Abstract
We introduce an informative labeling algorithm for the vertices of a family of Koch networks. Each label consists of two parts: the precise position and the time adding to Koch networks. The shortest path routing between any two vertices is determined only on the basis of their labels, and the routing is calculated only by few computations. The rigorous solutions of betweenness centrality for every node and edge are also derived by the help of their labels. Furthermore, the community structure in Koch networks is studied by the current and voltage characterizations of its resistor networks.
1. Introduction
The WS small-world models [1] and BA scale-free networks [2] are two famous random networks which stimulated an in-depth understanding of various physical mechanisms in empirical complex networks. The two main shortcomings are the uncertain creating mechanism and the huge computation in analysis. Deterministic models always have important properties similar to random models, such as being scale-free, having small-world behavior, and being highly clustered, and thus they could be used to imitate empirical networks appropriately. Hence, the study of the deterministic models of a complex network is increasing recently.
Inspired by the simple recursive operation and techniques of plane filling and generating processes of fractals, several deterministic models [3–18] have been created imaginatively and studied carefully. The lines in the famous Koch fractals [19] are mapped into vertices, and there is an edge between two vertices if two lines are connected; the generated novel networks were named Koch networks [20]. This novel class of networks incorporates some key properties which characterize the majority of real-life networked systems: a power-law distribution with exponent in the range between 2 and 3, a high clustering coefficient, a small diameter, and average path length and degree correlations. Besides, the exact numbers of spanning trees, spanning forests, and connected spanning subgraphs in the networks are enumerated by Zhang et al. in [20]. All these features are obtained exactly according to the proposed generation algorithm of the networks considered [21–29]. The deterministic models of the complex network have a fixed shortest path, but how to mark it only by their labels is rarely researched [30–34].
However, some important properties in Koch networks, such as vertex labeling, the shortest path routing algorithm, the length of the shortest path between arbitrary two vertices, the betweenness centrality, and the current and voltage properties of Koch resistor networks have not yet been researched. In this paper, we introduce an informative labeling and routing algorithm for Koch networks. By the intrinsic advantages of the labels, we calculate the shortest path distances between two arbitrary vertices in a couple of computations. We derive the rigorous solutions of betweenness centrality of every node and edge, and we also research the current and voltage characteristics of Koch resistor networks.
2. Koch Networks
The Koch networks are constructed in an iterative way. Let be the Koch networks after iterations, where is a structural parameter.
Definition 1. The Koch networks are generated as follows: initially , is a triangle. For , is obtained from by adding groups of vertices to each of the three vertices of every existing triangle in .
Remark 2. Each group consists of two new vertices, called son vertices. Both sons are connected to one another and to their father vertices; thus, the three vertices shape a new triangle.
That is to say, we can get from just by replacing each existing triangle in with the connected clusters on the right-hand side of Figure 1.
For the integrity of the article, we firstly introduce some important properties of Koch networks from [20]. The numbers of vertices and edges in networks are
By denoting as the numbers of nodes created at step , we obtain ; then, we also get that the degree distribution is ; by substituting , for , and then and , in the infinite limit, one can get
Then, the exponent of degree distribution is , which belongs to the interval . The average clustering coefficient of the whole network is given by . When increases from to infinity, increases from to . So, the Koch networks are highly clustered. The average path length (APL) approximates in the infinite , for APL is
This formula shows that Koch networks exhibit small-world behavior. These properties indicate that Koch networks incorporate some key properties characterizing a majority of empirical networks: simultaneously scale-free, small-world, and highly clustered [20].
3. Vertex Labeling and Routing Algorithm by the Shortest Path
Definition 3. All the vertices are located in three different subnetworks of the Koch network; the label , or is used to denote the subnetworks.
Remark 4. Denote the three symmetrical subnetworks in Koch networks as , , and . Then, is obtained just by linking the hub of three subnetworks directly. Therefore, the label , or is used to differentiate the vertices in the three different subnetworks .
A binary digits code is used to identify the precise position of a vertex in and the exact time which is linked to . The method is shown as follows.
Definition 5. Any vertex in is marked with binary digits , where when or when . The 0 (or in binary digits represents the notion that the new vertices are grown from a son vertex (or father vertex) in a triangle. The length of the binary digit is the time at which the vertex is linked to Koch networks.
Remark 6. Because the initial network is a triangle, all the three initial vertices in it have no father vertices, so that the new vertices adding to the initial vertices should be marked with at time ; that is, must be .
Then, we obtain the set , by processing the binary digit codes of each vertex in .
Remark 7. The element in implies that when , the length of is zero in .
Definition 5 implies that all the vertices, added to an existing vertex at step , have the same binary codes . Consequently, the number of vertices which are added to an existing father vertex at step is given by
So we need to mark the vertices of this group with an extra integer for they all have the same binary codes and the same group indicator .
Definition 8. An integer is used to identify the precise position, increasing by clockwise direction, of a vertex in the group which is added to a father vertex at the iteration .
Remark 9. Because increases from and is positioned after the binary blue code, a dot is needed between and the binary code to avoid confusion.
In sum, the arbitrary vertex which is added to at step is labeled with . The code denotes which subnetwork of the vertex belongs to; the binary digit indicates the father vertex which is linked to this new vertex; the positive integer , which increases in a clockwise manner, is used to mark the precise position around a father vertex.
Define the set as the label set of the vertices added to networks at step . It is apparent that and . Let the set represent the labels of all the vertices in ; we obtain
For example, Figure 2 demonstrates the vertex labeling of all the vertices in Koch network . In the following sections, we deduced some important properties of Koch networks just on the basis of the labels of their vertices.
Property 1. Each vertex has a unique label.
Suppose an arbitrary vertex is labeled with . Firstly, from the labeling algorithm, the labels of any pair of vertices are different from each other. Secondly, the size of equals the size of Koch networks. So, we deduced that any vertex has a unique label.
Assume that is the label of arbitrary vertex which is added to at step , and let the set denote the labels of all neighbor vertices of . By comparing the vertex’s degree between and its neighbors, can be divided into three subsets: , , and , the vertices in which sets have a degree equal to, lower than, and higher than the degree of , respectively. That is to say, .
Property 2. , where vertex degree if , or if .
According to the proposed algorithm, at each step, any father vertex adds groups of vertices, with each group consisting of two vertices. Thus, these latter, together with the father, are linked to each other and form a new triangle. Therefore, the two vertices in the same group are neighbors which are linking directly and have the same degrees. By the labeling method, the group vertices are labeled with the integers which increase from to in a clockwise manner. So, is the neighbor of with the same degrees if , or is the neighbor if .
Property 3.
From the labeling algorithm, the vertices with longer binary codes have lower degrees than the vertices with shorter binary codes. In addition, the or in binary codes indicates that the new vertex is growing from the two son vertices or father vertex in each triangle. Hence, we can understand that the vertices, adding to at steps , are labeled with .
Let be the function returning the biggest integer just smaller than real number .
Property 4. .
We remark that is the label of an arbitrary vertex . From construction, we obtain that the label of the only father vertex of depends on the composition in binary codes of vertex . Suppose the first in , from the right to the left side, is . By the construction method, it is clear that is linked to a vertex with higher degree which is labeled with . In particular, if the first of is , the vertex with a higher degree is exactly a hub of Koch networks which is labeled with or .
The deterministic models of the complex network have a fixed shortest path, but how to mark it only by their labels is rarely researched [15]. The following rules are used to determine the shortest path routing between any two vertices by the help of their labels. Let and be the labels of arbitrary pair of vertices in .
Algorithm 10. The shortest path routing algorithm in Koch networks is discussed.
If , find out, by Property 4, all the higher degree neighbors of the two vertices, until the hubs and . Thus, by linking all vertices of them, we obtained the shortest path.
If , the first step is to mark higher degree neighbors until the common highest degree vertex by Property 4 and then judge whether the two second highest degree vertices are neighbors or not by Property 3; if not, the shortest path is connected to all higher degree neighbors until the highest degree vertex; if so, the shortest path is just the same as above, only eliminating the highest degree vertex.
If , the two vertices are located in different subnetworks and . The routing by the shortest path between two vertices in different subsets is ascertained as follows. First, we obtain the neighbors which have higher degrees recursively by Property 4, until the hubs and . Then, we connect all of them in turn; this is the only shortest path between two vertices.
If , it is clear that the shortest path is located in the same subnetworks . We find out the neighbors with higher degree by using Property 4 repeatedly, until the common highest degree vertex. Then, we judge whether the two second highest degree vertices are neighbors or not by Property 3. If they are not neighbors, we determine the shortest path as above by linking all the higher degree vertices until the highest vertex, by the help of the construction method of Koch networks. Else, if they are neighbors, the shortest path is the same as above by excluding the highest degree vertex.
The shortest path between any pair of vertices in is obtained after no more than times of integral computations and modulo operations by the help of the labeling method and routing algorithm proposed in this research. That is to say, the shortest path routing and the shortest distance between an arbitrary pair of vertices in Koch networks can be dealt with in few computations.
4. Betweenness Centrality
Betweenness centrality is originated from the analysis of the importance of the individual in social networks, including the betweenness of any vertex and edge in networks. If the betweenness of a node/edge is bigger, then the node/edge in the social network is more important [2]. The betweenness of a vertex for undirected networks is given by the expression where is the number of the shortest paths passing through . The computation of betweenness is very difficult in most networks. Fortunately, the betweenness of Koch networks can be derived qualitatively and quantitatively by the help of their labels in Koch networks, which is shown as follows.
Suppose that an arbitrary vertex , which is adding to at time , is labeled with . The vertices in can be divided into three parts: the vertex , the offspring vertices which are connected to directly and indirectly after step (they all have lower degrees than ), and the other vertices in . Assume that the number of the second part vertices is , and it can be worked out that by (1). Apparently, the number of the third parts is . For the shortest path routing between any two vertices is unique, we get that . Substitute this equation and (1) into (8); the betweenness of a vertex which is labeled with is given by
For and , we obtain the formula which holds with . Therefore, the vertex betweenness in Koch networks is exponentially proportional to the vertex’s degree with an exponent belonging to the interval
The betweenness of edges can also be deduced in a similar way. Note as the edge between any two neighbor vertices and which are labeled with and . Without loss of generality, assume that vertex has higher degree than . So the label of belongs to the set by Property 4. Suppose a triangle is shaped by three vertices: , , and . Therefore, has the same degree as . Then, Koch network can be divided into three parts: the lower degree vertices linking to directly or indirectly, the vertices connected to directly or indirectly, and the lower degree vertices adding directly or indirectly, respectively. Correspondingly, the label set falls into three subsets: , , and . The relationship of these four label sets is shown as follows:
The size of is derived as . From the symmetry of vertices and , we obtain Then, we get that from the construction of Koch networks. For the shortest path between any two vertices is unique, then the betweenness of the edge is defined as follows:
Therefore, the betweenness centrality of the edge is given by Therefore, the edge betweenness holds , where . The edge betweenness is also exponentially proportional to the degree of the lower degree vertex , and the exponent is belonging to the interval . In a word, the betweenness of an edge is exponentially proportional to the time of adding to Koch networks.
5. Resistor Networks
The communities in networks are the groups of vertices within which the connections are dense but between which the connections are sparser. A community detection algorithm which is based on voltage differences in resistor networks is described in [35, 36]. The electrical circuit is formed by placing a unit resistor on each edge of the network and then applying a unit potential difference (voltage) between two vertices chosen arbitrarily. If the network is divided strongly into two communities and the vertices in question happen to fall in different communities, then the spectrum of voltages on the rest of the vertices should show a large gap corresponding to the border between the communities. For more work on resistance distance and resistor networks, the readers are referred to the recent papers [37, 38].
Moreover, the information in complex networks not always flows in the shortest path, so that the evaluation of the betweenness of nodes can also have other principles, such as the current-flow betweenness. Consider an electrical circuit created by placing a unit resistor on every edge of the network. One unit of current is injected into the network at a source vertex and one unit is extracted at a target vertex, so that the current in the network as a whole is conserved. Then, the current-flow betweenness of a vertex is defined as the amount of current that flows through in this setup; the average of the current flow over all source-target pairs is shown as follows: where is the current over vertex .
After placing a unit resistor on every edge in , insert one unit of current or voltage at source vertex labeled with , and further choose the target vertex with labels . Assume the shortest path is from to vertices until . Therefore, the shortest distance is . The property of Koch resister networks is described as follows.
Property 5. The voltages of vertices shape an arithmetic progression from to , and the step length is . The voltage of vertices decreases from to , but the step length is also .
If , from Algorithm 10, there are two hubs and with the highest degree in the shortest path. Hence, the vertices which are affected by unit voltage are , where , , is the neighbor of with the same degree, and apparently is the other hub. The edges between vertices formed triangles which are in series and the common vertices are , so that the unit current only passes through these edges in whole Koch networks .
If and there are two highest degree vertices, noting and , in the shortest path, hence the unit voltage can only affect vertices in , where , , and is the neighbor of with the same degree too, but is a higher degree neighbor which is linked with and directly; the unit current also flows through the edges in triangles which are in series.
If but there are the only highest degree vertices, denoting , in the shortest path, the unit voltage impacts vertices , where , , and is the neighbor of with the same degree; the behavior of unit current is the same as the two conditions above.
Property 6. The current stream from the edges which are linked to the vertices is , while the current passing though the edges linking to is the remaining .
The property can be proved similarly as the proof of Property 5 by the help of the forming mechanism of Koch resistor networks.
In brief, the spectrum of voltages on the vertices shows that Koch networks have no significant community structure in spite of having massive triangles between nodes. Also, the current flow can gauge well the importance of edges betweenness in Koch networks in information flowing which is not flowing only by the shortest path.
6. Conclusions
The family of Koch networks, with high clustering coefficient, scale-free, small diameter and average path length, and small-world properties, successfully reproduces some remarkable characteristics in many nature and man-made networks and has special advantages in the research of some physical mechanisms such as random walk in complex networks.
We provided an informative vertex labeling method and produced a routing algorithm for Koch networks. The labels include full information about any vertices precise position and the time adding to the networks. By the help of labels, we marked the shortest path routing and the shortest distance between any pair of vertices in Koch networks. The needed computation is just no more than times of integral computations and modulo operations. Moreover, we derived the rigorous solution of betweenness centrality of every vertex and edge in Koch networks, and we also researched the current and voltage characteristics in Koch networks on the basis of their labels.
By the help of our results, in contrast with more usually probabilistic approaches, the deterministic Koch models will have unique virtues in understanding the underlying mechanisms between dynamical processes (random walk, consensus, stabilization, synchronization, etc.) to the structure of complex networks by the new method of rigorous derivation.
Competing Interests
The authors declare that there are no competing interests regarding the publication of this paper.
Acknowledgments
This work was supported by the Natural Science Foundation of Guangdong Province under Grant no. 2016A030313703, Guangdong Science and Technology Plan under Grant no. 2016B030305002, the National Natural Science Foundation of China under Grants nos. 61471130 and 61473093, and the Project of Anhui Jianzhu University under Grant no. 2016QD116.