Fitness and Complexity in Two Mode Networks

In a highly cited piece, Tacchella et al. (2012) introduce a new prestige metric for two-mode networks that relies on the same “prismatic” model of status distribution we considered before.

Tacchella et al. (2012) called the prestige metrics they obtained using their approach “fitness” and “complexity” because they developed in the empirical context of calculating metrics for ranking nations based on their competitive advantage in exporting products, which means analyzing a two-mode country-by-product matrix (Hidalgo and Hausmann 2009).

However, when considered in the more general context of two-mode network link analysis (Fouss et al. 2016), it is clear that their approach is a prestige metric for two-mode networks that combines ideas from Bonacich Eigenvector scoring and PageRank scoring, as covered in the two-mode prestige lecture notes.

1 Fitness/Complexity Scores

Their basic idea is that when we are (asymmetrically) interested in determining the status or prestige of nodes in one particular mode (e.g., the row-mode nodes), we should not use summaries (e.g., sums or averages) of the scores for nodes in the other (e.g., column) mode in determining their status. Instead, we should deeply discount those nodes that connect to low-status nodes at the other end.

To understand what they are getting at, it helps to write down the Bonacich prestige scoring in equation form, as we go through each iteration of the two-mode status distribution game:

If you remember from the function tm.status, each iteration $q$, the vector of status scores for the row nodes $\mathbf{s}^R$ and the column nodes $\mathbf{s}^C$ is given by:

\[ s^R_i(q) = \sum_j\mathbf{A}_{ij}s^C_j(q-1) \tag{1}\]

\[ s^C_j(q) = \sum_i\mathbf{A}_{ij}s^R_i(q-1) \tag{2}\]

Where $\mathbf{A}$ is the two-mode network’s biadjacency matrix, and with the restriction that at the initial step $\mathbf{s}(0)^C = \mathbf{1}$ where $\mathbf{1}$ is the all ones vector of length equal to the number of columns of the biadjacency matrix $\mathbf{A}$.

At each iteration $q > 0$ we normalize both score vectors:

\[ \mathbf{s}^R(q) = \frac{\mathbf{s}^R(q)}{\left\langle\mathbf{s}^R(q)\right\rangle} \tag{3}\]

\[ \mathbf{s}^C(q) = \frac{\mathbf{s}^C(q)}{\left\langle\mathbf{s}^C(q)\right\rangle} \tag{4}\]

Where $\left\langle \mathbf{s} \right\rangle$ is the Euclidean vector norm, and the process continues iterating until the differences between the vectors across successive iterations are minimal.

So far, this is what we covered before. What Tacchella et al. (2012) propose is to substitute above with:

\[ s^C_j(q) = \left[\sum_i\mathbf{A}_{ij}\left(s^R_i(q-1)\right)^{-1}\right]^{-1} \tag{5}\]

Which means that first (inner parentheses), we take the reciprocal of the row-mode nodes’ status scores, sum them across column-mode nodes (such that column-mode nodes that connect to low-status row-mode nodes get a big score), and then take the reciprocal of the reciprocal to get back to a measure of status for column-mode nodes. This non-linear transformation heavily discounts the status scores assigned to column-mode nodes when they connect to lower-status row-mode nodes. Tacchella et al. (2012) also change the normalization step, using the mean value of the vector in the denominator of and .

2 Beyond Status as Popularity/Activity

How should we understand this modification? Recall that the basic principle of standard Bonacich prestige scoring is based on the equation of status/prestige and popularity/activity. In the canonical case of persons and groups (Breiger 1974), an event receives status from being attended by high-status individuals and an individual receives status from being affiliated with a high-status event; in each case, status from the point of view of the event means having highly active members, and from the point of view of the individual, it means being affiliated with popular events.

But status may not always work this way. Consider the world-economic network linking countries to the products they have a competitive advantage in producing (Hidalgo and Hausmann 2009). Analysts noticed that the most developed countries produce both “complex” (i.e., high-status) products that only a select few of other highly developed economies produce (like semiconductors) and also less “complex” (i.e., low-status, like extractive natural resources) products that the other less developed economies produce (Tacchella et al. 2012).

That means that the “complexity” (i.e., status score) of a product cannot be derived simply by taking a summary (e.g., sum or average) of the status scores of the countries that produce it, because high-status countries engage in both high and low-status forms of production. However, knowing that a product is produced by a low-status country is more informative (and should weigh more heavily in determining a product’s status score) because low-status countries produce only low-status products.

Applying the same reasoning to the aforementioned case of persons and groups (Breiger 1974), an equivalent situation would go as follows. Imagine there is a set of prestigious women and a set of prestigious events that only the prestigious women attend. However, prestigious women are also endowed with a spirit of noblesse oblige, which means that the most prestigious of them also attend low-prestige events.

This means that when determining the prestige of the events it is not very informative to know that prestigious women affiliate with them; rather, we should weigh more heavily whether low-status women affiliate with an event in determining an event’s status, such that as the number of low-status women who affiliate with an event increases, a given event’s status is downgraded in a non-linear way which feeds back into the computation of each woman’s prestige.

Let’s see how this would work in the Southern Women data. First, we load it up:

Code

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

And here’s a function called tm.fitness that modifies the old two-mode status distribution game function we played before—which, as you recall, was itself based on Kleinberg’s (1999) HITS algorithm for directed one-mode networks—to compute the fitness and complexity prestige scores for persons and groups:

Code

 tm.fitness <- function(w, iter = 1000) {
 y <- matrix(1, ncol(w), 1) #initial group status column vector set to a constant
 z <- t(w)
 k <- 0
 while (k < iter) {
 o.y <- y 
 x <- w %*% o.y #fitness status scores for people
 x <- x/mean(x) #normalizing new people status scores 
 y <- (z %*% x^-1)^-1 #complexity status scores for groups
 y <- y/mean(y) #normalizing new group status scores 
 k <- k + 1
 }
 return(list(p.s = x, g.s = y, k = k))
 }

Note that we use a number-of-iterations approach to indicate convergence (governed by the iter argument) rather than a successive-differences approach, due to the non-linear nature of the algorithm’s scoring.

We then apply the tm.fitness function to the SW data:

Code

 fc <- tm.fitness(A, iter = 50)

We also calculate the usual Bonacich eigenvector scores using the svd function for comparison purposes:

Code

 eig <- svd(A)
 p.s <- eig$u[, 1] * -1
 g.s <- eig$v[, 1] * -1
 names(p.s) <- rownames(A)
 names(g.s) <- colnames(A)

And we put them in a table:

Table 1: Status Scores

(a) Persons.

	Bonacich	Fitness
EVELYN	0.903	1.000
LAURA	0.834	0.842
NORA	0.712	0.781
SYLVIA	0.748	0.763
KATHERINE	0.594	0.746
THERESA	1.000	0.669
BRENDA	0.845	0.666
CHARLOTTE	0.454	0.294
FRANCES	0.564	0.126
HELEN	0.542	0.111
MYRNA	0.504	0.091
ELEANOR	0.616	0.066
VERNE	0.589	0.055
RUTH	0.637	0.050
PEARL	0.486	0.026
DOROTHY	0.355	0.007
OLIVIA	0.188	0.006
FLORA	0.188	0.006
^* Scores normalized by dividing by the maximum.

(b) Groups.

	Bonacich	Complexity
3/2	0.297	1.000
6/27	0.280	0.998
11/21	0.223	0.937
8/3	0.223	0.937
9/26	0.347	0.498
4/12	0.499	0.223
6/10	0.336	0.154
4/7	0.400	0.088
2/25	0.635	0.071
5/19	0.647	0.054
3/15	0.757	0.051
9/16	1.000	0.014
2/23	0.177	0.011
4/8	0.749	0.006
^* Scores normalized by dividing by the maximum.

Each table sorts persons and groups by fitness/complexity score. We can see that the status order changes once we introduce the fitness/complexity scoring mode. While {Theresa} is the top person according to the usual dual Bonacich prestige score, once we heavily discount the status of events that include low-status people, {Evelyn} becomes the top person, with {Theresa} dropping to the sixth spot. In the same way, while {Nora} is ranked sixth by the Bonacich prestige, her standing improves to third in the fitness scoring. Meanwhile, {Brenda} ranked third according to the Bonacich score, but her prestige drops to seventh place in the fitness/complexity scoring.

The status of groups changes even more dramatically once complexity is calculated by heavily discounting the status of groups that include lower-status people. While {9/16} is the top event by the usual Bonacich prestige scoring, this event has minimal status according to the complexity scoring, ending up third from the bottom. Instead, the top event by complexity is {3/2}, a relatively low-status event according to the Bonacich score. In fact, all other top events, according to the complexity scoring, were ranked minimally by the Bonacich scoring, except for event {2/23}, which is a low-status event on both accounts. This means that the Bonacich prestige and complexity scores for events have a strong negative correlation (r = -0.63). This is different from the person ranks, which agree more closely (r = 0.77).

To help us make sense of the differences between the Bonacich and the fitness/complexity prestige scoring, -1 shows the original SW biadjacency matrix, with rows and columns ordered by each node’s degree. This means that highly attended events appear on the left, and highly active women appear at the top.

gives us some insight as to why event {9/16} drops so much in the fitness/complexity rankings. While it is the most prestigious (and well-attended) event by the Bonacich score (leftmost in -1), this also means it is attended by many low-status women (those at the bottom of -2). The same thing happens to event {4/8}, which is third according to Bonacich prestige, but dead last according to complexity, given its heavy bulk of low fitness attendees (e.g., {Verne, Ruth, Pearl, Dorothy, Olivia, Flora}).

Event {3/2} on the other hand, while having relatively sparse attendance (only three women), and thus low Bonacich prestige (which rewards volume), has a membership composed of exclusively high-status persons (ones at the top of -2: {Evelyn, Laura, Theresa}). Event {2/23} gets ranked similarly by the two measures because it has both sparse attendance and the majority (two-thirds) of its attendees are low-status women ({Olivia, Flora}).

also helps us make sense of the {Theresa} versus {Evelyn} contrasts across prestige rankings. Both are highly active women, but Evelyn’s membership in the top two events according to fitness/complexity {3/2, 6/27} puts her towards the top. The same goes for Laura’s membership in the same events, which move her from fourth to second in the fitness/complexity ranks. Nora’s membership in the third- and fourth-most-highly ranked events according to complexity, {11/21, 8/3}, also helps improve her fitness ranks in -2.

3 An Eigenvector-Based Approach

Sciarra et al. (2020, 8–9) describe an eigenvector-based (and thus linear and non-iterative) method to derive values that are close (but not exactly) to the fitness/complexity scores. Their idea is to produce a degree-normalized biadjacency matrix that equals:

\[ \tilde{\mathbf{A}} = \mathbf{D}_p \mathbf{A} \mathbf{D'}_g \]

Where $\mathbf{D}_p$ is the diagonal matrix of people’s degrees and $\mathbf{D'}_g$ is a diagonal matrix with each entry $d'_{gg}$ in the diagonal equal to:

\[ d'_{gg} = \sum_g \frac{A_{pg}}{k_p} \]

Where $k_p = \sum_p A_{pg}$ is the degree of each person.

In R we can obtain these matrices as follows:

Code

 D.p <- diag(1/rowSums(A))
 k.g <- colSums(A/(rowSums(A)))
 D.g <- diag(1/k.g)
 A.n <- D.p %*% A %*% D.g

Once we have the normalized biadjacency matrix, we compute the normalized Breiger projections:

\[ \tilde{\mathbf{P}} = \tilde{\mathbf{A}}\tilde{\mathbf{A}}^T \]

\[ \tilde{\mathbf{G}} = \tilde{\mathbf{A}}^T\tilde{\mathbf{A}} \]

The linear fitness/complexity scores $\mathbf{x}$ and $\mathbf{y}$ are then obtained as the solution to the following, now familiar, eigenvector problem:

\[ \lambda \mathbf{x} = \tilde{\mathbf{P}}\mathbf{x} \]

\[ \lambda \mathbf{y} = \tilde{\mathbf{G}}\mathbf{y} \]

Which in R we obtain as follows:

Code

 P.n <- A.n %*% t(A.n)
 G.n <- t(A.n) %*% A.n
 eig.p <- abs(eigen(P.n)$vectors[,1])
 eig.g <- abs(eigen(G.n)$vectors[,1])
 names(eig.p) <- rownames(A)
 names(eig.g) <- colnames(A)

We can then compare the scores for both persons and groups, with the proviso that, to maximize the correlation with their linear approximations, we normalize the fitness scores for people by dividing by the person degrees $k_p$, and the complexity scores for groups by multiplying by the corresponding $d'_{gg}$ score.

We can see that they are pretty close:

Code

 cor(fc$p.s/rowSums(A), eig.p)

          [,1]
[1,] 0.9814337

Code

 cor(fc$g.s*k.g, eig.g)

          [,1]
[1,] 0.9350304

(a) Rows/Columns Reordered by Bonacich Prestige.

Table 2: Status Scores

(a) Persons.

	Fitness	Linear Fitness
EVELYN	1.000	0.840
KATHERINE	0.994	1.000
LAURA	0.962	0.829
SYLVIA	0.872	0.897
NORA	0.781	0.836
BRENDA	0.762	0.690
THERESA	0.669	0.623
CHARLOTTE	0.589	0.584
FRANCES	0.253	0.362
MYRNA	0.183	0.384
HELEN	0.177	0.382
ELEANOR	0.132	0.275
VERNE	0.111	0.262
RUTH	0.099	0.216
PEARL	0.069	0.196
DOROTHY	0.028	0.122
OLIVIA	0.024	0.154
FLORA	0.024	0.154
^* Scores normalized by dividing by the maximum.

(b) Groups.

	Complexity	Linear Complexity
6/27	1.000	0.853
11/21	0.993	1.000
8/3	0.993	1.000
3/2	0.958	0.834
9/26	0.781	0.724
4/12	0.564	0.669
6/10	0.332	0.704
2/25	0.266	0.538
4/7	0.243	0.611
3/15	0.232	0.544
5/19	0.197	0.531
9/16	0.104	0.442
4/8	0.051	0.359
2/23	0.035	0.275
^* Scores normalized by dividing by the maximum.

Table 2 shows the linear fitness/complexity scores compared to the non-linear ones for persons and groups, and -3 shows the biadjacency matrix with rows and columns re-ordered according to the linearized fitness/complexity scores, which we can see is pretty close to the non-linear version in -2.

References

Breiger, Ronald L. 1974. “The Duality of Persons and Groups.” Social Forces 53 (2): 181–90.

Fouss, François, Marco Saerens, and Masashi Shimbo. 2016. Algorithms and Models for Network Data and Link Analysis. Cambridge University Press.

Hidalgo, César A, and Ricardo Hausmann. 2009. “The Building Blocks of Economic Complexity.” Proceedings of the National Academy of Sciences 106 (26): 10570–75.

Kleinberg, Jon M. 1999. “Authoritative Sources in a Hyperlinked Environment.” Journal of the ACM (JACM) 46 (5): 604–32.

Sciarra, Carla, Guido Chiarotti, Luca Ridolfi, and Francesco Laio. 2020. “Reconciling Contrasting Views on Economic Complexity.” Nature Communications 11 (1): 3352.

Tacchella, Andrea, Matthieu Cristelli, Guido Caldarelli, Andrea Gabrielli, and Luciano Pietronero. 2012. “A New Metrics for Countries’ Fitness and Products’ Complexity.” Scientific Reports 2 (1): 723.

--- title: "Fitness and Complexity in Two Mode Networks" --- In a highly cited piece, @tacchella_etal12 introduce a new prestige metric for two-mode networks that relies on the same "prismatic" model of status distribution we considered [before](tm-prest-eigen.qmd). @tacchella_etal12 called the prestige metrics they obtained using their approach "fitness" and "complexity" because they developed in the empirical context of calculating metrics for ranking nations based on their competitive advantage in exporting products, which means analyzing a two-mode country-by-product matrix [@hidalgo_hausmann09]. However, when considered in the more general context of two-mode network link analysis [@fouss_etal16], it is clear that their approach is a prestige metric for two-mode networks that combines ideas from Bonacich Eigenvector scoring and PageRank scoring, as covered in the [two-mode prestige lecture notes](tm-prest-eigen.qmd). ## Fitness/Complexity Scores Their basic idea is that when we are (asymmetrically) interested in determining the status or prestige of nodes in one particular mode (e.g., the row-mode nodes), we should not use summaries (e.g., sums or averages) of the scores for nodes in the other (e.g., column) mode in determining their status. Instead, we should deeply discount those nodes that connect to low-status nodes at the other end. To understand what they are getting at, it helps to write down the Bonacich prestige scoring in equation form, as we go through each iteration of the two-mode [status distribution game](tm-prest-eigen.qmd): If you remember from the function `tm.status`, each iteration $q$, the vector of status scores for the row nodes $\mathbf{s}^R$ and the column nodes $\mathbf{s}^C$ is given by: $$ s^R_i(q) = \sum_j\mathbf{A}_{ij}s^C_j(q-1) $${#eq-iter1} $$ s^C_j(q) = \sum_i\mathbf{A}_{ij}s^R_i(q-1) $${#eq-iter2} Where $\mathbf{A}$ is the two-mode network's biadjacency matrix, and with the restriction that at the initial step $\mathbf{s}(0)^C = \mathbf{1}$ where $\mathbf{1}$ is the all ones vector of length equal to the number of columns of the biadjacency matrix $\mathbf{A}$. At each iteration $q > 0$ we normalize both score vectors: $$ \mathbf{s}^R(q) = \frac{\mathbf{s}^R(q)}{\left\langle\mathbf{s}^R(q)\right\rangle} $${#eq-norm1} $$ \mathbf{s}^C(q) = \frac{\mathbf{s}^C(q)}{\left\langle\mathbf{s}^C(q)\right\rangle} $${#eq-norm2} Where $\left\langle \mathbf{s} \right\rangle$ is the Euclidean vector norm, and the process continues iterating until the differences between the vectors across successive iterations are minimal. So far, this is what we [covered before](tm-prest-eigen.qmd). What @tacchella_etal12 propose is to substitute above with: $$ s^C_j(q) = \left[\sum_i\mathbf{A}_{ij}\left(s^R_i(q-1)\right)^{-1}\right]^{-1} $${#eq-gen2} Which means that first (inner parentheses), we take the reciprocal of the row-mode nodes' status scores, sum them across column-mode nodes (such that column-mode nodes that connect to low-status row-mode nodes get a big score), and then take the reciprocal of the reciprocal to get back to a measure of status for column-mode nodes. This non-linear transformation heavily discounts the status scores assigned to column-mode nodes when they connect to lower-status row-mode nodes. @tacchella_etal12 also change the normalization step, using the mean value of the vector in the denominator of and . ## Beyond Status as Popularity/Activity How should we understand this modification? Recall that the basic principle of standard Bonacich prestige scoring is based on the equation of status/prestige and *popularity*/*activity*. In the canonical case of persons and groups [@breiger74], an event receives status from being attended by high-status individuals and an individual receives status from being affiliated with a high-status event; in each case, status from the point of view of the event means having highly active members, and from the point of view of the individual, it means being affiliated with popular events. But status may not always work this way. Consider the world-economic network linking countries to the products they have a competitive advantage in producing [@hidalgo_hausmann09]. Analysts noticed that the most developed countries produce both "complex" (i.e., high-status) products that only a select few of other highly developed economies produce (like semiconductors) and also less "complex" (i.e., low-status, like extractive natural resources) products that the other less developed economies produce [@tacchella_etal12]. That means that the "complexity" (i.e., status score) of a product cannot be derived simply by taking a summary (e.g., sum or average) of the status scores of the countries that produce it, because high-status countries engage in both high and low-status forms of production. However, knowing that a product is produced by a low-status country is more informative (and should weigh more heavily in determining a product's status score) because low-status countries produce only low-status products. Applying the same reasoning to the aforementioned case of persons and groups [@breiger74], an equivalent situation would go as follows. Imagine there is a set of prestigious women and a set of prestigious events that only the prestigious women attend. However, prestigious women are also endowed with a spirit of *noblesse oblige*, which means that the most prestigious of them *also* attend low-prestige events. This means that when determining the prestige of the *events* it is not very informative to know that prestigious women affiliate with them; rather, we should weigh more heavily whether *low-status* women affiliate with an event in determining an event's status, such that as the number of low-status women who affiliate with an event increases, a given event's status is downgraded in a non-linear way which feeds back into the computation of each woman's prestige. Let's see how this would work in the *Southern Women* data. First, we load it up: ```{r} library(networkdata) library(igraph) g <- southern_women A <- as.matrix(as_biadjacency_matrix(g)) ``` And here's a function called `tm.fitness` that modifies the old two-mode status distribution game function [we played before](tm-prest-eigen.qmd)---which, as you recall, was itself based on Kleinberg's [-@kleinberg99] HITS algorithm for directed one-mode networks---to compute the fitness and complexity prestige scores for persons and groups: ```{r} tm.fitness <- function(w, iter = 1000) { y <- matrix(1, ncol(w), 1) #initial group status column vector set to a constant z <- t(w) k <- 0 while (k < iter) { o.y <- y x <- w %*% o.y #fitness status scores for people x <- x/mean(x) #normalizing new people status scores y <- (z %*% x^-1)^-1 #complexity status scores for groups y <- y/mean(y) #normalizing new group status scores k <- k + 1 } return(list(p.s = x, g.s = y, k = k)) } ``` Note that we use a number-of-iterations approach to indicate convergence (governed by the `iter` argument) rather than a successive-differences approach, due to the non-linear nature of the algorithm's scoring. We then apply the `tm.fitness` function to the SW data: ```{r} fc <- tm.fitness(A, iter = 50) ``` We also calculate the usual Bonacich eigenvector scores using the `svd` function for comparison purposes: ```{r} eig <- svd(A) p.s <- eig$u[, 1] * -1 g.s <- eig$v[, 1] * -1 names(p.s) <- rownames(A) names(g.s) <- colnames(A) ``` And we put them in a table: ```{r} #| echo: false #| label: tbl-bfc #| tbl-cap: "Status Scores" #| tbl-subcap: #| - "Persons." #| - "Groups." #| layout-nrow: 1 library(kableExtra) library(dplyr) s.tab <- data.frame(b = p.s/max(p.s), fc = fc$p.s/max(fc$p.s)) |> arrange(desc(fc)) kbl(s.tab, format = "pipe", digits = 3, align = c("l", "c", "c"), col.names = c(" ", "Bonacich", "Fitness")) %>% kable_styling(bootstrap_options = c("hover", "condensed", "responsive")) %>% column_spec(1, bold = TRUE) |> footnote(symbol = c("Scores normalized by dividing by the maximum.")) s.tab <- data.frame(b = g.s/max(g.s), fc = fc$g.s/max(fc$g.s)) |> arrange(desc(fc)) kbl(s.tab, format = "pipe", digits = 3, align = c("l", "c", "c"), col.names = c(" ", "Bonacich", "Complexity")) %>% kable_styling(bootstrap_options = c("hover", "condensed", "responsive")) %>% column_spec(1, bold = TRUE) |> footnote(symbol = c("Scores normalized by dividing by the maximum.")) ``` Each table sorts persons and groups by fitness/complexity score. We can see that the status order changes once we introduce the fitness/complexity scoring mode. While *\{Theresa\}* is the top person according to the usual dual Bonacich prestige score, once we heavily discount the status of events that include low-status people, *\{Evelyn\}* becomes the top person, with *\{Theresa\}* dropping to the sixth spot. In the same way, while *\{Nora\}* is ranked sixth by the Bonacich prestige, her standing improves to third in the fitness scoring. Meanwhile, *\{Brenda\}* ranked third according to the Bonacich score, but her prestige drops to seventh place in the fitness/complexity scoring. The status of groups changes even more dramatically once complexity is calculated by heavily discounting the status of groups that include lower-status people. While *\{9/16\}* is the top event by the usual Bonacich prestige scoring, this event has minimal status according to the complexity scoring, ending up third from the bottom. Instead, the top event by complexity is \{3/2\}, a relatively low-status event according to the Bonacich score. In fact, all other top events, according to the complexity scoring, were ranked minimally by the Bonacich scoring, except for event *\{2/23\}*, which is a low-status event on both accounts. This means that the Bonacich prestige and complexity scores for events have a strong negative correlation (*r* = `r round(cor(g.s, fc$g.s),2)`). This is different from the person ranks, which agree more closely (*r* = `r round(cor(p.s, fc$p.s),2)`). To help us make sense of the differences between the Bonacich and the fitness/complexity prestige scoring, -1 shows the original SW biadjacency matrix, with rows and columns ordered by each node's degree. This means that highly attended events appear on the left, and highly active women appear at the top. gives us some insight as to why event *\{9/16\}* drops so much in the fitness/complexity rankings. While it is the most prestigious (and well-attended) event by the Bonacich score (leftmost in -1), this also means it is attended by many low-status women (those at the bottom of -2). The same thing happens to event *\{4/8\}*, which is third according to Bonacich prestige, but dead last according to complexity, given its heavy bulk of low fitness attendees (e.g., *\{Verne, Ruth, Pearl, Dorothy, Olivia, Flora\}*). Event *\{3/2\}* on the other hand, while having relatively sparse attendance (only three women), and thus low Bonacich prestige (which rewards volume), has a membership composed of exclusively high-status persons (ones at the top of -2: *\{Evelyn, Laura, Theresa\}*). Event *\{2/23\}* gets ranked similarly by the two measures because it has both sparse attendance and the majority (two-thirds) of its attendees are low-status women (*\{Olivia, Flora\}*). also helps us make sense of the *\{Theresa\}* versus *\{Evelyn\}* contrasts across prestige rankings. Both are highly active women, but Evelyn's membership in the top two events according to fitness/complexity *\{3/2, 6/27\}* puts her towards the top. The same goes for Laura's membership in the same events, which move her from fourth to second in the fitness/complexity ranks. Nora's membership in the third- and fourth-most-highly ranked events according to complexity, *\{11/21, 8/3\}*, also helps improve her fitness ranks in -2. ## An Eigenvector-Based Approach @sciarra_etal20 [p. 8-9] describe an eigenvector-based (and thus linear and non-iterative) method to derive values that are close (but not exactly) to the fitness/complexity scores. Their idea is to produce a degree-normalized biadjacency matrix that equals: $$ \tilde{\mathbf{A}} = \mathbf{D}_p \mathbf{A} \mathbf{D'}_g $$ Where $\mathbf{D}_p$ is the diagonal matrix of people's degrees and $\mathbf{D'}_g$ is a diagonal matrix with each entry $d'_{gg}$ in the diagonal equal to: $$ d'_{gg} = \sum_g \frac{A_{pg}}{k_p} $$ Where $k_p = \sum_p A_{pg}$ is the degree of each person. In `R` we can obtain these matrices as follows: ```{r} D.p <- diag(1/rowSums(A)) k.g <- colSums(A/(rowSums(A))) D.g <- diag(1/k.g) A.n <- D.p %*% A %*% D.g ``` Once we have the normalized biadjacency matrix, we compute the normalized Breiger projections: $$ \tilde{\mathbf{P}} = \tilde{\mathbf{A}}\tilde{\mathbf{A}}^T $$ $$ \tilde{\mathbf{G}} = \tilde{\mathbf{A}}^T\tilde{\mathbf{A}} $$ The linear fitness/complexity scores $\mathbf{x}$ and $\mathbf{y}$ are then obtained as the solution to the following, now familiar, eigenvector problem: $$ \lambda \mathbf{x} = \tilde{\mathbf{P}}\mathbf{x} $$ $$ \lambda \mathbf{y} = \tilde{\mathbf{G}}\mathbf{y} $$ Which in `R` we obtain as follows: ```{r} P.n <- A.n %*% t(A.n) G.n <- t(A.n) %*% A.n eig.p <- abs(eigen(P.n)$vectors[,1]) eig.g <- abs(eigen(G.n)$vectors[,1]) names(eig.p) <- rownames(A) names(eig.g) <- colnames(A) ``` We can then compare the scores for both persons and groups, with the proviso that, to maximize the correlation with their linear approximations, we normalize the fitness scores for people by dividing by the person degrees $k_p$, and the complexity scores for groups by multiplying by the corresponding $d'_{gg}$ score. We can see that they are pretty close: ```{r} cor(fc$p.s/rowSums(A), eig.p) cor(fc$g.s*k.g, eig.g) ``` ```{r} #| label: fig-sw #| fig-cap: "Southern Women Data Biadjacency Matrix." #| echo: false #| fig-cap-location: margin #| fig-subcap: #| - "Rows/Columns Reordered by Bonacich Prestige." #| - "Rows/Columns Reordered by Fitness/Complexity." #| - "Rows/Columns Reordered by Linearized Fitness/Complexity." #| layout-nrow: 2 #| fig-width: 8 #| fig-height: 8 #| library(corrplot) A.reord <- A[names(sort(p.s, decreasing = TRUE)), names(sort(g.s, decreasing = TRUE))] corrplot(A.reord, is.corr = FALSE, method = "color", tl.col = "black", tl.cex = 1, tl.srt = 45, addCoef.col = "black", cl.pos = "n", number.cex = 1.5, number.digits = 0, addgrid.col = "darkgray", col=colorRampPalette(c("white","white","red"))(200)) a <- names(fc$p.s[order(fc$p.s[, 1]*-1), ]) b <- names(fc$g.s[order(fc$g.s[, 1]*-1), ]) A.reord <- A[a, b] corrplot(A.reord, is.corr = FALSE, method = "color", tl.col = "black", tl.cex = 1, tl.srt = 45, addCoef.col = "black", cl.pos = "n", number.cex = 1.5, number.digits = 0, addgrid.col = "darkgray", col=colorRampPalette(c("white","white","red"))(200)) A.reord <- A[names(sort(eig.p, decreasing = TRUE)), names(sort(eig.g, decreasing = TRUE))] corrplot(A.reord, is.corr = FALSE, method = "color", tl.col = "black", tl.cex = 1, tl.srt = 45, addCoef.col = "black", cl.pos = "n", number.cex = 1.5, number.digits = 0, addgrid.col = "darkgray", col=colorRampPalette(c("white","white","red"))(200)) ``` ```{r} #| echo: false #| label: tbl-fclc #| tbl-cap: "Status Scores" #| tbl-subcap: #| - "Persons." #| - "Groups." #| layout-nrow: 1 nfc.ps <- fc$p.s/rowSums(A) s.tab <- data.frame(fc = nfc.ps/max(nfc.ps), lfc = eig.p/max(eig.p)) |> arrange(desc(fc)) kbl(s.tab, format = "pipe", digits = 3, align = c("l", "c", "c"), col.names = c(" ", "Fitness", "Linear Fitness")) %>% kable_styling(bootstrap_options = c("hover", "condensed", "responsive")) %>% column_spec(1, bold = TRUE) |> footnote(symbol = c("Scores normalized by dividing by the maximum.")) nfc.gs <- fc$g.s * k.g s.tab <- data.frame(fc = nfc.gs/max(nfc.gs), lfc = eig.g/max(eig.g)) |> arrange(desc(fc)) kbl(s.tab, format = "pipe", digits = 3, align = c("l", "c", "c"), col.names = c(" ", "Complexity", "Linear Complexity")) %>% kable_styling(bootstrap_options = c("hover", "condensed", "responsive")) %>% column_spec(1, bold = TRUE) |> footnote(symbol = c("Scores normalized by dividing by the maximum.")) ``` @tbl-fclc shows the linear fitness/complexity scores compared to the non-linear ones for persons and groups, and -3 shows the biadjacency matrix with rows and columns re-ordered according to the linearized fitness/complexity scores, which we can see is pretty close to the non-linear version in -2.