Bonacich Centrality and the Method of Reflections

reflections
two-mode networks
centrality
eigenvectors
Author

Omar Lizardo

Published

May 5, 2024

In a previous post, I showed how the “method of reflections” introduced by Hidalgo & Hausmann (2009)—hereafter HH—works in the case of two-mode network data. While reading that, some of you might have thought that there is a well-established network centrality metric for two-mode networks that is supposed to work just like the method of reflections: That is, defining the centrality of entities in one mode (e.g., persons) according to centralities of the entities in the other mode (e.g., groups). I refer to the Bonacich “eigenvector” centrality—see Bonacich (1991) and Faust (1997) for discussion.

Table 1: Southern Women Data.
E1 E2 E3 E4 E5 E6 E7 E8 E9 E10 E11 E12 E13 E14
BRENDA 1 0 1 1 1 1 1 1 0 0 0 0 0 0
CHARLOTTE 0 0 1 1 1 0 1 0 0 0 0 0 0 0
DOROTHY 0 0 0 0 0 0 0 1 1 0 0 0 0 0
ELEANOR 0 0 0 1 1 1 1 1 0 0 0 0 0 0
EVELYN 1 1 1 1 1 1 0 1 1 0 0 0 0 0
FLORA 0 0 0 0 0 0 0 0 1 0 1 0 0 0
FRANCES 0 0 1 1 1 1 0 1 0 0 0 0 0 0
HELEN 0 0 0 0 0 0 1 1 0 1 1 1 0 0
KATHERINE 0 0 0 0 0 0 0 1 1 1 0 1 1 1
LAURA 1 1 1 0 1 1 1 1 0 0 0 0 0 0
MYRNA 0 0 0 0 0 0 0 1 1 1 0 1 0 0
NORA 0 0 0 0 0 0 1 0 1 1 1 1 1 1
OLIVIA 0 0 0 0 0 0 0 0 1 0 1 0 0 0
PEARL 0 0 0 0 0 1 0 1 1 0 0 0 0 0
RUTH 0 0 0 1 1 0 1 1 1 0 0 0 0 0
SYLVIA 0 0 0 0 0 0 1 1 1 1 0 1 1 1
THERESA 0 1 1 1 1 1 1 1 1 0 0 0 0 0
VERNE 0 0 0 0 0 0 1 1 1 0 0 1 0 0

In this post I show how the Bonacich centrality fits into the reflection idea, and why this means it is a different metric from that yielded by HH’s reflection method. This also the difference between Bonacich and Correspondence Analysis (CA), because the HH reflection method is equivalent to CA.

Consider an affiliation matrix \(\mathbf{A}\) of dimensions \(|P| \times |G|\) where \(|P|\) is the cardinality of the person set and \(|G|\) the cardinality of the group set in the two-mode network. Each entry in the matrix \(a_{pg} = 1\) if person p affiliates with group \(g\) and zero otherwise. An affiliation matrix of this type, representing Davis et al.’s (1941) famous Southern Women data, is shown as Table 1.

In this case, the degree centrality of people is given by:

\[ C^B_p(1) = \sum_g a_{pg} \tag{1}\]

And for groups:

\[ C^B_g(1) = \sum_p a_{pg} \tag{2}\]

The right hand sides of Equation 1 and Equation 2 are just the first order degree centralities of people and groups (Faust, 1997). People are central when they have many memberships and groups are central when they have many members.

Once we have this information, we can define a second order “Bonacich-reflection” on both the persons and groups using the formulas:

\[ C^B_p(2) = \sum_g a_{pg}C^R_g(1) \tag{3}\]

\[ C^B_g(2) = \sum_p a_{pg}C^R_p(1) \tag{4}\]

Equation 3 says that people are central when the sum of the number of members of the groups they belong to is large. Equation 4 says groups are central when the sum of the number of memberships of the people who belong to them is large.

We can keep on going and define a third order reflection using the formulas:

\[ C^B_p(3) = \sum_g a_{pg}C^R_g(2) \tag{5}\]

\[ C^B_g(3) = \sum_p a_{pg}C^R_p(2) \tag{6}\]

Equation 5 says that people are central when the sum of the sum of the number of memberships held by the people who belong to the groups they belong to is large. Equation 5 says that groups are central when the sum of the sum of the number of members of the groups their members belong to is large.

Once again, we can keep going and define fourth order, fifth order, and higher reflections \(C^R_p(4), C^R_p(4) \ldots C^R_p(q)\).

More generally, the reflective Bonacich centralities for persons and groups are given by:

\[ C^B_p(q) = \sum_g a_{pg}C^R_g(q-1) \tag{7}\]

\[ C^B_g(q) = \sum_p a_{pg}C^R_p(q-1) \tag{8}\]

For all q > 1.

Actually, the preceding set of equations won’t quite work, because we are computing sums of sums, of sums which means that both the person and group reflective Bonacich centralities would diverge toward infinity as \(q\) increases.

To prevent this divergence and guarantee convergence to a set of manageable scores after a few iterations, we need to normalize the sum of centralities at each step \(q > 1\) before feeding them to the subsequent sum at step \(q+1\). We do this by dividing the computed centrality vectors for both persons and group at each step q by a vector norm, which in this case is the Euclidean vector norm, as follows:

\[ C^B_p(q) = \frac{C^B_p(q)}{||C^B_p(q)||_2} \tag{9}\]

\[ C^B_p(q) = \frac{C^B_p(q)}{||C^B_p(q)||_2} \tag{10}\]

With the Euclidean norm for each centrality vector defined as:

\[ ||C^B_p(q)||_2 = \sqrt{\sum_i^{|P|}(C^B_p(q)_i)^2} \tag{11}\]

\[ ||C^B_g(q)||_2 = \sqrt{\sum_j^{|G|}(C^B_g(q)_j)^2} \tag{12}\]

Let’s see how that looks like with real data. The results of the reflection equations for the Southern Women data, for people up to twenty reflections, are shown in Table 2. The corresponding results for groups are shown in Table 3.

Table 2: Bonacich reflections on Southern Women Data (Persons)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
EVELYN 0.35 0.32 0.33 0.33 0.34 0.34 0.34 0.34 0.34 0.34 0.34 0.34 0.34 0.34 0.34 0.34 0.34 0.34 0.34 0.34
LAURA 0.30 0.27 0.29 0.29 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30
THERESA 0.35 0.36 0.37 0.37 0.37 0.37 0.37 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38
BRENDA 0.30 0.30 0.31 0.31 0.32 0.32 0.32 0.32 0.32 0.32 0.32 0.32 0.32 0.32 0.32 0.32 0.32 0.32 0.32 0.32
CHARLOTTE 0.17 0.17 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.19 0.19 0.19 0.19 0.19 0.19 0.19 0.19 0.19 0.19
FRANCES 0.22 0.23 0.23 0.24 0.24 0.24 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
ELEANOR 0.22 0.25 0.25 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.26
RUTH 0.22 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27 0.27
VERNE 0.17 0.23 0.21 0.22 0.21 0.21 0.21 0.21 0.21 0.21 0.21 0.21 0.21 0.21 0.21 0.21 0.21 0.21 0.21 0.21
MYRNA 0.17 0.20 0.18 0.19 0.18 0.18 0.18 0.18 0.18 0.17 0.17 0.17 0.17 0.17 0.17 0.17 0.17 0.17 0.17 0.17
KATHERINE 0.26 0.23 0.22 0.22 0.21 0.21 0.21 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20
SYLVIA 0.30 0.28 0.28 0.27 0.27 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.26
NORA 0.30 0.23 0.22 0.21 0.20 0.20 0.20 0.20 0.20 0.20 0.19 0.19 0.19 0.19 0.19 0.19 0.19 0.19 0.19 0.19
HELEN 0.22 0.21 0.20 0.20 0.19 0.19 0.19 0.19 0.19 0.19 0.19 0.19 0.19 0.19 0.19 0.19 0.19 0.19 0.19 0.19
OLIVIA 0.09 0.09 0.07 0.07 0.07 0.07 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06
FLORA 0.09 0.09 0.07 0.07 0.07 0.07 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06
PEARL 0.13 0.18 0.17 0.17 0.17 0.17 0.17 0.17 0.17 0.17 0.17 0.17 0.17 0.17 0.17 0.17 0.17 0.17 0.17 0.17
DOROTHY 0.09 0.14 0.13 0.13 0.13 0.13 0.13 0.13 0.13 0.13 0.13 0.13 0.13 0.13 0.13 0.13 0.13 0.13 0.13 0.13
Table 3: Bonacich reflections on Southern Women Data (Groups)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
E1 0.11 0.14 0.13 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14
E2 0.11 0.15 0.14 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15
E3 0.22 0.25 0.24 0.25 0.25 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.26 0.26
E4 0.26 0.27 0.28 0.28 0.29 0.29 0.29 0.29 0.29 0.29 0.29 0.29 0.29 0.29 0.29 0.29 0.29 0.29 0.29 0.29
E5 0.29 0.32 0.32 0.33 0.33 0.33 0.33 0.34 0.34 0.34 0.34 0.34 0.34 0.34 0.34 0.34 0.34 0.34 0.34 0.34
E6 0.26 0.28 0.28 0.29 0.29 0.29 0.29 0.29 0.29 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30
E7 0.36 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38 0.38
E8 0.51 0.49 0.51 0.51 0.51 0.51 0.51 0.51 0.51 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50 0.50
E9 0.44 0.37 0.38 0.37 0.37 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.36 0.36
E10 0.18 0.19 0.17 0.16 0.16 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15
E11 0.15 0.10 0.09 0.08 0.08 0.08 0.08 0.08 0.08 0.07 0.07 0.07 0.07 0.07 0.07 0.07 0.07 0.07 0.07 0.07
E12 0.22 0.21 0.20 0.19 0.19 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18
E13 0.11 0.13 0.11 0.11 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10
E14 0.11 0.13 0.11 0.11 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10

We can see that the results for both persons and groups freeze into values after a few iterations. The resulting final ranks of persons and groups are shown in Table 4 (a) and Table 4 (b).

Table 4: Final rankings of persons and groups according to Bonacich reflections.

(a) Final rank of persons
THERESA 0.38
EVELYN 0.34
BRENDA 0.32
LAURA 0.30
RUTH 0.27
ELEANOR 0.26
SYLVIA 0.26
FRANCES 0.25
VERNE 0.21
KATHERINE 0.20
CHARLOTTE 0.19
NORA 0.19
HELEN 0.19
MYRNA 0.17
PEARL 0.17
DOROTHY 0.13
OLIVIA 0.06
FLORA 0.06
(b) Final rank of groups
E8 0.50
E7 0.38
E9 0.36
E5 0.34
E6 0.30
E4 0.29
E3 0.26
E12 0.18
E2 0.15
E10 0.15
E1 0.14
E13 0.10
E14 0.10
E11 0.07

The final rankings of both persons and groups should look familiar, for they are equivalent to the “dual” Bonacich centralities that can be calculated directly using methods from linear algebra as detailed in Bonacich (1991). That is, consider the standard one-mode person and group projections, which can be obtained from the original affiliation matrix as discussed in Breiger (1974).

These are given by:

\[ B_p = AA^T \tag{13}\]

\[ B_g = A^TA \tag{14}\]

Where \(B_p\) is the \(|P| \times |P|\) person projection matrix (with off-diagonal cells indicating the number of groups two people have in common and diagonal cells indicting each person’s number of memberships), \(B_g\) is the \(|G| \times |G|\) group projection matrix (with off-diagonal cells indicating the number of people two groups have in common and diagonal cells indicting each group’s number of members), and \(A\) is the original affiliation matrix. For both persons and groups the projection matrices are shown in Table 5 (a) and Table 5 (b).

Table 5: One mode projections for Southern Women Data.

(a) One mode projection of persons
EVELYN 8 6 7 6 3 5 4 4 2 2 2 2 1 1 1 1 3 2
LAURA 6 7 6 6 3 4 4 3 2 1 1 2 1 2 0 0 2 1
THERESA 7 6 8 6 4 5 5 5 3 2 2 3 2 2 1 1 3 2
BRENDA 6 6 6 7 4 5 5 4 2 1 1 2 1 2 0 0 2 1
CHARLOTTE 3 3 4 4 4 3 3 3 1 0 0 1 1 1 0 0 0 0
FRANCES 5 4 5 5 3 5 4 3 1 1 1 1 0 1 0 0 2 1
ELEANOR 4 4 5 5 3 4 5 4 2 1 1 2 1 2 0 0 2 1
RUTH 4 3 5 4 3 3 4 5 3 2 2 3 2 2 1 1 2 2
VERNE 2 2 3 2 1 1 2 3 4 3 3 4 3 3 1 1 2 2
MYRNA 2 1 2 1 0 1 1 2 3 4 4 4 3 3 1 1 2 2
KATHERINE 2 1 2 1 0 1 1 2 3 4 6 6 5 3 1 1 2 2
SYLVIA 2 2 3 2 1 1 2 3 4 4 6 7 6 4 1 1 2 2
NORA 1 1 2 1 1 0 1 2 3 3 5 6 7 4 2 2 1 1
HELEN 1 2 2 2 1 1 2 2 3 3 3 4 4 5 1 1 1 1
OLIVIA 1 0 1 0 0 0 0 1 1 1 1 1 2 1 2 2 1 1
FLORA 1 0 1 0 0 0 0 1 1 1 1 1 2 1 2 2 1 1
PEARL 3 2 3 2 0 2 2 2 2 2 2 2 1 1 1 1 3 2
DOROTHY 2 1 2 1 0 1 1 2 2 2 2 2 1 1 1 1 2 2
(b) One mode projection of groups
E1 3 2 3 2 3 3 2 3 1 0 0 0 0 0
E2 2 3 3 2 3 3 2 3 2 0 0 0 0 0
E3 3 3 6 5 6 5 4 5 2 0 0 0 0 0
E4 2 2 5 7 7 5 5 6 3 0 0 0 0 0
E5 3 3 6 7 8 6 6 7 3 0 0 0 0 0
E6 3 3 5 5 6 7 4 7 3 0 0 0 0 0
E7 2 2 4 5 6 4 10 8 5 3 2 4 2 2
E8 3 3 5 6 7 7 8 14 9 4 1 5 2 2
E9 1 2 2 3 3 3 5 9 12 4 3 5 3 3
E10 0 0 0 0 0 0 3 4 4 5 2 5 3 3
E11 0 0 0 0 0 0 2 1 3 2 4 2 1 1
E12 0 0 0 0 0 0 4 5 5 5 2 6 3 3
E13 0 0 0 0 0 0 2 2 3 3 1 3 3 3
E14 0 0 0 0 0 0 2 2 3 3 1 3 3 3

Bonacich shows that vectors that will order people and groups in the same way as that shown in Table 4, can be obtained by solving the following system of linear equations:

\[ B_p c_p = \lambda c_p \tag{15}\]

\[ B_g c_g = \lambda c_g \tag{16}\]

Which means that \(c_p\) is an eigenvector of the projection matrix \(B_p\) and \(c_g\) is an eigenvector of the projection matrix \(B_g\). In fact, the person and group vectors of centrality scores that will order people and groups exactly as in Table 4, are the leading (first) eigenvectors of each respective projection matrix (Bonacich, 1991; Faust, 1997). These are shown in Table 6. As we can see, the (absolute) value for the eigenvectors of persons and groups in Table 6 is identical to that shown in Table 4. The reason for this is that the reflective iterative method, is actually a way of computing the eigenvectors of a matrix! (It’s called the “power” method and was first discovered by von Mises)

Table 6: Leading Eigenvectors for projection matrices of Southern Women Data.

(a) Eigenvector scores of persons
THERESA -0.38
EVELYN -0.34
BRENDA -0.32
LAURA -0.30
RUTH -0.27
ELEANOR -0.26
SYLVIA -0.26
FRANCES -0.25
VERNE -0.21
KATHERINE -0.20
CHARLOTTE -0.19
NORA -0.19
HELEN -0.19
MYRNA -0.17
PEARL -0.17
DOROTHY -0.13
OLIVIA -0.06
FLORA -0.06
(b) Eigenvector scores of groups
E8 -0.50
E7 -0.38
E9 -0.36
E5 -0.34
E6 -0.30
E4 -0.29
E3 -0.26
E12 -0.18
E2 -0.15
E10 -0.15
E1 -0.14
E13 -0.10
E14 -0.10
E11 -0.07

References

Bonacich, P. (1991). Simultaneous group and individual centralities. Social Networks, 13(2), 155-168.

Breiger, R. L. (1974). The duality of persons and groups. Social Forces, 53(2), 181-190.

Davis, A., Gardner, B. B., & Gardner, M. R. (1941). Deep South: A social anthropological study of caste and class. University of Chicago Press

Faust, K. (1997). Centrality in affiliation networks. Social Networks, 19(2), 157-191.