In graph theory, the Katz centrality or alpha centrality of a node is a measure of centrality in a network. It was introduced by Leo Katz in 1953 and is used to measure the relative degree of influence of an actor (or node) within a social network. Unlike typical centrality measures which consider only the shortest path (the geodesic) between a pair of actors, Katz centrality measures influence by taking into account the total number of walks between a pair of actors. It is similar to Google's PageRank and to the eigenvector centrality.
Measurement
Katz centrality computes the relative influence of a node within a network by measuring the number of the immediate neighbors (first degree nodes) and also all other nodes in the network that connect to the node under consideration through these immediate neighbors. Connections made with distant neighbors are, however, penalized by an attenuation factor α {\displaystyle \alpha } . Each path or connection between a pair of nodes is assigned a weight determined by α {\displaystyle \alpha } and the distance between nodes as α d {\displaystyle \alpha ^{d}} . For example, in the figure on the right, assume that John's centrality is being measured and that α = 0.5 {\displaystyle \alpha =0.5} . The weight assigned to each link that connects John with his immediate neighbors Jane and Bob will be ( 0.5 ) 1 = 0.5 {\displaystyle (0.5)^{1}=0.5} . Since Jose connects to John indirectly through Bob, the weight assigned to this connection (composed of two links) will be ( 0.5 ) 2 = 0.25 {\displaystyle (0.5)^{2}=0.25} . Similarly, the weight assigned to the connection between Agneta and John through Aziz and Jane will be ( 0.5 ) 3 = 0.125 {\displaystyle (0.5)^{3}=0.125} and the weight assigned to the connection between Agneta and John through Diego, Jose and Bob will be ( 0.5 ) 4 = 0.0625 {\displaystyle (0.5)^{4}=0.0625} .
Mathematical formulation Let A be the adjacency matrix of a network under consideration. Elements ( a i j ) {\displaystyle (a_{ij})} of A are variables that take a value 1 if a node i is connected to node j and 0 otherwise. The powers of A indicate the presence (or absence) of links between two nodes through intermediaries. For instance, in matrix A 3 {\displaystyle A^{3}} , if element ( a 2 , 12 ) = 1 {\displaystyle (a_{2,12})=1} , it indicates that node 2 and node 12 are connected through some walk of length 3. If C K a t z ( i ) {\displaystyle C_{\mathrm {Katz} }(i)} denotes Katz centrality of a node i, then, given a value α ∈ ( 0 , 1 ) {\displaystyle \alpha \in (0,1)} , mathematically:
C K a t z ( i ) = ∑ k = 1 ∞ ∑ j = 1 n α k ( A k ) j i {\displaystyle C_{\mathrm {Katz} }(i)=\sum _{k=1}^{\infty }\sum _{j=1}^{n}\alpha ^{k}(A^{k})_{ji}}
Note that the above definition uses the fact that the element at location ( i , j ) {\displaystyle (i,j)} of A k {\displaystyle A^{k}} reflects the total number of k {\displaystyle k} degree connections between nodes i {\displaystyle i} and j {\displaystyle j} . The value of the attenuation factor α {\displaystyle \alpha } has to be chosen such that it is smaller than the reciprocal of the absolute value of the largest eigenvalue of A. In this case the following expression can be used to calculate Katz centrality:
C → K a t z = ( ( I − α A T ) − 1 − I ) I → {\displaystyle {\overrightarrow {C}}_{\mathrm {Katz} }=((I-\alpha A^{T})^{-1}-I){\overrightarrow {I}}}
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