Spectral dimensions of hierarchical scale-free networks with shortcuts
The spectral dimension has been widely used to understand transport properties on regular and fractal lattices. Nevertheless, it has been little studied for complex networks such as scale-free and small world networks. Here we study the spectral dimension and the return-to-origin probability of random walks on hierarchical scale-free networks, which can be either fractals or non-fractals depending on the weight of shortcuts. Applying the renormalization group (RG) approach to the Gaussian model, we obtain the spectral dimension exactly. While the spectral dimension varies between and for the fractal case, it remains at , independent of the variation of network structure for the non-fractal case. The crossover behavior between the two cases is studied through the RG flow analysis. The analytic results are confirmed by simulation results and their implications for the architecture of complex systems are discussed.
The problem of random walks (RWs) on complex networks have attracted much attention as a model to study diffusion processes on complex systems such as fad or disease spreading over social networks, data packets transport in the Internet, data mining on the web, and so on hughesbook (); havlinbook (); rednerbook (); albert02 (); dorogovtsev03a (); newman03 (). Such importance in theoretical and application aspects has led to the extensive studies on the problem sven01 (); noh04 (); masuda04 (); sood05 (); bollt05 (); condamin07 (); baronchelli08 (); zhang09 (); tejedor09 (), but it still remains unclear how RW properties depend on network structure. In this paper, we investigate the effect of shortcut links on RW motion. Shortcuts may be classified into two types: Long-range shortcuts, utilized in a skeleton or superhighways, and local shortcuts, complementary to the skeleton or local roads dhkim04 (); goh06 (); wu06 (). It was recently shown systematically rozenfeld10 () that long-range shortcuts can change the type of networks from fractal to non-fractal. The fractal (non-fractal) network is the one in which the mean separation between two nodes scales with system size in a power-law (logarithmic) manner. We are particularly interested in how RW properties are changed as long-range shortcuts are added and thereby the network changes from a fractal to a non-fractal network.
The Gaussian model is useful to study RW problems analytically gaussian (), which we use to understand the RW motion on hierarchical scale-free networks berker79 (); berker06 (); rozenfeld07a (); rozenfeld07b (). In this artificial network, we can control the degree distribution and the link weight of shortcuts. We consider RWs on this hierarchical network, and obtain the exact solution of the return-to-origin (RTO) probability by applying the renormalization group (RG) approach to the Gaussian model. The RTO probability is related to the free energy of the Gaussian model and the spectral density function of the Laplacian matrix, all of which are characterized by the spectral dimension hughesbook (); gaussian (). The hierarchical network is recursively organized, so that one can obtain analytically the spectral density function by using the decimation method of the RG transformation. We find that when the network structure is a fractal, the spectral dimension varies in the range ; however, it is fixed at for the non-fractal networks. These analytic results are confirmed by numerical simulations. Moreover, we perform the numerical simulations of RWs on various real-world networks and check the relationship between the network fractality and the spectral dimension.
The paper is organized as follows. In Section II, the hierarchical network model and the formalism to derive the spectral density function of the Laplacian matrix from the Gaussian model is briefly introduced along with their relation to the RTO probability of RWs. In Section III, we apply the RG approach to the Gaussian model and obtain the spectral density function explicitly. In Section IV, numerical results of the RTO probability are presented and compared with the analytic solutions. We summarize our findings and discuss their implications in Section V.
Ii Model and formalism
The network we use in the paper is a modified version of an existing hierarchical network model berker79 (); berker06 (); rozenfeld07a (); rozenfeld07b (), in which shortcuts are assigned weights. We call this modified model the weighted flower (WF) network. The WF network contains two types of links, type I and II, which evolve differently. The WF network is constructed iteratively as follows. We start from a single type-I link between two nodes, which is the 0-th generation. At the next generation, two chains are added between the two nodes at both ends of the type-I link. On one chain, nodes are located and thereby links are placed. On the other chain, nodes are located and thereby links are placed. Those links are assigned as type-I links, and the type of the seed link is changed to type II. These steps - adding nodes and links and assigning link types - are performed at every generation. Every link is assigned weight: for each type-I link and for each type-II link. The link weights will be used for the transition rate of RWers. The growth of the WF network with and is depicted in Fig. 1 (a). We consider only the case and throughout this paper. The type-II links play a role as shortcuts in the system and the parameter represents the transition rate across a type-II link. We point out that in the previous model of the hierarchical network berker79 (); berker06 (); rozenfeld07a (); rozenfeld07b (), the parameter was used as the occupation probability of the type-II links, but here, it is used as the weight of the type-II links. Thus, the two network models are equivalent if or . Otherwise, they are different.
|number of type-I links|
|number of type-II links|
|number of nodes|
The link weights of the WF network at the -th generation are represented in a symmetric matrix form
where and denote the set of type-I and type-II links at the -th generation, respectively, and represents the link connecting node and . Basic structural properties of the WF networks can be obtained analytically berker06 (); rozenfeld07a (); rozenfeld07b () and are summarized in Table 1. Here we defined the degree of node as . The distance between two nodes was defined as the length of the minimum-cost path with the link cost given by a decreasing function of satisfying as noh02 (). While the degree exponent depends on and but remains the same under the variation of the weight , the mean distance between nodes depends on . When , the type-II links are not considered in determining the minimum-cost paths. Then the mean distance between nodes in the WF networks is related to the total number of nodes as berker06 (); rozenfeld07a (); rozenfeld07b ()
Here is the fractal dimension song06 () of the WF networks with . When , shortcuts may participate in the minimum-cost paths and the mean distance is proportional to the number of nodes logarithmically berker06 (); shortestpath ()
Let us consider a RWer located initially at node and going around in the network with the transition probability from node to . Then the probability to find the RWer at node after time is given by , where is the identity matrix and is the Laplacian matrix defined as with .
The RTO probability is determined by the eigenvalues ( of the Laplacian matrix as
where we introduced the spectral density function in the thermodynamic limit perron (). For large , the term in Eq. (LABEL:eq:PRTO) is dominant as and . The contribution from , however, cannot be larger than that from since , but can cause an oscillatory behavior in depending on whether is even or odd. For example, in -dimensional regular lattices, the eigenvalues are symmetrically distributed about , and for small . Then, it follows from Eq. (LABEL:eq:PRTO) that when is even and when is odd.
For fractal lattices, the spectral density function and the RTO probability are characterized by the spectral dimension as havlinbook ():
We use these formulae in further discussion later.
The mean first passage time between two nodes is related to the eigenvalues of the Laplacian matrix via hughesbook (), which scales with system size as for and for bollt05 (). For the fractal WF network with in the -th generation, it was shown that based on the scaling behavior of the mean first passage time between two hub nodes rozenfeld07a (). Thus, it follows that for the case . On the other hand, the mean first passage time for a non-fractal WF network with , and was calculated, which scales as bollt05 (); rozenfeld07a (). Thus, the spectral dimension is in this case. These results suggest that the spectral dimension can be different depending on whether a network is fractal or non-fractal. Motivated by these previous works, we study the spectral dimension of the hierarchical network as it crosses over from a fractal to a non-fractal by varying the parameter in the range .
To derive the spectral density function of the matrix , we use the Gaussian model of which the partition function is given as
where is the number of nodes, , and . is a similarity-transformed Laplacian matrix given in Appendix A. is a parameter introduced in the Gaussian model. The spectral density function is derived from the free energy as
where is assumed to possess a positive infinitesimal imaginary part.
Iii Renormalization group approach to Gaussian model
iii.1 Fixed points and RG flow
The parameter in Eq. (6) is renormalized differently for the two types of links in the WF networks. Thus, we first rescale as in Eq. (6) and introduce distinct parameters and for the respective link types. The Hamiltonian then takes the form , where is given in Eq. (1). The partition function in the n-th generation is written as
Note that initially. The RG transformations for the shortcut weight , and the parameters , and are carried out to obtain the spectral dimension . As we show later, is invariant under the RG transformation and we investigate the fixed points and the RG flow in the plane. If there is only one fixed point at , then the spectral dimension will be the same regardless of . On the other hand, more than one fixed points will indicate a crossover or a transition of depending on .
The WF network in the -th generation contains leaves and links, and the partition function may be written as
Here, each leaf contains nodes, among which are the youngest nodes, and the two remaining old nodes are connected by a type-II link as shown in Fig. 1(b). Integrating out the variables on the youngest nodes, we find that takes a similar form to as
The derivation of Eq. (9) and the properties of , and for small are given in Appendices B and C. In order for in Eq. (9) to take the same functional form as , all the variables should be rescaled as so that the coefficient of be unity. With the introduction of , , and given by
the partition function is related to as
where the function is defined as
As this RG transformation is repeated, the parameters and are renormalized successively according to Eq. (11). The parameter remains at its initial value . On the other hand, the parameter is renormalized in a non-trivial way.
For small , one finds two fixed points in the plane located at
These two fixed points meet when . Let us first consider the case . Near the fixed point , the RG equation, (11), is linearized as
and therefore, after RG transformations, and take the following values:
Since and are not smaller than for and , the fixed point is unstable. Around the other fixed point (, the linearized RG equation is written as
The matrix has two eigenvalues and and the parameters after RG transformations are given by
If , there is only one fixed point at for small . The renormalized value can be calculated from Eq. (11) as
To understand the renormalization of and , which depends on their initial values as well as the the network model parameters and , we consider the following three cases:
(I) If initially, remains at zero and increases according to Eq. (16). (II) If initially and , both and increase by Eq. (16) and the RG flow runs outward from along the curve as obtained from Eq. (16). The RG flow then runs towards the other fixed point , driven by one eigenvalue of in Eq. (17). Once gets sufficiently close to , the RG flow is bent outward along the line , parallel to the other eigenvector of (see Fig. 2 (a)). (III) When , the two fixed points coincide each other. The RG flow along the axis runs quite slowly as the corresponding eigenvalue of is . If , and are renormalized in the same way as in (I). If , the RG flow runs along the curve as obtained from Eq. (20), which is depicted in Fig. 2(b).
iii.2 Spectral density function and spectral dimension
where we used for large . Using the expansion of the functions and given in Appendix B, we can obtain the expansion of up to the first order in and as
where we used and introduced a constant
For the case (I), the shortcuts have no weight and the network is a fractal as long as and . The parameter remains at zero under the RG transformation and is renormalized by Eq. (16). Inserting Eq. (16) in Eqs. (21) and (22) with , we obtain the free energy and then the spectral density function via Eq. (7), which is calculated as
This is consistent with the previous result obtained for the mean first passage time studied in Ref. rozenfeld07a (). The spectral dimension in Eq. (25) is reduced to when , and ranges between and when and differ from . Note that it is different from the fractal dimension given in Eq. (2).
For the case (II), the RG flow starting from a point with non-zero values of and first approaches the fixed point , and then it is repelled in the direction of one eigenvector of the matrix in Eq. (17) as shown in Fig. 2(a). Using Eq. (18) and setting in Eqs. (21) and (22), we obtain the spectral density function
Thus, the spectral density function for the case (II) is a constant for small , indicating that
This means that the spectral dimension is changed by the addition of shortcuts. Remarkably remains at regardless of or .
We remark that Eq. (28) with is reduced to Eq. (24). On the other hand, for , the spectral density function takes a logarithmic form, for . In the scaling regime where is finite, the spectral density function displays the following crossover behavior:
Iv The Return-to-origin probability
In this section, we derive the RTO probability of RWs from the spectral density function obtained in the previous section. The behavior of around , which should be known for Eq. (LABEL:eq:PRTO), can be understood by the symmetry property of the RG equation (11). First let us consider two cases. (i) When both and are odd, and are symmetric and are anti-symmetric with respect to . Thus, the variable and are renormalized in the same manner as Eqs. (16) and (18) around the fixed points and , respectively. This leads to . (ii) When and both and are even, are symmetric and and are anti-symmetric with respect to . Interestingly, the parameter initially close to is renormalized by one RG transformation to a value near zero by Eq. (11), and then follows the same RG flow as given in Eqs. (16) and (18). Therefore, the spectral density function is symmetric about . In these two cases (i) and (ii), it follows that
For other cases than (i) or (ii), there is no fixed point on the line and thus the contribution of around to for large is subdominant, leading to
Taking account of different behaviors of for small in the cases (I), (II), and (III), we obtain from Eq. (LABEL:eq:PRTO)
where is the Gamma function and the spectral dimension in the case (I) is given in Eq. (25). Note that for the case (III), a logarithmic term appears. As discussed above, the coefficient is when both and are odd or when and both and are even. Otherwise, it is .
We performed numerical simulations of RWs on the WF networks. In the simulation, independent RWers moved around on the networks with the transition probability in Eq. (1). The RTO probability is measured as the fraction of the walkers who are found in their own initial positions at . The measured RTO probabilities for various values of and (,) are shown in Fig. 3.
The spectral dimension is obtained from the power-law decay of . When , the measured values of for various and are in good agreement with Eq.(25) (see Fig. 3 (a)). On the other hand, when , the numerical values of are somewhat larger than the theoretical value 2, weakly dependent on and ranging as shown in Fig. 3 (b). Given the logarithmic term for in Eq. (32), this slight deviation as ( approaches can be a finite-size effect. The faster decay of for than is caused by shortcuts. Other structural factors such as the degree distribution, however, do not affect the spectral dimension, unlike the case .
The case is also interesting. For , we obtain that increases rapidly with as shown in the case