Topologies and Laplacian spectra of a deterministic uniform recursive tree

Topologies and Laplacian spectra of a deterministic uniform recursive tree

Abstract

The uniform recursive tree (URT) is one of the most important models and has been successfully applied to many fields. Here we study exactly the topological characteristics and spectral properties of the Laplacian matrix of a deterministic uniform recursive tree, which is a deterministic version of URT. Firstly, from the perspective of complex networks, we determine the main structural characteristics of the deterministic tree. The obtained vigorous results show that the network has an exponential degree distribution, small average path length, power-law distribution of node betweenness, and positive degree-degree correlations. Then we determine the complete Laplacian spectra (eigenvalues) and their corresponding eigenvectors of the considered graph. Interestingly, all the Laplacian eigenvalues are distinct.

pacs:
89.75.HcNetworks and genealogical trees and 02.10.YnMatrix theory and 02.10.UdLinear algebra and 89.75.FbStructures and organization in complex systems

1 Introduction

The flexibility and generality in the description of natural and social systems have made complex networks become an area of tremendous recent interest (1); (2); (3); (4). Among many interesting aspects, topological characterization is very significant for the study in network field. In the past ten years, there has been a considerable interest in characterizing and understanding the topological properties of networked systems (5). A lot of network measurements have been proposed, among which degree distribution, average path length (APL), betweenness, and degree correlations have been extensively studied, since they have profound effects on the dynamical processes taking place on networks, such as robustness (6); (7); (8); (9), epidemic spreading (10); (11), synchronization (12); (13); (14), and games (15); (16).

The above mentioned topological characteristics focus on direct measurements of structural properties of networks. Apart from these investigations there exists a vast literature related to (Laplacian) spectrum of complex networks (17); (18); (19); (20); (21), which provides useful insight into the relevant structural properties of graphs. In fact, topological features capture the static structural properties of complex networks, while spectrum provides global measures of the network properties (17). In the past years, graph spectrum has found many important applications in physics and other fields (3); (4). For example, the ratio of the maximum eigenvalue to the smallest nonzero one of Laplacian matrix determines the synchronizability of the network (12); (13); (14). On the other hand, the eigenvectors of Laplacian matrix have also been successfully used to detect community structure of networks (22). While the Laplacian eigenvalues and eigenvectors have high influence on the structural properties of networks and dynamics running on them, until now, most analysis of Laplacian spectra and eigenvectors has been confined to approximate or numerical methods, the latter of which is prohibitively time and memory consuming for large networks (17).

On the other hand, in order to describe real systems and study their structural properties, a wide variety of network models have been presented (1); (2); (3); (4), among which the uniform recursive tree (URT) is perhaps one of the most widely studied models (23). It is now established that the URT is one of the two principal models (24); (25) of a random graph (the second one is the famous Erdös-Rényi model (26)). The URT is perhaps the simplest tree and is built in the following way: at each time step, we attach each new node to an existing node which is chosen uniformly at random. It has found applications in several areas. For example, it has been suggested as models for the spread of epidemics (27), the family trees of preserved copies of ancient or medieval texts (28), chain letter and pyramid schemes (31), to name but a few. Recently, a deterministic version (32) of the URT has been proposed to mimic real-life systems whose number of nodes increases exponentially with time. This kind of deterministic models have drawn much attention from the scientific communities and have turned out to be a useful tool  (33); (34); (35); (36); (37); (38); (39); (40); (41); (42); (43); (44); (45); (46); (47). Although uniform recursive tree is well understood (23); (24); (25); (27); (28); (29); (30), less is known about the topologies and other nature of the deterministic uniform recursive tree (DURT) (32).

In this paper, we offer a detailed analysis of the deterministic uniform recursive tree (DURT) (32) from the viewpoint of complex networks. We first determine accurately relevant topological characteristics of DURT, such as degree distribution, average path length, betweenness distribution, and degree correlations. We then use methods of graph theory and algebra to calculate or estimate the eigenvalues and eigenvectors of the Laplacian matrix. We present that there is a strong relationship between the eigenvalues and the eigenvectors of the Laplacian matrix.

2 The deterministic uniform recursive tree

The deterministic uniform recursive tree is one of the simplest models. It is constructed in an iterative way (32). We denote the tree (network) after steps by (). Then the network at step is built as follows. For , is an edge connecting two nodes. For , is obtained from . We attach a new node to each node in . This iterative process is repeated, then we obtain a deterministic tree with an exponential decreasing spectrum of degrees as shown below. The definition of the model is illustrated schematically in Figure 1.

Figure 1: Illustration of the deterministic uniform recursive tree, showing the first five steps of growth process.

We first compute the total number of nodes and the total number of edges in the tree . Let and denote the numbers of nodes and edges created at step , respectively. Then, and . By construction, we have , thus . Considering the initial condition , we obtain and . Since is a tree, we have the following relation . On the other hand, at arbitrary step , the addition of each new node leads to only new edge, thus for all .

3 Topological properties

Topological features of a network are of fundamental significance to understanding the complex dynamics taking place on it. Here we focus on four important characteristics of the tree , i.e., degree distribution, average path length, betweenness distribution, and degree correlations.

3.1 Degree distribution

The degree is the simplest and most intensively studied characteristic of an individual node. The degree of a node is the number of edges in the whole network connected to . The degree distribution is defined as the probability that a randomly selected node has exactly edges. Let denote the degree of node at step . If node is added to the network at step , then by construction, . In each of the subsequent time steps, a new node will be created connected to . Thus the degree of node satisfies the relation

(1)

Considering the initial condition , we obtain

(2)

Since the degree of each node has been obtained explicitly as in Eq. (2), we can get the degree distribution via its cumulative distribution (3)

(3)

which is the probability that the degree is greater than or equal to . An important advantage of the cumulative distribution is that it can reduce the noise in the tail of probability distribution. Moreover, for some networks whose degree distributions have exponential tails: , the cumulative distribution also has an exponential expression with the same exponent:

(4)

This makes exponential distributions particularly easy to detect experimentally, by plotting the corresponding cumulative distributions on semilogarithmic scales.

Using Eq. (2), we have . Hence

(5)

which decays exponentially with . Thus the deterministic uniform recursive tree is an exponential network, and has a similar form of degree distribution as its stochastic version— the random uniform recursive tree (25).

3.2 Average path length

Average path length (APL) means the minimum number of edges connecting a pair of nodes, averaged over all node pairs. It is defined to be:

(6)

where denotes the sum of the total distances between two nodes over all pairs, that is

(7)

where is the shortest distance between node and .

Let and represent the set of nodes created at step or earlier, respectively. Then one can write the sum over all shortest paths in network as

(8)

where the third term is exactly , i.e.,

(9)

By construction, we can obtain the following relations for the first and second terms:

(10)
(11)

Substituting Eqs. (9), (10) and (11) into Eq. (8) and considering , the total distance is obtained to be

(12)

Inserting Eq. (3.2) into Eq. (6), we have

(13)

In the infinite network size limit (),

(14)

which means that the average path length shows a logarithmic scaling with the size of the network, indicating a similar small-world behavior as the URT (30) and the Watts-Strogatz (WS) model (48).

3.3 Betweenness distribution

Betweenness of a node is the accumulated fraction of the total number of shortest paths going through the given node over all node pairs (49); (50). More precisely, the betweenness of a node is

(15)

where is the total number of shortest path between node and , and is the number of shortest path running through node .

Since the considered network here is a tree, for each pair of nodes there is a unique shortest path between them (51); (52); (53); (54). Thus the betweenness of a node is simply given by the number of distinct shortest paths passing through the node. Then at time , the betweenness of a -generation-old node , which is created at step , denoted as becomes

(16)

where denotes the total number of descendants of node at time , where the descendants of a node are its children, its children¡¯s children, and so on. Note that the descendants of node exclude itself. The first term in Eq. (16) counts shortest paths from descendants of to other vertices. The second term accounts for the shortest paths between descendants of . The third term describes the shortest paths between descendants of that do not pass through .

To find , it is necessary to explicitly determine the descendants of node , which is related to that of children via (53); (54)

(17)

Using , we can solve Eq. (17) inductively,

(18)

Substituting the result of Eq. (18) and into Eq. (16), we have

(19)

which is approximately equal to for large . Then the cumulative betweenness distribution is

(20)

which shows that the betweenness distribution exhibits a power law behavior with exponent , the same scaling has been also obtained for the URT (29) and the case of the Barabási-Albert (BA) model (55) describing a random scale-free treelike network (51); (52). Therefore, power-law betweenness distribution is not an exclusive property of scale-free networks.

3.4 Degree correlations

An interesting quantity related to degree correlations (56) is the average degree of the nearest neighbors for nodes with degree , denoted as (57); (58); (59). When increases with , it means that nodes have a tendency to connect to nodes with a similar or larger degree. In this case the network is defined as assortative (60); (61). In contrast, if is decreasing with , which implies that nodes of large degree are likely to have near neighbors with small degree, then the network is said to be disassortative. If correlations are absent, .

For the deterministic uniform recursive tree, we can exactly calculate . Except for the initial two nodes generated at step 0, no nodes born at the same step, which have the same degree, will be linked to each other. All links to nodes with larger degree are made at the creation step, and then links to nodes with smaller degree are made at each subsequent steps. This results in the expression for ()

(21)

where represents the degree of a node at step , which was generated at step . Here the first sum on the right-hand side accounts for the links made to nodes with larger degree (i.e. ) when the node was generated at . The second sum describes the links made to the current smallest degree nodes at each step .

After some algebraic manipulations, Eq. (21) is simplified to

(22)

Writing Eq. (22) in terms of , it is straightforward to obtain

(23)

Thus we have obtained the degree correlations for those nodes born at . For the initial two nodes, each has a degree of , and it is easy to obtain

(24)

From Eqs. (23) and (24), it is obvious that for large network (i.e., ), is approximately a linear function of , which shows that the network is assortative.

4 Eigenvalues and eigenvectors of the Laplacian matrix

As known from section 2, there are vertices in . we denote by the vertex set of , i.e., . Let be the adjacency matrix of network , where if nodes and are connected, otherwise, then the degree of vertex is defined as . Let represent the diagonal degree matrix of , then the Laplacian matrix of is defined by . For an arbitrary graph, it is generally difficult to determine all eigenvalues and the corresponding eigenvectors of it Laplacian matrix, but below we will show that for one can settle this problem.

4.1 eigenvalues

We first study the Laplacian spectra of making use of an iterative method (62). By construction, it is easy to find that the adjacency matrix and diagonal degree matrix obeys the following relations:

(25)

and

(26)

where each block is a matrix and is identity matrix. Thus, according to the definition of Laplacian matrix, we have the recursive relation between and as

(27)

Then, the characteristic polynomial of is

where the elementary column operations of matrix have been used. According to the results in (63), we have

(29)

Thus, can be written recursively as follows:

(30)

where . This recursive relation given by Eq. (30) is very important, from which we will determine the complete Laplacian eigenvalues of and their corresponding eigenvectors. Notice that is a monic polynomial of degree , then the coefficient of in is 1, and hence 1 is the constant term of . Consequently, 1 is never an eigenvalue of .

Note that has Laplacian eigenvalues, and all these eigenvalues are distinct, which will be shown below. Let these eigenvalues are , , …, , respectively. For convenience, we presume and denote by the set of these eigenvalues of , i.e., ={, , …, }.

From Eq. (30), we have that for an arbitrary element in , say , both solutions of are in . In fact, equation is equivalent to

(31)

We use notations and to represent the two solutions of Eq. (31), since they provide a natural increasing order of the Laplacian eigenvalues of , which can be seen from below argument. Solving this quadratic equation, its roots are obtained to be and , where the function and satisfy

(32)
(33)

Substituting each Laplacian eigenvalue of into Eqs. (32) and (33), we can obtain the set of Laplacian eigenvalues of . Since , by recursively applying the functions provided by Eqs. (32) and (33), the Laplacian eigenvalues of can be determined completely.

It is obvious that both and are monotonously increasing functions. On the other hand, since the independent variables here are greater than or equal to zero, so and . Furthermore, , where the term is clearly less than zero, thus . Similarly, we can show that . Thus, for arbitrary fixed , holds for all . Then we have the following conclusion: If the Laplacian eigenvalues set of is , then solving Eq. (31) one can obtain the set of the Laplacian eigenvalues of to be where . Therefore, all the Laplacian eigenvalues of are different, which has never been previously reported in other network models thus may have some far-reaching consequences.

4.2 eigenvectors

Similarly to the eigenvalues, the eigenvectors of follow directly from those of . Assume that is an arbitrary Laplacian eigenvalue of , whose corresponding eigenvectors is v, then we can solve equation ( to find the eigenvector v. This equation can be also written as

(34)

where and are two components of v. Eq. (34) leads to the two following equations:

(35)
(36)

Resolve Eq. (36) to find

(37)

Substituting Eq. (37) into Eq. (35) we have

(38)

which indicates that is the solution of Eq. (35) while is uniquely decided by via Eq. (37).

From Eq. (30) in preceding subsection, it is clear that if is an eigenvalue of Laplacian matrix , then must be one eigenvalue of . (Recall that if , then for , or for ). Thus, Eq. (38) together with Eq. (30) shows that is an eigenvector of matrix corresponding to the eigenvalue determined by , while

(39)

is an eigenvector of corresponding to the eigenvalue .

Since for the initial graph , its Laplacian matrix has two eigenvalues 0 and 2 with respective eigenvectors and . By recursively applying the above process, we can obtain all the eigenvectors of the deterministic uniform recursive tree .

5 Conclusion and discussion

We have investigated a deterministic model for the uniform recursive tree, which is constructed in a recursive way. The model is actually a deterministic variant of the intensively studied random uniform recursive tree. We have presented an exhaustive analysis of many properties of the considered model, and obtained the analytic solutions for most of the topological features, including degree distributions, average path length, betweenness distribution, and degree correlations. Aside from their deterministic structures, the obtained statistical characteristics are equivalent with its corresponding random URT. Consequently, the DURT may provide useful insight to the practices as URT.

Also, we have performed a detailed analysis of the complete Laplacian spectra and eigenvectors of DURT by using the methods of linear algebra and graph theory. We have fully characterized the spectral properties and eigenvectors for DURT. We have shown that all the eigenvalues and eigenvectors of the Laplacian for DURT can be directly determined from those for the initial graph. Interestingly, all the Laplacian eigenvalues have only one multiplicity. To the best of knowledge, this property has never been reported for other previously studied model. Finally, it should be mentioned that although we have examined only a specific model, the methods used here can be applicable to a larger type of networks.

Acknowledgment

We 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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