Firstly, let’s delve into the overarching network architecture of the T-fractal, which is categorically known as tree-like fractals [26]. These fractals are constructed through an iterative process and possess a fundamental positive integer parameter, m, which plays a pivotal role in their formation. Let $F_t$ signify the tree-like fractals at a particular generation t (where t is a non-negative integer).

The methodology employed in their creation commences with a rudimentary starting point involving two nodes that are linked by a solitary edge, which is denoted as $F_0$. This foundational configuration serves as the building block for more complex iterations that follow. For t values that are greater than or equal to 1, the fractal $F_t$ is pieced together by inserting a new vertex at the midpoint of each existing edge within $F_{t-1}$ . This innovative vertex is then seamlessly connected to both terminal points of the original initial edge. The process does not end there, as m additional edges are appended to the nodes that were recently introduced into the structure.

For visual clarity, Figure 1 provides a graphical representation of the initial generation network’s structure when various values of m are chosen. Figure 2 further elucidates the composition of each generation within the tree-like fractals when m is set to 1, a scenario also referred to as the T-fractal. In this figure, the nodes that are generated at disparate generations are depicted using a array of colors, which serves as a helpful tool for comprehending the composite nature of the T-fractal. In this vibrant depiction, the color black is utilized to designate nodes that are born in the zero generation, while the color red is employed to mark the nodes from the first generation. The hue blue is assigned to represent the nodes from the second generation, followed by green for the third generation, and so forth, thereby providing a vivid and clear illustration of the fractal’s generational growth.

It is worth mentioning that there is another way to construct the tree-like fractals. If we look at it from the first generation, the central node O of the first generation is regarded as the innermost node, and the node farthest from the center is regarded as the outermost node. Then can be regarded as m+2 copies of , connecting the outermost vertex and the innermost vertex, as shown in Figure 3.

According to the tree-like fractals construction method, the number of edges in each generation of the network is $m+2$ times that of the previous generation. Therefore, we can easily conclude that the number of edges in the t-th generation of the tree-like fractals is $E_t={(m+2)}^t$. According to the fact that the number of vertices in the tree structure is 1 more than the number of edges, we can conclude that the number of vertices in the t-th generation of the network is $N_t={(m+2)}^t+1$ .

In addition, after each iteration, the diameter of the fractals (i.e., the distance from the innermost node to the outermost node) doubles. Since the number of network edges increases by $m+2$ times, we can conclude that the tree dimension is $d_f={\displaystyle \frac{ln(m+2)}{ln2}}$. In addition, one found that [27] in the spectral for any two nodes i and j in the current generation, the mean first passage time (MFPT) from node i to node j will increase by $2(m+2)$ times in the next generation. Therefore, the dimension of the tree walk is and the spectral dimension is

In this subsection, we study the properties of the first passage time (FPT) from A to B (see Figure 2). In order to calculate the generating function of FPT, we set A as the starting point and B as the absorption domain, assuming that the FPT from A to B is a random variable $T_{A\to B}\left(t\right)$, and let $P(t,n)$ be the first passage probabilities (FPP). Suppose the generating function of $T_{A\to B}\left(t\right)$ is ${\Phi }_T^{A\to B}(t,z)$. Therefore, according to the definition of the generating function (see Appendix A), we can get the probability generating function of $T_{A\to B}\left(t\right)$ as: Where t represents the generation of the T-fractal.

Now, we denote $\Omega$ the set of node in generation 1, which includes nodes $A,B,C,O$. For any path $\pi$ starting at node A and reaches B, we use $v_i$ denote the node in $F_t$ that reaches at time i. In that way, we can denote the way as: $\pi =(v_0=\mathrm{A},v_1,\cdots ,v_{T_{A\to B}}\left(t\right)=\mathrm{B})$. Also, we introduce the observable ${\tau }_i={\tau }_i\left(\pi \right)$ to represent the time taken to reach for the i-th time any node in $\Omega$ along with the path $\pi$. The time can be defined as the following way: And we call $N=min\{i:v_{{\tau }_i}=\mathrm{B}\}$. In fact, N stands for the first passage time in the random walk on the generation 1 (or set $\Omega$). So if we only consider the random walk in the set $\Omega$, the path $\pi$ can be simplified as: The simplified path $\sigma \left(\pi \right)$ only includes nodes that generated in generation 1, and of course, the interval time between the two steps in $\sigma \left(\pi \right)$ is stochastic. So we can denote the random variable ${\eta }_i$ the interval time between $v_{{\tau }_{i-1}}$ and $v_{{\tau }_i}$, as to say, According to the second T-fractal construction method, $F_t$ can be regarded as composed of three $F_{t-1}$, and there are two points in A, B, C, O as the endpoints of $F_{t-1}$. Therefore, we can infer that ${\eta }_i$ has the same distribution, and they have the same distribution as $T_{A\to B}(t-1)$, and the generating function is ${\Phi }_T^{A\to B}(t-1,z)$. Moreover, N is the FPT wandering in the first generation, and its generating function is ${\Phi }_T^{A\to B}(1,z)$. According to the properties of the generating function (see Appendix A) and Equation (5), we can deduce that the generating function of $T_{A\to B}\left(t\right)$ satisfies: According to the second T-fractal construction method, $F_t$ can be regarded as composed of three $F_{t-1}$, and there are two points in A, B, C, O as the endpoints of $F_{t-1}$. Therefore, we can infer that ${\eta }_i$ has the same distribution, and they have the same distribution as $T_{A\to B}(t-1)$, and the generating function is ${\Phi }_T^{A\to B}(t-1,z)$. Moreover, N is the FPT wandering in the first generation, and its generating function is ${\Phi }_T^{A\to B}(1,z)$. According to the properties of the generating function (see Appendix A) and Equation (5), we can deduce that the generating function of $T_{A\to B}\left(t\right)$ satisfies:As for the initial condition ${\Phi }_T^{A\to B}(1,z)$, it can be derived by transition probability matrix for random walks on T-fractal in generation 1, see exactly in Appendix B, the result is:By solving the Equation (6) with initial condition Equation (7), we get:Now we turn to calculate the mean first passage time (MFPT) of the random walk Starting from node A and absorbing at node B, we denote it as $\mathbb{E}\left(T_{A\to B}\left(t\right)\right)$, which is the mathematical expectation for the random variable $T_{A\to B}\left(t\right)$. We can derive it through taking derivatives on both sides of the Equation (6), and posing $z=1$ (see exactly in Appendix C), we get:

We now use the similar method to calculate the mean first passage time from node D to E. We suppose the random variable $T_{D\to E}\left(t\right)$ denotes the first passage time from node D to E in generation t ($t\ge 2$), where the starting point is D, and the absorbing domain is E. Let ${\Omega }_2$ denotes the node in generation 2. Like the method to calculate MFPT from A to B, we can only consider the random walk in generation 2, and we use $v_i$ to denote the interval time between the random walk in ${\Omega }_2$. So the $T_{D\to E}\left(t\right)$ can also be written as:We suppose the generating function of the random variable $T_{D\to E}\left(t\right)$ is ${\Phi }_T^{D\to E}(t,z)$. So in this situation, the generating function of random variable N is ${\Phi }_T^{D\to E}(2,z)$, since N stands for the random walk from node D to E in generation 2. Now, we analyze the generating function of the random variable $v_i$. It is obvious that $v_i(i=1,2,\cdots )$ has the identical distribution. For each random walk in generation t, the interval time between the nodes in generation 2 can be seen as the random walk from A to B in generation $t-2$. Because no matter what two nodes in generation 2 are, they have as only as two nodes on the edge of the random walk. So we can conclude that the generating function of $v_i(i=1,2,\cdots )$ are ${\Phi }_T^{A\to B}(t-2,z)$. So by using the Equation (10) and the properties of generating function (See Appendix A, Equation (A7)), we can conclude that:As for the initial condition ${\Phi }_T^{D\to E}(2,z)$, it can also be calculated by using the generating function of the matrix, also see in Appendix B. The result is:Similar to the above method, we take derivation to both sides of the Equation (11), see exactly in Appendix C, we can get the MFPT of the random walk from D to E:

Similar to the analysis from the above two situations, we can also use the generating function to calculate the MFPT from node O to B, where O as the innermost node is the starting point of the random walk and B is the absorbing domain. We suppose the random variable $T_{O\to B}\left(t\right)$ denote the FPT from O to B in generation t. And the generating function of $T_{O\to B}\left(t\right)$ denote as ${\Phi }_T^{O\to B}(t,z)$, similar to the above method, we have:Where ${\eta }_i$ is the interval time between random walks in generation 1. The generating function of random variable N is ${\Phi }_T^{O\to B}(1,z)$, and the generating function of ${\eta }_i$ is ${\Phi }_T^{A\to B}(t-1,z)$. So we can get the following recursion:The initial condition is (see exactly in Appendix B):By taking derivation to both sides of Equation (15), we get (see in Appendix C):

As a comparison of the random walk from O to B, we now discuss the random walk, starting at node O, and absorbing at node E (see in Figure 2). We denote $T_{O\to E}\left(t\right)$ as the FPT from O to E in generation t. And the generating function of it denotes as ${\Phi }_T^{O\to E}(t,z)$. With the same method as the random walk from O to B, we get the recursion:With the initial condition:By taking derivation to both sides of Equation (18), we also get:

In this section, we will discuss a new problem, about the random walk from any point to a given absorbing point. The starting point is selected based on the stationary distribution. Here we first introduce the stationary distribution.

In the context of a random walk, the stationary distribution (also known as the invariant distribution or equilibrium distribution) refers to a probability distribution over the states of the system that remains unchanged as time progresses, assuming the process reaches equilibrium.

For a random walk on a finite state space (such as a Markov chain), the stationary distribution $\Pi$ satisfies the following condition:Here, P is the transition matrix of the Markov chain, where each entry $P_{ij}$ represents the probability of transitioning from i to j.

In the random walk on network, we have the conclusion that:Where $d_k$ stands for the degree of the vertex k.

Then, we discuss the random walk to B with the starting site randomly selected according to the stationary distribution $\Pi$ in Equation (22). We denote the first passage time (FPT) of this random walk in T-fractal as $T_{\Pi \to B}\left(t\right)$ in generation t. And the ${\Phi }_T^{\Pi \to B}(t,z)$ denotes the generating function of $T_{\Pi \to B}\left(t\right)$. And we also introduce a new concept in random walk, the return time (RT). It is the time that the walker returns to the initial node. We denote the return time to B as $T_{B\to B}\left(t\right)$ in generation t of the T-fractal. And the generating function of the $T_{B\to B}\left(t\right)$ is ${\Phi }_T^{B\to B}(t,z)$. It has been found that there is a relationship between ${\Phi }_T^{\Pi \to B}(t,z)$ and ${\Phi }_T^{B\to B}(t,z)$ (See [30] exactly):Where $d_B$ stands for the degree of node B, and E is the total number of edges of the network. Replacing $d_B=1$ and $E\left(t\right)=3^t$ into Equation (23), we get:About the ${\Phi }_T^{B\to B}(t,z)$, we have the following equations (See Appendix ): with:With the initial condition ${\Phi }_T^{B\to B}(0,z)={\displaystyle \frac{1}{1-z^2}}$ and ${\Phi }_T^{B\to B}(1,z)={\displaystyle \frac{3-2z^2}{3(1-z^2)}}$ (see in Appendix B), we get:Combine with Equation (24) (25) (27), we get:With the initial condition:By calculating the derivatives to both sides of the Equation (29) and posing $z=1$, we get the MFPT of this random walk:

Random walk with stochastic resetting are characterized by restarts occurring at a random time. We suppose the resetting was happened on random time R. Let $T_R$ refers to the first passage time under resetting, and T refers to first passage time without resetting. One obtain [28,29]:Then, in this section, we will focus on the discrete-time random walk on network with resetting. We will investigate the relationship between the mean of $T_R$ and the generating function of T and R. Then, we will calculate the mean of $T_R$ by using the generating function. The method was presented by Ref. [29]. Let So we can rewrite Equation (18) as:Let $\mathbb{E}\left(\xi \right)$ denote the first moment (i.e., the mathematical expectation) of the random variable $\xi$ and $Pr\left(A\right)$ denote the probability the event A occurs. So we obtain:Note that the random variable $T_R$ and $T_R^{\prime }$ are independent and identically distributed. Of course, their first moments are equivalent. So, we can get:By the definition of the first moment and some properties, we can get:

Now we discuss the discrete random walk on network with stochastic resetting at a fixed rate. At each step, there is a fixed probability $\gamma$ to reset to the initial site, we call $\gamma$ the resetting rate. While, there are probability $1-\gamma$ for the walker to walk to the adjacent nodes that the walker currently occupied. So, the random variable R discuss above, which is the time that takes walker to restart to the initial site, follows the geometric distribution with the parameter $\gamma$, that is to say, for any $l\ge 1$:This can be explained as, if the walker resets at time l, which means that he doesn’t reset in the $(l-1)$ times before, with the probability ${(1-\gamma )}^{l-1}$. And he reset at time l, with the probability $\gamma$. Therefore, for any $m\ge 0$, Substituting Equation (26) in Equation (23) (24), we get:

Also, we get:Where ${\Phi }_T\left(z\right)$ is the generating function of the random variable T, can be written as:

Inserting the Equation (27) (28) into Equation (22), we get an important conclusion:

This formula can be used to calculate mean first passage time (MFPT) with stochastic resetting. Through this formula, we can transform the problem of solving MFPT with resetting into the problem of solving generating function without resetting. It should be pointed out, for any random variable R, if R follows the geometric distribution with parameter $\gamma$, the result shown above are also true.

In section II, we have found some recursions about the generating function of the random walk on T-fractal without resetting. And in this section, we have found a relationship between the MFPT of the random walk with stochastic resetting at a fixed resetting rate and the generating function without resetting. So in the next section, we will combine the two findings above and conclude the MFPT of the random walk on T-fractal with stochastic resetting.

We now discuss the random walk on T-fractal with resetting from O to B. The walker starts the random walk from node O, and the absorbing node is B. At each step, there is a probability $\gamma$ for the worker to reset to the initial site O, and there is probability $1-\gamma$ for the walker to walk through the adjacent node. By solving the Equation (15) with the initial condition (16), we can get the generating function for the first passage time from node O to B without resetting. The result is:Therefore, Let $\mathbb{E}\left(T_{O\to B}^R\left(t\right)\right)$ denote the first moment of the random walk from O to B with resetting in generation t. So, by the Equation (30), we have:

By taking the first-order derivative with respect to $\gamma$ on both sides of Equation (46), and setting ${\displaystyle \frac{\partial }{\partial \gamma }}\mathbb{E}\left(T_{O\to B}^R\left(t\right)\right)=0$ , we get:So we can get the root (the negative root was removed):

By plotting the ${\gamma }_{O\to B}^{*}$ as a function of t (see Figure 4), we find for at any generation t, the optimal solution of the Equation (47) ${\gamma }^{*}\in (0,1)$. We also find that from $t=20$, the ${\gamma }^{*}$ decreases, and from $t=27$, ${\gamma }^{*}$ increase again. And the minimum of the ${\gamma }^{*}$ is 0.33 when t=27. And when $t\to \infty$, ${\gamma }^{*}\to \sqrt{2}-1\approx 0.41$.

And Figure 5 shows that when $t=30$, the MFPT of the random walk from O to B with the resetting rate $\gamma$. We find the optimal $\gamma$ is about 0.40 where the MFPT becomes the smallest. By performing numerical calculations, we find the ratio of MFPT in random walks with and without resetting when $t\to \infty$:This shows that when the network size is very large, resetting can improve the efficiency of random search from the central O to the outer most side .

We now discuss the random walk with resetting from O to E. At each step, there is probability $\gamma$ for the walker to reset to the initial position O. And there is probability $1-\gamma$ for the walker to walk to the adjacent node from the current position. By solving the Equation (18) with the initial condition (19), we get the generating function of $T_{O\to E}\left(t\right)$ without resetting:Therefore, So, the MFPT with resetting can be calculated as:

By taking the derivative to Equation (51), and letting ${\displaystyle \frac{d}{d\gamma }}\mathbb{E}\left(T_{O\to E}^R\left(t\right)\right)=0$, we get:By solving this Equation, we can get the optimal ${\gamma }^{*}$. The plot of ${\gamma }^{*}$ vs t see Figure 6. And Figure 7 shows that when $t=10$, the MFPT vs $\gamma$ on the interval $[0,1)$.

And by taking the numerical calculations, we find when $t\to \infty$, the ratio of the MFPT between the random walk with resetting with the optimal resetting rate and the random walk without resetting:So, in this situation, resetting can also improve the efficiency of random search.

In this section, we discuss the random walk from a random starting node to the fixed node B. In each step, there is probability $\gamma$ for the walker to reset to the initial position. The starting node is selected according to the stationary distribution, as mentioned in Section III. By solving Equation (28) with the initial condition (29), we get:So, we get:Therefore, the MFPT under resetting is:

By taking the first-order derivative of both sides of Equation (57), and setting ${\displaystyle \frac{d}{d\gamma }}\mathbb{E}\left(T_{\Pi \to B}^R\left(t\right)\right)=0$, we get:By solving Equation (58), we get the expression for the optimal ${\gamma }_{\Pi \to B}^{*}$. And Figure 8 plots the optimal ${\gamma }_{\Pi \to B}^{*}$ as a function of generation t. We found that from generation 4, the optimal ${\gamma }_{\Pi \to B}^{*}$ decreases monotonically with the increase of the generation t. And when $t\to \infty$, ${\gamma }_{\Pi \to B}^{*}\to 0$.

By calculating the ratio of MFPT at the optimal resetting rate and the MFPT without resetting, in the T-fractal with a sufficiently large size, we find:So, we can conclude that by taking resetting to a randomly drawn from the stationary distribution is a good strategy for the random research. The conclusion in this part provides insights into optimizing research in T-fractal and other large networks.

Let $\xi$ be a discrete random variable which takes only non-negative integer values. Suppose the probability distribution of the random variable $\xi$ is $p_n$, which means that:And the probability generating function of $\xi$ is defined as:

As shown in the above formula, the generating function is determined by the probability distribution. Therefore, we can infer the probability distribution based on the Taylor expansion of the generating function at $z=0$:

Also, we can derive the mathematical expectation of the random variable $\xi$, by derivatives of ${\Phi }_{\xi }\left(z\right)$ calculated at $z=1$:

Finally, we list two important properties of the generating function, which are helpful for the calculation of the generating function of the random walk.

If ${\xi }_1$ and ${\xi }_2$ are two random variables whose generating functions are ${\Phi }_{{\xi }_1}\left(z\right)$ and ${\Phi }_{{\xi }_2}\left(z\right)$ respectively, then the generating function of ${\xi }_1+{\xi }_2$ is:

Let N, ${\xi }_1$, ${\xi }_2$, ⋯ be independent random variables, and ${\xi }_i(i=1,2,\cdots ,N)$ are identically distributed, the generating function of ${\xi }_i$ are all ${\Phi }_{\xi }\left(z\right)$, and the generating function of N is ${\Phi }_N\left(z\right)$, the random variable $S_N$ defined as:

Then the generating function of $S_N$ is:

For a small network, the probability generating function can be calculated directly by using the symbolic toolbox of MATLAB. Here are the general methods to calculate the initial condition.

First, let $\mathbf{M}=\left(P_{ij}\right)$ be the transition matrix of the random walk on the network, where

Then, the probability generating function of the matrix $\mathbf{M}$ can be obtained by :

Where $\mathbf{I}$ is the identity matrix, and $\Pi \left(z\right)=\left({\Phi }_{ij}\left(z\right)\right)$, where ${\Phi }_{ij}\left(z\right)$ represents the random walk starting at node i and ends at node j. And ${\Phi }_{ii}\left(z\right)$ refers to the return time to node i. If j is in the absorbing domain, then ${\Phi }_{ij}\left(z\right)$ can be referred to as the generating function of the first passage time from i to j. And ${\Phi }_{ii}\left(z\right)$ refers to the first return time to node i.

In the T-fractal network, in generation 1, we now study the generating function of the first passage time from node A to B, denoted as ${\Phi }_T^{A\to B}(1,z)$. First, the transition matrix can be written as (where node B is the absorbing node):

Where the nodes A, B, C, O can be denoted by the numbers 1, 2, 3, 4. Then we use MATLAB to calculate $\Pi \left(z\right)$ and the generating function of the random walk from A to B is:

Also, we can calculate the generating function from O to B:

In order to calculate the return time to node B without a trap, we can rewrite the matrix as the situation without a trap:

Also, by calculating the $\Pi \left(z\right)$ in Equation (A9), we get:By the definition of the generating function in Equation (A2), it is obvious that:

For the following calculation, we define:

As for generation 2, we can also use the same method to calculate ${\Phi }_T^{D\to E}(2,z)$. First, we write the transition matrix of generation 2 where E is the absorbing domain:So, we can derive the initial condition as: and

Let’s first introduce some notations to facilitate our subsequent discussion. First, we denote $\mathbb{E}\left(T_{A\to B}\left(t\right)\right)$ as the first moment of the random variable FPT from node A to B in generation t. According to Equation (A4), it is obvious that:

In the section on the random walk from A to B, we have found the recursion:

Now, we take the derivative on both sides of the equation and combine it with Equation (A4), setting $z=1$. We obtain:As to the random walk from D to E, the recursion is: