Network centrality and sampling effort

Degree, strength and eigenvector centrality for an animal social network in R: each measures something different, and all three rise with sighting effort.

You have a weighted social network and you want a number per individual: who is central, who is peripheral, who holds the group together. That number is a centrality metric, and there are several. This post computes three of them by hand, shows that they disagree because they measure different things, and then delivers the caveat that undoes a lot of published rankings: raw centrality climbs with how often you saw the animal.

Three notions of central

Start from the association matrix, the same simple ratio index network as before. Three metrics fall straight out of it.

• Degree counts an individual’s distinct associates: how many others it has a non-zero edge to. It ignores edge weight.
• Strength sums the edge weights: total association across all partners. A few strong bonds and many weak ones can give the same strength.
• Eigenvector centrality rewards being connected to individuals who are themselves well connected. It is the leading eigenvector of the matrix, so an individual scores highly by sitting inside a dense, well-linked cluster, not merely by having many edges.

Here they are, each a one-liner.

sri_from_gbi <- function(gbi) {
  co <- crossprod(gbi); n <- diag(co)
  a <- co / (outer(n, n, "+") - co); diag(a) <- 0; a
}
node_degree   <- function(a) rowSums(a > 0)
node_strength <- function(a) rowSums(a)
eigen_cent    <- function(a) { e <- eigen(a, symmetric = TRUE); v <- abs(e$vectors[, 1]); v / max(v) }

Now a population to try them on: three communities of differing cohesion, sighting effort that varies mildly across individuals, and one animal (number seven) that lives in the first community but also joins the second, a bridge between them.

set.seed(4)
n_ind <- 21; n_grp <- 55
community <- rep(1:3, each = 7)
effort <- runif(n_ind, 0.85, 1.3)
cohesion <- c(0.70, 0.40, 0.55)[community]
bridge <- 7
gbi <- matrix(0L, n_grp, n_ind)
for (k in 1:n_grp) {
  fc <- sample(1:3, 1)
  base <- ifelse(community == fc, cohesion, 0.05) * effort
  if (fc == 2) base[bridge] <- 0.45
  gbi[k, ] <- rbinom(n_ind, 1, pmin(base, 0.95))
}
gbi <- gbi[rowSums(gbi) > 0, ]; keep <- colSums(gbi) > 0; gbi <- gbi[, keep]
community <- community[keep]; count <- colSums(gbi)

net <- sri_from_gbi(gbi)
deg <- node_degree(net); str <- node_strength(net); eig <- eigen_cent(net)

They do not agree

If the three metrics ranked individuals the same way, you would only ever need one. They do not.

c(degree_vs_strength = round(cor(deg, str, method = "spearman"), 3),
  degree_vs_eigen    = round(cor(deg, eig, method = "spearman"), 3),
  strength_vs_eigen  = round(cor(str, eig, method = "spearman"), 3))

degree_vs_strength    degree_vs_eigen  strength_vs_eigen 
             0.355              0.211              0.874 

Strength and eigenvector centrality track each other fairly closely. Degree parts company with both: its rank correlation with eigenvector centrality is only about 0.21. Degree and eigenvector centrality are answering different questions, and on this network the answers barely line up.

The bridging individual is where the disagreement becomes a story.

ranks <- data.frame(
  id = seq_len(ncol(gbi)), community = community, count = count,
  degree_rank = rank(-deg), strength_rank = rank(-str), eigen_rank = rank(-eig))
ranks[order(ranks$strength_rank), ][1:6, ]

   id community count degree_rank strength_rank eigen_rank
20 20         3    23         3.5             1          1
5   5         1    11        12.0             2          6
7   7         1    20         1.0             3         11
17 17         3    22         3.5             4          3
4   4         1     9        15.5             5          9
21 21         3    25         9.0             6          2

Individual twenty tops both strength and eigenvector centrality: a well-embedded member of a cohesive community, connected to others who are themselves connected. Individual seven, the bridge, is a different animal. It has the highest degree in the network (it associates with the most distinct others, because it moves between two communities) yet its eigenvector centrality is only middling: its partners are split across two groups rather than concentrated in one dense core, so it does not sit at the heart of any single cluster. Degree calls it the most central animal in the population; eigenvector centrality calls it ordinary. Both are right about their own question.

df <- data.frame(id = seq_len(ncol(gbi)), degree = deg, eigen = eig,
                 community = factor(community), count = count)
ggplot(df, aes(degree, eigen, colour = community)) +
  geom_point(aes(size = count), alpha = 0.85) +
  annotate("text", x = df$degree[bridge], y = df$eigen[bridge],
           label = "bridge (id 7)", vjust = -1, size = 3.4) +
  scale_colour_manual(values = c("#2b5d34", "#c46a1b", "#3a6ea5")) +
  scale_size_continuous(name = "sightings", range = c(2, 6)) +
  labs(title = "Degree and eigenvector centrality measure different things",
       x = "degree (number of associates)", y = "eigenvector centrality")

Figure 1: Eigenvector centrality against degree, one point per individual, coloured by community. The relationship is weak, and the bridging individual (labelled) sits far below the trend: many associates, but low embeddedness.

The number that spoils the ranking

Now the caveat. Pick any of these metrics and correlate it with how many times each individual was seen.

c(cor_strength_count = round(cor(str, count), 3),
  count_range_low  = min(count),
  count_range_high = max(count))

cor_strength_count    count_range_low   count_range_high 
             0.719              3.000             25.000 

Strength correlates with sighting count at about 0.72. Individuals here were seen between three and twenty-five times, and the ones seen most have the highest strength. Some of that is real (a genuinely gregarious animal turns up more and associates more), but some of it is pure sampling: watch an animal more and its measured associations accumulate, whatever its true sociality. Degree and eigenvector centrality carry the same passenger. A centrality ranking, taken at face value, can be a ranking of who your field protocol happened to record most.

df$strength <- str
ggplot(df, aes(count, strength, colour = community)) +
  geom_point(size = 3, alpha = 0.85) +
  scale_colour_manual(values = c("#2b5d34", "#c46a1b", "#3a6ea5"), guide = "none") +
  labs(title = "Raw strength rises with how often you looked",
       x = "number of sightings", y = "network strength")

Figure 2: Strength against sighting count. The upward trend means part of what the ranking calls sociality is really observation effort.

What to do about it

Do not read a raw centrality ranking as a sociality ranking until you have looked at this scatter. Two honest routes forward. You can control for effort directly: regress the metric on sighting count and work with the residual, the part of centrality that is not explained by how often the animal was seen. Or you can test centrality against the data-stream null from the permutation post, which rebuilds the network under random association with the same sighting counts, so the effort component sits in the null instead of in your result. Both say the same thing in different dialects: separate the sociality from the sampling before you rank anyone.

Which metric you report should follow your question, not habit. Degree for reach, how many others an individual contacts. Strength for total association. Eigenvector centrality for being embedded among the well connected. They are not substitutes, and the bridge above is the proof: it is the most central animal or an unremarkable one depending only on which you pick.

Where to go next

The checking post makes this effort diagnostic step one of a short routine, then adds two more: whether your data even has the power to detect real structure, and whether the group definition underneath the whole network holds up. The permutation post is where the data-stream null comes from, and the association-index post covers how the edge weights were built in the first place.

Related tutorials

• Association indices for social networks
• Permutation tests for social networks
• Checking a social network analysis
• Bipartite network metrics

References

Farine DR, Whitehead H 2015 Journal of Animal Ecology 84(5):1144-1163 (10.1111/1365-2656.12418)

Whitehead H 2008 Analyzing Animal Societies: Quantitative Methods for Vertebrate Social Analysis. University of Chicago Press. ISBN 978-0226895215

Croft DP, James R, Krause J 2008 Exploring Animal Social Networks. Princeton University Press. ISBN 978-0691127521