Exact solution of mean geodesic distance for Vicsek fractals

Exact solution of mean geodesic distance for Vicsek fractals

Zhongzhi Zhang zhangzz@fudan.edu.cn    Shuigeng Zhou sgzhou@fudan.edu.cn    Lichao Chen    Ming Yin    Jihong Guan Department of Computer Science and Engineering, Fudan University, Shanghai 200433, China Shanghai Key Lab of Intelligent Information Processing, Fudan University, Shanghai 200433, China Department of Computer Science and Technology, Tongji University, 4800 Cao’an Road, Shanghai 201804, China
July 4, 2019
Abstract

The Vicsek fractals are one of the most interesting classes of fractals and the study of their structural properties is important. In this paper, the exact formula for the mean geodesic distance of Vicsek fractals is found. The quantity is computed precisely through the recurrence relations derived from the self-similar structure of the fractals considered. The obtained exact solution exhibits that the mean geodesic distance approximately increases as an exponential function of the number of nodes, with the exponent equal to the reciprocal of the fractal dimension. The closed-form solution is confirmed by extensive numerical calculations.

pacs:
83.80.Rs, 05.10.-a, 89.75.-k, 61.43.Hv

The concept of fractals plays an important role in characterizing the features of complex systems in nature, since many objects in the real world can be modeled by fractals Ma82 (). In the last two decades, a great deal of activity has been concentrated on the studies of fractals HaBe87 (); BeHa00 (). It has been shown BeOs79 (); GeMaAh80 (); GeAhMaKi81 (); GrKa82 (); ScSc89 (); HiBe06 (); Hi07 (); RoAv07 () that regular fractals capture important aspects of critical percolation clusters, aerogels, amorphous solids, and unusual phase transition in the Ising model. Among various regular fractals, the Vicsek fractals Vi83 () are a class of typical candidates for exact mathematical ones and have received much attention. A variety of structural and dynamical properties of Vicsek fractals have been investigated in much detail, including eigenvalue spectrum JaWu94 (), eigenstates WeGr85 (), Laplacian spectrum BlJuKoFe03 (), random walks Vi84 (), diffusion WaLi92 (), and so on. The results of these investigations uncovered many unusual and exotic features of Vicsek fractals.

A central issue in the study of complex systems is to understand how their dynamical behaviors are influenced by underlying geometrical and topological properties Ne03 (); BoLaMoChHw06 (). Among many fundamental structural characteristics CoRoTrVi07 (), mean geodesic distance is an important topological feature of complex systems that are often described by graphs (or networks) where nodes (vertices) represent the component units of systems and links (edges) stand for the interactions between them AlBa02 (); DoMe02 (). Mean geodesic distance is defined as the mean length of the shortest paths between all pairs of nodes. It has been well established that mean geodesic distance directly relates to many aspects of real systems, such as signal integrity in communication networks, the propagation of beliefs in social networks or of technology in industrial networks. Recent studies indicated that a number of other dynamical processes are also relevant to mean geodesic distance, including disease spreading WaSt98 (), random walks CoBeTeVoKl07 (), navigation Robe0607 (), to name but a few. Thus far great efforts have been made to valuate and understand the mean geodesic distance of different systems ChLu02 (); CoHa03 (); HoSiFrFrSu05 (); DoMeOl06 (); ZhChZhFaGuZo08 (); ZhCoFeRaRoZh08 ().

Despite the importance of this structural property, to the best of our knowledge, the rigorous computation for the mean geodesic distance of Vicsek fractals has not been addressed. To fill this gap, in this present paper we investigate this interesting quantity analytically. We derive an exact formula for the mean geodesic distance characterizing the Vicsek fractals. The analytic method is on the basis of an algebraic iterative procedure obtained from the self-similar structure of Vicsek fractals. The obtained precise result shows that the mean geodesic distance exponentially with the number of nodes. Our research opens the way to theoretically study the mean geodesic distance of regular fractals and deterministic networks BaRaVi01 (); DoGoMe02 (); ZhZhZoChGu07 (). In particularly, our exact solution gives insight different from that afforded by the approximate solution of stochastic fractals.

Figure 1: Illustration of a particular Vicsek fractal .

The classical Vicsek fractals are constructed iteratively Vi83 (); BlJuKoFe03 (). We denote by (, ) the Vicsek fractals after generations. The construction starts from () a star-like cluster consist of nodes arranged in a cross-wise pattern, where peripheral nodes are connected to a central node. This corresponds to . For , is obtained from . To obtain , we generate replicas of and arrange them around the periphery of the original , then we connect the central structure by additional links to the corner copy structures. These replication and connection steps are repeated times, with the needed Vicsek fractals obtained in the limit , whose fractal dimension is . In Fig. 1, we show schematically the structure of . According to the construction algorithm, at each time step the number of nodes in the systems increase by a factor of , thus, we can easily know that the total number of nodes (network order) of is .

After introducing the Vicsek fractals, we now investigate analytically the mean geodesic distance between all the node pairs in the fractals. We represent all the shortest path lengths of as a matrix in which the entry is the geodesic distance from node to node , where geodesic distance is the path connecting two nodes with minimum length. The maximum value of is called the diameter of . A measure of the typical separation between two nodes in is given by the mean geodesic distance defined as the mean of geodesic lengths over all couples of nodes:

(1)

where

(2)

denotes the sum of the geodesic distances between two nodes over all pairs.

Figure 2: A schematic illustration of the iterative construction for , which is obtained by joining copies of denoted as , , , , and , respectively.

We continue by exhibiting the procedure of the determination of the total distance and present the recurrence formula, which allows us to obtain of the generation from of the generation. By construction, the fractal is obtained by the juxtaposition of copies of that are consecutively labeled as , , , , see Fig. 2. This obvious self-similar structure allows us to calculate analytically. It is easy to see that the total distance satisfies the recursion relation

(3)

where is the sum over all shortest path length whose endpoints are not in the same branch. The solution of Eq. (3) is

(4)

Thus, all that is left to obtain is to compute .

The paths that contribute to must all go through at least one of the edge nodes (such as , , , and in Fig. 2) at which the different branches are connected. The analytical expression for , named the crossing path length, can be derived as below.

Denote as the sum of all shortest paths with endpoints in and . For convenience, we denote by the central branch of . According to whether or not the two branches are adjacent, we sort the crossing path length into two classes: (), (, , and ). For any two crossing paths in the same class, they have identical length. Therefore, in the following computation of , we will only consider and . The total sum is then given by

(5)

To calculate the crossing path length and , we give the following definition and notations. We define external nodes of as the nodes that will be linked to one of its copes at step to form . Let denote the sum of length of the path from an external node of to all nodes in including the external node itself. We assume that the two branches and are connected at two nodes and , which separately belong to and , and that and are linked to each other at two nodes and that are in and , respectively.

In order to determine , we should compute the diameter of first. By construction, one can see that the diameter equals the path length between arbitrary pair of external nodes of . Thus, we have the following recursive relation:

(6)

Considering the initial condition , Eq. (6) is solved inductively to obtain

(7)

which is independent of .

We now calculate the quantity . Let denote the external node of , which is in the branch . By definition, can be given by the sum

(8)

We denote the second and third terms in Eq. (8) by and , respectively. Thus, . The quantity is evaluated as follows:

(9)

where and were used. Analogously,

(10)

With Eqs. (9) and  (10), Eq. (8) becomes

(11)

Using , and , Eq. (11) is resolved by induction

(12)

With above obtained results, we can determine the length of crossing paths and , which can be expressed in terms of the previously explicitly determined quantities. By definition, is given by the sum

(13)

where we have used the equivalence relation .

Proceeding similarly,

(14)

Inserting Eqs. (13) and (14) into Eq. (5), we have

(15)

Substituting Eq. (15) into Eq. (4) and using the initial value , we can obtain the exact expression for the total distance

(16)

Then the analytic expression for mean geodesic distance can be obtained as

(17)

In the infinite system size, i.e.,

(18)

where the exponent is equal to the reciprocal of the fractal dimension. Thus, the mean geodesic distance grows exponentially with increasing size of the system. In contrast to many recently studied network models mimicking real-life systems in nature and society AlBa02 (); DoMe02 (), the Vicsek fractals are not small worlds despite of the fact that these fractals show similarity (fractality) observed in many real-world systems.

We have checked our analytic result against numerical calculations for different and various . In all the cases we obtain a complete agreement between our theoretical formula and the results of numerical investigation, see Fig. 3.

Figure 3: Mean geodesic distance versus network order on a log-log scale. The solid lines are guides to the eyes.

To sum up, in complex systems the mean geodesic distance plays an important role. It has a profound impact on a variety of crucial fields, such as information processing, disease or rumor transmission, network designing and optimization. In this paper, we have derived analytically the solution for the mean geodesic distance of Vicsek fractals which have been attracting much research interest. We found that in the infinite network size limit the mean geodesic distance scales exponentially with the number of nodes. Our analytical technique could guide and shed light on related studies for deterministic fractals and network models by providing a paradigm for calculating the mean geodesic distance. Moreover, as a guide to and a test of approximate methods, we believe our vigorous solution can prompt the studies on random fractals.

We would like to thank Yichao Zhang for preparing this manuscript. This research was supported by the National Basic Research Program of China under grant No. 2007CB310806, the National Natural Science Foundation of China under Grant Nos. 60496327, 60573183, 90612007, 60773123, and 60704044, the Shanghai Natural Science Foundation under Grant No. 06ZR14013, the China Postdoctoral Science Foundation funded project under Grant No. 20060400162, the Program for New Century Excellent Talents in University of China (NCET-06-0376), and the Huawei Foundation of Science and Technology (YJCB2007031IN).

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