$, we can calculate ${\overline{\mathcal{R}}}_{sum}\left(g\right)$ and $\overline{\mathcal{F}}\left(g\right)$ for the weighted FSFTs ${\mathcal{N}}_{w2}\left(g\right)$. Then, ${\mathcal{T}}_{global}^2\left(g\right)$ can be obtained from Equation (7).

4.1. Exact Result of ${\overline{\mathcal{R}}}_{sum}\left(g\right)$ for the Weighted FSFTs ${\mathcal{N}}_{w2}\left(g\right)$

Here, we analyze the weighted FSFTs where each edge $(x,y)$ is assigned a weight $w_{xy}={\left(d_x{d}_y\right)}^{-1}$, and we calculate ${\overline{\mathcal{R}}}_{sum}\left(g\right)$, which totals the effective resistance between any pair of nodes within the equivalent resistance network, where each edge $(x,y)$ is assigned a resistance $r_{xy}={\displaystyle \frac{1}{w_{xy}}}$ and defined in Equation (9).

Similar to weighted FSFTs ${\mathcal{N}}_w(g+1)$, the weighted FSFTs ${\mathcal{N}}_{w2}(g+1)$ also consists of $2q+2$ subunits, labeled by ${\mathcal{N}}_{w2}^{\left(1\right)}\left(g\right)$, ${\mathcal{N}}_{w2}^{\left(2\right)}\left(g\right)$, …, ${\mathcal{N}}_{w2}^{(2q+2)}\left(g\right)$, which have the same structure as ${\mathcal{N}}_{w2}\left(g\right)$. For any $k=1,2,\cdots ,2q+2$, let ${\overline{\mathcal{R}}}_{sum}^{\left(k\right)}\left(g\right)$ denote the total of effective resistances for all node pairs of ${\mathcal{N}}_{w2}^{\left(k\right)}\left(g\right)$ and ${\overline{\mathcal{D}}}_R^{\left(k\right)}\left(g\right)$ represent the difference between ${\overline{\mathcal{R}}}_{sum}^{\left(k\right)}\left(g\right)$ and ${\overline{\mathcal{R}}}_{sum}\left(g\right)$.

Analogous to the derivation process of ${\mathcal{R}}_{sum}\left(g\right)$ for weighted FSFTs ${\mathcal{N}}_w\left(g\right)$, we find that ${\overline{\mathcal{R}}}_{sum}\left(1\right)=8{(q+1)}^3$, and for any $g\ge 1$, ${\overline{\mathcal{R}}}_{sum}(g+1)$ satisfies

$$\begin{array}{c}\hfill {\overline{\mathcal{R}}}_{sum}(g+1)=(2q+2){\overline{\mathcal{R}}}_{sum}\left(g\right)+2q{\overline{\mathcal{D}}}_R^{\left(1\right)}\left(g\right)+2{\overline{\mathcal{D}}}_R^{(2q+1)}\left(g\right)+{\overline{\mathcal{L}}}_g.\end{array}$$

(23)

where

$${\overline{\mathcal{L}}}_g={\displaystyle \frac{1}{2}}\sum _{k_1\ne k_2}\sum _{{\displaystyle \genfrac{}{}{0pt}{}{x\in {\mathcal{N}}_w^{k_1}\hfill }{y\in {\mathcal{N}}_w^{k_2}\hfill }}}{\overline{\mathcal{R}}}_{xy},$$

and

$$\begin{array}{ccc}\hfill {\overline{\mathcal{D}}}_R^{\left(1\right)}\left(g\right)& =& {\displaystyle \frac{2q(6q^3+7q^2+3q){(2q+2)}^{2g}{(q+1)}^g}{8q^3+18q^2+13q+3}}\hfill \\ & -& {\displaystyle \frac{(q^2-2q){(2q+2)}^g{(q+1)}^{2g}}{q+1}}-{\displaystyle \frac{2q^2{(2q+2)}^g{(q+1)}^g}{2q^2+3q+1}}\hfill \\ & +& {\displaystyle \frac{2q(q^2+3q){(q+1)}^{2g}}{(4q+3)(q+1)}},\hfill \end{array}$$

(24)

$$\begin{array}{ccc}\hfill {\overline{\mathcal{D}}}_R^{(2q+1)}\left(g\right)& =& {\displaystyle \frac{2(6q^3+7q^2+3q){(2q+2)}^{2g}{(q+1)}^{g+1}}{8q^3+18q^2+13q+3}}\hfill \\ & -& {\displaystyle \frac{q(q-1){(2q+2)}^g{(q+1)}^{2g}}{q+1}}-{\displaystyle \frac{2q{(2q+2)}^g{(q+1)}^{g+1}}{2q^2+3q+1}}\hfill \\ & +& {\displaystyle \frac{2(q^2+3q){(q+1)}^{2g+1}}{(4q+3)(q+1)}}.\hfill \end{array}$$

(25)

Using Equation (23) recursively, we can obtain

$$\begin{array}{ccc}\hfill {\overline{\mathcal{R}}}_{sum}\left(g\right)& =& {(2q+2)}^{g-1}{\overline{\mathcal{R}}}_{sum}\left(1\right)+2q{\sum }_{k=1}^{g-1}{(2q+2)}^{g-1-k}{\overline{\mathcal{D}}}_R^{\left(1\right)}\left(k\right)\hfill \\ & +& 2{\sum }_{k=1}^{g-1}{(2q+2)}^{g-1-k}{\overline{\mathcal{D}}}_R^{\left(0\right)}\left(k\right)+{\sum }_{k=1}^{g-1}{(2q+2)}^{g-1-k}{\overline{\mathcal{L}}}_k.\hfill \end{array}$$

(26)

Similarly, for any integer $k\ge 1$, we find

$$\begin{array}{ccc}\hfill {\overline{\mathcal{L}}}_k& =& {\displaystyle \frac{4q(6q^4+19q^3+27q^2+19q+7){(2q+2)}^{2k}{(q+1)}^k}{(q-1)(2q+1)}}\hfill \\ & -& {\displaystyle \frac{4(6q^2+6q+1)(q+1)2^k{(2q+2)}^{2k}}{q-1}}\hfill \\ & +& {\displaystyle \frac{(12q^3+14q^2-4q-4){(2q+2)}^{2k}}{q-1}}\hfill \\ & -& {\displaystyle \frac{2q(2q+1)(q^2+q+1){(2q+2)}^k{(q+1)}^{2k}}{q-1}}\hfill \\ & -& {\displaystyle \frac{4(q^2+q+1){(2q+2)}^k{(q+1)}^k}{2q+1}}.\hfill \end{array}$$

(27)

Replacing ${\overline{\mathcal{D}}}_R^{\left(1\right)}\left(g\right)$, ${\overline{\mathcal{D}}}_R^{(2q+1)}\left(g\right)$, and ${\overline{\mathcal{L}}}_k$ from Equations (24), (25), and (27), respectively, in Equation (23), we get

$$\begin{array}{ccc}\hfill {\overline{\mathcal{R}}}_{sum}\left(g\right)& =& {\displaystyle \frac{\left[2q(2q^2+3q+2)(6q^3+19q^2+18q+9)\right]{(2q+2)}^{2g}{(q+1)}^g}{(q-1){(q+1)}^2(4q+3)(2q^2+4q+1)}}\hfill \\ & +& {\displaystyle \frac{2(6q^2+6q+1)2^g{(2q+2)}^{2g}}{3+q-4q^2}}+{\displaystyle \frac{(3q^2+2q-2){(2q+2)}^{2g}}{q^2-1}}\hfill \\ & -& {\displaystyle \frac{2[1+q^2(q^3+3q^2+2q+2)]{(2q+2)}^g{(q+1)}^{2g}}{q(q-1)(q+2){(q+1)}^2}}\hfill \\ & -& {\displaystyle \frac{(2q^2+2q+2){(2q+2)}^g{(q+1)}^g}{q{(q+1)}^2}}\hfill \\ & -& {\displaystyle \frac{2(q+1)(4q^5+16q^4+27q^3+11q^2+2q+3){(2q+2)}^g}{q(q-1)(q+2)(4q+3)(2q^2+4q+1)}}\hfill \\ & +& {\displaystyle \frac{4q(q^2+q+1)(q+3){(q+1)}^{2g}}{(4q+3)(q-1){(q+1)}^2}}.\hfill \end{array}$$

(28)

4.2. Derivation of $\overline{\mathcal{F}}\left(g\right)$ for the Weighted FSFTs ${\mathcal{N}}_{w2}\left(g\right)$

Here, we calculate $\overline{\mathcal{F}}\left(g\right)$, which totals the weight for all edges of ${\mathcal{N}}_{w2}\left(g\right)$. Let ${\overline{\mathcal{D}}}_F^{\left(k\right)}\left(g\right),(k=1,2,\cdots 2q+2)$ denote the differences between $\overline{\mathcal{F}}\left(g\right)$ and the total weight sum of all edges of subunit ${\mathcal{N}}_{w2}^{\left(k\right)}\left(g\right)$. Analogous to the derivation process of ${\overline{\mathcal{R}}}_{sum}\left(g\right)$, we can get $\overline{\mathcal{F}}\left(1\right)=4-{\displaystyle \frac{2}{q+1}}$, and for $g\ge 1$, $\overline{\mathcal{F}}(g+1)$ satisfies

$$\begin{array}{c}\hfill \overline{\mathcal{F}}(g+1)=(2q+2)\overline{\mathcal{F}}\left(g\right)+2q{\overline{\mathcal{D}}}_F^{\left(1\right)}\left(g\right)+2{\overline{\mathcal{D}}}_F^{(2q+1)}\left(g\right),\end{array}$$

(29)

where

$$\begin{array}{c}\hfill {\overline{\mathcal{D}}}_F^{\left(1\right)}\left(g\right)=-{\displaystyle \frac{q(2q+1){(q+1)}^{g-1}}{{(q+1)}^{g+1}}},\end{array}$$

(30)

$$\begin{array}{c}\hfill {\overline{\mathcal{D}}}_F^{(2q+1)}\left(g\right)=-{\displaystyle \frac{(2q+1)(3q+1){(q+1)}^{g-1}}{2{(q+1)}^{g+1}}}.\end{array}$$

(31)

Replacing ${\mathcal{D}}_F^{\left(1\right)}\left(g\right)$ and ${\mathcal{D}}_F^{(2q+1)}\left(g\right)$ from Equations (30) and (31), respectively, in Equation (29), we get

$$\begin{array}{c}\hfill \overline{\mathcal{F}}\left(g\right)={\displaystyle \frac{(2q+1){(2q+2)}^g+(2q+1)(2q+2)}{2{(q+1)}^2}}.\end{array}$$

(32)

4.3. GMFPT for Degree-Dependent Biased Random Walk Strategy 2 on the FSFTs

Replacing ${\mathcal{R}}_{sum}\left(g\right)$ and ${\mathcal{F}}_g$ from Equations (28) and (32) in Equation (7), we can get the exact result for the GMFPT, ${\mathcal{T}}_{global}^2\left(g\right)$.

We also find, for large networks, i.e., $g\to \infty$,

$$\begin{array}{ccc}\hfill {\mathcal{T}}_{global}^2\left(g\right)& \approx & 2^g{(q+1)}^{2g}\times [{\displaystyle \frac{q(2q^2+3q+2)(6q^3+19q^2+18q+9)}{(q-1)(4q+3)(2q^2+4q+1){(q+1)}^4}}\hfill \\ & +& {\displaystyle \frac{2q^2(2q^2+3q+2)(6q^3+19q^2+18q+9)}{(q-1)(4q+3)(2q^2+4q+1){(q+1)}^4}}]\hfill \\ & -& 2^{2g}{(q+1)}^g\times {\displaystyle \frac{(6q^2+6q+1)(2q+1)}{(q-1)(4q+3){(q+1)}^2}}\hfill \\ & +& {(2q+2)}^g\times {\displaystyle \frac{(3q^2+2q-2)(2q+1)}{2(q-1){(q+1)}^3}}\hfill \\ & -& {(q+1)}^{2g}\times {\displaystyle \frac{(2q+1)(q^5+3q^4+2q^3+2q^2+1)}{q(q-1)(q+2){(q+1)}^4}}.\hfill \end{array}$$

(33)

Therefore, as $g\to \infty$,

$$\begin{array}{c}\hfill {\mathcal{T}}_{global}^2\left(g\right)\sim 2^g{(q+1)}^{2g}\sim {{\mathcal{V}}_g}^{1+{\displaystyle \frac{ln(q+1)}{ln(2q+2)}}},\end{array}$$

(34)

with the exponent greater than 1 and related to the factor of the node degree $(q+1)$. Thus, for strategy 2, the GMFPT grows super-linearly with the number of network nodes.

5. Comparison

In this section, we compare the GMFPT for the two degree-dependent random walk strategies on the FSFTs with the GMFPT for unbiased random walk strategy on FSFTs, as found in Ref. [11]. Recalling that, for unbiased random walk on FSFTs, the GMFPT

$${\mathcal{T}}_{global}\left(g\right)\sim {{\mathcal{V}}_g}^{1+{\displaystyle \frac{ln2}{ln(2q+2)}}}.$$

(35)

By comparing the exact results of GMFPTs for the three random walk strategies on the FSFTs, in the case $q>1$, we find

$${\mathcal{T}}_{global}\left(g\right)<{\mathcal{T}}_{global}^1\left(g\right)<{\mathcal{T}}_{global}^2\left(g\right).$$

Figure 5 shows the plots of the GMFPTs for the three random walk strategies on the FSFTs versus the network size ${\mathcal{V}}_g$ on the FSFTs with $q=2,4$. Thus, for random walks on FSFTs, if the walker is more inclined to move to neighbors with a higher (resp. lower) degree at each step, the global transport (resp. search) efficiency will decrease, and the strategy where the walker is more inclined to move to neighbors with a higher degree is more efficient than the strategy where the walker is more inclined to move to neighbors with a lower degree.

Figure 5. The GMFPTs for three different random walk strategies are drawn as functions of the network size on the FSFTs with (a) $q=2$, (b) $q=4$.

Calculating $Rati{o}^1={\displaystyle \frac{{\mathcal{T}}_{global}^1\left(g\right)}{{\mathcal{T}}_{global}\left(g\right)}}$, $Rati{o}^2={\displaystyle \frac{{\mathcal{T}}_{global}^2\left(g\right)}{{\mathcal{T}}_{global}\left(g\right)}}$, and $Rati{o}^3={\displaystyle \frac{{\mathcal{T}}_{global}^2\left(g\right)}{{\mathcal{T}}_{global}^1\left(g\right)}}$, we find, for large networks, i.e., $g\to \infty$,

$$\begin{array}{c}\hfill Rati{o}^1\sim Rati{o}^2\sim {{\mathcal{V}}_g}^{{\displaystyle \frac{ln(q+1)-ln2}{ln(2q+2)}}}\to \infty ,\end{array}$$

(36)

which shows that both $Rati{o}^1$ and $Rati{o}^2$ increase monotonically as the network size, ${\mathcal{V}}_g$, increases. Thus, in terms of global transport (resp. search) efficiency on FSFTs, the two degree-dependent biased strategies are much lower than the classical unbiased strategy. Figure 6 shows the semi-log plots for $Rati{o}^1$ and $Rati{o}^2$ v.s. ${\mathcal{V}}_g$ on the FSFTs with different choices of q.

Figure 6. The semilog plots for $Rati{o}^1$ and $Rati{o}^2$ v.s. ${\mathcal{V}}_g$ on the FSFTs with different choices of q. (a) $q=2$, (b) $q=4$.

We also find that, for larger networks,

$$\begin{array}{c}\hfill Rati{o}^3\approx f\left(q\right),\end{array}$$

(37)

where $f\left(q\right)={\displaystyle \frac{2q{(2q+1)}^2(2q^2+3q+2)(6q^3+19q^2+18q+9)}{(q+1)(q+2)(q^2+q+1)(6q^2+6q+1)(2q^2+4q+1)}}$ is a constant that is independent of network size, ${\mathcal{V}}_g$ (see Figure 7a). Note that $f\left(2\right)\approx 5.17$, and for any $q>1$, $f^{\prime }\left(q\right)>0$. Therefore, $f\left(q\right)$ increases monotonically as the q increases, and $f\left(q\right)\ge f\left(2\right)>5$ for any $q\ge 2$ (see Figure 7b), which shows ${\mathcal{T}}_{global}^2\left(g\right)$ is at least 5 times ${\mathcal{T}}_{global}^1\left(g\right)$, and strategy 2 is less efficient than strategy 1.

Figure 7. (a) Plot of $Rati{o}^3$ v.s. ${\mathcal{V}}_g$ on the FSFTs with different choices of q in $q=2,4,6$. (b) $Rati{o}^3$ is drawn as a function of $q(2\le q\le 20)$ on FSFTs with generation $g=50$.

6. Conclusions

In this paper, we have introduced two degree-dependent biased random walk strategies on FSFTs. The GMFPT for the two degree-dependent biased random walk strategies are evaluated analytically by exploring the connection between the fist-passage time for the degree-dependent biased random walk strategy and the biased random walk strategy on the weighted network. We obtained the exact results for the GMFPT of two degree-dependent biased random walk strategies on FSFTs and compared the GMFPTs for the degree-dependent biased random walk strategies and the GMFPT for the classical unbiased random walk strategy on FSFTs. The results show that the GMFPT for the two degree-dependent biased random walk strategies grows super-linearly with the network size and is far larger than the GMFPT for the classical unbiased random walk strategy, and the GMFPT for biased strategy 2, where the walker is biased to the node with the lower degree at each step, is at least 5 times the GMFPT for biased strategy 1, where the walker is biased to the node with the higher degree at each step. Thus, strategy 2 is less efficient than strategy 1. Our work not only presents a way to evaluate the GMFPT for degree-dependent biased random walk strategies on general networks but also offers valuable insights to enrich the controlling of chemical kinetics, network transport, random search, and pathway elaboration.