Academic Journal of Science and Technology 
ISSN: 2771-3032 | Vol. 13, No. 2, 2024 

 

242 

The	Mean	Capture	Time	on	Horizontal	Divided	Nested	
Networks	
Zuodong Xiang1, * 

1 School of Mathematical Sciences, Jiangsu University, Zhenjiang, 212013, China 
* Corresponding author: Zuodong Xiang (Email: zuodongxiang24@163.com) 

 

Abstract: In this paper, we consider the division of nested networks with a horizontal division line 𝑙 , where 𝑘 is the division 
coefficient. The problem of capture on the surplus network obtained after the division is studied. In addition, by studying the 
structure of the surplus network, we obtain the relationship between the transmission efficiency on the surplus network and the 
division coefficient 𝑘. When 𝑘 is larger, the capture time is shorter and the network transmission efficiency is higher. At the 
same time, we also solved the mean capture time of each node on the bottom edge of the surplus network. 

Keywords: Nested network; horizontal division line; division coefficient; mean capture time. 
 

1. Introduction 
In recent years, complex networks have been studied by 

scholars in different fields. They have been applied more and 
more widely, such as sociology, computer science, biology, 
physics, and so on[1-4]. For example, in power systems, 
applications of complex network theory include design 
optimisation of power grids, fault detection and recovery, and 
power demand forecasting[5-8]. In logistics and supply chain 
management, complex network theory can be used to 
optimise the design and management of logistics networks 
and improve the efficiency of supply chains[9,10]. The study of 
dynamic processes has gradually become the focus of 
research, and random walk is the basis and important tool of 
our research. Many scholars in various fields have conducted 
in-depth research on random walk[11-14]. As we know, the 
mean capture time (MCT) as an important index obtained 
through the random walk model on the network. MCT can be 
defined as average of the mean first passage time (MFPT) of 
all nodes on the network of the wanderer to reach the trap 
node. MFPT is defined as the average time for a walker to 
reach the trap node for the first time from the node. 

Complex networks are networks with some or all of the 
properties in self-similarity, self-organisation, attractors, 
small worlds, and scale-free. Therefore, fractal networks with 
the important property of self-similarity have attracted a great 
deal of attention in various fields, such as flower network[15,16], 
Koch curve[17,18], Sierpinski Gasket triangle network[19], 
Vicsek fractal[20] and so on. Among the many fractal networks, 
Sierpinski Gasket network is one of the most popular network 
models. Many existing research papers have explored the 
MCT on Sierpinski Gasket network and its derivative network. 

Wu et al.[21,22] studied the MCT on three-level Sierpinski 
gasket network and half-Sierpinski gasket. Zhang et al.[23,24] 
studied the MCT on three-dimensional horizontally 
segmented Sierpinski gasket network and three-dimensional 
Sierpinski gasket gasket network. Hu et al.[25] studied the 
MCT of a three-level horizontally segmented Sierpinski 
gasket network. Sun et al.[26] studied the MCT on a network 
extended by a Sierpinski gasket. Han et al.[27] studied the 
MCT when performing non-nearest-neighbour jumps on a 
nested network derivative by a Sierpinski gasket network. On 
the basis of nested networks, we consider a horizontal 
division of them to obtain a surplus network model (SSN ). 
Solve the analytical expression of MCT on the SSN, which 
contains the number of iterations and the division coefficients. 
At the same time, we analysed the effect of changes in 
division coefficients on the efficiency of the random walk. In 
addition, we solve the MCT when each node on the bottom 
edge of the SSN is used as a trap node.  

The rest of the paper is as follows: in Section 2, we present 
the method for constructing SSN  and auxiliary networks 
(ASN) by nested networks. In Section 3, we solve the exact 
expression for the MCT on the SSN. In Section 4, we solve 
for the MCT of each node on the bottom edge of the SSN. 
Finally, in Section 5, we conclude the full paper.  

2. Preliminaries  
In this section, we provide a brief description of the nested 

network model and introduce the definition and construction 
of the horizontal division line 𝑙  , the surplus network SSN 
and the auxiliary network (ASN).  

 

 
Figure 1. The structure of S N. 



 

243 

 

2.1. Structure of 𝐒𝐠𝐍 

S N is an extension of the Sierpinski Gasket, also with self-
similarity. First, we denote the equilateral triangle and the 
three-regular graph as S and N, respectively. where S is a 
structural graph of S by taking a trisection on each side of the 
triangle, and connecting all pairs of trisections parallel to the 
initial side. Then, six copies of the three-regular graph N are 
embedded into S to form the 1st generation network, denoted 
as S N. When 𝑛 2, the six S N are embedded into S to 
form S N, as shown in figure 1. 

According to the construction method of S N , we can 
divide S N  into six parts, denoted as Γ 𝑔 1  , 𝑖
1,2, … ,6 . We label the nodes on the network in the top-to-

bottom order, and the three corner nodes are denoted as 
denoted as 1, L, and R. Where each region consists of many 
S N of the 0th generation, and we denote the number of S N 
of the gth generation as Δ . It is easy to get the number of 
S N as 6 . By the structure of the nested network, we denote 
the set of all nodes as N(g) and the set of all edges as E(g), 
can be calculated. 

 
1

12 8
| ( ) | 6 ,

5 5
| ( ) | 6 .

g

g

N g

E g 

   

 

           (1) 

Here we use the symbol | | for the number of elements in 

the set. 

2.2. Structures of 𝐒𝑺𝐍 𝐠, 𝒌  and 𝐀𝐒𝐍 𝐠, 𝐧  
First, we define the horizontal division line 𝑙   as the 

horizontal line where nodes 7, 8 and 9 are located (at  of the 

line segment between corner nodes 1 and L). The network 
S N  is divided into two parts by the horizontal line 𝑙  , 
preserving the lower network of the division line. Notice that 
the lower network preserves the points on the division line. 
We call the lower network the surplus network notated as 
SSN g, 1 . Similarly, the second horizontal division line 𝑙  is 
defined as the line segment identified through corner nodes 1 
and L at   (near node L) and parallel to 𝑙  . The retained 

surplus network is therefore denoted as SSN g, 2 . Based on 
the above surplus network construction method, we denote 
the division coefficients by the integration number 𝑘. When 
the network is divided using the division line 𝑙 , SSN g, k  
is obtained as shown in Figure 2. Note that the division 
coefficients here need to satisfy: 1 𝑘  𝑔. It is easy to see 
that SSN g, k  consists of 3  S N g k . 

 
Figure 2. The structure of SSN g, k .

Therefore, we give here the definition of the auxiliary 
network ASN g, n  , that is, 𝑛  S N  can be obtained by 
connecting them in series in order to obtain ASN g, n .  

For a clearer representation, we denote these 𝑛 regions as 
Γ 𝑔  , Γ 𝑔  ,…, and Γ 𝑔   from right to left. For any 

region Γ g 1 𝑗 n  , we denote the corner nodes as 
(A,j), (B,j), (C,j), as shown in Figure 3. Note that since some 
of the nodes connect two regions, there exists (B,i) = (C,i+1), 
where 1 𝑖 n. 

 

 
Figure 3. The structure of ASN g, n . 



 

244 

 

Then, similar to the definitions of the set of nodes and the 
set of edges given earlier. We denote the set of all nodes and 
the set of all edges of SSN g, k  and ASN g, n  as N g, k , 
N g, n  and E g, k , E g, n  respectively. Based on the 
structure of each of the two networks, we can obtain: 

 

1

12 3
| ( , ) | ( ) ( 1) 6 1,

5 5
| ( , ) | ( ) 6 ;

A g

A g

N g n n N g n n n

E g n n E g n 

        

    

  (2) 

 

1

3
| ( , ) | 3 ( ) (3 1) (12 6 3) 1,

5
| ( , ) | ( ,3 ) 3 6 .

k
S k k g k

S A k k g k

N g k N g k

E g k E g k



 


        


    

  (3) 

 

3. Analytic Expression for The Mean 
Capture Time on Network 𝐒𝐒𝐍 𝐠, 𝐤  

In this section, we will first compute the MCT on the 
network S N  with corner node L as the trap node. Then, 
compute the MCT on ASN g, n  when node (C,1) is used as 
a trap node. Finally, the equivalence between the network 
ASN g, n   and SSN g, k   is obtained through the network 
ASN g, n . This results in an analytic expression for the MCT 
of SSN g, k . 

3.1. MCT on the 𝐒𝐠𝐍 

In this section, we will fix the trap node as the corner node 
L, thus considering the MCT on S N. We consider unbiased 
Markov random roaming on the network. During the roaming, 
the directions are random and the step sizes are equal. First, 
the random walker starts from any non-trap node. It wanders 
along the network only one unit step at a time until it is 
captured by a trap node and the wandering ends. The transfer 
probability at each roam is the inverse of the degree of that 
node. Here, we denote the degree of node 𝑢 as 𝑑 . Denoting 
the transfer probability from node 𝑢  to 𝑣  as 𝑝 ,  , we 
obtain: 

𝑝 ,

1
𝑑

, 𝑖𝑓 𝑢~𝑣 𝑎𝑛𝑑 𝑢 2

0,            𝑜𝑡ℎ𝑒𝑟𝑠.                      
 

Then, we define the mean first passage time (MFPT), that 
is, the mean time for a random wanderer to reach a trap node 
𝑣  for the first time from any node 𝑢  is denoted as 𝑇 ,  . 
When node 𝑢 is not a trap node and there exists a trap node 
𝛼 , the MPFT from node 𝑢  to the trap node, is denoted as 
𝑇 𝑇 , . In particular, if the initial node is the same as the 
trap node, the we have 𝑇 , 0. Denoting 𝑇  as the sum 
of the MFPT of the wanderers starting from all nodes, we 
obtain the following equation: 

 

 ,
( ) ( )

.sum u u
u N g u N g

T T T 
 

         (4) 

 

From this, we obtain the MCT on S N , denoted 𝑇  , 

which can be obtained: 

 .
| ( ) | 1

sum
sum

T
T

N g



         (5) 

 

According to previous scholars we can get a MCT on S N 

containing 𝜆 𝑞 ∈ , 1  [27]: 

1

1

611 ( ) 135
( ) ( ) 6

640 7
3993 ( ) 135 4011135

( ) ( ) 6
155 7 1984 7

11
6 .

4

g g
sum

g g

g

q
T g

q
q



 





 

  

 

 

1

1

( )
3968(12 6 3)

135 135
[18941 ( )( ) 6 511104 ( )( )

7 7

5(10912 25785 ( ))6 .

1

]

sum g

g g g

g

T g

q q

q

 








 

 

 

 

When we make 𝑞 0 that is, 𝜆 𝑞 1, what is going on 
at this point is what we define as a random wandering, and we 
can get: 

 

1

1

611 135
( ) ( ) 6

640 7
3993 135 77355 11

( ) 6 6 .
155 7 1984 4

g g
sum

g g g

T g 



 

   

 (6) 

 

 

1

1

135
( ) [18941( ) 6

3968(12 6 3) 7

135
511104( ) 74365 6 .

1

]
7

g g
sum g

g g

T g 




 

  

 (7) 

3.2. MCT on the 𝐀𝐒𝐍 𝐠, 𝐧  
The above discussion of MCT on S N are based on when 

corner nodes L are used as trap nodes, but are not sufficient to 
support the computation of MCT on ASN g, n . We also need 
to obtain parsing expressions for MCT when two and three 
corner nodes are used as traps, which can be used to compute 
parsing expressions for MCT on the network ASN g, n . 

Here, we use the corner nodes L and R as trap nodes on the 
network S N. At this point any node 𝑢 reaches the MFPT of 
the absorbing node, and the total capture time and MCT are 
denoted as 𝑇 𝑔  , 𝑇 𝑔  , and 𝑇 𝑔  , respectively. 
Similarly, we use corner nodes 1, L and R as trap nodes on the 
network S N. At this point any node 𝑢 reaches the MFPT of 
the absorbing node, and the total capture time and MCT are 
denoted as 𝑇 𝑔 , 𝑇 𝑔 , and𝑇 𝑔 , respectively. 

Based on the symmetry of the nested network S N., we can 
obtain the following relation: 

 
2

1

135
( ) ( ).

7
T g T g  (8) 



 

245 

 

2 171 135 749 135
( ) ( ) 6 ( )

128 7 31 7
44619

6 .
1984

g g g
sum

g

T g  



       (9) 

 

From this, we can obtain the MCT when there are two trap 
nodes on S N, that is: 

 
2

2

1

( )
( )

| ( ) | 2

355 135 3745 135 223095
( ) 6 ( ) 6

128 7 31 7 1984 .
12 6 2

sum
sum

g g g g

g

T g
T g

N g






 


 

 

The above conclusions are solved in detail in Appendix A. 
Next, we compute the parsing expression of the MCT on 

the auxiliary network ASN g, n   with node (C,1) as the 
capture node. We let the set of all nodes on Γ g 1 𝑗 n  

be 𝑁 . Denote the set of nodes containing only (B,j) and (C,j) 

as  𝑁 . Furthermore, let 𝑁 𝑁 ∪ 𝑁 … ∪ 𝑁 , 𝑁 𝑁 /
𝑁  and 𝑁 𝑁 /𝑁 . We let the MFPT from node (a,b) to 
node (c,d) on the auxiliary network ASN g, n  be denoted as 
𝑇 , , , 𝑔, 𝑛  . Let the sum capture time on the auxiliary 
network ASN g, n   be 𝑇 𝑔, 𝑛   and the mean capture 
time be 𝑇 𝑔, 𝑛 . 

Based on the above classification of nodes, we divide the 
path from any node 𝑖, 𝑗 ∈ 𝑁 to the capture node (C,1) into 
two steps: 

(1) (i,j) reaches node (B,j) or (C,j) after time 𝑇 𝑔 . 
(2) Departing from node (B,j) or (C,j) is finally captured 

at node (C,1). 
So, we have the following equation: 
 

( , ),( ,1)
( , ) ( )

( , ),( ,1)
1 ( , )

( , ),( ,1)
ˆ( , )

( , ) ( , )

( , )

( , ).

j
A

A

A A
sum i j C

i j N g

n
A
i j C

j i j N

A
i j C

i j N

T g n T g n

T g n

T g n



 











 



    (10) 

 

The nodes (B,j) and (C,j) in 𝑁  are symmetric in the 
network 𝛤 𝑔 , so the following equation can be obtained 
in any 𝑗 ∈ 1, 𝑛 : 

 

( , ),( ,1)
( , )

2
( , )

( , )

( , ),( ,1) ( , ),( ,1)
( , )

2

( , ),( ,1) ( , ),( ,1)

( , )

( )

1 1
[ ( , ) ( , )]
2 2

1
( ) (| ( ) | 2)

2

[ ( , ) ( , )].

j
A

j
A

j
A

A
i j C

i j N

i j
i j N

A A
B j C C j C

i j N

sum

A A
B j C C j C

T g n

T g

T g n T g n

T g N g

T g n T g n









 

  








 (11) 

 

So substituting Eq.(11) into Eq.(10) yields the following 
equation: 

 

 

2

( , ),( ,1) ( , ),( ,1)
1

( , ),( ,1)
ˆ( , )

( , )

1
( ) (| ( ) | 2)

2

[ ( , ) ( , )]

( , ).
A

A
sum

sum

n
A A
B j C C j C

j

A
i j C

i j N

T g n

n T g N g

T g n T g n

T g n





   

 



 (12) 

 

Since we labelled the nodes (B,j) and (C,j) in the corner 
nodes (B,j) and (C,j) on the network ASN g, n  in such a way 
as to know that (B,j) = (C,j+1), where 1 𝑗 𝑛 , we can 
convert Eq.(12) into the following equation: 

 

 

2

( , ),( ,1) ( , ),( ,1)
ˆ( , )

( , ),( ,1)
ˆ( , )

2

( , ),( ,1)
ˆ( , )

( , ),( ,1)

( , ) ( ) (| ( ) | 2)

1
[ ( , ) ( , )]

2

( , )

( )

(| ( ) | 1) ( , )

1
(| ( ) | 2) ( , ).

2

A

A

A

A
sum sum

A A
i j C B n C

i j N

A
i j C

i j N

sum

A
i j C

i j N

A
B n C

T g n n T g N g

T g n T g n

T g n

n T g

N g T g n

N g T g n







   





 

 

 







 (13) 

 
Next, we consider the computation 𝑇 , , , 𝑔, 𝑛  , 

𝑖, 𝑗 ∈ 𝑁 , where 𝑇 , , , 𝑔, 𝑛 0 when (i,j)=(C,1). So 
we only need to consider nodes (B,j) , 𝑗 ∈ 1, 𝑛 . The random 
walks we consider are equal probability walks, so a walk 
starting at node (B,j) 1 𝑗 𝑛 will reach node (B,j-1) or 
node (B,j+1) for the first time after time 𝑇 , 𝑔  with equal 
probability. Thus, computing 𝑇 , , , 𝑔, 𝑛 𝑖, 𝑗 ∈ 𝑁  
reduces the model to a random walk model on a one-
dimensional finite lattice with the node (C,1) as the capture 
node, denoted 𝐿 𝑛 , with a length of 𝑛 and a length of unit 
length for each edge. We label it from right to left as 0 to 𝑛, 
a total of 𝑛 1 nodes. Denote the commuting time from any 
of these nodes 𝑎  to node 𝑏  as 𝑇 ↔ 𝑛  . So, by the finite 
resistance principle, we can get the following equation[28]: 

 

, , ,( ) ( ) ( ) 2 .L L
a b a b b a a bT n T n T n nR     

where 𝑇 , 𝑛   denotes the MFPT from node 𝑎  to node 
𝑏  on 𝐿 𝑛   and 𝑅 ,   denotes the effective resistance from 
node 𝑎  to node 𝑏  on 𝐿 𝑛  . So we can easily get 𝑅 ,

𝑏 𝑎 when 𝑎 0: 

0 0, ,0 0,( ) ( ) ( ) 2 2 .L L
b b b bT n T n T n nR nb      

Since 𝑏 𝑛 , we have: 𝑇 , 𝑛 𝑇 , 𝑏 𝑇 , 𝑏  . 
From this, we we can obtain: 

2
0, 0, 0

1
( ) ( ) ( ) .

2
L L
b b bT n T b T n b    



 

246 

On this basis, we can get: 
2

,0 0,( ) 2 ( ) 2 .L L
b bT n nb T n nb b     

So from the above results and analyses on the one-
dimensional finite lattice, we can get the following relation: 

 

 
2

( , ),( ,1) ,0 1,

135
( , ) ( ) 2 ( ) .

7
A L g
B n C n LT g n T T g n    (14) 

 

 

ˆ 1
( , ),( ,1) ,0 1,

( , )

2

1

( , ) ( ) ( )

135
2 ( ) (2

7

1 135
( 1)(4 1)( ) .

3

)

7

A

A L
i j C b L

i j N

g

g

n

b

n

b

T g n T n T g

nb b

n n n









  

  

 

 (15) 

 
Therefore, substituting Eq.(9), Eq.(14) and Eq.(15) into 

Eq.(13), we get: 
 

 

3

3 2

16 809 135
( , ) ( ) 6 ( )

5 320 7
4 3714 135

( ) ( )
5 155 7
44619

6 .
1984

A g g
sum

g

g

T g n n n

n n n

n

   

   

  

 (16) 

 
Hence we can solve for MCT on the auxiliary network 

ASN g, n : 
 

 
2

2

( , )
( , )

| ( , ) | 1

1 809 135
[(16 )6 ( )

12 6 3 64 7
3714 135 223095

(4 5 ) ( ) 6 ].
31 7 1984

A
A sum

sum A

g g
g

g g

T g n
T g n

N g n

n

n n




   
 

    

 (17) 

 

3.3. MCT on the 𝐒𝐒𝐍 𝐠, 𝐤  
In the previous section we have solved the analytic 

expression for MCT on the auxiliary network ASN g, n  . 
Through the equivalence between the surplus network 
SSN g, k   and the auxiliary network ASN g, n  , we can 
easily solve the following equation: 

 

 

3

3 2

( , ) ( ,3 )

16 809 135
( 3 3 ) 6 ( )

5 320 7
4 3714 135

( 3 3 3 )( )
5 155 7
44619

3 6 .
1984

S A k
sum sum

k k g k g k

k k k g k

k g k

T g k T g k

 





 

     

    

  

 (18) 

 
From this, we can solve the analytic expression for MCT 

on the surplus network SSN g, k  with corner nodes L as trap 
nodes: 

 
2

2

( , )
( , )

| ( , ) | 1

1 809 135
{(16 3 )6 ( )

12 6 3 64 7
3714 135 223095

(4 3 5 3 )( ) 6 }.
31 7 1984

S
S sum

sum S

k g k g k
g k

k k g k g k

T g k
T g k

N g k

 


 




   
 

      

(19) 

 
 
Make 𝑘 0  to verify the above equation and we get 

𝑇 𝑔, 0 𝑇 𝑔  . We take the principal term of 
𝑇 𝑔, 𝑘  to satisfy the following equation: 

 

 

2 809 135
(16 3 ) 6 ( )

64 7( , ) ~
12 6 3

3 63 135
~ ( ) ( ) .

4 135 7

k g k g k

S
sum g k

k g

T g k

 



   

 

 

 (20) 

 
Thus, we can know that 𝑇 𝑔, 𝑘  is proportional to the 

number of iterations, which is inversely proportional to the 
division coefficient. 

In order to be more intuitive we carry out numerical 
simulation of 𝑇 𝑔, 𝑘   with respect to 𝑔  and 𝑘 , see 
Figure. 4. 

 
Figure 4. Numerical simulation plot of 𝑇 𝑔, 𝑘  with respect to 

variables 𝑔 and 𝑘. 

 

In order to better see the trend of 𝑇 𝑔, 𝑘  with respect 
to the number of iterations 𝑔 and the division coefficient 𝑘, 
we take the time when 𝑘=10, 30 and 50, respectively. The 
trend of 𝑇 𝑔, 𝑘  with respect to the number of iterations 
𝑔 is shown in Figure. 5.  

 

 
Figure 5. It shows that  𝑇 𝑔, 𝑘  increases as 𝑔 increases, that 

is, it is positively correlated, by fixing 𝑘 = 10, 30 or 50, 
respectively. 



 

247 

When we take 𝑔=10, 30, and 50, respectively, the trend of 
𝑇 𝑔, 𝑘  relative to the division coefficient 𝑘 is shown in 
Figure 6. 

 

 
Figure 6. It is fixed 𝑔 10, 30 or 50 respectively, which gives 

𝑇 𝑔, 𝑘  decreases with increasing 𝑘, that is, a negative 
correlation. 

 

In order to see that with the change of division coefficients 
𝑘, the surplus network SSN g, k  on MCT versus MCT on 
the uncut front network S N . The trend of 𝑇 𝑔, 𝑘  
versus 𝑇 𝑔  with the number of iterations 𝑔 is shown in 
Figure. 7 when we take 𝑘=5, 25 and 50. It is easy to find that 
as the division coefficients 𝑘  gets smaller, the network 
retains a higher degree of completeness 𝑇 𝑔, 𝑘   which 
converges to 𝑇 𝑔 . 

 

 
Figure 7. Numerical simulation plots of MCT with iteration 

coefficients 𝑔 for the network SSN g, k  versus the network 
S N, which are taken to be𝑘 = 5, 25 or 50. 

 

Moreover, it is easy to know that 𝑇 𝑔, 𝑘   takes the 
minimum value when 𝑘 𝑔, which we can obtain: 

2

2 3
135

2 7

4 1 4
( , ) 3 3

3 15 3

~| ( , ) | ~ [ ( )] .

S g g
sum

ln

lnS
sum

T g g

N g g T g

    
 

Similarly, 𝑇 𝑔, 𝑘    takes the maximum value when 
𝑘 0, which we can obtain: 

135

7
6

1833 135 3869 135 223095
( ) 6 ( ) 6

64 7 31 7 1984( , )
12 6 3

~| ( ,0) | .

g g g g

S
sum g

ln

S ln

T g g

N g

 


   

4. MCT of the Other Trap Nodes on 
𝐒𝐒𝐍 𝐠, 𝐤  

In the previous section, we solved the MCT on the auxiliary 
network ASN g, n  with node (C,1) as the trap node. In this 
section, instead of considering only node (C,1) as a trap node, 
we consider the parsing expression for MCT when all nodes 
on the bottom edge of ASN g, n   are individually used as 
trap nodes. Based on the previous way of labelling the nodes 
on ASN g, n , we know that (B,j) and (C,j+1) denote a single 
node, so we unify the labelling on the bottom edge of 
ASN g, n   and set it to (C,j), where 𝑗 ∈ 1, 𝑛 1  . At this 
time, we set the trap node as (C,h) where 1 ℎ 𝑛 1 . 
Hence, based on the network structure of the auxiliary 
network ASN g, n , we denote the network to the left of the 
node (C,h) as ASN g, n h 1  , and the network to the 
right as ASN g, h 1 , see Figure 8. 

 

 
Figure 8. The structure of ASN g, n h 1  and ASN g, h 1  

 

Immediately, we can get the total capture time of the 
auxiliary network ASN g, n   on the auxiliary network 
ASN g, n  containing the trap node (C,h) by Eq.(16), denoted 
as 𝑇 𝑔, 𝑛, ℎ , so that we can get: 

 

3 2 2 2

3 2 2 2

2

( , , ) ( , 1) ( , 1)

16 809
[ ( 3 3 3 6 3 ) ]

5 320
135 4 12 12 17

6 ( ) (
7 5 5 5 5

34 4396 135
2 4 2) ( )

5 155 7
44619

6 .
1984

A A A
sum sum sum

g g

g

g

T g n h T g n h T g h

n n h nh n nh n n

n n h nh n

nh n h h

n

    

       

         

       

  

(21) 

 
Based on the equivalence between the network ASN g, n  

and the network SSN g, k , we can solve for the total capture 
time of the other trap nodes on the network SSN g, k , which 
is denoted as 𝑇 𝑔, 𝑘, ℎ , and so we get: 

 



 

248 

3 2 1 1 2 2 1 1

3 2 2 2

2

( , , ) ( ,3 , )

16 809
[ (3 3 3 3 6 3 3 ) 3 ]

5 320
135 4 12 12 17

6 ( ) ( 3 3 3 3
7 5 5 5 5

34 4396
3 3 2 4 2)

5 155
135 44619

( ) 3 6 .
7 1984

S A k
sum sum

k k k k k k k

g k g k k k k k

k k

g k k g k

T g k h T g k h

h h h

h h

h h h

   

 

 

 

        

        

      

   

 (22) 

 
From this, we can obtain analytic expressions for the MCT 

of the other trap nodes on the network SSN g, k : 
 

 2 1 2 1
( )

2 2

2

( , , )
( , , )

| ( , ) | 1

1 809
{[16(3 3 3 3 6 3) ]

12 6 3 64
135

6 ( ) (4 3 12 3 12 17 3 34
7

4396 135 223095
(10 20 10)3 ) ( ) 6 }.

31 7 1984

S
S sum

sum S

k k k
g k

g k g k k k k

k g k g k

T g k h
T g k h

N g k

h h h

h h h

h h

 


 

  




      
 

          

      

 (23) 

 
We can find that the parsed expression of 𝑇 𝑔, 𝑘, ℎ  is 

still positively correlated with the number of iterations 𝑔 and 
negatively correlated with the division coefficient 𝑘. 

5. Conclusion  
In this paper, we obtain the surplus network SSN g, k  

based on nested networks, which are divided by a horizontal 
division coefficient 𝑘 . Firstly, we obtain the mean capture 
time of the fixed nodes on the nested network. Secondly, we 
solve for the mean capture time of the fixed nodes on the 
auxiliary network ASN g, n  , and then we obtain the mean 
capture time on the surplus network SSN g, k  based on the 
equivalence between the surplus network SSN g, k   and 
ASN g, n  . We perform analytical numerical simulations of 
the MCT on SSN g, k  . The results show that the network 
SSN g, k    is only locally self-similar, and the larger the 
division coefficient 𝑘 is, the smaller the MCT is, that is, the 
network becomes more and more efficient in transmission 
during successive destructions. Finally, we also solve the 
analytical expression for MCT when any node on the bottom 
edge of the surplus network SSN g, k  is used as a trap node.  

Appendix A. MCT of Two Trap Nodes 
on 𝐒𝐠𝐍 

First, we consider the iterative relationship between the 
nodes corresponding to the first generation of the network in 
the network S N as they wander at random. According to the 
labels in Figure 1, the average first pass time of node 1 in S N 
when it arrives at a node with node L or R as the trap node is 
denoted as 𝑇 ′. Then, L and R in the network S N are set as 
trap nodes. 

Based on the definition of unbiased Markov random walk 
and the symmetry of the nested network, the MFPT of each 
node is denoted as 𝑇  𝑖 1,2, … ,16  , where 𝑇 𝑇  , 
𝑇 𝑇  , 𝑇 𝑇  , 𝑇 𝑇  , 𝑇 𝑇  , and 𝑇
𝑇 0, and we can set up the following equation: 

2 2 2
1 2 1

2 2 2
2 1 1

2 2 2 2 2 2 2
3 1 2 4 5 7 8

2 2 2 2
5 3 7 8

2 2 2 2
7 3 5 8 1

1 2
( ) ( )

3 3
1 2

( ) ( )
3 3
1 1 1 1 1 1

( ) ( ) ( ) ( ) ( ) ( )
6 6 6 6 6 6
1 1 1

( ) ( ) ( )
3 3 3
1 1 1 1

( ) ( ) ( ) (
6 6 6 6

T T T T T

T T T T T

T T T T T T T T T T T T T

T T T T T T T

T T T T T T T T T

 

 

     

  

   

   

   

           

     

        2 2
0 14

2 2 2 2 2 2 2
8 3 4 5 7 11 14

2 2 2
10 7 14

2 2 2
11 8 14

2 2 2 2 2
14 7 8 10 11

1 1
) ( )

6 6
1 1 2 2 1 2

( ) ( ) ( ) ( ) ( ) ( )
9 9 9 9 9 9
1 1 1

( ) ( )
3 3 3
1 2

( ) ( )
3 3
1 1 1 1

( ) ( ) ( ) ( )
6 6 6 6

T T T

T T T T T T T T T T T T T

T T T T T T

T T T T T

T T T T T T T T T

 

     

  

 

   

  

           

    

   

        2
15

1 1
( )

6 6
T T T 



























  
 

Solving the system of equations gives: 

2
1

135
.

7
T T    

Here 𝜆  is the time difference between the nodes of two 
consecutive generations to reach the trap node. 

Since the network S N  has self-similarity, 𝑇 ′ 𝑔
𝑇 𝑔 1 . The MFPT to reach the trap node L or R from 
node 1 in the network S N is 1, denoted as 𝑇 0 1. So 
we can get the following equation: 

 
2

1

135
( ) ( ) .

7
gT g   (A.1) 

Then, by the symmetry of the network S N, we know that 
nodes L and R have the same capture probability. Therefore, 
the average first pass time for a wanderer to reach node L from 
node 1, that is, the time it takes for a walker to reach either L 
or R after 𝑇 𝑔  from node 1. It ends if it reaches node L, 
and vice versa, it reaches R after another 𝑇 𝑔  time. The 
following relation is satisfied: 

2 2
1, 1 1 ,

1 1
( ) ( ) ( ( ) ( )).

2 2L R LT g T g T g T g    

From the symmetry of the network S N , we know that: 
𝑇 , 𝑔 𝑇 , 𝑔 . From this, we get the following equation: 

2
1, , 1

135
( ) ( ) 2 ( ) 2 ( ) .

7
g

L R LT g T g T g     

In the following we will consider the relationship between 
𝑇 𝑔  and 𝑇 𝑔  when node 1, nodes L and R are all 
used as capture nodes. When only the corner node L is used 
as the capture node, it takes two steps to reach the capture 
node L from any node 𝑖 ∈ 𝑁 𝑔 / 1, 𝐿, 𝑅 : 

(1) The walker reaches the corner node for the first time, 
and the walk ends if the node is node L, otherwise, the second 
step is performed. 

(2) The first step is repeated with a random wander starting 
from the corner node when the first reached node is 1 or R. 

When 1, 𝐿, 𝑅   is used as a capture node, 𝑖 ∈ 𝑁 𝑔 /
1, 𝐿, 𝑅  is captured by 1, 𝐿, 𝑅  with equal probability. We 

can obtain the following equation: 



 

249 

( )

3
, 1,

( )

3
1,

( )

1

sum i,L

3 2

T

.

(g)= T

[ ( ) (1 ) ( )]

1
( 2[ (| ( ) | 3) 1] ( )

3

4
( ) | ( ) )

(g)

)

| (
3

i N g

i q L L
i N g

i L
i N g

sum

T g T g

T g N g T g

T g N g T g








 

  

 










 

where 𝑞 ∈ 1, 𝐿, 𝑅   serves as the capture node and the 
random walk starts from any node i, we can get the following 
relation: 

,

,

1,

0, 1/ .

q L

q L

q L

q R





 
  

 

Since 𝑇 𝑔   is known, we can obtain the following 
equation: 

3 2
1

1

4
( ) ( ) | ( ) | ( )

3
809 135 10987 135 44619

( ) 6 ( ) 6 .
1920 7 465 7 1984

sum sum

g g g g

T g T g N g T g



 

  

 

Similarly, with the symmetry of the network S N, we get: 
2

( )

3 2
1

( )

3 2
1

1

2( ) (

1
( [ (| ( ) | 3) 1] ( )

3

1
( ) | ( ) | ( )

3
71 135 749 135 44619

( ) 6 ( ) 6 .

)

128 7 31

)

7 1984

sum i
i N g

i
i N g

sum

g g g g

T g T g

T g N g T g

T g N g T g









  

 

 










(A.2)

2
2

1

( )
( )

| ( ) | 2

355 135 3745 135 223095
( ) 6 ( ) 6

128 7 31 7 1984 .
12 6 2

sum
sum

g g g g

g

T g
T g

N g






 


 

 

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