A Method for Measuring Node Importance in Hypernetwork Model

This study is aiming at the central challenge for network analysis and application of how to identify key nodes in hypernetwork representations. Although there are many measuring methods based on traditional network, most of them use single criteria, which is often confronted with the problem of incomplete or inaccurate information on the structure of hyper network. In this study, a new multi-criteria measuring method with adjustable parameters is proposed, which takes the node degree, star degree and betweenness into considerations. According to the experimental results with a numerical example of hypernetwork, our method shows good performance on discrimination and precision to evaluate the importance of nodes as compared with several existing measuring methods for nodes in traditional network.


INTRODUCTION
It is no exaggeration to say that we are living in kinds of networks every minute due to the development of information technology.Social network, computer network, traffic network, knowledge network and even contagion transmission network are surrounding us, making us unable to escape (Newman, 2010).Based on different network, researchers have been studying gradually from the shallower to the deeper and there occurred a lot of important developments in many fields including mathematics, physics, computer and information sciences, biology and the social sciences.For instance, "how does knowledge spread and deliver between humans", "why rumors diffuses faster than the truth", "how does regional or global financial crisis emerge and sprawl", "what causes traffic jams in metropolises and how to avoid them" are some of the hot topics recently discussed.From these topics we can see that on the one hand, humans are thinking how to utilize network to promote the flowing of persons, goods, capitals, information and finally the efficiency of production and management, while on the other hand, it is widely considered how to avoid and reduce hazards of pandemic diseases or big electricity cut caused by the out-of-control of networks.
Along with the rapid development of network theories, the constructed networks specific to different application background are becoming progressively more complex over time (Slotte-Kock and Coviello, 2010): the number of nodes increases, their types vary and the connections between nodes are diversified.Additionally, nodes and edges may have complicated dynamic behaviors such as aggregation, nesting, or reduction etc.In this situation, the relations between objects described by nodes are not limited to be binary, but might be in the form of multi-elemental subset clusters.While sometimes, the isomerism, hierarchy and granularity of objects in network need to be considered.The increasing complexity of network makes it difficult to describe the network feature of real world by ordinary graph in more and more cases.Taking scientific collaboration network for example, by ordinary graph, we can easily represent whether two authors have co-author relationships by two nodes and a line join them up.But, it is difficult to describe the situation that more than three authors write one paper collaboratively.Bipartite graph may be a feasible representation, but it breaks the homogeneity of nodes through the distinction between "author nodes" and "paper nodes", which will result in the ineffectiveness of some parameters by the generated ambiguity when calculating connectivity, centrality and other topological properties (Estrada and Rodriguez-Velazquez, 2005).
Hypernetwork makes it possible for researching on the relationships in and between complex idiosyncratic networks, the concept of which was first proposed by Sheffi (1985) and hypernetwork is regarded as "network above the existing network".By extending network to hypernetwork, it is able to describe the multi-elemental, multilayer, multilevel and multiattribute of objects and significantly enriches the represent ability of the traditional network.For the present, supply chain hypernetwork, financial hypernetwork, power supply hypernetwork, population migration hypernetwork, interpersonal hypernetwork and knowledge hypernetwork has been proposed and studied by many researchers (Wu and Cai, 2008).
Whether in the traditional network or in hypernetwork, it has been an issue worth studying in the fields of network analysis and system science to evaluate the importance of nodes and discover the critical ones.For example, who takes the authoritative position in a research field in scientific collaboration (hyper) network?How to identify and protect the hidden danger points in (hyper) network security?How to distinguish the person (s) whose dimission may bring about great losses of knowledge in knowledge sharing (hyper) network and then make policies to retain the talented person (s).In the control process of virus and disease, the infection of which individuals or groups will lead to pandemic diseases?The theoretical solutions of the problems above can be equivalent to the designing of the method for measuring node importance in (hyper) network aiming at different system features.The objective of this study is to investigate the method for measuring node importance in hypernetwork model, which may help answering the questions above and furthermore make some contributions to hypernetwork theories and applications.Corley and Sha (1982) firstly propose the problem of "most vital node" (MVN).They define the MVN between two nodes as the node that results in the maximum increment of the distance between the two nodes when removed.But regrettably, this method is incapable of the evaluation of node importance in the whole network globally.In recent years, the blossom of empirical study in network, especially the discovery of "small world" and "scale free", makes it reasonable to evaluate the importance of network nodes on the basis of their otherness.The method of measuring node importance in network arisen at present can be classified into two types as follows.

Social Network Analysis (SNA) based methods:
Major methods for measuring node importance based on SNA are under the assumption that the importance of a node is equivalent to its notability when connected to other nodes (Knoke and Burt, 1983).These methods generally study the metrics while maintaining the integrity of network, i.e., do not break the connectivity.The basic idea is to find some useful attributes (such as quantity of information contain in degree or shortest path) to highlight diversity between nodes, in other words, to adequately display the positional traits of nodes in a network and to "magnify" the significance to define the importance.Proposed metrics based on SNA include two categories as "centrality" and "prestige", while indicators include degree, closeness, betweenness, eigenvector, cumulated nomination etc.Many representative works have been done about the SNA based methods by Bonacich (1987), Altmann (1993) and Poulin et al. (2000).Typically, Lee et al. (2010) studied the suitability of degree and betweenness as indicators for the influence of individual customers on the behavior of the entire customer base.Gneiser et al. (2010) analyzed different centrality measures and propose a measure which is based on the Page Rank algorithm.Jun et al. (2010) proposed a unify multiple metrics for evaluating framework of node importance with non-conflict equivalent classes.In a word, these methods can be considered as topology-based.

Removing based methods:
To evaluate the importance of a node in network by measuring the destructiveness to the network performance when remove it is another thinking that is "destructiveness equals to importance".The greater the extent of severity when we remove a node is, the more important it is.Because the maintenance of network connectedness or system function depends on its existence.Network efficiency and contraction are typical removing based metrics for node importance measuring.Li et al. (2004) took the opinion that to destroy node with different distance would bring different losses to network and assumed the losses to vary inversely with the distance between a pair of node.They took the inverse of distance between node pair as weight value and then calculated the weighted sum of all disconnected node pairs to measure the destructiveness to the network connectivity.This study made a quantized contribution to the "removing based" thinking.An et al. (2006) analyzed the requirements of an effective importance measurement for nodes in a network and modeled the comprehensive importance measure for nodes in a node-weighed network based on the approach of deleting nodes.With the help of the procedure of calculating the distances among nodes on a graph, a new algorithm for this model is designed.Leung and Tanbeer (2012) proposed a novel algorithm to mine social networks for significant friend groups, which first constructs a significant friend-tree (SF-tree) to capture important information about linkage between users in the social networks and then uses the SF-tree to find significant friend groups among all friends of users in the social networks by removes friends with low climp values.In general, it is necessary for removing based methods to settle the problems of topological changes even the segmentation of network arise by the nodes removed.

Literature analysis:
In different methods for measuring node importance, common metrics include: degree, betweenness, cohesion, subgraph, spanning tree count and average shortest distance etc.Each indicator focuses on a particular feature of network, e.g., the position of node in network structure, the influence and control capability of node to information spreading, or the contribution of node in substructure.Degree is the simplest method, but it only emphasizes the number of edges connected with adjacent nodes, which ignores the indirect effects from them.Betweenness is used to describe the informational control capability of a node to the connection between other nodes, but it is unable to mediate its partial contribution.Cohesion prefers the geometrical position in a network, but the distinction between nodes in the same hierarchy is not very obvious.Sub graph based methods reflect the contribution to the partial network structure, but the stability of sub graph under different weighting cannot be guaranteed.Removing based methods is difficult to measure the node importance when the influence of removing a node to the network performance is too little or too much.Through the analysis above, we conclude that measuring the node importance under a single indicator will rely seriously on certain aspect of the network characteristics, which ignores the role of other indicators.Moreover, the measuring results on the basis of static network are unlikely to take account of the dynamic relationship between network nodes.Different evaluation methods in some cases provide considerably different results in terms of the importance of nodes (Landherr et al., 2010).Consequently, during the process of practices and studies, usually multiple approaches are adopted to measure the importance of the same node corporately.

Related definitions and notations:
To facilitate the description of our method, we give the definitions to be appeared in the measuring model first as bellows: Hyper network: There are two different definitions to hyper network.The first one defines hyper network as "network in network", while the other one considers hyper network as "network represented by hypergraph".We take the latter one in this study and refer Berge's basic definition of hypergraph (Berge, 1987) to define hyper network as follows: Definition 1: A hypernetwork HN = (V, E) on a set ܸ is a family ݁ of non-empty subsets of ܸ called hyperedges, where V = {V 1 , V 2 , … , V n } is a finite set of nodes, E = {e 1 , e 2 , … , e m } is a family of subset of ܸ, s.t. e ୨ ≠ ∅, j = 1, 2, ⋯ , m and ⋃ e ୨ = V ୨ .Fig. 1 shows an example of hypernetwork, where, In the hypernetwork diagram, the edge e j is represented by a solid line surrounding its nodes; if |e j | = 2 e j can be represented by a solid line connecting the two nodes.The hypernetwork (V, E) can also be denoted by (V; e ଵ , e ଶ , … , e ୫ ).The hypernetwork discussed in this study is undirected hypernetwork without isolated nodes.
Besides the forms of set and diagram, in scientific or engineering calculation tasks, hypernetwork is often handled as matrix.Adjacency matrix ‫ܯܣ‬ × reflects the adjacent relations between hypernetwork nodes, where the element am ୧୨ in ‫ܯܣ‬ is the adjacent value of nodes v i and v ୨ .Incidence matrix IM ୫×୬ describes the relations between nodes and hyperedges, where im ୧୨ = 1 if node v ୨ is an element in ݁ , or else im ୧୨ = 0.

Node degree and star degree:
We give the definition of node degree and star degree as follows: Definition 2: The node degree d (v ୧ ) of node v ୧ ∈ V in hypernetwork HN = (V, E) is the number of nodes connected with ‫ݒ‬ .The definition is the same as that in traditional network and node degree can be calculated as: d ሬሬሬሬԦ = AM ୬×୬ × e ୬×ଵ ሬሬሬሬሬሬሬሬԦ, where e ୬×ଵ ሬሬሬሬሬሬሬሬԦ = {1,1, ⋯ ,1} which is normalized as:

୬ିଵ
Definition 3: For node ‫ݒ‬ ∈ ܸ in hypernetwork HN = (V, E), the star S(v ୧ ) of v ୧ is the partial hypergraph derived from all hyperedges ݁ ∈ ‫ܧ‬ that contain v ୧ .The star degree d ୗ (v ୧ ) of v ୧ is the number of hyperedges in S(v ୧ ), which can be calculated as: where, e ଵ×୫ ሬሬሬሬሬሬሬሬሬԦ = {1,1, ⋯ 1} and element denotes the star degree of v ୧ , which is normalized as: For instance, the star of node ‫ݒ‬ ଶ in Fig. 1 is Hyperpath: Definition 4: Declare hyperpath ‫ܲܪ‬൫v ଵ , v ୯ାଵ ൯ = {v ଵ , e ଵ , v ଶ , e ଶ , ⋯ , e ୯ , v ୯ାଵ } as an alternating sequence of node and hyperedge in hypernetwork ‫ܰܪ‬ = (ܸ, ‫)ܧ‬ with length ‫ݍ‬ when the conditions below are satisfied: • {e 1 , e 2 , … , e q } is different from each other in HN = (V, E) • v k , v k +1 ε e k , k = 1, 2, … , q Betweenness: Definition 5: Betweenness is the fraction of shortest hyperpaths between node pairs that pass through the node of interest.The betweenness ‫ݒ(ܤ‬ ) of node ‫ݒ‬ in hypernetwork HN = (V, E) is calculated as follows: • For each pair of nodes (v ୨ , v ୩ ), compute all shortest hyperpaths between them • For each pair of nodes (v ୨ , v ୩ ), determine the fraction of shortest hyperpaths that pass through v ୧ • Sum this fraction over all pairs of nodes ‫ݒ(‬ , ‫ݒ‬ ) More compactly the normalized betweenness can be represented as: where, ݃ = Total number of shortest hyperpaths from node v ୨ to v ୩ g ୩ (i) = The number of those hyperpaths that pass through node v ୧ Measuring method: From a social capital view, the amount of social capital owned by individual is determined by two factors: • The first one is the scale of connections accessible for actor.• The second one is the occurrence frequency of an individual in the social exchange activities between different peoples (Bourdieu, 1986).
Regard Fig. 1 as a social hypernetwork, the importance of node v ଶ is not only directly bound up with its neighbors as v ଵ , v ଷ , v ସ , v ହ , but as well closely related to its associated hyperedges as e ଵ , e ଶ , e ଷ .Besides, if v ସ , v ହ intent to communicate with v ଵ , v ଷ , or v , it is inevitable to ask v ଶ for agency.Hence, there is indirect relevancy between the importance of v ଶ and the number of individuals that exchange frequently via v ଶ .When we measuring the importance of individual v ଶ in the whole hypernetwork, it is insufficient to count on only one indicator, but should take its adjacent nodes, associated hyperedges and its betweenness in the exchange of hypernetwork into account.
For these reasons, we propose a method for measuring the node importance in hypernetwork with adjustable parameters, which include node degree, star degree and betweenness.Denote I(v) as the importance of node ‫ݒ‬ in hypernetwork and it is calculated as: where, α, β, γ are adjustable parameters, which satisfy α + β + γ = 1.
In order to facilitate contrast between different hypernetworks and eliminate the influence of hypernetwork size on numerical values, it is necessary to normalize indicators to make them in the domain of (0, 1).The normalize importance value of node v is computed as: where,

NUMERICAL EXAMPLE AND DICUSSION
To better illustrate the differentiation between the typical methods for measuring node importance in network, we take the hypernetwork in Fig. 2 for example to discuss the effectiveness and applicability of our measuring method proposed in this study.The experiment is conducted in the environment of Java, in April, 2012 at Jiangxi University of Finance and Economics.
From the calculated results above we find that there are slight differences in the importance value and sort of nodes among these methods, due to their different emphasis on evaluation.Actually, there are certain correlations between them.In traditional network, a node has large degree usually has larger betweenness centrality, closeness centrality, or page rank values as well.However, in hypernetwork, the situation to be considered should make some difference.
In the hypernetwork of Fig. 2, nodes v ଷ and v ସ are adjacent to 5 nodes separately, which have the largest node degree value, yet they locate in relatively marginal positions.The node degree of v , v ଼ and v ଵଵ are all 4, but structurally v ଵଵ is obviously more important than v and v ଼ are the alternative to each other to some extent.Consequently, measuring node importance in hypernetwork solely relying on node degree is impracticable.
Taking the star degree into accounts, the star degree of node v ଵ and v ଷ is 3 equally and this can be considered as they appear in three different groups respectively.But for the reasons of adjacent nodes and intermediary positions, the importance of ‫ݒ‬ ଵ should be more than v ଷ .
Page Rank is a link analysis algorithm, named after Larry Page and used by the Google Internet search engine, which assigns a numerical weighting to each element of a hyperlinked set of documents, such as the World Wide Web, with the purpose of measuring its relative importance within the set (WikiPedia, 2012).In traditional network, PageRank is an effective method for measuring the authority of web pages and individuals.But in hypernetwork, the calculated result by PageRank method shows that ‫ݒ‬ ଷ and ‫ݒ‬ ସ are the most important nodes and the distinctions between node importance values are not obvious.That means PageRank method might not be appropriate for hypernetwork.
Betweenness centrality reflects the significance of node in social/information exchanges.The result shows that the importance value measured solely by betweenness centrality differs over-obviously and the BC value of 7 nodes are 0.For this reason, betweenness centrality should be utilized together with other metrics.
Closeness centrality is also a frequently-used method which indicates the distance between a node to others.But the calculated value shows insufficient differentiations.
According to the method proposed in this study, the sorting of node importance value in Fig. 2 from high to low is: v ଵ , v ଵଵ , v ଼ , v ଷ , v ସ , v , v ଵଷ , v ଵସ , v ଵଶ , v ଽ , v ଵ , v ଶ , v ହ and v .In contrast to the different measuring method above, the calculated result indicates that the proposed method for measuring node importance in hypernetwork model with adjustable parameters is able to highlight the differentiation between nodes, while reflecting the influence of node on the entire hypernetwork objectively.Moreover, the adjustable parameters expand the applied scope of the measuring method, to achieve instructive value to practical application under different hypernetwork modeling circumstances.

CONCLUSION
In this study, we propose a method for measuring node importance in hypernetwork model with adjustable parameters, which takes node degree, star degree and betweenness into considerations.Through comparing among the different node importance measuring methods, our measuring method has the advantages as follows: Firstly, it is capable of depicting nuance of different nodes meticulously.Secondly, the parameters in the computational model can be changed according to different hypernetwork environment, which make our method more flexible.Finally, the method synthetically takes three typical indicators into account that avoids the one-sidedness of measuring based on only one indicator.
Although we hold the point that the method for measuring node importance in hypernetwork model proposed in this study is feasible and effective, there are still many issues that need further researches.Such as: the influence of different values selected for the adjustable parameters on the measuring result and how to judge and weigh the relations of these parameters, etc.All in all, for different hypernetwork application background, to choose suitable parameter combination rationally, this method will develop a greater value in practical hypernetwork applications