Topological Properties of a 3-Regular Small World Network

Complex networks have seen much interest from all research fields and have found many potential applications in a variety of areas including natural, social, biological, and engineering technology. The deterministic models for complex networks play an indispensable role in the field of networkmodel.The construction of a networkmodel in a deterministic way not only has important theoretical significance, but also has potential application value. In this paper, we present a class of 3-regular network model with small world phenomenon.We determine its relevant topological characteristics, such as diameter and clustering coefficient.We also give a calculation method of number of spanning trees in the 3-regular network and derive the number and entropy of spanning trees, respectively.


Introduction
In recent years, research on complex networks is in the ascendant.Complex networks by using graph theory and some methods of statistical physics can be used to capture and describe the evolution of the system mechanism, evolution pattern, and the overall behavior, which is one of the main causes of vigorous development on complex networks research [1].
Construction of small-world networks that conform to the real system features not only has important theoretical significance but also has potential application value in deterministic way [2].In this paper, we present 3-regular small-world network.We determine the relevant topological characteristics of the regular network, such as diameter and clustering coefficient.The results show that our model has a discrete exponential degree distribution, high clustering, and small diameter, which appears in a small-world effect.It is known that the number of spanning trees is an important quantity characterizing the reliability of a network.Because of the diverse applications in a number of fields, a lot of efforts have been devoted to the study of spanning trees.For example, in [3,4] the author gave the accurate number of spanning trees in regular lattices.The number of spanning trees in a network can be obtained by directly calculating a related determinant corresponding to the network.We also give a method to calculate the number of spanning trees of the 3-regular network and derive the formulas.Based on the number of spanning trees we determine the entropy of its spanning trees.

Network Construction
The degree (V) of a vertex V in  is the number of edges of  incident with V, each loop counting as two edges [5].A graph  is -regular if (V) =  for all V ∈ .In this section we describe a model of growing network, which is constructed in an iterative manner.We denote our network after  steps by   ( ≥ 1).Then the network is constructed in the following way.For  = 1,  1 is a graph that two nodes connect three edges.For  ≥ 2,   is obtained from  −1 by adding two new nodes at step  = 1; then two edges are increased; replicate it and connect four new nodes.The process is repeated till the desired graph order is reached; see Figure 1.Now we compute the size and order of   .Denote by  V () the total number of nodes in the network and by   (V) the total number of edges in the network.It is easy to get that Hindawi Publishing Corporation Since  V (1) = 2 and Δ V (2) = 6, it follows that Note that the addition of four new nodes leads to two new edges.Then As   (1) = 3, we have

Relevant Characteristics
Due to the determinacy, the relevant characteristics of our model described above can be solved exactly.In the following we concentrate on the diameter and clustering coefficient.

Diameter.
Small-world networks describe many real-life networks; that is, there is a relatively short distance between most pairs of nodes in most real-life networks and their average path length (APL) does not increase linearly with the system size but grows logarithmically with the number of nodes or slower.The average path length is the smallest number of links connecting a pair of nodes, averaged over all pairs of nodes [2].The longest shortest path between all pairs of nodes is called diameter, which is one of the most important evaluation index because it characterizes the maximum communication delay in the network [6].For most network models, it is hard to obtain the analytic solution of the average path length.So we will study the diameter instead of average path length.We denote the diameter at iteration  as (); then according to Figure 1, we can clearly know that (1) = 1 and (2) = 3.At each iteration for  > 2, one can see that the diameter always lies between a pair of newly created nodes at this iteration.Thus the diameter for the network proposed has the following formula: So, the diameter () grows logarithmically with the number of nodes.Because the average path length is smaller than diameter, the APL should increase more slowly.So our model satisfies the property for small-world networks.

Clustering Coefficient.
Clustering is another important property of a complex network, which provides a measure of the local structure within the network [2].The clustering coefficient indicates the connection relationship between a node and its neighborhood nodes.By definition, clustering coefficient   of a node  with degree   is the ratio of the total number   of existing edges between all its   nearest neighbors and the number   (  − 1)/2 of all possible edges between them; that is,   = 2  /  (  − 1).The clustering coefficient  of the whole network is the average of all individual    ; that is,  = (1/) ∑  =1   , where  is the number of nodes of the network.So the clustering coefficient for our network has the following simple formula.We know  = 1 at step  = 1.At each iteration  ≥ 2, we have Obviously,  ≈ 1/3 when  → ∞; thus the clustering is high and also satisfies the properties for small-world networks.Based on the above discussion, we can conclude that our model is a deterministic small-world network, because it is sparse with small diameter and average path length and high clustering coefficient.

Enumeration of Spanning Trees
A spanning tree of any connected network is defined as a minimal set of edges that connect every node.The enumeration of spanning trees in networks is a fundamental issue in mathematics [7][8][9], physics [3,10], and other disciplines [11].The problem of spanning trees is relevant to various aspects of networks, such as reliability [12,13], optimal synchronization [14], and random walks [15].In what follows we will examine the number of spanning trees in the 3-regular network.
In order to calculate the number of spanning trees accurately, at first we build a new model   shown in Figure 2. We list the relationship between   and   in Figure 3.The number of the spanning trees and the spanning forests with two components such as  and V belongs to distinct components of   that are denoted by ST(  ) and SF(  ), respectively.Then the number of spanning trees of the original model () can be obtained from the number of the spanning trees of   .Then we have the following relationship: We calculate the number of spanning trees of   in the following.It is easy to obtain the relationship of ST(  ) and ST( −1 ) from (7).From Figure 4, one can see that the shape of the spanning trees of   includes two cases.
Figure 3: The network models   and   at step .And the shape of the spanning forests with two components such as  and V belongs to distinct components of   that have five cases; see Figure 5.
By considering the symmetric, it is not hard to get the recurrence relation of the number of spanning trees and spanning forests of   and  −1 in Figures 4 and 5: Let   = ST(  )/SF(  ); then   = 2 −1 /(2 −1 +1) with  1 = 1.
So, the entropy of the spanning trees of   is given by We can compare the entropy of the spanning trees of   with other networks [17,18].The entropy of the spanning trees in Koch networks with an average degree of 3 is 0.5493 [17], while the entropy of the spanning trees of our model is 0.6931.

Conclusions
In this paper, we proposed a class of deterministic regular small-world model which is constructed in an iterative manner and presented an exhaustive analysis of many properties of considered model.Then we obtained the analytic solutions for most of the topological features, including diameter and clustering coefficient.We also determined the number of spanning trees in the 3-regular small-world network.In addition, using the algorithm, we obtained the entropies of spanning trees.

Figure 1 :
Figure 1: Construction of the deterministic 3-regular network   , showing three steps of the iterative progress.

Figure 2 :
Figure 2: Construction of the deterministic 3-regular network   , showing three steps of the iterative progress.

Figure 4 :
Figure 4: The number of spanning trees of   .

1 Figure 5 :
Figure 5: The number of spanning forests with two components of   .