Calculating Information centrality for complex network

Question

I want to add the information centrality metric to my analysis of complex network. Is there any in built function or library that I can use in mathematical. I did not come across any algorithm with clear idea how to implement this.

Update: If I use the algorithm as mentioned in the literature above I get a situation that the inverse of the matrix cannot be calculated (mathematica displays as singular matrix ) as shown below

Data is:

adjCitationData = {
    {18, 5, 2,  4 ,   9,  0,  0,  5, 0, 3,   1,   5,    0,   0},
    {  3, 6, 0,  2 ,   2,  0,  0,  0, 0, 0,   0,   1,   0,    0},
    {  1, 3, 3,  4 ,   3,  0,  0,  0, 0, 0,   0,   1,   0,    0},
    {  9, 0, 0, 68,  25,  0,  0,  0, 0, 6,  12,   0,   6,    0},
    {19, 4, 1, 57, 139,  0,  0, 0, 0, 7,  62,  0,  44,  0},
    {  1, 0, 0,  0 ,   0,  5,  4,  0, 0,  0,   0,    0,   0,   0},
    {  1, 0, 0,  0 ,   0,  3,  2,  0, 0,  0,   0,    0,   0,   0},
    {  6, 0, 0,  0 ,   1,  0,  0,  2, 0,  0,   0,    3,   0,   0},
    {  0, 0, 0,  0 ,   0,  0,  0,  0, 0,  0,   0,    0,   0,   0},
    {  0, 0, 0,  0 ,   0,  0,  0,  0, 0,  0,   0,    0,   0,   0},
    {  8, 2, 0, 44 , 85, 0,  0,  0, 0,  4,  53,   0,  35,  0},
    {  8, 1, 0,  0 ,   1,  1,  0,  2, 0,  0,   0,    6,   0,   0},
    {  1, 0, 0, 25 , 59,  0,  0, 0, 0,  1,  47,   0,  37,  0},
    {  0, 0, 0,  0 ,   0,   0,  0,  0, 0,  0,    0,    0,   0,   0} }


DCr = DegreeCentrality@IGWeightedAdjacencyGraph@adjCitationData;
n = 14;
diagD = DiagonalMatrix[DCr];
matJ = Table[1, {i, n}, {j, n}];
matB = diagD - adjCitationData + matJ;
Inverse[matB];
invB = Inverse[matB] // N;
infoC[i_, j_] := invB[[i, i]] + invB[[j, j]] - 2*invB[[i, j]];
Inverse[matB];(*Inverse cannot be calculated as determinant is zero*)
Dimensions[matB]

IGWeightedAdjacencyGraph is from IGraph http://szhorvat.net/pelican/igraphm-a-mathematica-interface-for-igraph.html

I am I doing this correctly.

Answer 1

@Tugrul Temel`s answer followed the paper instructions and works fine. Here is the fine-tuned version of that code which is around one order of magnitude faster:

ClearAll[informationCentrality];
informationCentrality[g_] := 
 Block[{b = 
    Inverse[1. + DiagonalMatrix[DegreeCentrality[g]] - 
      AdjacencyMatrix[g](*KirchhoffMatrix[g]+1.*)], bDiagonal},

  bDiagonal = Diagonal[b];
  Outer[Plus, bDiagonal, bDiagonal] - 2 * b
  ]

Comparison

(* Tugrul Temel`s answer *)
TugrulTemel[g_] := Block[{n = VertexCount[g], invB},
  invB = Inverse[
    1. + DiagonalMatrix[DegreeCentrality[g]] - AdjacencyMatrix[g]];
  Table[invB[[i, i]] + invB[[j, j]] - 2*invB[[i, j]], {i, n}, {j, n}]
  ]

g = RandomGraph[{70, 260}]

informationCentrality[g] == TugrulTemel[g]
(* Out: True *)

informationCentrality[g]; // RepeatedTiming
(* Out: {0.000533871, Null} *)

TugrulTemel[g]; // RepeatedTiming
(* Out: {0.0123265, Null} *)

Answer 2

Here is an example with a 5 by 5 matrix.

ClearAll[adjA, adjG, degD, diagD, matJ, matB];
SeedRandom[11];
adjA = RandomInteger[{0, 1}, {5, 5}];  (*an example Adjaceny Matrix*)
adjG = AdjacencyGraph[adjA, VertexLabels -> "Name"]; (* Directed graph of adjA *)
degD = DegreeCentrality[adjG]; (* Degree vector of adjG *)
diagD = DiagonalMatrix[degD]; (* Diagonal Degree Matrix of degD *)
matJ = Table[1, {i, 5}, {j, 5}];  (* matrix with all elements 1 *)
matB = diagD - adjA + matJ; (* matrix B *)
Inverse[matB] // N; (* inverse of matB *)

infoC[i_, j_] := invB[[i, i]] + invB[[j, j]] - 2*invB[[i, j]]; (* definition of informational centrality *)

Example: i=2, j=4;

infoC[2,4]
(* 0.320202 *)

You can then calculate any centrality value using infoC[i,j].

UPDATE A generalization of the centrality measure for any directed graph with n vertices;

ClearAll[n, adjA, adjG, degD, diagD, matJ, matB, invB];
SeedRandom[11];
n = 10; (* number of vertices *)
adjA = RandomInteger[{0, 1}, {n, n}];
adjG = AdjacencyGraph[adjA, VertexLabels -> "Name"];
degD = DegreeCentrality[adjG];
diagD = DiagonalMatrix[degD];
matJ = Table[1, {i, n}, {j, n}];
matB = diagD - adjA + matJ;
invB = Inverse[matB] // N;

(* informational centrality measure for individual vertices i and j *)
infoC[i_, j_] := invB[[i, i]] + invB[[j, j]] - 2*invB[[i, j]]

(* full matrix of informational centrality measures *)
Table[infoC[i, j], {i, n}, {j, n}] // MatrixForm