Average trapping time on weighted directed Koch network

Numerous recent studies have focused on random walks on undirected binary scale-free networks. However, random walks with a given target node on weighted directed networks remain less understood. In this paper, we first introduce directed weighted Koch networks, in which any pair of nodes is linked by two edges with opposite directions, and weights of edges are controlled by a parameter θ . Then, to evaluate the transportation efficiency of random walk, we derive an exact solution for the average trapping time (ATT), which agrees well with the corresponding numerical solution. We show that leading behaviour of ATT is function of the weight parameter θ and that the ATT can grow sub-linearly, linearly and super-linearly with varying θ . Finally, we introduce a delay parameter p to modify the transition probability of random walk, and provide a closed-form solution for ATT, which still coincides with numerical solution. We show that in the closed-form solution, the delay parameter p can change the coefficient of ATT, but cannot change the leading behavior. We also show that desired ATT or trapping efficiency can be obtained by setting appropriate weight parameter and delay parameter simultaneously. Thereby, this work advance the understanding of random walks on directed weighted scale-free networks.


Results
construction and properties of directed weighted Koch network. The Koch network is derived from the Koch curve, a well-known fractal, in which each node in the network corresponds to one triangle in the curve. The generation procedure of binary Koch networks was detailed in 44 . Let K(g) denotes the network after g iterations. For g = 0, K(0) is a triangle containing three nodes. For g ≥ 1, K(g) can be obtained from K(g −1 ) as follows: For each existing node, two new nodes are created and attached to the existing node to form a triangle 44 . Figure 1 shows the construction process of the first three generations. We note that L(g), the number of triangles in K(g), equals to 4 g . The variable L v (g), the number of nodes created at g th generation, equal to 6 × 4 g−1 . The variable L e (g), the number of edges created at g th generation, equal to 9 × 4 g−1 . Besides, the variable N g , the number of the nodes in K(g), equals to 2 × 4 g + 1 44 .
Based on the undirected and unbiased Koch network, we propose a weighted directed Koch network by introducing a weight parameter θ quantifying the reciprocating relation between edges. In other words, the edge in the binary network is replaced by two opposite directed edges with different weights.
In the directed weighted network, we define the out-strength and in-strength of node i at generation g as respectively. Based on the evolution rule of network, it can be shown where g i represents the generation at which the node i is created.
numerical formulations of Att in directed weighted Koch network. In this paper, we address a special biased random walks with a given target node. For convenience, we label all the nodes in the following way: the initial three nodes in → K (0) are labeled as 1, 2 and 3, respectively. Let N g−1 + 1, N g−1 + 2, …, N g denote the new nodes created at g th generation in → K g ( ), which are shown in Fig. 2. We assume that the target node is placed at any node in K(0). Without loss of generality, let node 1 be target node. This stochastic process of random walks is characterized by the transition matrix P g . The element of P g named p ij (g) denotes the transition probability of jumping from i to j in each time step. It satisfies ∑ = = p g ( ) 1 Let T i g ( ) denote the mean first passage time for a walker moving from node i to target node in → K g ( ). Then, it obeys the equation which can be rewritten in matrix form as g g g g where T g and P g are formed by deleting the first row of T g and P g and respectively. The variable e g is formed by removing the first element of e g , which is the N g -dimensional vector of ones. From Eq. (2), we obtain By definition, 〈T〉 g is the average trapping time (ATT) after g iterations. Note that Eq. (4) is valid only if node 1 is designated as absorbing node, or p 1,1 (g) = 1 and p 1,j Equation (4) shows that in order to compute 〈T〉 n or ATT, we only need to compute the inverse of transition matrix. However, computing the inverse of a matrix leads to heavy computational burden when network's size is large. Therefore, alternative approach is needed to efficiently compute the ATT and equation 4 can be applied to verify the solutions derived by alternative approach.
Analytical solutions for Att in weighted directed Koch network. Now, we analytically derive the ATT with a given target node on directed weighted Koch network. Let Ω g denote the set of nodes in generation g and Ω g be the set of new nodes created at generation g, then |Ω | = |Ω | + |Ω | −ḡ g g 1 . It is obviously that Let k i (g) represents the degree of node i at gth generation. According to network's construction rule, the iteration rule of k i (g) is k i (g) = 2k i (g − 1). Furthermore, among the 2k i (g − 1) neighbors of node i at generation g, there are k i (g − 1) neighbors created before generation g(old neighbor) and there are k i (g − 1) neighbors created just at generation g(new neighbor). We use Z to denote mean first passage time for a walker moving from node i to any of its k i (g) old neighbors, and let X represent the mean first passage time for a walker going from any of the new neighbors of i to one of its k i (g) old neighbors. By the iteration rule of the directed weighted Koch network, we have from which we obtain Z = (1 + θ) 2 , which is just the iteration rule of the average trapping time. In other words, We define the intermediary quantities for 1 ≤ m ≤ g.
Then, we obtain the relation Thus, the problem of computing 〈T〉 g reduces to computing T g tot g , ( ) . Taking the iteration rule of the average trapping time into consideration, we have , ( ) Hence, in order to get T g tot g , ( ) , we must first obtain T g tot g , ( ) . When g = 1, Thus, we have  T  T  T   T  T  T  T  T  T   T  T  T  T  T  T  T  T  T   T From Eqs (10) and (11), we obtain g tot g g g tot g g tot g g tot (1 ) g tot g g g tot g g tot g g tot ( 1) g Multiplying Eq. (13) with (1 + θ) 2 and subtracting the result from Eq. (14), we find Using , Eq. (16) can be resolved inductively to yield Thus, according to the Eq. (9), we obtain  www.nature.com/scientificreports www.nature.com/scientificreports/ when g → ∞ and θ ≠ 1, we have Equation (21) shows that 〈T〉 g grows sub-linearly with the network size if θ < 1, while 〈T〉 g grows super-linearly with the network size if θ > 1.
We have compared our analytical result with numerical result obtained from Eq. (4). They agree well with each other for different values of θ and network order g, which are summarized in Fig. 3.

Expressions for ATT with delay parameter on weighted directed Koch network.
We now introduce a delay parameter p to explore the impact of time-delay on random walk.
By the construction of network, we define the random walk process in the network with delay as follows: The walker in → − K g ( 1) is allowed to move to any neighbor either in → − K g ( 1) or → K g ( ) with probability p and 1 − p, respectively, where 0 ≤ p ≤ 1. The transition probability p ij for the walker jumping from node i to node j is defined by is the out-strength of node i. If the walker is at nodes newly created in generation g, it will do random walk on → K g ( ). When p = 0, the walker moves from its current location (node i) to any neighboring node j with transition probability = . In other words, random walk with delay reduces to random walk in → K g ( ). When p = 1, the walker jumps from current position (node i) to any neighboring node j in if node i is created before generation g. Otherwise, the walker moves from current position to any neighbor in → K g ( ) with transition probability = . By the definition above, the quantities of Z and X obeys the relation  (19) or (20).
from which we have = θ θ θ θ In this section, we define D i (g) as the MFPT in → K g ( ) with the delay state and 〈D〉 g as the average trapping time in → K g ( ). According Eq. (23), we have θ θ θ θ          Then from Eq. (25), we have We compared analytical expressions in Eqs (34) and (38) with numerical results from Eq. (4). As shown in Fig. 4, the analytical results agree well with numerical results for different values of θ, p and n.
It can be seen from Eqs (35) and (39) that 〈D〉 g grows sub-linearly, linearly, super-linearly with network size when θ < 1, θ = 1, θ > 1 respectively. However, Eqs (35) and (39) also show that the delay parameter p can only (2019) 9:14609 | https://doi.org/10.1038/s41598-019-51229-2 www.nature.com/scientificreports www.nature.com/scientificreports/ change the coefficient of the ATT. Although the delay parameter p has less pronounced influence on the transport efficiency, both it and the weight parameter are needed to tune for getting desired average trapping time.

Discussion
In this paper, we first proposed a weighted directed Koch network, in which the weight of edge is controlled by a parameter θ. Then, we derived the closed form solution of average trapping time (ATT) of a random walk with a given trap node. The solution shows that the weighted directed construction has a significant effect on trapping efficiency and the leading scale of ATT can be sub-linear, linear and super-linear with varying weight parameter. Finally, when a delay is introduced such that new extended structures do not always affect the system immediately, then we introduce a delay parameter p to steer the random walk on the weighted directed network. The exact solution of the ATT in this scenario shows that the delay parameter can only change the prefactor of the ATT. Our results also show that we can obtain desired ATT with appropriate values of weight parameter θ and delay parameter p. It is noteworthy that the procedure used here is limited to deriving average trapping time of random walks with trap located at node 1,2 or 3. For next steps, we will generalize our work to average trapping time with trap located at any nodes.