Counting spanning trees in self-similar networks by evaluating determinants
Spanning trees are relevant to various aspects of networks. Generally, the number of spanning trees in a network can be obtained by computing a related determinant of the Laplacian matrix of the network. However, for a large generic network, evaluating the relevant determinant is computationally intractable. In this paper, we develop a fairly generic technique for computing determinants corresponding to self-similar networks, thereby providing a method to determine the numbers of spanning trees in networks exhibiting self-similarity. We describe the computation process with a family of networks, called -flowers, which display rich behavior as observed in a large variety of real systems. The enumeration of spanning trees is based on the relationship between the determinants of submatrices of the Laplacian matrix corresponding to the -flowers at different generations and is devoid of the direct laborious computation of determinants. Using the proposed method, we derive analytically the exact number of spanning trees in the -flowers, on the basis of which we also obtain the entropies of the spanning trees in these networks. Moreover, to illustrate the universality of our technique, we apply it to some other self-similar networks with distinct degree distributions, and obtain explicit solutions to the numbers of spanning trees and their entropies. Finally, we compare our results for networks with the same average degree but different structural properties, such as degree distribution and fractal dimension, and uncover the effect of these topological features on the number of spanning trees.
pacs:89.75.Hc, 05.50.+q, 05.20.-y, 04.20.Jb
Counting spanning trees in networks is a fundamental issue in both theory and applications, which has recently attracted increasing attention from mathematics (1); (2); (3); (4), physics (5); (6); (7); (8); (9); (10), and other fields (11). As a network invariant, spanning tree is a crucial measure of the network reliability (12); (13); (14). The notion is also closely related to various aspects of networks. For example, the number of spanning trees of a connected network is exactly the number of recurrent configurations of the Abelian sandpile model on the network (15); (16), which is equivalent to the chip-firing game (17) under certain restrictions and is a paradigm for self-organized criticality (18). As another example, spanning trees offer useful insights into understanding the origin and mechanism of fractality in fractal scale-free networks (19). In addition, spanning trees are relevant to other interesting problems on networks, such as transport (20), electrical networks (21), unbiased random walks (22); (23), loop-erased random walks (24), and so on.
In view of their relevance to diverse aspects of networks and a wide range of applications (25), spanning trees in networks have become a focus of some recent research (26); (27); (28); (29); (30); (31); (32). Particularly, in the physics literature a lot of efforts have been devoted to enumerating spanning trees in specific networks by using different techniques according to their special structures. Examples include regular lattices (5); (6); (7); (9), Sierpinski gaskets (8); (10), Erdsö-Rényi random graphs (4), the pseudofractal scale-free web (33), fractal scale-free networks (34), and so on. These works have provided some effective methods for determining spanning trees in special networks, and have revealed some nontrivial effects of network structural properties on spanning trees.
Most existing methods for counting the number of spanning trees are applicable only to particular networks. Due to the complexity and diversity of networks, a general approach for counting spanning trees of a generic network is not available, with the exception of the Kirchhoff Matrix Tree Theorem (35). This well-known theorem provides a universal algorithm for determining the number of spanning trees of an arbitrary connected graph, in terms of a determinant (36). However, the computational complexity for evaluating the determinant of a general network is very high (37). For large networks, it is difficult and even impossible to obtain their numbers of spanning trees. For this reason, seeking an efficient and somewhat general technique for calculating the determinant of a network is a matter of exceptional importance.
On the other hand, it has been observed that by nature most real networks are fractal (38) as well as scale-free (39). Both striking properties have significant influences on other structural properties, e.g., average distance (40); (41); (42) and degree correlations (43). In particular, existing results have shown that in some special networks, the scale-free property may decrease the number of spanning trees (33), while fractality can increase the number of spanning trees (34). Thus, some interesting questions rise naturally: Can the power-law behavior alone determine the number of spanning trees in a scale-free network? What is the relation between the fractal dimension and the spanning trees in a fractal scale-free network? Both questions remain unsolved today.
In this paper, we present an approach for explicitly determining the number of spanning trees by evaluating a related determinant in a network, which is expected to hold true for a broad class of self-similar graphs. Our method is based on the recursion relations of determinants of submatrices of the Laplacian matrix associated with the network at different iterations, which avoids the laborious computation of the largest determinant as needed by classic algebraic schemes. To the best of our knowledge, this constitutes the first work in exploring fast evaluation of the determinant of a complex network, especially for networks with scale-free (39), small-world (44), and fractal (38) properties.
To demonstrate the computation process of our technique, we apply it to find the number of spanning trees in the family of self-similar networks, called -flowers (45); (46), which display some remarkable properties observed in real-life networks (47); (48); (49) and include the extensively studied pseudofractal scale-free web (50); (51) and fractal hierarchical lattices (52); (53); (54); (55) as their limiting cases. Using the proposed technique, we derive the exact number of spanning trees in the whole family of -flowers, based on which we obtain their entropies of spanning trees. In addition, in order to exhibit the generality of our method, we apply it to some other self-similar networks and obtain closed-form solutions for the numbers of spanning trees in such networks. Finally, we present a detailed analysis of the obtained results, and show that power-law distribution alone do not suffice to characterize the numbers of spanning trees in scale-free networks, and that the increasing fractal dimensions of fractal scale-free networks may lead to decrements of the numbers of spanning trees.
Ii Model and properties of -flowers
In this section, we introduce the construction and structural properties of the -flowers (45); (46), which are built in an iterative way. Let () denote the -flowers after generations of evolution. In what follows it is assumed that and without loss of generality. The -flowers are constructed as follows. For , is an edge connecting two nodes, called the initial nodes hereafter. For , is derived from through replacing each exiting edge in by two parallel paths of lengths and , see Fig. 1. For illustration, in Figs. 2 and 3, we present the growing processes of two particular networks: -flower and -flower. According to the generating algorithm for the -flowers, it is easy to see that the number of edges in is
Alternatively, the -flowers can be constructed by using the following equivalent algorithm (45), which highlights the self-similarity of the networks. From , the next generation is obtained by joining copies of at the two initial nodes of each replica. Figure 4 provides an illustration for the second construction approach for the -flower. According to this construction, the number of nodes in satisfies the recursive relation , which together with the initial condition yields
Thus, the average degree of the -flowers is
which approaches as .
The deterministic construction also allows to determine the full degree distribution of the -flowers. In , the degree spectra of all nodes are discrete, i.e., nodes have only degree in the form of (). Let be the number of nodes with degree in . Then, for , and for . Thus, in the family of , all networks with the same have an identical degree sequence, hence an identical degree distribution.
The -flowers exhibit rich behavior in their architecture (45); (46). They obey a power-law degree distribution with the exponent lying in the interval . For , the networks are small-world but non-fractal; while for , they are “large-world” and fractal with the fractal dimension . These topological properties are not shared by any other modeling networks, which make the -flowers unique within the family of networks.
Iii Spanning trees in -flowers
After introducing the construction and some characteristics of the -flowers, we now attempt to determine analytically the number of spanning trees in , denoted by . According to the Matrix Tree Theorem (35); (36), can be evaluated by computing the determinant of a submatrix of the Laplacian matrix corresponding to , which is obtained by removing a row and a column corresponding to an arbitrary node of . The elements of the Laplacian matrix associated with , represented by , are defined as follows: the non-diagonal entry is (or ) if nodes and are (or not) directly connected by an edge, while the diagonal entry equals the degree of node .
To find , we denote by the submatrix of obtained by removing from the row and column corresponding to an initial node of . Then, we have
Our aim is to determine . Although for a general network of large size, the evaluation of its determinant is very hard and even impossible, below we show that for and other self-similar networks, we can evaluate the associated determinants and derive closed-form expressions for the number of their spanning trees.
First, we define another submatrix of , which is obtained by deleting the two rows and columns from corresponding to the two initial nodes, and denote it by . In what follows, we show the recursive relations for and , thereby, expressing and in terms of and , based on which we offer explicit solutions for these two quantities. For simplicity, let , , and represent , , and , respectively. Obviously, for all , since the sum of each row in is zero.
It should be noted that the technique we develop for evaluating the determinants can be adapted to a wide class of self-similar networks other than the -flowers. Here, we illustrate the computation process with the -flowers. Moreover, we only consider in detail two particular and representative networks in the -flower family, namely, -flower and -flower. For the general -flowers we can treat them analogously although the computation is somewhat complex. In so doing, for notational simplicity, we only provide the recursion relations governing and for the general -flowers, but omit their detail computations.
iii.1 The -flower
According to the second construction algorithm of the networks, see Fig. 4, the Laplacian matrix of , denoted by in the case without confusion, can be expressed in terms of as
In Eq. (5), the first two rows and two columns correspond to the two initial nodes of ; is the degree of either of initial nodes at generation ; and are two row vectors with order describing respectively the interaction between the two initial nodes and the other nodes in different replicas of that are joined together forming ; and the superscript of a vector represents its transpose. The third and forth rows and columns describe the connection relation of the two nodes created at (each of which has the same degree as the two initial nodes) and other nodes in .
Based on Eq. (5), we can also represent and in terms of , as
With the expressions for and given in Eqs. (6) and (7), we can evaluate the determinants of and , namely, and , on the basis of their recursive relations that will be derived. We first show a detailed computation process for , while the evaluation for can be dealt with in an analogous way.
Note that the first row of can be viewed as the sum, i.e., a linear combination with scalars being 1, of two vectors; that is,
Analogously, the second and third rows can also be regarded separately as the sum of two vectors, namely,
Thus, according to the properties of determinants, can be decomposed as the sum of eight determinants, as follows:
Therefore, the evaluation of is reduced to finding the values of the eight determinants on the right-hand side of Eq. (III.1), which are denoted sequentially as . Using some elementary matrix operations, we can verify that and . Having in terms of the quantities of , the next step is to explicitly determine these quantities. In view of the equivalence of some , we only compute and .
For and , utilizing some elementary matrix operations, we can easily obtain
Plugging these expressions into Eq. (III.1), we have
Next, we derive the recursion relation for . Through a computational process similar to that for , we obtain the following relation:
Having obtained the above recursive expressions for and , we are now in a position to explicitly determine . For this purpose, we examine a new quantity , which obviously satisfies the following recursive relation:
Considering the initial condition , the above equation is solved by induction to yield
It is easy to express in terms of the network size , so as to obtain the relation governing the two quantities. Recalling that for flower, , hence we have and . These relations allow us to recast as a function of :
which indicates that approximately increases exponentially in .
iii.2 The -flower
We proceed to determine the number of spanning trees in the -flower. In the computation process, we use the same notations as those defined for the -flower. Similarly to the -flower, we can derive the recursion relations for and , as
Based on the above two relations, we obtain the following relations for the determinants corresponding to and by using some elementary matrix operations:
These two recursion relations, coupled with the initial conditions and , yield
which can be further expressed in terms of the network size as
Thus, for the -flower, the number of spanning trees also scales exponentially with the network size. And its entropy of spanning trees is
a value larger than that corresponding to the -flower.
iii.3 The -flowers
The above process for computing spanning trees in the -flower and the -flower can be used to determine the number of spanning trees in the whole family of -flowers. Since for the general -flowers, the recursive relations for and are lengthy and the computation process for and is more complex, we only provide the recursive relations for and , leaving out the derivation detail, which is similar to those for the two particular cases considered above. The recursive relations of and for the -flowers are given by
For and , Eq. (16) recovers the result previously obtained in (33); for and , Eq. (16) reduces to Eq. (10); while for and , Eq. (16) agrees with the result given by Eq. (13). The consistency confirms the validity of the formula for spanning trees in the -flowers.
Since the size for the -flowers is , it is evident from Eq. (16) that scales exponentially with . This permits to get the entropies of spanning trees for the -flowers as follows:
It is worthwhile to mention that the above technique and process for finding spanning trees in the -flowers by evaluating determinants can be adapted to other self-similar networks. In order to show the universality, in the Appendix we apply the approach to determine the number of spanning trees and their entropies for a small-world network (56) with an exponential degree distribution and a scale-free Apollonian network (57); (58), and obtain explicit solutions to the corresponding quantities.
iii.4 Analysis of the results
The entropy for spanning trees is a very important quantity characterizing the network structure. Thus, it is interesting to show its dependence on the network structure. Since, for the whole family of the -flowers, its architecture is controlled by the two parameters and , it is expected that the entropy is dependent on and . In Fig. 5, we show how the entropy varies with and . The intrinsic dependence relation of entropy on and is somewhat complex. Here we are mainly concerned about how the entropy is reliant on and when the sum is fixed with . The main reason why we are interested in this case lies in that for this case it is meaningful to compare the entropy. As shown in section II, for the -flowers with fixed , they have an identical average degree. In addition, for fixed , the networks have the same degree sequence and thus the same degree distribution. Particularly, for with fixed , the networks are fractal having the fractal dimension monotonously decreasing in .
Figure 5 shows that when is fixed, is determined by the difference between and : the larger the difference, the smaller the entropy. Thus, in the case of fixed , is an increasing function of parameter . For example, as shown above, when , the entropy for the -flower is less than that for the -flower. Thus, we can conclude that the power-law degree distribution alone can not determine the entropies for spanning trees in scale-free networks. In fact, other than scale-free networks, the degree distributions of other networks do not suffice to characterize the number of spanning trees either. For instance, both the small-world network (56) and the Koch network family (59); (60), to be addressed in the Appendix, have the same average degree and the same entropy of spanning trees, but their degree distributions are disparate: the former is exponential while the latter is scale-free.
As pointed out above, degree distribution alone cannot determine the number of spanning trees. Then an important question arises: which structural property has the decisive effect on the number of spanning tress in the -flower family with an identical value of ? Since, for the whole family of -flowers with the same , they have the same average node degree, the same degree sequence and thus the same degree distribution, and the same (zero) clustering coefficient, we argue that the difference in the number of spanning trees seen in Fig. 5 is attributed solely to the fractality. Specifically, the fractal property of networks significantly increases the number of spanning trees, which is related to the fractal dimension: the larger the fractal dimension, the less the number of spanning trees. This is obvious from Fig. 5. For those -flowers with the same value of , when grows from 1 to its maximum, their fractal dimension decreases from to its minimum, while the entropies of spanning trees increase from their minimum to maximum. Particularly, when , the network is non-fractal, it has the minimum number of spanning trees.
Before closing this section, we discuss the case of different values of . Figure 5 shows that for two network classes with different values of , the class with smaller has a larger entropy than that of the other class. This can be easily understood. Since for two networks with distinct values of , their average node degrees are also disparate: the network with smaller has a larger average node degree than that the other. Thus, the edge density of the former is greater than that of the latter, which is responsible for the difference of entropies for spanning trees in the two networks.
In this paper, we have developed a rather generic technique for evaluating determinants used to compute the numbers of spanning trees in self-similar networks. Our method is grounded on the recursive relations for the determinants of submatrices of the Laplacian matrix corresponding to the network at two consecutive generations, which are established based on the structure of the network. An advantage of our approach is that it avoids the laborious computation of a determinant and reduces the computational complexity that a standard method needs. We have illustrated our technique by applying it to the family of -flowers, which exhibits rich and special structural properties, as well as some remarkable features of many real systems. Moreover, to demonstrate the universality of our approach, we have applied it to some other networks with self-similarity. We obtained explicit formulas for the numbers of spanning trees in different networks. Our results show that degree distribution alone does not suffice to characterize the number of spanning trees in a network, and that fractal dimension has a predominant influence on the number of spanning trees in fractal scale-free networks with the same average degree.
Acknowledgements.This work was supported by the National Natural Science Foundation of China under Grant No. 61074119 and by the Hong Kong Research Grants Council under the GRF Grant CityU 1117/10E.
Appendix A Spanning trees in other self-similar networks
To illustrate the university of our proposed method, we apply it to calculate the number of spanning trees in some other self-similar networks with different degree distributions, exponential or scale-free.
a.1 Small-world exponential network
We first discuss a self-similar small-world network with an exponential form of degree distribution (56), which is observed from some real-life systems (61). The small-world network is constructed iteratively (56), denoted by after iterations. The network starts from a triangle , with three nodes called initial nodes. For , is obtained from : for each existing node in , two new nodes are generated, which and their mother node together form a new triangle. Figure 6 shows the structure of . It is easy to derive that the total number of nodes in is . The resultant network has a degree distribution decaying exponentially with the degree. In addition, it is small-world with the average distance growing logarithmically with its size.
We now begin to evaluate the number of spanning trees in . In the case without any confusion, we use to denote the Laplacian matrix for , and use to denote a submatrix of , which is obtained by removing from the row and column corresponding to one of three initial nodes of . We have
where is the degree of an initial node in , and is a row vector representing the interaction between an initial node and the other non-initial nodes.
According to the Matrix Tree Theorem, the number of spanning trees in is . Using a similar iterative process as that in the main text, Eq. (19) leads to the following relation: . Considering the initial condition of , we obtain the exact solution to the number of spanning trees in the small-world network as
and the entropy of spanning trees as
The above computational process for can also be applied to the scale-free Koch networks (59); (60) with the same average node degree 3 as that of . Interestingly, the spanning tree entropy for the Koch networks is identical to despite the fact that they have totally different types of degree distributions.
a.2 Small-world scale-free Apollonian network
We apply our technique to determine the spanning trees in the Apollonian network with both small-world and scale-free properties (57); (58). The Apollonian network is derived from the classic Apollonian packing. This well-known packing problem (62) starts with three mutually touching disks and their interstice is filled by a disk that touches all the three initial disks, forming three smaller interstices to be filled, as shown in the left panel of Fig. 7. Let each disk denote a node located at the center of the corresponding disk, linking the centers of touching disks by lines leads to the Apollonian network, as displayed in the right panel of Fig. 7. The Apollonian network can also be constructed iteratively and has the small-world effects (58). After iterations, the size of the Apollonian network is .
Making use of our technique for evaluating the corresponding determinant, we can determine the number of spanning trees in the Apollonian network. Since the computational process is complicated and the relation equations are lengthy, we only provide the main results. The exact number of spanning trees in the Apollonian network after iterations is
Consequently, the entropy for spanning trees in the Apollonian network is
which is smaller than 1.9902, the entropy for spanning trees in three-dimensional lattices (7) having the same mean degree 6 as this Apollonian network.
- preprint: APS/123-QED
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