Status and Prestige in Two Mode Networks

Two-Mode Networks as Prisms

We can use the same “prismatic” mode of status distribution to rank nodes in two-mode networks that we used in the standard one-mode case. The difference is that in the one-mode case you gain status by connecting to entities of the same “type” as you (e.g., other people). In the two-mode case, your status reflects the status of the “other type” entities that you connect to. And vice versa.

For instance, if the two entities are people and groups (Breiger 1974), then people gain status by connecting to high status groups and groups gain status by connecting to high-status people. In other words, people distribute status points to the groups they belong to and groups distribute status points to the people that belong to them.

When it comes to eigenvector-style measures, the neat idea is that people are central if they belong to central groups and groups and central if their members are central people (with people centrality defined by membership in central groups) can be effectively captured by these metrics (Bonacich 1991).

For this reason, measures of status and prestige are particularly applicable to two-mode networks. The reason is that the reflective principle behind these measures interacts nicely with the duality principle.

Bonacich Prestige in the Southern Women Data

Let’s see an example using the classic Southern Women dataset. We can load it from the trusty networkdata package, and extract the bi-adjacency matrix from the igraph object:

   library(networkdata)
   library(igraph)
   g <- southern_women
   A <- as.matrix(as_biadjacency_matrix(g))

Below is a quick function that plays a two-mode version of the status distribution game that we described at the beginning, and which is really just a modification of the one-mode HITS algorithm:

   tm.status <- function(A1, A2) {
      y <- matrix(1/ncol(A1), ncol(A1), 1) #initial group status column vector set to a constant
      delta <- 1 
      k <- 0
      while (delta > 1e-10) {
         o.y <- y #old group status scores
         x <- A1 %*% o.y #new people scores a function of people matrix and old group scores 
         x <- x/norm(x, type = "E") #normalizing new people status scores 
         y <- A2 %*% x #new group scores a function of group matrix and new people scores
         y <- y/norm(y, type = "E") #normalizing new group status scores 
         if (k > 1) {
            delta <- abs(sum(abs(y) - abs(o.y))) #diff. between new and old group status scores
            }
         k <- k + 1
         }
   return(list(p.s = x, g.s = y, k = k))
   }

Line 2 initializes the original status scores for each group stored in the y object, which are just set to \(|G|^{-1}\) where \(|G|\) is the number of groups (the number of columns of the bi-adjacency matrix \(\mathbf{A}\)). Line 3 initializes the \(\delta\) value, which determines when the while loop stops. Then, inside the while loop starting on line 5, we assign status scores to the people equal to the sum of the status scores of the groups they belong to in line 6, which is just \(\mathbf{A}\) post-multiplied by the group status vector \(\mathbf{y}\), then we normalize the people status score vector in line 8 using the Euclidean norm. In line 9 we calculate the new group status vector, which, for each group, is given by the sum of the status scores of the people that belong them that we computed in line 7. Then we normalize the new group status scores in line 13 and compute the difference between these new group scores and the old one to calculate \(\delta\). When \(\delta\) is small (\(\delta \leq 10^{-10}\)) the while loop stops as the old and new scores have achieved convergence.

We can now use the tm.status function to estimate the status scores for people and groups:

   tm.hits <- tm.status(A, t(A))

Here are the scores for the people:

   round(tm.hits$p.s/max(tm.hits$p.s), 3)
           [,1]
EVELYN    0.903
LAURA     0.834
THERESA   1.000
BRENDA    0.845
CHARLOTTE 0.454
FRANCES   0.564
ELEANOR   0.616
PEARL     0.486
RUTH      0.637
VERNE     0.589
MYRNA     0.504
KATHERINE 0.594
SYLVIA    0.748
NORA      0.712
HELEN     0.542
DOROTHY   0.355
OLIVIA    0.188
FLORA     0.188

And for the groups:

   round(tm.hits$g.s/max(tm.hits$g.s), 3)
       [,1]
6/27  0.280
3/2   0.297
4/12  0.499
9/26  0.347
2/25  0.635
5/19  0.647
3/15  0.757
9/16  1.000
4/8   0.749
6/10  0.336
2/23  0.177
4/7   0.400
11/21 0.223
8/3   0.223

Here, we can see that Theresa is the top person (attending the most central events) closely followed by Evelyn, with Flora and Nora toward the bottom. The event held at 9/16 is the top event (attended by the most central people).

When we use the usual affiliation matrix A and its transpose t(A) as inputs, the tm.status function implements the biHITS algorithm described in Liao et al. (2014). One thing to note in this regard is that the biHITS algorithm is just a rediscovery of the two-mode prestige scoring described much earlier by Bonacich (1991).

Interestingly as Bonacich (1991) noted in that paper, the eigenvector status scores can also be obtained by playing the one-mode version of the status game over the Breiger-style one-mode projections of the two-mode network.

   status1 <- function(w) {
      x <- rep(1, nrow(w)) #initial status vector set to all ones of length equal to the number of nodes
      d <- 1 #initial delta
      k <- 0 #initializing counter
      while (d > 1e-10) {
          o.x <- x #old status scores
          x <- w %*% o.x #new scores a function of old scores and adjacency matrix
          x <- x/norm(x, type = "E") #normalizing new status scores
          d <- abs(sum(abs(x) - abs(o.x))) #delta between new and old scores
          k <- k + 1 #incrementing while counter
      }
   return(as.vector(x))
   }

To see this, let’s play the usual status game defined by the function status1 over the one mode projections. For people this would be:

   p.s <- status1(A %*% t(A))
   names(p.s) <- rownames(A)
   round(p.s/max(p.s), 3)
   EVELYN     LAURA   THERESA    BRENDA CHARLOTTE   FRANCES   ELEANOR     PEARL 
    0.903     0.834     1.000     0.845     0.454     0.564     0.616     0.486 
     RUTH     VERNE     MYRNA KATHERINE    SYLVIA      NORA     HELEN   DOROTHY 
    0.637     0.589     0.504     0.594     0.748     0.712     0.542     0.355 
   OLIVIA     FLORA 
    0.188     0.188

And for groups:

   g.s <- status1(t(A) %*% A)
   names(g.s) <- colnames(A)
   round(g.s/max(g.s), 3)
 6/27   3/2  4/12  9/26  2/25  5/19  3/15  9/16   4/8  6/10  2/23   4/7 11/21 
0.280 0.297 0.499 0.347 0.635 0.647 0.757 1.000 0.749 0.336 0.177 0.400 0.223 
  8/3 
0.223

Lo and behold, these are the same status scores we obtained via the biHITS approach.

Also like eigenvector style measures for two-mode networks, the iterative status game scores can be obtained as a solution to an linear algebra eigenvector problem involving the relevant matrices over which the game is played.

In the case of the two-mode eigenvector scores, as Bonacich (1991) also noted, it turns out they can be computed by figuring out the leading eigenvector (what our status game does for any matrix) of the Breiger projection matrices:

\[ \lambda x = (\mathbf{A}\mathbf{A}^T)x \]

\[ \lambda y = (\mathbf{A}^T\mathbf{A})y \]

In R we can do this using the eigen function:

   eig.p <- eigen(A %*% t(A))
   eig.g <- eigen(t(A) %*% A)
   p.s <- eig.p$vector[, 1] * -1
   g.s <- eig.g$vector[, 1] * -1
   names(p.s) <- rownames(A)
   names(g.s) <- colnames(A)

And for the big reveal:

   round(p.s/max(p.s), 3)
   EVELYN     LAURA   THERESA    BRENDA CHARLOTTE   FRANCES   ELEANOR     PEARL 
    0.903     0.834     1.000     0.845     0.454     0.564     0.616     0.486 
     RUTH     VERNE     MYRNA KATHERINE    SYLVIA      NORA     HELEN   DOROTHY 
    0.637     0.589     0.504     0.594     0.748     0.712     0.542     0.355 
   OLIVIA     FLORA 
    0.188     0.188 
   round(g.s/max(g.s), 3)
 6/27   3/2  4/12  9/26  2/25  5/19  3/15  9/16   4/8  6/10  2/23   4/7 11/21 
0.280 0.297 0.499 0.347 0.635 0.647 0.757 1.000 0.749 0.336 0.177 0.400 0.223 
  8/3 
0.223

Neat! These are the same scores we obtained by playing our status game on the affiliation matrix (biHITS) using tm.status or in the one-mode projection (Bonacich) using the status1 function.

The scores are also readily interpretable: The most central people belong to the most central (largest membership) groups and the most central groups are the ones that attract the most central (highest activity) members.

Duality and Two-Mode Bonacich Prestige

Another way of thinking of the eigenvector status score of each node in this context is as a weighted sum1 of the eigenvector centralities on the nodes in the other mode they are connected to (Faust 1997, 170).

For instance the unnormalized eigenvector status scores for persons and groups are:

   p.s
   EVELYN     LAURA   THERESA    BRENDA CHARLOTTE   FRANCES   ELEANOR     PEARL 
0.3347340 0.3092247 0.3705636 0.3130088 0.1682137 0.2089673 0.2283461 0.1800298 
     RUTH     VERNE     MYRNA KATHERINE    SYLVIA      NORA     HELEN   DOROTHY 
0.2360402 0.2183986 0.1867900 0.2202912 0.2771746 0.2639043 0.2006669 0.1314354 
   OLIVIA     FLORA 
0.0695706 0.0695706 
   g.s
      6/27        3/2       4/12       9/26       2/25       5/19       3/15 
0.14194312 0.15047999 0.25285306 0.17599173 0.32173361 0.32761921 0.38350293 
      9/16        4/8       6/10       2/23        4/7      11/21        8/3 
0.50663271 0.37949240 0.17040087 0.08954622 0.20279505 0.11293095 0.11293095

So for any person, let’s say {EVELYN}, their eigenvector centrality is equal to:

   sum(A["EVELYN", ] * g.s) * 1/sqrt(eig.p$values[1])
[1] 0.334734

Which is indeed Evelyn’s Eigenvector score.

The same goes for the eigenvector score of groups, which are just a weighted sum of the Eigenvector centralities of the people who belong to them:

   sum(A[, "6/27"] * p.s) * 1/sqrt(eig.p$values[1])
[1] 0.1419431

Which is indeed the eigenvector score for the event held on 6/27.

Duality at work!

Two-Mode PageRank

Like in the case of the usual eigenvector approach to calculating status, the distribution game in the function tm.status assumes that persons and groups distribute the same amount of status regardless of their own degree (which you recall for people is the number of memberships and for groups is the number of members).

But just like with regular PageRank for one-mode networks, we can change this assumption in the two-mode case by presuming that people get more status points to distribute when they belong to exclusive (smaller) groups, and groups get more status to distribute when their members are more selective in their affiliation behavior (they have a smaller number of memberships).

To do that, we create new versions of the affiliation matrix \(\mathbf{A}\) and its transpose \(\mathbf{A}^T\) normalized by the group and persons degrees (respectively).

To do this, imagine that \(\mathbf{D}_p\) is a matrix containing the inverse of the degrees of each person (number of memberships) along the diagonals. This matrix can be created in R as follows:

   D.p <- diag(1/rowSums(A))
   round(D.p, 2)
      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13]
 [1,] 0.12 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00  0.00  0.00  0.00  0.00
 [2,] 0.00 0.14 0.00 0.00 0.00 0.00 0.00 0.00 0.00  0.00  0.00  0.00  0.00
 [3,] 0.00 0.00 0.12 0.00 0.00 0.00 0.00 0.00 0.00  0.00  0.00  0.00  0.00
 [4,] 0.00 0.00 0.00 0.14 0.00 0.00 0.00 0.00 0.00  0.00  0.00  0.00  0.00
 [5,] 0.00 0.00 0.00 0.00 0.25 0.00 0.00 0.00 0.00  0.00  0.00  0.00  0.00
 [6,] 0.00 0.00 0.00 0.00 0.00 0.25 0.00 0.00 0.00  0.00  0.00  0.00  0.00
 [7,] 0.00 0.00 0.00 0.00 0.00 0.00 0.25 0.00 0.00  0.00  0.00  0.00  0.00
 [8,] 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.33 0.00  0.00  0.00  0.00  0.00
 [9,] 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.25  0.00  0.00  0.00  0.00
[10,] 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00  0.25  0.00  0.00  0.00
[11,] 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00  0.00  0.25  0.00  0.00
[12,] 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00  0.00  0.00  0.17  0.00
[13,] 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00  0.00  0.00  0.00  0.14
[14,] 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00  0.00  0.00  0.00  0.00
[15,] 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00  0.00  0.00  0.00  0.00
[16,] 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00  0.00  0.00  0.00  0.00
[17,] 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00  0.00  0.00  0.00  0.00
[18,] 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00  0.00  0.00  0.00  0.00
      [,14] [,15] [,16] [,17] [,18]
 [1,]  0.00   0.0   0.0   0.0   0.0
 [2,]  0.00   0.0   0.0   0.0   0.0
 [3,]  0.00   0.0   0.0   0.0   0.0
 [4,]  0.00   0.0   0.0   0.0   0.0
 [5,]  0.00   0.0   0.0   0.0   0.0
 [6,]  0.00   0.0   0.0   0.0   0.0
 [7,]  0.00   0.0   0.0   0.0   0.0
 [8,]  0.00   0.0   0.0   0.0   0.0
 [9,]  0.00   0.0   0.0   0.0   0.0
[10,]  0.00   0.0   0.0   0.0   0.0
[11,]  0.00   0.0   0.0   0.0   0.0
[12,]  0.00   0.0   0.0   0.0   0.0
[13,]  0.00   0.0   0.0   0.0   0.0
[14,]  0.12   0.0   0.0   0.0   0.0
[15,]  0.00   0.2   0.0   0.0   0.0
[16,]  0.00   0.0   0.5   0.0   0.0
[17,]  0.00   0.0   0.0   0.5   0.0
[18,]  0.00   0.0   0.0   0.0   0.5

Recalling that the row sums of \(\mathbf{A}\) give us the person degrees.

And then we do the same for groups by creating a matrix \(\mathbf{D}_g\) containing the inverse of the degrees of each group (number of members) along the diagonals.

   D.g <- diag(1/colSums(A))
   round(D.g, 2)
      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13]
 [1,] 0.33 0.00 0.00 0.00 0.00 0.00  0.0 0.00 0.00   0.0  0.00  0.00  0.00
 [2,] 0.00 0.33 0.00 0.00 0.00 0.00  0.0 0.00 0.00   0.0  0.00  0.00  0.00
 [3,] 0.00 0.00 0.17 0.00 0.00 0.00  0.0 0.00 0.00   0.0  0.00  0.00  0.00
 [4,] 0.00 0.00 0.00 0.25 0.00 0.00  0.0 0.00 0.00   0.0  0.00  0.00  0.00
 [5,] 0.00 0.00 0.00 0.00 0.12 0.00  0.0 0.00 0.00   0.0  0.00  0.00  0.00
 [6,] 0.00 0.00 0.00 0.00 0.00 0.12  0.0 0.00 0.00   0.0  0.00  0.00  0.00
 [7,] 0.00 0.00 0.00 0.00 0.00 0.00  0.1 0.00 0.00   0.0  0.00  0.00  0.00
 [8,] 0.00 0.00 0.00 0.00 0.00 0.00  0.0 0.07 0.00   0.0  0.00  0.00  0.00
 [9,] 0.00 0.00 0.00 0.00 0.00 0.00  0.0 0.00 0.08   0.0  0.00  0.00  0.00
[10,] 0.00 0.00 0.00 0.00 0.00 0.00  0.0 0.00 0.00   0.2  0.00  0.00  0.00
[11,] 0.00 0.00 0.00 0.00 0.00 0.00  0.0 0.00 0.00   0.0  0.25  0.00  0.00
[12,] 0.00 0.00 0.00 0.00 0.00 0.00  0.0 0.00 0.00   0.0  0.00  0.17  0.00
[13,] 0.00 0.00 0.00 0.00 0.00 0.00  0.0 0.00 0.00   0.0  0.00  0.00  0.33
[14,] 0.00 0.00 0.00 0.00 0.00 0.00  0.0 0.00 0.00   0.0  0.00  0.00  0.00
      [,14]
 [1,]  0.00
 [2,]  0.00
 [3,]  0.00
 [4,]  0.00
 [5,]  0.00
 [6,]  0.00
 [7,]  0.00
 [8,]  0.00
 [9,]  0.00
[10,]  0.00
[11,]  0.00
[12,]  0.00
[13,]  0.00
[14,]  0.33

Recalling that the column sums of \(\mathbf{A}\) give us the group degrees.

Now we can construct degree-normalized versions of the original bi-adjacency matrix and its transpose like this:

   P.n <- A %*% D.g 
   colnames(P.n) <- colnames(A)
   G.n <- t(A) %*% D.p
   colnames(G.n) <- rownames(A)

Let’s see what’s in the matrix P.n:

   round(P.n, 2)
          6/27  3/2 4/12 9/26 2/25 5/19 3/15 9/16  4/8 6/10 2/23  4/7 11/21
EVELYN    0.33 0.33 0.17 0.25 0.12 0.12  0.0 0.07 0.08  0.0 0.00 0.00  0.00
LAURA     0.33 0.33 0.17 0.00 0.12 0.12  0.1 0.07 0.00  0.0 0.00 0.00  0.00
THERESA   0.00 0.33 0.17 0.25 0.12 0.12  0.1 0.07 0.08  0.0 0.00 0.00  0.00
BRENDA    0.33 0.00 0.17 0.25 0.12 0.12  0.1 0.07 0.00  0.0 0.00 0.00  0.00
CHARLOTTE 0.00 0.00 0.17 0.25 0.12 0.00  0.1 0.00 0.00  0.0 0.00 0.00  0.00
FRANCES   0.00 0.00 0.17 0.00 0.12 0.12  0.0 0.07 0.00  0.0 0.00 0.00  0.00
ELEANOR   0.00 0.00 0.00 0.00 0.12 0.12  0.1 0.07 0.00  0.0 0.00 0.00  0.00
PEARL     0.00 0.00 0.00 0.00 0.00 0.12  0.0 0.07 0.08  0.0 0.00 0.00  0.00
RUTH      0.00 0.00 0.00 0.00 0.12 0.00  0.1 0.07 0.08  0.0 0.00 0.00  0.00
VERNE     0.00 0.00 0.00 0.00 0.00 0.00  0.1 0.07 0.08  0.0 0.00 0.17  0.00
MYRNA     0.00 0.00 0.00 0.00 0.00 0.00  0.0 0.07 0.08  0.2 0.00 0.17  0.00
KATHERINE 0.00 0.00 0.00 0.00 0.00 0.00  0.0 0.07 0.08  0.2 0.00 0.17  0.33
SYLVIA    0.00 0.00 0.00 0.00 0.00 0.00  0.1 0.07 0.08  0.2 0.00 0.17  0.33
NORA      0.00 0.00 0.00 0.00 0.00 0.12  0.1 0.00 0.08  0.2 0.25 0.17  0.33
HELEN     0.00 0.00 0.00 0.00 0.00 0.00  0.1 0.07 0.00  0.2 0.25 0.17  0.00
DOROTHY   0.00 0.00 0.00 0.00 0.00 0.00  0.0 0.07 0.08  0.0 0.00 0.00  0.00
OLIVIA    0.00 0.00 0.00 0.00 0.00 0.00  0.0 0.00 0.08  0.0 0.25 0.00  0.00
FLORA     0.00 0.00 0.00 0.00 0.00 0.00  0.0 0.00 0.08  0.0 0.25 0.00  0.00
           8/3
EVELYN    0.00
LAURA     0.00
THERESA   0.00
BRENDA    0.00
CHARLOTTE 0.00
FRANCES   0.00
ELEANOR   0.00
PEARL     0.00
RUTH      0.00
VERNE     0.00
MYRNA     0.00
KATHERINE 0.33
SYLVIA    0.33
NORA      0.33
HELEN     0.00
DOROTHY   0.00
OLIVIA    0.00
FLORA     0.00

The P.n matrix is of the same dimensions as the original affiliation matrix A. However, the \({ij}^{th}\) cell is now equal to \(1/k_j\) where \(k_j\) is the number of members of group \(j\). That means that if group \(j\) has a lot of members then the \({ij}^{th}\) cell will contain a small number (as in column in the non-zero entries in the column corresponding to event 9-16) but when the group is small, the \({ij}^{th}\) cell will contain a bigger number (as in the non-zero entries in the column corresponding to event 11-21). This means the people who belong to smaller groups will have more centrality points to distribute.

The same goes for groups, as we can see by checking out the G.n matrix:

   round(G.n, 2)
      EVELYN LAURA THERESA BRENDA CHARLOTTE FRANCES ELEANOR PEARL RUTH VERNE
6/27    0.12  0.14    0.00   0.14      0.00    0.00    0.00  0.00 0.00  0.00
3/2     0.12  0.14    0.12   0.00      0.00    0.00    0.00  0.00 0.00  0.00
4/12    0.12  0.14    0.12   0.14      0.25    0.25    0.00  0.00 0.00  0.00
9/26    0.12  0.00    0.12   0.14      0.25    0.00    0.00  0.00 0.00  0.00
2/25    0.12  0.14    0.12   0.14      0.25    0.25    0.25  0.00 0.25  0.00
5/19    0.12  0.14    0.12   0.14      0.00    0.25    0.25  0.33 0.00  0.00
3/15    0.00  0.14    0.12   0.14      0.25    0.00    0.25  0.00 0.25  0.25
9/16    0.12  0.14    0.12   0.14      0.00    0.25    0.25  0.33 0.25  0.25
4/8     0.12  0.00    0.12   0.00      0.00    0.00    0.00  0.33 0.25  0.25
6/10    0.00  0.00    0.00   0.00      0.00    0.00    0.00  0.00 0.00  0.00
2/23    0.00  0.00    0.00   0.00      0.00    0.00    0.00  0.00 0.00  0.00
4/7     0.00  0.00    0.00   0.00      0.00    0.00    0.00  0.00 0.00  0.25
11/21   0.00  0.00    0.00   0.00      0.00    0.00    0.00  0.00 0.00  0.00
8/3     0.00  0.00    0.00   0.00      0.00    0.00    0.00  0.00 0.00  0.00
      MYRNA KATHERINE SYLVIA NORA HELEN DOROTHY OLIVIA FLORA
6/27   0.00      0.00   0.00 0.00   0.0     0.0    0.0   0.0
3/2    0.00      0.00   0.00 0.00   0.0     0.0    0.0   0.0
4/12   0.00      0.00   0.00 0.00   0.0     0.0    0.0   0.0
9/26   0.00      0.00   0.00 0.00   0.0     0.0    0.0   0.0
2/25   0.00      0.00   0.00 0.00   0.0     0.0    0.0   0.0
5/19   0.00      0.00   0.00 0.12   0.0     0.0    0.0   0.0
3/15   0.00      0.00   0.14 0.12   0.2     0.0    0.0   0.0
9/16   0.25      0.17   0.14 0.00   0.2     0.5    0.0   0.0
4/8    0.25      0.17   0.14 0.12   0.0     0.5    0.5   0.5
6/10   0.25      0.17   0.14 0.12   0.2     0.0    0.0   0.0
2/23   0.00      0.00   0.00 0.12   0.2     0.0    0.5   0.5
4/7    0.25      0.17   0.14 0.12   0.2     0.0    0.0   0.0
11/21  0.00      0.17   0.14 0.12   0.0     0.0    0.0   0.0
8/3    0.00      0.17   0.14 0.12   0.0     0.0    0.0   0.0

Note that both matrices P.n and G.n are column stochastic which means that its columns sum to one. We can check for this property as follows:

   colSums(P.n)
 6/27   3/2  4/12  9/26  2/25  5/19  3/15  9/16   4/8  6/10  2/23   4/7 11/21 
    1     1     1     1     1     1     1     1     1     1     1     1     1 
  8/3 
    1 
   colSums(G.n)
   EVELYN     LAURA   THERESA    BRENDA CHARLOTTE   FRANCES   ELEANOR     PEARL 
        1         1         1         1         1         1         1         1 
     RUTH     VERNE     MYRNA KATHERINE    SYLVIA      NORA     HELEN   DOROTHY 
        1         1         1         1         1         1         1         1 
   OLIVIA     FLORA 
        1         1

So now, to calculate the two-mode PageRank status score, we just play our two-mode status game on these matrices:

   tm.pr <- tm.status(P.n, G.n)

Note that because the bi-adjacency matrices that go into the tm.status function are already normalized, we can set the norm argument of the function to FALSE.

Here are the resulting scores for the people:

   round(tm.pr$p.s/max(tm.pr$p.s), 3)
           [,1]
EVELYN    1.000
LAURA     0.875
THERESA   1.000
BRENDA    0.875
CHARLOTTE 0.500
FRANCES   0.500
ELEANOR   0.500
PEARL     0.375
RUTH      0.500
VERNE     0.500
MYRNA     0.500
KATHERINE 0.750
SYLVIA    0.875
NORA      1.000
HELEN     0.625
DOROTHY   0.250
OLIVIA    0.250
FLORA     0.250

And for the groups:

   round(tm.pr$g.s/max(tm.pr$g.s), 3)
       [,1]
6/27  0.214
3/2   0.214
4/12  0.429
9/26  0.286
2/25  0.571
5/19  0.571
3/15  0.714
9/16  1.000
4/8   0.857
6/10  0.357
2/23  0.286
4/7   0.429
11/21 0.214
8/3   0.214

The resulting scores are the same ones returned by the “co-HITS” algorithm of Deng, Lyu, and King (2009).

On this ranking, the top persons are Theresa and Nora (who wasn’t as highly ranked according to the Bonacich prestige) and Dorothy, Olivia, and Flora are the least prestigious. The top ranked event is the one that took place on 9/16.

Double Normalized Two-Mode PageRank

Recall that the main reason for using the PageRank normalization in the two-mode case (and preferring the PageRank scoring over the eigenvector scoring) is to give more status points to discerning people, so that people who belong to more exclusive clubs distribute more centrality in the system. So to do that we divided each person’s centrality points according to the size of the group they belong to, so that evenly gets 0.33 centrality points for attending event 6/27 because that event was attended by only three people, which is more than she gets for attending event 4/8 which was attended by twelve people (and the same for events).

However, the PageRank centrality rankings can still be affected by the sheer number of events attended by a person (or the sheer number of members of a group) even if that person is not particularly selective. Thus, a person who attends a whole bunch of not-very selective (well-attended) events still contributes as much centrality to the system as a person who attends fewer but more selective events.

To do that, it would be useful to come up with versions of P.n and G.n that normalize (for persons) by the exclusivity of the groups they attend and by the number of events they go to, so as not to give an advantage to those who attend a lot of events. In the same way, it would be also be useful normalize (for groups) by the discernment of the people who are its members and by their number of members they go to, so as not to give an advantage to well-attended events.

In the case of P.n we can do that by pre-multiplying this matrix, which already adjusts for group size by D.p which is the diagonal matrix containing each person’s number of memberships:

   P.n2 <- D.p %*% P.n
   rownames(P.n2) <- rownames(A)

Let’s see what’s inside the new P.n2 matrix:

   round(P.n2, 2)
          6/27  3/2 4/12 9/26 2/25 5/19 3/15 9/16  4/8 6/10 2/23  4/7 11/21
EVELYN    0.04 0.04 0.02 0.03 0.02 0.02 0.00 0.01 0.01 0.00 0.00 0.00  0.00
LAURA     0.05 0.05 0.02 0.00 0.02 0.02 0.01 0.01 0.00 0.00 0.00 0.00  0.00
THERESA   0.00 0.04 0.02 0.03 0.02 0.02 0.01 0.01 0.01 0.00 0.00 0.00  0.00
BRENDA    0.05 0.00 0.02 0.04 0.02 0.02 0.01 0.01 0.00 0.00 0.00 0.00  0.00
CHARLOTTE 0.00 0.00 0.04 0.06 0.03 0.00 0.03 0.00 0.00 0.00 0.00 0.00  0.00
FRANCES   0.00 0.00 0.04 0.00 0.03 0.03 0.00 0.02 0.00 0.00 0.00 0.00  0.00
ELEANOR   0.00 0.00 0.00 0.00 0.03 0.03 0.03 0.02 0.00 0.00 0.00 0.00  0.00
PEARL     0.00 0.00 0.00 0.00 0.00 0.04 0.00 0.02 0.03 0.00 0.00 0.00  0.00
RUTH      0.00 0.00 0.00 0.00 0.03 0.00 0.03 0.02 0.02 0.00 0.00 0.00  0.00
VERNE     0.00 0.00 0.00 0.00 0.00 0.00 0.03 0.02 0.02 0.00 0.00 0.04  0.00
MYRNA     0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.02 0.02 0.05 0.00 0.04  0.00
KATHERINE 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.01 0.01 0.03 0.00 0.03  0.06
SYLVIA    0.00 0.00 0.00 0.00 0.00 0.00 0.01 0.01 0.01 0.03 0.00 0.02  0.05
NORA      0.00 0.00 0.00 0.00 0.00 0.02 0.01 0.00 0.01 0.03 0.03 0.02  0.04
HELEN     0.00 0.00 0.00 0.00 0.00 0.00 0.02 0.01 0.00 0.04 0.05 0.03  0.00
DOROTHY   0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.04 0.04 0.00 0.00 0.00  0.00
OLIVIA    0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.04 0.00 0.12 0.00  0.00
FLORA     0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.04 0.00 0.12 0.00  0.00
           8/3
EVELYN    0.00
LAURA     0.00
THERESA   0.00
BRENDA    0.00
CHARLOTTE 0.00
FRANCES   0.00
ELEANOR   0.00
PEARL     0.00
RUTH      0.00
VERNE     0.00
MYRNA     0.00
KATHERINE 0.06
SYLVIA    0.05
NORA      0.04
HELEN     0.00
DOROTHY   0.00
OLIVIA    0.00
FLORA     0.00

We can see that now, people with lots of memberships (like Evelyn) get their centrality points reduced relative to people with not that many memberships like Charlotte.

This is easy to see if we compare the row sums of the two matrices, which give us the total amount of centrality points each person has to distribute:

   rowSums(P.n)
   EVELYN     LAURA   THERESA    BRENDA CHARLOTTE   FRANCES   ELEANOR     PEARL 
1.4880952 1.2547619 1.2547619 1.1714286 0.6416667 0.4880952 0.4214286 0.2797619 
     RUTH     VERNE     MYRNA KATHERINE    SYLVIA      NORA     HELEN   DOROTHY 
0.3797619 0.4214286 0.5214286 1.1880952 1.2880952 1.5916667 0.7880952 0.1547619 
   OLIVIA     FLORA 
0.3333333 0.3333333 
   rowSums(P.n2)
    EVELYN      LAURA    THERESA     BRENDA  CHARLOTTE    FRANCES    ELEANOR 
0.18601190 0.17925170 0.15684524 0.16734694 0.16041667 0.12202381 0.10535714 
     PEARL       RUTH      VERNE      MYRNA  KATHERINE     SYLVIA       NORA 
0.09325397 0.09494048 0.10535714 0.13035714 0.19801587 0.18401361 0.19895833 
     HELEN    DOROTHY     OLIVIA      FLORA 
0.15761905 0.07738095 0.16666667 0.16666667

Note that while in the original P.n matrix Evelyn has way more centrality points to distribute than Charlotte (a ratio of 2.32), this discrepancy is much smaller in the new P.n2 double-normalized matrix (a ratio of 1.16).

Now to compute our double-normalized PageRank status scores, we just play the status game on this new matrix and its transpose:

   tm.bgrm <- tm.status(P.n2, t(P.n2))

Which implements the “BGRM” algorithm of Rui et al. (2007, 589, eq. 8).2

Here are the resulting scores for the people:

   round(tm.bgrm$p.s/max(tm.bgrm$p.s), 3)
           [,1]
EVELYN    0.081
LAURA     0.061
THERESA   0.081
BRENDA    0.059
CHARLOTTE 0.055
FRANCES   0.050
ELEANOR   0.057
PEARL     0.116
RUTH      0.100
VERNE     0.141
MYRNA     0.188
KATHERINE 0.235
SYLVIA    0.212
NORA      0.407
HELEN     0.471
DOROTHY   0.145
OLIVIA    1.000
FLORA     1.000

And for the groups:

   round(tm.bgrm$g.s/max(tm.bgrm$g.s), 3)
       [,1]
6/27  0.032
3/2   0.034
4/12  0.037
9/26  0.037
2/25  0.045
5/19  0.067
3/15  0.102
9/16  0.111
4/8   0.396
6/10  0.183
2/23  1.000
4/7   0.173
11/21 0.140
8/3   0.140

We can see that the most highly ranked women are now Olivia and Flora (in contrast to Evelyn, Theresa, and Nora in the standard PageRank scores), who used to be toward the bottom when using the traditional Bonacich prestige or PageRank scoring. In the same way, the top event is now the one held on 2/23, which wasn’t particularly distinguished using the other ranking algorithms.

A summary of the three ranking models goes as follows:

Person Prestige Scores Ordered by Degree
hits pr bgrm k
EVELYN 0.903 1.000 0.081 8
THERESA 1.000 1.000 0.081 8
NORA 0.712 1.000 0.407 8
LAURA 0.834 0.875 0.061 7
BRENDA 0.845 0.875 0.059 7
SYLVIA 0.748 0.875 0.212 7
KATHERINE 0.594 0.750 0.235 6
HELEN 0.542 0.625 0.471 5
CHARLOTTE 0.454 0.500 0.055 4
FRANCES 0.564 0.500 0.050 4
ELEANOR 0.616 0.500 0.057 4
RUTH 0.637 0.500 0.100 4
VERNE 0.589 0.500 0.141 4
MYRNA 0.504 0.500 0.188 4
PEARL 0.486 0.375 0.116 3
DOROTHY 0.355 0.250 0.145 2
OLIVIA 0.188 0.250 1.000 2
FLORA 0.188 0.250 1.000 2
Group Prestige Scores Ordered by Degree
hits pr bgrm k
9/16 1.000 1.000 0.111 14
4/8 0.749 0.857 0.396 12
3/15 0.757 0.714 0.102 10
2/25 0.635 0.571 0.045 8
5/19 0.647 0.571 0.067 8
4/12 0.499 0.429 0.037 6
4/7 0.400 0.429 0.173 6
6/10 0.336 0.357 0.183 5
9/26 0.347 0.286 0.037 4
2/23 0.177 0.286 1.000 4
6/27 0.280 0.214 0.032 3
3/2 0.297 0.214 0.034 3
11/21 0.223 0.214 0.140 3
8/3 0.223 0.214 0.140 3

References

Bonacich, Phillip. 1991. “Simultaneous Group and Individual Centralities.” Social Networks 13 (2): 155–68.
Breiger, Ronald L. 1974. “The Duality of Persons and Groups.” Social Forces 53 (2): 181–90.
Deng, Hongbo, Michael R Lyu, and Irwin King. 2009. “A Generalized Co-Hits Algorithm and Its Application to Bipartite Graphs.” In Proceedings of the 15th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, 239–48.
Faust, Katherine. 1997. “Centrality in Affiliation Networks.” Social Networks 19 (2): 157–91.
Liao, Hao, Rui Xiao, Giulio Cimini, and Matúš Medo. 2014. “Network-Driven Reputation in Online Scientific Communities.” PLoS One 9 (12): e112022.
Rui, Xiaoguang, Mingjing Li, Zhiwei Li, Wei-Ying Ma, and Nenghai Yu. 2007. “Bipartite Graph Reinforcement Model for Web Image Annotation.” In Proceedings of the 15th ACM International Conference on Multimedia, 585–94.

Footnotes

  1. Where the weight (for obscure technical reasons) is the inverse of the square root of the first eigenvalue obtained from the eigen analysis.↩︎

  2. The acronym stands for “Bipartite Graph Reinforcement Model.”↩︎