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General Centrality in a Hypergraph

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General Centrality in a Hypergraph General Centrality in a hypergraph Evo Busseniers March 20, 2014 The goal of this paper is to present a centrality measurement for the nodes of a hypergraph, by using existing literature which extends eigenvector cen- trality from a graph to a hypergraph, and literature which give a general centrality measurement for a graph. We’ll use this measurement to say more about the number of communications in a hypergraph, to implement a learn- ing mechanism, and to construct certain networks. 1Introduction Network theory is a well established discipline. Usually the presumptions are that there is a one-to-one connection between nodes, by edges. But this is sometimes too simple for reality. We meet up with other peopleingroups, which has other dynamics then each of its members meeting everyone else separately. On the internet, we post messages on forums, which lead to other results then emailing someone. Ahypergraphformalizesthisidea.Itisageneralizationofagraphwhere an edge can connect more then two edges. A hyperedge leads to different behavior then a clique (where all the nodes of the hyperedge would be con- nected one on one to each other). For example, a message can only be send to all or none of the members of a hyperedge, depending on whether it is passed trough the hyperedge or not. The reason I am interested in a centrality measurement is to know which nodes play an important role. I want to see whether there is hierarchy in the network, whether there are nodes who have more to say or more control. IlikedthepaperofBonacichaboutgeneralcentrality[1]because it gives 1 one parameter which corresponds to different sorts of centrality. Usually, a centrality measurement looks either to the local structure (bythedegree) or to the global, the whole network (by the eigenvector centrality). The pa- rameter in the general centrality measurement says how localorglobalyou look. In the eigenvector centrality, the more central your neighbors are, the more central you will be. In some real life situations however, we see the opposite. In a market for example, the more central your neighbors are, the more resources they will already have, and thus the more difficult it will be for you to trade with them. Thus you will be less central. This is also taken into account in the general centrality measurement, it corresponds to the case when the parameter is negative. This general centrality measurement also has a nice intuitive explanation. The centrality of a node can be seen as the number of communications start- ing from that node. The parameter is then equal to the chance a message is passed. Thus the bigger the parameter, the further in the network you will look. Iwillnowdefineahypergraph,tobeabletodefinealltheseconcepts. Hypergraph The idea of a hypergraph is to extend a graph so that edges can connect more than two nodes. It is defined as follows: 1.1 Definition. An (undirected) hypergraph is a couple of two sets (V,E), with an element Ej ∈ E asubsetofV . An element vi ∈ V is called node, an element Ej ∈ E is called (hyper)edge. We can represent this by a |V | × |E|-matrix R, where Rij =1ifvi ∈ Ej, and 0 otherwise. All edges have the same strength this way, if wewant the edges to have weights, we’ll work with a matrix of weights W ,with Wij ∈ [0, 1] the weight of vertex vi in edge Ej.Inanundirectedhypergraph, this is the same as the weight from Ej to vi.Thiswon’tbethecaseina directed hypergraph, which is defined as follows: 1.2 Definition. Adirectedhypergraph is a couple of two sets (V,E), with an element Ej ∈ E acouple(Ij,Oj)oftwosubsetsofV . Ij is called the set of input nodes and Oj is called the set of output nodes. 2 Aweighteddirectedhypergraphcanberepresentedbytwomatrices: a |V |×|E|-matrix W ,givingtheweightsfromverticestoedges,anda|V |×|E|- matrix Z,whichrepresentstheweightsfromedgestonodes.Inthefollowing we will work with an undirected hypergraph, note that this caneasilybe extented to a directed hypergraph by using Z instead of W T . Ahypergraphcanbepulledbacktoastandardgraph.Onewayisto repre- sent the hypergraph by a bipartite graph with 2 kind of nodes: one sort is the nodes of the hypergraph, the other sort consists of the hyperedges of the hypergraph. Two nodes of different sorts are connected if the node is in the hyperedge in the hypergraph structure. Another way to build a standard graph out of a hypergraph is to put an edge between all nodes which are in the same hyperedge. Consider two nodes vi and vk which are in the same hyperedge Ej.ThecontributionofEj to the weight aik between vi and vk is wijwkj.Thetotalweightisobtainedby summing over all the hyperedges which contain both vi and vk,thus: |E| a = w w ik ! ij kj j=1 or in matrix notation: A = WWT For a directed hypergraph, we use Z instead of W T ,thus A = WZ Note that there is some information lost: it is impossible to know in this representation which nodes are in the same hyperedge. Ahypergraphcanhavedifferent topologies. To differentiate between them, we can extrapolate from the work already done in standard graphs. A common way to differentiate between different topologies, is to take different distributions of a property of the nodes. We’ll look at two properties: the degree and the centrality. First, we’ll look at the eigenvector centrality, next, we’ll introduce a more general measurement c(α, β), which depends on some parameters. Then we’ll explain the link between the general centrality and the number of communications send by a node. 3 2Degree What is mostly done first, is to look at the degree. A hypergraphhastwo sorts of degree: the degree of a node, which is the number of hyperedges it is contained in, and the degree of a hyperedge, which is the number of nodes it contains. 3Centrality 3.1 Eigenvector centrality What we will represent here, basicly comes from a paper of Volpentesta [3]. The idea behind the eigenvector centrality is that a node is more central as his neighboring nodes are more central. In a standard graph, the eigenvector centrality ei of a node i is: λe = W e i ! ij j j with W the matrix of the weights of the edge between two nodes (0 if there is no edge). λ is just a factor so that the equations have a solution. In matrix notation this looks as follows: λe = W e This is an eigenvector equation: the solutions e are the eigenvectors of W ,withλ its eigenvalue. Due to the theorem of Perron-Frobenius, the eigen- vector associated with the largest eigenvalue has only positive entries. To extend this to a hypergraph, we will assign a centrality both on the nodes and the edges. A node is more central as the edges it is contained in are more central, and analog for the edges. Thus, the centrality xi of node i is: c1x = W y i ! ij j j while the centrality yj of an edge j is: c2y = W x . j ! ij i i 4 Or, in matrix notation: c1x = W y T c2y = W x or, written in equations with only x or y: WWT x = λx W T W y = λy with λ = c1c2 3.2 General centrality In a graph, a general centrality measurement for a node i is defined as follows [1]: c (α, β)=α W +β c W i ! ij ! j ij j j "degree#$ % eig." #$ centr.% or in matrix notation: c(α, β)=αW 1 + βW c(α, β) ⇒ (I − βW )c(α, β)=αW 1 ⇒ c(α, β)=α(I − βW )−1W 1 (1) The above formula is only defined if the inverse (I − βW )−1 is defined, thus if det(I − βW ) =0.Wehave,if% β =0:% 1 det(I − βW )=det(β( I − W )) β 1 =det(−β(W − I)) β 1 =(−β)ndet(W − I) β Since the eigenvalues of W are the solutions of the equation det(W −λI)= 0, 1 (−β)ndet(W − I)=0 β 5 1 1 if β is an eigenvalue. This is equal to β = λ ,withλ an eigenvalue. If β =0, we simpy have I−1 = I,thustheinverseisdefined.Thiscasemeansthe centrality of a node is just α times his degree. We will use this result further on, so we put it in a property: −1 1 3.1 Property. (I − βW ) is defined, if and only if β =% λ ,withλ an eigenvalue. 1 If β = λ ,wegottheeigenvectorcentralityifα =0. The idea behind the formula is that the centrality partially depends on the local situation, measured by the degree, and partially on theglobalsituation, measured by the eigenvector centrality. The bigger the absolute value of β, the more globally the centrality is. The sign of β tells whether the neighbours of a node has a positive or a negative effect on that node: if β is positive, the more central your neighbours are, the more central you will be, while if β is negative, the more central your neighbours are, the less central you will be. An example of the second case is a market, where you’re punished if you have a central neighbour, who is more difficult to trade with. As can be seen in the above formula, α is just a scaling factor for the centrality, thus it has no effect on the distribution. Usually α is chosen such that c (α, β)2 = |V | ! i i Thus, ci(α, β)=1meanspositioni has an average centrality. The extension to a hypergraph is similar to the previous subsection: the centrality of a node i is defined as: x = α1 W + β1 W y i ! ij ! ij j j j And for an edge j: y = α2 W + β2 W x j ! ij ! ij i i i In matrix notation: x = α1W 1 + β1W y (2) T T y = α2W 1 + β2W x 6 Or notated with x and y in seperate equations: T −1 T x =(I − β1β2WW ) W (α11 + β1α2W 1)(3) T −1 T y =(I − β1β2W W ) W (α21 + β2α1W 1) 1 T T If β1β2 =% λ ,withλ an eigenvalue of WW or W W , the inverse is defined (3.1).
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