Scaling of average receiving time on weighted polymer networks with some topological properties

In this paper, a family of the weighted polymer networks is introduced depending on the number of copies f and a weight factor r. The topological properties of weighted polymer networks can be completely analytically characterized in terms of the involved parameters and/or of the fractal dimension. Moreover, assuming that the walker, at each step, starting from its current node, moves to any of its neighbors with probability proportional to the weight of edge linking them, namely weight-dependent walk. Then, we calculate the average receiving time (ART) with weighted-dependent walks, which is the sum of mean first-passage times (MFPTs) for all nodes absorpt at the trap located at the central node as a recursive relation. The obtained remarkable results display that when 1f+1<r<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{1}{f+1} < r < 1$$\end{document}, the ART grows sublinearly with the network size; when r=1f+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r=\frac{1}{f+1}$$\end{document}, ART grows with increasing size Ng as ln2Ng\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm{ln}}^{2}{N}_{g}$$\end{document}; when 0<r<1f+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0 < r < \frac{1}{f+1}$$\end{document}, ART grows with increasing size Ng as ln Ng. In the treelike polymer networks, ART grows with linearly with the network size Ng when r = 1. Thus, the weighted polymer networks are more efficient than treelike polymer networks in receiving information.


Weighted treelike networks
In this section, a family of weighted polymer networks are introduced. Intuited by polymer networks 24,25 and Weighted Fractal Networks (WFN for short) 26 , a family of weighted polymer networks are constructed in a deterministically iterative way.
Let r(0 < r < 1) be a positive real numbers, and f(f ≥ 1) be a positive integer. Denote by G g the weighted polymer networks after g iterations, and the following is the iterative algorithm to create weighted polymer networks: (1) For g = 0, G 0 consists of an isolated node, called the central node. For g = 1, f new nodes are generated connecting the central node to form G 1 . Let G 1 be our base graph, composed by f + 1 nodes and f edges with unit weight. The f + 1 nodes in G 1 are all the attaching nodes, labeled by  f 0, 1, 2, , .
(2) For g = 2, G 2 is obtained from G 1 : , , f ( ) be f + 1 replicas of G 1 , whose weighted edges have been scaled by the weight factor r. For =  i f 0, 1, 2, , , let us denote by i′ the central node in G i 1 ( ) . Then merge the central node i′ in G i 1 ( ) and Node i in G 1 into a single new node, still labelled by =  i i f ( 0, 1, , ). Figure 1 illustrates the iterative construction processes of a particular network from g = 1 to g = 3 for the The weighted polymer networks is one of type of WFN. According to Carletti and Righi 26 , WFN are scale-free, the exponent being the fractal dimension. WFN exhibit the "small-world" property (i.e. slow (logarithmic) increase of the average shortest path with the network size) and large average clustering coefficient. Thus, the fractal dimension of weighted polymer networks is completely characterized by two main parameters: the number of copies f ≥ 1 and the weight factor 0 < r < 1. We have that the fractal dimension of the weighted polymer networks is According to the construction approach, it is easy to derive that at each iterative step g i (g i ≥ 1), the number of newly generated nodes is Then the total number of nodes at each generation g is g g g i g 1 i and the total number of edges in G g is E g = N g − 1 = (f + 1) g − 1.

Topological properties of weighted polymer networks
The aim of this section is to characterize the topology of weighted polymer networks, by analytically studying their properties such as the average degree, the average node strength, the node strength distribution, and the average weighted shortest path.
Average degree and average node strength. The degree of a node i in a network, that is, the number of connections or edges the node i has to other nodes, is denoted by deg(i). The average degree of the weighted polymer networks G g , denoted by ad(G g ), is defined as Hence in the limit of large g, the average degree ad(G g ) is finite and it is asymptotically given by In the weighted polymer networks G g , a weight w ij is assigned to the edge connecting the nodes i and j, and the strength of node i can be defined as where the sum index j runs over the set ν(i) of neighbors of i. The strength of a node integrates the information concerning its connectivity and the weights of its links 33,34 . Then using the recursive construction, we can explicitly compute the total node strength = ∑ ∈ S s g i G i g , and, provided ≠ + r f 1 1 , easily show that Because r < 1, we trivially find that the average node strength goes to zero as g increases: ; in this case, in fact S g = 2fg grows linearly with g, thus slower than N g .
Node strength distribution. Let n(s) denote the number of nodes in the weighted polymer networks G g that have strength s. Let s i (g i ) be the strength of any one of newly generated nodes i at each iterative step g i (g i > 0). Assume that node i entered the networks at generation g(g > 0), then s i (g) = r g−1 . By construction, the strength of node i entered the networks at generation g i (0 < g i < g) For g i = 0, the strength of the initial central node labeled by 0, equals to . Using the property of iterative construction method, we can conclude: − n s g f f a nd n s ( ( )) ( 1) , ( (0)) 1 . And therefore, for large g, n(s) can be obtained as where c is constant. Thus, n(s) also follows a power-law distribution.
Average weighted shortest path. By definition the average weighted shortest path(AWSP) of the weighted networks G g 35 is given by The modular recursive construction of G g allows us to calculate the exact value of S tot (g). At step g + 1, we incise G g into f + 1 branches, which we label as is a copy of G g and has the same structure as G g , while their edge weights have been scaled by a weight factor r. The central are all connected to central node 0′ of G g (0) by f edges with unit weight. Thus, the total of shortest distances S tot (g) satisfies the following recursion: where Ω g is the sum over all weighted shortest paths whose nodes are not in the same copy of Note that the weighted paths that contribute to Ω g must all go through central node 0′ of G g branches are connected. This recursive relation can be elaborated as follows: The first term on the rhs of (5) describes the sum of the weighted shortest path linking nodes i and j in , respectively, i.e., Using the scaling mechanism for the edges, the above sum can be easily identified with One can prove (see Method) that Considering S tot (1) = f 2 , we can solve Eq. (4) recursively to yield We find that if r = 1 then S tot (g) = (fg − 1)(f + 1) 2g−1 + (f + 1) g−1 , which coincides with the S tot (g) in ref. 24. Therefore which provides the following asymptotic behavior in the limit of large g (see Fig. 4). When g → ∞, Thus the network grows unbounded but with the logarithm of the network size, while the weighted shortest distances stay bounded.
Recalling N g = (f + 1) g as given in Eq. (1), we have = We can also compute the average shortest path (ASP), d g , formally obtained by setting r = 1. Hence, when the network size is large enough, we have g g

Average receiving time on weighted polymer networks
The purpose of this section is to determine explicitly the average receiving time 〈T〉 g and show how it scales with network order. We aim at a particular case on G g with the perfect trap being located at the central node, labelled by 0. The process of biased walks is that the particle (walker), at each time step, starting from its current Node i, jumps to its neighbor Node j with probability → p i j w (see Eq. (11)).
Scientific RepoRts | 7: 2128 | DOI:10.1038/s41598-017-02036-0 For weight-dependent walk, a walker chooses one of its nearest neighbors with probability proportional to the weight of edge linking them 36,37 . The transition probability from node i to its neighbor j is where s i denotes the strength of node i (see Eq. (2)).
For convenience of description, let us denote by  f 0, 1, 2, , the f + 1 attaching nodes in G g , and by −  N 4, 5, , 2 g and N g − 1 all other nodes except for the f + 1 attaching nodes. Let F ij (g) be the mean first-passage time (MFPT) for a walker starting from Node i to Node j. Let F i (g) be the MFPT from Node i to the trap. 〈T〉 g is the average receiving time (ART), which is defined as the average of F i (g) over all starting nodes other than the trap. 〈T〉 g is the key question considered in this section.
By definition, 〈T〉 g is given by Here we denote by T tot (g) the sum of MFPTs for all nodes to absorption at the trap located the central Node 0, i.e., Thus, the problem of determining 〈T〉 g is reduced to finding T tot (g). We will compute T tot (g) by segmenting G g . From the iterative construction method of G g , G g can be regarded as merging f + 1 groups, sequentially denoted by ( ) are f + 1 replicas of G g−1 , whose weighted edges have been scaled by the weight factor r.  (  0, 1, , ). The process is described in Fig. 2. Through this division we could rewrite the sum T tot (g) as follows: We now elaborate Eq. (12). The first term on the rhs of Eq. (12) describes the sum of MFPTs for all nodes in 1, , ) is a copy of G g−1 and the scaling mechanism for edges, the first term in the rhs of Eq. (12) can be identified with (f + 1)T tot (g − 1); the second term describes the sum of MFPTs for all nodes in − G g j 1 ( ) from Node =  j j f ( 1, 2, , ) to Central Node 0.
Because of the symmetry of nodes  f 1, 2, , , . Eq. (12) can be simplified as Thus, the problem of determining T tot (g) is reduced to finding F 1 (g). Using the construction of the weighted polymer networks G g and the scaling mechanism for edges, we obtain www.nature.com/scientificreports/ 7 Scientific RepoRts | 7: 2128 | DOI:10.1038/s41598-017-02036-0  From Eqs (14) and (15), we can further have Considering the initial condition  We find that if r = 1 then Recalling N g = (f + 1) g and g = ln N g /ln(f + 1), we have  According to Eqs (19) and (20), ART 〈T〉 g versus g for the range of g ≤ 50 on a semilogarithmic scale is shown in Figs 5, 6 and 7. From Eq. (22), we can have draw the conclusions as follows: Case 1: When r = 1, ART grows linearly with the network size N g . Figure 5 shows that ART increases with the increase of the values of f. That is to say, the smaller the value of f is, the more efficient the trapping process is. (1) When f is kept fixed, the exponent θ(f, r) is an increasing function of r(0 < r < 1) in Fig. 5. When r grows from 0 to 1, the exponent increases from 0 and approaches 1, indicating that ART grows sublinearly with the network size N g . This also means that the efficiency of the trapping process depends on the parameter r: the smaller the value of r, the more efficient the trapping process is. (2) When r(r ≠ 1) is kept fixed, ART grows sublinearly with the network size N g and the exponent θ(f, r) is an increasing function of the values of f in Fig. 5. That is to say, the smaller the value of f is, the more efficient the trapping process is. is an increasing function of the values of d fract , which means that the smaller the value of d fract is, the more efficient the trapping process is. , ART grows with increasing size N g as ln 2 N g according to Eq. (22). Figure 6 shows the smaller the value of f is, the more efficient the trapping process is. , ART grows with increasing size N g as ln N g in Fig. 7. When f is kept fixed, the smaller the value of r, the more efficient the trapping process is. When r(r ≠ 1) is kept fixed, the smaller the value of f is, the more efficient the trapping process is. If 0 < r < 1, then  (1) (1) (1) ( 2) where d 10 = 1, d 12 = 2 have been used. Substituting Eqs (24) and (25)  If 0 < r < 1, then