Modified box dimension and average weighted receiving time on the weighted fractal networks.

In this paper a family of weighted fractal networks, in which the weights of edges have been assigned to different values with certain scale, are studied. For the case of the weighted fractal networks the definition of modified box dimension is introduced, and a rigorous proof for its existence is given. Then, the modified box dimension depending on the weighted factor and the number of copies is deduced. Assuming that the walker, at each step, starting from its current node, moves uniformly to any of its nearest neighbors. The weighted time for two adjacency nodes is the weight connecting the two nodes. Then the average weighted receiving time (AWRT) is a corresponding definition. The obtained remarkable result displays that in the large network, when the weight factor is larger than the number of copies, the AWRT grows as a power law function of the network order with the exponent, being the reciprocal of modified box dimension. This result shows that the efficiency of the trapping process depends on the modified box dimension: the larger the value of modified box dimension, the more efficient the trapping process is.

Recently, self-similar fractals have attracted much attention. The renormalization procedure tiles a network according to the box-covering algorithm. Self-similarity is then obtained if the network structure remains invariant under the renormalization. Gallos et al. reviewed the findings of self-similarity in complex networks. Using the box-covering technique, it was shown that many networks present a fractal behavior, which is seemingly in contrast to their small-world property1. Then they used scaling theory to quantify the degree of correlations in the particular case of networks with a power-law degree distribution2. Starting from the fractal network, Rozenfeld et al.3 applied renormalization group theory to study complex networks using the box covering technique, which is useful to classify network topologies into universality classes in the space of configurations. After defining a unified mathematical framework for both immunization and spreading, Morone and Makse provided its optimal solution in random networks by mapping the problem onto optimal percolation and found that the top influencers are highly counterintuitive4.

Motivated by the hierarchial and scale-free networks56, Komjáthy and Simon7 introduced deterministic the scale-free graphs derived from a graph directed self-similar fractal. Chen et al.8 constructed a class of scale-free networks with fractal structure based on the subshift of finite type and base graphs. When embedding the growing network into the plane, its image is a graph-directed self-affine fractal, whose Hausdorff dimension is related to the power law exponent of cumulative degree distribution.

Unfortunately, many previous works have focused on the un-weighted networks. In real networks, the relations between two nodes have been affected by specific physical properties of network elements, including the number of passengers traveling yearly between two airports in airport networks9, to the intensity of predator-prey interactions in ecosystems10 or the traffic measured in packets per unit time between routers in the Internet11. So weighted networks commendably represent the natural framework to describe natural, social, and technological systems, in which the intensity of a relation or the traffic between elements is an important parameter1213. In general terms, weighted networks are extension of networks or graphs1415, in which each edge between nodes i and j is associated with a variable , called the weight.

A key quantity related to weighted networks is the mean weighted first-passage time (MWFPT), that is, the expected weighted first time for the walker starting from a source node to a given target node. The average weighted receiving time (AWRT) is the sum of mean weighted first-passage times (MFPTs) for all nodes absorpt at the trap located at a given target node161718. In 2013, Dai et al. introduced the non-homogenous weighted Koch networks depending on the three weight factors19. They defined the average weighted receiving time (AWRT) for the first time and studied the AWRT on random walk. Recently, fractals have also attracted an increasing attention in physics and other scientific fields, owning to the striking beauty intrinsic in their structures and the significant impact of the idea of fractals. These structures have been a focus of research objects and many underlying properties have been found. So it makes sense to combining weighted networks with fractals which are called weighted fractal networks. Daudert and Lapidus20 studied weighted graphs and random walks on the Koch snowflake. Carletti and Righi21 defined a class of weighted complex networks whose topology can be completely analytically characterized in terms of the involved parameters and of the fractal dimension.

This paper is organized as follow. Based on weighted fractal networks21, we introduce a family of the weighted fractal networks depending on the number of copies s and the weight factor r in the next section. In Section 3, the definition of modified box dimension and a rigorous proof for its existence are given in the case of the weighted fractal networks. In Section 4, the average weighted receiving time (AWRT) on random walk is obtained by recursive formulas for and . When the weight factor is larger than the number of copies, we show that the efficiency of the trapping process depends on the modified box dimension: the larger the value of modified box dimension, the more efficient the trapping process is. In the last section we draw conclusions.

Weighted fractal networks

In this section a family of weighted fractal networks are introduced.

Let be a positive real numbers, and be a positive integer.

(1) Let be our base graph, composed by nodes . We partition into two non-empty sets , labeled attaching node, all other nodes except for the attaching node, satisfying the symmetry of nodes in . The edge set of is denoted by . If the pair is connected by an edge, then this edge is denoted by . Each of with unit weight.

Remark: The symmetry of nodes in means that the network is invariable no matter how two arbitrary nodes i and j are exchanged .

(2) For any , is obtained from (see Fig. 1): has one attaching node labelled by . Let be s copies of . is obtained by the union of s copies . Let be the set of nodes in , which is . If the pair is connected by an edge, then this edge is denoted by . Let be the set of edges in . For let us denote by the node in image of the labeled node . Let , then link all those label nodes to the attaching node , each of the edges assigns weight .

Figure 1

Take the ‘Cantor dust’ weighted fractal networks for example.

The weighted fractal networks are set up.

According to the construction of the weighted fractal networks, one can see that , the weighted fractal networks of n- th generation, is characterized by three parameters n, s and r: n being the number of generations, s being the number of copies, and r representing the weight factor. The total number of nodes in is as follows.

Modified box dimension

Definition 3.1. The weighted shortest path of nodes i and j in the weighted graphs is given by

where Γ is the set of paths linking i and j in 21.

The self-similar property of real-world networks, box-counting method turns to be practical22. The method works as follows: we partition the nodes into boxes of size . The maximal distance between vertices within a box is at most . The resulting number of boxes needed to tile the networks denoted by . Then the box dimension is defined by .

Modified box dimension was motivated by the fact that in the case of the weighted fractal networks the original definition of box dimension is infinite. It is worth mentioning, our new concept of dimension does exist and is finite for this model as Theorem 3.3 shows.

Definition 3.2. The modified box dimension is defined by

where and denotes the minimal number of boxes of size that we need to cover .

Theorem 3.3. For the weighted fractal networks the modified box dimension:

where s is the number of copies, r is the weighted factor.

For convenience of description, we recall the following notations.

(i) Let be the set of nodes in , which is where , and be the set of edges in .

(ii) Given , we denote the common prefix by s.t. and .

(iii) We fix an arbitrary self-map p of such that for , , i.e., .

For a word , we define

Then is an edge in .

The diameter of G

Lemma 3.4. The diameter of is

Proof. We will prove this from two respects.

(1) Considering the worst case scenario, i.e., choosing and such that (i) . (ii) , yields that

(2) We construct a path between two arbitrary nodes x and y that is no longer than . Let where and , where .

Starting from x the first half of the path is as follows:

Starting from the first half of the path is as follows.

In this way

Clearly,

Lower bound of modified box dimension

Lemma 3.5. The following inequality holds for ,

Proof. It is easy to see that we need one l1-box to cover . It follows from the weighted structure of that contains copies of and nodes. This implies that we can cover with l1-boxes. #

Lemma 3.6.

proof. Suppose that and two arbitrary nodes in contained by the same l1-box, i.e., the distance between x and y is not greater than . If we blow them up, we get two cylinder sets of nodes:

and

Next, we calculate the maximal distance between the elements of and . Considering the worst case scenario , and . Namely that

and

Starting from it at most takes steps to reach the . Similarly, starting from we need at most steps to reach .

Thus the distance between and is not greater than . Therefor, the same -boxing that we have used in is an appropriate -boxing for . #

From Eqs (4) and (5), we can see that Then from Eqs (1, 2, 3), we obtain

Upper bound of modified box dimension

Lemma 3.7. The following inequality holds for

Proof. For every digit , we define the cylinder set of words with .

Let . Now we give a lower bound on the shortest path between and thus we need at least steps on any path between and . These witness must be in distinct boxes, so we need at least l1-boxes to cover . #

Lemma 3.8. The following inequality holds

Proof. We have constructed nodes in whose pairwise distance is greater than . It is enough to show that we can find the same number of nodes (i.e., ) in , such that the pairwise distances between them are greater than , this implies

Let

where the cylinder set of nodes

Now we give a lower bound on the shortest path between and where . We need at least steps on any path between and . Hence these witness must be in distinct boxes. So we need at least -boxes to cover . i.e., substitutily and yields that

From Eq. (7) we can see that . Then from Eqs (1, 2, 3), we obtain

Proof of Theorem 3.3. Combining lower bound and upper bound of modified box dimension i.e., Eqs (6) and (8) yields Theorem 3.3, hence:

The average weighted receiving time on random walk

The purpose of this section is to determine explicitly the average weighted receiving time (AWRT) and to show how scales with network order. We aim at a particular case on with the trap placed on the attaching node , let us denote by 0. All other nodes, except for the attaching node, are denoted by .

Assuming that the walker, at each step, starting from its current node, moves uniformly to any of its nearest neighbors.

For two adjacency nodes i and j, the weighted time is defined as the corresponding edge weight . The mean weighted first-passing time (MWFPT) is the expected first arriving weighted time for the walks starting from a source node to a given target node. Let be the mean weighted first-passage time (MWFPT) for a walker starting from Node i to Node j. Let be the MWFPT from Node i to the trap. is the average weighted receiving time (AWRT), which is defined as the average of over all starting nodes other than the trap. is the key question concerned in this paper.

Theorem 4.1. For a large system, i.e., ,

(1) if , we have the following expression for the dominating term of :

where ;

(2) if , we have the following expression for the dominating term of :

(3) if , we have the following expression for the dominating term of :

Remark. This confirms that in the large limit, if then the AWRT grows as a power law function of the network order with the exponent, represented by , being the reciprocal of . When grows from 0 to 1, the exponent decreases from approaches 1. This also means that the efficiency of the trapping process depends on the modified box dimension: the larger the value of modified box dimension, the more efficient the trapping process is.

Proof. By definition, is given by

Here, we denote by the sum of MWFPTs for all nodes to absorption at the trap located the attaching node , i.e.,

Thus, the problem of determining is reduced to finding . We will compute by segmenting .

From the self-similarity construction method of , can be regarded as merging groups, sequentially denoted by . The groups are obtained as follows. includes the central Node 0 and s nodes denoted by , Each node in s nodes is linked to the central Node 0 through the weighted time ; is a copy of . In order to completely explain the division of the general weighted fractal networks, we present the special division of the ‘Sierpinski’ weighted fractal networks when (see Fig. 2).

Figure 2

Take the ‘Sierpinski’ weighted fractal networks G, for example, G2 is regarded as merging , , , .

Through this division, we can rewrite the sum as follows:

where .

Thus, the problem of determining is reduced to finding . Note that the strength of Node is according to the construction of . Using the division of , we have

Through the reduction of Eq. (13), we obtain

In the given initial network , let be the the mean weighted first-passage times (MWFPTs) for a walker from Node i in to the attaching node 0 in . Here, we denote by the sum of MWFPTs for all nodes to the attaching node 0, i.e., . Because of the symmetry of nodes , and . is a constant number for the given initial network . Considering the initial network , one can prove

Through the simplifications of Eq. (15), we obtain

From Eq. (16), we can solve Eq. (14) recursively to yield

Using the construction of , we have

When from Eqs (17) and (18), we can solve Eq. (10) inductively to yield

Hence, , which we are concerned about, could be expressed as follows:

(1) If , the dominating term of is written as follows:

For a large system, i.e., , from Eq. (1) we have the following expression for the dominating term of

where .

(2) If , the dominating term of is written as follows:

For a large system, i.e., , from Eq. (1) we have the following expression for the dominating term of :

(3) If r = s, from Eqs (17) and (18), we can solve Eq. (12) inductively to yield

For a large system, i.e., , from Eq. (1) we have the following expression for the dominating term of :

Conclusions

In this paper, we introduced a family of weighted fractal networks with weight factor r. We mainly studied its modified box dimension and AWRT on the weighted fractal networks. For the case of , the AWRT grows as a power law function of the network order with the exponent, being the reciprocal of . We found that when grows from 0 to 1, the exponent decreases from approaches 1. This result showed that the efficiency of the trapping process depends on the modified box dimension: the larger the value of modified box dimension, the more efficient the trapping process is. Otherwise, for the case of , the AWRT grows linearly with the network size , and for the case of , the AWRT grows with increasing order as .

It should be mentioned that we only studied a particular family of weighted fractal networks, whether the conclusion also holds for other more general networks, which needs further investigation.

Additional Information

How to cite this article: Dai, M. et al. Modified box dimension and average weighted receiving time on the weighted fractal networks. Sci. Rep. 5, 18210; doi: 10.1038/srep18210 (2015).