Research on the Shortest Hyperpath Algorithm of Hypernetwork and Its Application

2018 International Conference on Computational, Modeling, Simulation and Mathematical Statistics (CMSMS 2018) ISBN: 978-1-60595-562-9

 

Research on the Shortest Hyperpath Algorithm of

Hypernetwork and Its Application

Zheng GUO

1,2,*

, Dan JI

3

and

Fu-hong WANG

1,2.

1_{Shanghai Association for Quality, China}

2_{Shanghai Zhong-de Quality Technology Research Center Shanghai, 200052, China}

3_{Shanghai JiaoTong University, 200240, China}

*_{Corresponding author}

Keywords: Hypernetwork, Shortest hyperpath algorithm, Efficiency of hypernetwork, Betweenness of nodes and hyperedges.

Abstract. Hypernetwork based on the theory of hypergraph can better describe some problems which complex network cannot describe. Many scholars pay more attention to hypernetwork theory study. From a literature review, There are many researches on the theory of hypernetwork, but the research on the shortest hyperpath analysis algorithm of hypernetwork is a blank at present. In this paper, based on the incidence matrix of hypergraph, we use the combination of iterative algorithm and breadth-first algorithm and depth-first algorithm to calculate the shortest distance between any two points in the hypernetwork, and then calculates the maximum diameter of hypernetwork, average of hyperpath length, the propagation efficiency of hypernetwork, nodes and hyperedges of betweenness of hypernetwork are further calculated according to the shortest distance between any two points. These studies are the basis of exploring the dynamic mechanism of hypernetwork.

Introduction

In the theory of complex networks, a adjacency matrix based a graph is used to calculate the

shortest distance between any two points using Dijkstra[1-2] _{or Floyd}[3-4]_{algorithm. Thus the network}

diameter, average of path length and network efficiency can be further calculated and analyzed according to the shortest distance. The research and optimization on shortest path algorithm has wide application. for example: resource allocation and management, flow analysis, evaluation of the importance of nodes and edges and so on.

But complex network can only express the relationship between the two nodes, and cannot well description of the relationship between nodes with more than two elements. Although bipartite graphs can describe relationships with more than two elements. the homogeneity of nodes and

edges are lost. Complex networks have some limitations when portraying the real world[5-7]

With the development of hypergraph theory, hypernetwork theory based on Hypergraph has attracted the attention of scholars both at home and abroad[8-10]. The mathematical basis of hypernetworks is the incidence matrix. However, how to calculate the shortest distance between any two points in the hypernetwork based on incidence matrix is a research gap now.

According to the research of Feng Keqin [11] and Ma [12], first, we need to map the node set and edge set in the hypergraph into the bipartite graph, and then define the adjacency matrix of the hypergraph according to bipartite graph. Based on this adjacency matrix, we get the shortest path between any nodes of hypergraph and get the diameter of hypergraph. But this transformation is sometimes rather complicated, and not reasonably reflect the interaction and restriction among nodes of hypernetwork. Furthermore, the adjacency matrix of the hypernetwork cannot fully express the entire relationship between the nodes and edges of the hypernetwork. Therefore, this kind of research has certain limitations in theory.

In this paper, we propose a new algorithm, which based on the incidence matrix of the hypernetwork, the shortest path of any two points is calculated directly. Furthermore, we can use this algorithm to calculate the maximum diameter of the whole hypernetwork, the average of shortest hyperpath, the efficiency of the whole hypernetwork and so on. This new the shortest hyperpath algorithm of hypernetwork is also different from the shortest hyperpath algorithm proposed by document [12] to transform hypergraphs into graphs.

Related Concepts

The Definition of Hypernetwork

Suppose V {v₁,v₂,...,v_n}is a finite set, whereV {v₁,v₂,...,v_n}is the set of nodes (or vertices),

and _Eh {_E1,_E2,...,_Em}_{is the set of the hyperedges, which contain an arbitrary number of nodes.} And it is easily to realized that E_i 



(i1,2,..,m)and_im_1

E

_i 

V

, We Call H (V,Eh)is a

hypergraphH[13]_{. If two nodes belong to the same hyperedge, we consider the two nodes are}

adjacent; If the intersection of two hyperedge is not empty, we say the two edges are adjacent. As shown in Figure 1, A hypergraph is H (V,E) , a vertices set is V 



v₁,v₂,v₃,v₄,v₅,v₆



_,

hyperedge is as follow:

1

E =



v₄,v₂,v₆



,E₂ 



v₁,v₄



,E₃ 



v₁,v₃,v₅



,E₄ 



v₃,v₅,v₆



,E₅ 



v₂,v₅,v₆



, E6 



v1,v6



If the number of vertices in each hyperedge are equal, it is called uniform hypergraph, if

V _and Eh _{are both finite, call} H _{is finite hypergraph., as shown in figure 1.A} k _-uniform

hypergraph is a hypergraph that all hyperedges are formed with k nodes. In this way, the

2-uniform hypergraph as shown in figure 2 is just the regular graph as shown in figure 3. Hypergraph in figure1 can be transformed into a bipartite graph of complex network theory, as shown in figure 4.

       

Figure 1. A non-uniform Hypergraph and hypernetwork simple.

[image:3.612.131.498.71.224.2]

Figure 3. Regular graph. Figure 4. A bipartite graph.  

The incidence matrix of hypergraph ( as shown in Figure 1 is expressed byBm n , If node Vis the

element of edge E, then we call A and B are associated. We define

 

_{i j,}

n m

B b



 , if and only if

i j

v



E

, where B _{i j,} 1, Otherwiseb_{i j,} 0 Such that

1

_{2 3 4 5 6} 1

2 3 4 5 6

0 1 0 1 0 1 1 0 0 1 0 0 1 0 1 0 1 0 0 0 1 0 1 1 0 1 0 0 1 1 1 0 0 0 0 1

m n

v v v v v v

E

E

E B

E

E

E



 

 

 

 

 _{ }

 

 

 

 

Based on the above definition, Guo[14] give mathematical definition of the hypernetwork. Suppose_{ }



(_V,_Eh),(_V,_Eh)



_{is a finite hypergraph and G is a map from T }

 

_{0, into;}

for any given

t

_ 0,

G

(

t

)_ (

V

(

t

),

E

h(

t

))_{is a finite hypergraph. The index tis often interpreted}

as time. A hypernetwork



G t t( ), T



is a collection of hypergraphs.

The Definition of Characteristic Parameters of Hypernetwork

In the hypergraph H, if v_ie_i,then v_i is associated with the hyperedge e_i. If v_ie_f and

i R

v e , It is called e_f and e_Ris adjacent. Suppose the vertex hyperedge interleave sequence:



v1,e1,v2,e2,es,es1



,

a. v v1, ,2 vs is different from each other in H.

b. e1,e2,,es is different from each other in H.

C. vR,vR1eR , R1, 2, , , g this sequence is a hyperpath. If v1vg1_, It is called a

hyper-loop.

The number of hyperedges between two nodes passing a hyperpath with the least number of hyperedges is called the hyper-distance between two nodes. The diameter of the hypernetwork is defined as the maximum of all hyper-distances.

1 ,maxi j N ij

D d

 

 (1) WhereNis the number of all nodes in the hypernetwork and d_ijis the length of the hyperpath

V3 V4

V1 V2

V5

between the node iand the nodej.

Average of hyperpath length of hypernetwork:





2 1 ij i j D D N N    



(2)

Global efficiency of hypernetwork:

1 E

D



  (3) Hyper-betweenness of node:

 

,

jl i

j l V ij j l N i B N  





(4)

V represents a node set,  N _jl

 

i is the number of shortest hyperpath between nodejand nodel

passing through nodei, andN_ijis the total number of shortest hyperpaths between all nodes in the

hypernetwork.

Hyper-betweenness of edge:

 

 

 

1 , , ,, i i m i gl j

j _gl j

E E

g l E g l i V

N E B E N E    





(5)

 

gl i

N E is the number of shortest hyperpath passing through the edge E_i, and N_gl

 

E_j is the

total number of shortest hyperpath passing through all hyperedges in the hypernetwork.

Graph Algorithm

The traversal algorithm for graphs and special subgraphs (such as trees and hypertrees) is one of the most useful graph algorithms. There are two common approaches to searching a graph: depth-first search and breadth-first search.

Depth First Searching Algorithm. Start with one nodev_ifirst, and then visting any node, v_jin its neighbor node collectionO v

 

i , vjO v

 

i , Performing DFS on the node vj, in turn visiting

other neighbor nodes in O v

 

_i . DFS searches for a node as deep as possible and then traces back to

other neighbor nodes.

Breadth-First Searching Algorithm. Starting from a node in the graph, first visiting all its directly neighbors and then move to their second-level nodes by traversing their neighbors. BFS searches for a node as widely as possible in the graph and then moves to the next layer of nodes for accesse.

In this paper, we combines two traversal algorithms: depth first search (DFS) and breadth first searching (BFS) to analysis the shortest hyper-distance between any two points of incidence matrix based on hypernetwork.

The Shortest Hyperpath Algorithm of Hypernetwork Based on Incidence Matrix

An algorithm of finding the shortest path from the source node V_s to the target node V_d based on

the incidence matrix.

The "OnePath" variable represents a path, for example: onePath=V_sE₁E₂V_d. The "MinPath" variable represents the shortest path.

The "OnePathExist" variable represents whether a path is found or not. BEGIN

Step1:Initialize the variables, including current searching path: CurrentSearchPath=V_s,shortest path, MinPath=””, OnePathExist=false

Step2:Get all routes E1n that links the vertices of the end of CurrentSearchPath Step3: Get all vertices V_1m connected to E_i, E_i is in E_1n.

Step4: Check if V_d is in V_1m?, if “Yes” we get a path: CurrentSearchPath+E_i+ V_d,so we set OnePathExist=true, .if MinPath=”” then assign the path to MinPath, else if Len(OnePath)<len(MinPath), we set the path to MinPath too, else goto Step 6 . if V_d is not in

1 m

V_{ , goto Step 5}

Step5: i i 1 , if in goto Step3, else if OnePathExist=true, goto Step6, else CurrentSearchPath=CurrentSearchPath E_i V i_j, 1n j, 1m and goto Step2.

Step6: END .Of course we have gotten thes shortest path: MinPath.

The flow chart of the shortest hyperpath algorithm of hypernetwork is shown in Figure 5.

CurrentSearchPath=Vs  MinPath=””  OnePathExist=false

Get all verticesV1...m connected to Ei，Ei  is In E1...n   

OnePath= CurrentSearchPath+Ei+Vd  OnePathExist=true

MinPath=OnePath

Is  Vd  in V1…m ?

MinPath=””？

Len(OnePath)<len(MinPath)？

i=i+1 i<=n？

OnePathExist=true？

CurrentSearchPath=CurrentSearchPath+Ei+Vj i=1…n j=1…m

End

NO  Yes 

NO 

Yes Yes 

Yes 

Get all routes E1...n that links the vertices of the end of  CurrentSearchPath

NO 

NO 

No

Yes Yes 

[image:5.612.186.442.322.641.2]

 

Figure 5. Flow chart of the shortest hyperpath algorithm of hypernetwork. 

China High Speed Rail Hypernetwork Model Based on Simple Hypergraph Data Source Description

China High Speed Rail Hypernetwork Model Based on Simple Hypergraph

[image:6.612.226.367.118.267.2]

We use each station as a node in the hyper network, and each high-speed railway line is regarded as a edge of the hypernetwork, we establish a high-speed rail network model as shown figure 6.

Figure 6. A schematic illustration of hypernetwork structure of high-speed railway in China.

In figure 6, the black dot represents the site for the node, E1, E9 is treated as hyperedges representing the lines, Considering the simplicity of the study, we do not consider the repeating route. Each lines is regarded as an hyperedge which mean spans all the stations that the train visits. The shortest hyperpath algorithm based on the hypernetwork’s incidence matrix is presented in this paper. Taking Shanghai Station as an example, the shortest hyperpath between Shanghai and any other station is shown in Table 1.

Table 1. Some of shortest hyperpath from Shanghai station to any other station.

any two stations shortest hyperpath

Shanghai-Anyang Shanghai-Jinghu line-Beijing-Jinggang line-Anyang

ShangHai-Baiyin Shang hai-Jinghu line-Beijing-Shenlan line-Baiyin

Shanghai-Baotou Shang hai-Jinghu line-Beijing-Shenlan line-BaoTou

Shanghai-Baoji Shang hai-Hahu line-Yancheng-Yanxi line-Baoji

Shanghai-Baoding Shang hai-Jinghu line-Beijing –Jinggang line-Baoding

Shanghai-Beihai Shang hai-Hunan line-Nanning –Xizhan line-Beihai

Shanghai-Dazhou Shang hai-Hunan line-Nanning –Xizhan line-Dazhou

According to the formula (1) and(2), China's high-speed rail hypernetwork with 19 hyperedges and 192 stations, which is a connected hypernetwork with a maximum diameter of 4, which means any two points can be reached through a turnover of four lines.According to formula (2), the average minimum distance is 2. 338, which means any two points can be reached by minimum average of 2.338 line. According to the formula (3), the maximum efficiency of the whole hypernetwork is 0.4227.According to the formula(4), the node with the largest hyper-betweeness is Shanghai Station, and the edge with the largest hypere-betweeness is the Jin-Kun Line.

Conclusions and Future Research

[image:6.612.113.501.391.496.2]

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