Centrality vs. power

One common goal in network analysis is to score or rank nodes based on the whether their network position is advantageous. Doing so requires specifying advantageous for what? Different metrics identify nodes whose positions are advantageous for different outcomes [5]. For example, in the context of a communication network, closeness centrality identifies nodes whose position is advantageous for disseminating a message quickly, while betweenness centrality identifies nodes whose position is advantageous for intercepting a message.

In the context of seeking to maximize one’s resources, degree centrality simply counts a node’s number of neighbors, which are each a potential source of resources. For example, in a social network, a high-degree person might be advantaged because they have many friends, each of whom is a potential source of social support. Although neighbors are potential sources of resources, some neighbors may be better sources than others. Following this logic, eigenvector centrality extended degree centrality to identify nodes that are connected to well-connected neighbors [3]. Returning to the social network example, a high-eigenvector person is advantaged because they have ‘friends in high places’ who are potential sources of more or better resources.

However, there are also cases where being connected to well-connected others is not advantageous. Specifically, if one’s goal is to exploit one’s neighbors, then having well-connected neighbors is a liability. Exploiting poorly-connected neighbors is easier because they have fewer alternative sources for resources themselves (i.e., they are dependent). Indeed, exchange experiments have demonstrated that traders who are connected to well-connected others generate less profit than those who are connected to poorly-connected others [8]. Accordingly, such a position is often described not as a position of ‘centrality,’ but instead as a position of ‘power.’

Distinctiveness centrality

Distinctiveness centrality aims to identify nodes that occupy positions of power, that is, whose neighbors are poorly-connected [1]. Although distinctiveness centrality was proposed as a set of five closely-related measures, here I focus on ‘weighted distinctiveness centrality’ (D1), because it is the root measure from which the others were derived, and because despite the name can be computed in both weighted and unweighted networks.

The weighted distinctiveness centrality of node i is defined as (1) where w_ij is the weight of the edge between nodes i and j, n is the number of nodes in the network, g_j is the degree of j, and α is a tuning parameter. Like degree centrality, D1 sums the weight of a node’s connections, but penalizes connections to highly connected neighbors. The α tuning parameter adjusts the severity of this penalty. The computation of D1 has been implemented in packages available for both Python [9] and R [10].

Fronzetti Colladon and Naldi imply that the allowable range of the tuning parameter is 1 ≤ α < ∞, but do not offer an explanation. In their analyses, they examine the performance of D1 for 1 ≤ α ≤ 15, but also do not explain why they chose these values [1]. In the analyses below, I instead investigate −1 < α < 1 for two reasons. First, as I show below, D1 has a meaningful interpretation when α < 1. Second, it is unclear what very large values of α would mean, or how one would select a value of α from such a wide range.

In their original analysis, Fronzetti Colladon and Naldi compared D1 to four classic measures of centrality: degree, closeness, betweenness, eigenvector. It is unclear why they chose these four for comparison. None of these are intended to identify nodes that are connected to poorly-connected neighbors, and therefore there is no reason to expect that they would be similar D1 [2, 3]. In the analyses below, I instead compare D1 to beta [6] and gamma [11] centrality, which do identify nodes that are connected to poorly-connected neighbors, and therefore are similar to D1.

Beta centrality

When exchange experiments demonstrated, unexpectedly, that highly central nodes do not generate the most profit [8], ‘beta centrality’ (BC) was proposed as a centrality measure that could identify advantageous positions in exchange networks [6]. A vector of beta centrality scores are defined as: (2) where A is an adjacency matrix, I is an identity matrix, 1 is a column vector of 1s, inv is the matrix inverse function, and β is a tuning parameter.

The tuning parameter can range , where λ₁ is the largest eigenvalue of A. When β = 0, BC is equal to degree. When β > 0, BC assigns higher scores to nodes that are connected to well-connected neighbors, and converges on eigenvector centrality as β approaches from below. Finally, when β < 0, BC assigns higher scores to nodes that are connected to poorly-connected neighbors. Accordingly, BC is conceptually similar to D1 when β < 0.

Computing BC does not require a specialized software package. In R, it can be computed using:

B <- 0 #Set value of beta
I <- diag(nrow(A)) #Define identity matrix
O <- matrix(1, nrow = nrow(A)) #Define vector of ones
BC <- solve(I - (B * A)) %*% A %*% O #Compute

Gamma centrality

Beta centrality’s computation can be opaque and the allowable range of β can be a source of confusion for users. Gamma centrality aims to overcome these challenges by offering a simplified alternative. This alternative to beta centrality evolved in a series of initial papers in the urban studies [7], psychology [12], and network [13] literatures, but here I focus on its final form, known as ‘gamma centrality’ (GC) [11]. A vector of gamma centrality scores are defined as: (3) where A is an adjacency matrix, 1 is a column vector of 1s, and γ is a tuning parameter.

In principle γ can take any real value, however in practice the narrower range −1 < γ < 1 is sufficient for distinguishing nodes’ positions [7, 11–13]. The tuning parameter γ controls the interpretation of GC in the same way that β controls the interpretation of BC. When γ = 0, GC is equal to degree. When γ > 0, GC assigns higher scores to nodes that are connected to well-connected neighbors, Finally, when γ < 0, GC assigns higher scores to nodes that are connected to poorly-connected neighbors. Accordingly, GC is conceptually similar to D1 when γ < 0.

Computing GC does not require a specialized software package. In R, it can be computed using:

G <- 0 #Set value of gamma
O <- matrix(1, nrow = nrow(A)) #Define vector of ones
GC <- A %*% ((A %*% O)^G) #Compute

Harmonizing tuning parameters

One challenge comparing these tunable centrality measures is that their tuning parameters—α, β, and γ—each have different scales. Rather than directly specify the value of each measure’s tuning parameter, I instead define a meta-parameter p, and use it to define the tuning parameters according to a set of harmonizing functions.

First, for D1: (4) Eq 4 simply reverse-scores the α parameter.

Second, for BC: (5) Eq 5 applies the logistic function so that as p approaches ∞, β approaches its maximum allowable value, while as p approaches −∞, β approaches its minimum allowable value.

Finally, for GC: (6) Eq 6 uses p as the value of γ.

Collectively, these harmonizing functions ensure that for all centrality measures, when p > 0 the measure functions as a centrality index by assigning higher scores to nodes that are connected to well-connected neighbors, while when p < 0 the measure functions as a power index by assigning higher scores to nodes that are connected to poorly-connected neighbors. In principle, p can take any real value, however in the analyses below I consider only −1 ≤ p ≤ 1 because it is within this range that all three measures perform as identical centrality and power indices.

Networks

To evaluate the similarities between D1, BC, and GC, I compare them in five networks. First, I compare them in the toy network (see Fig 1A) from the exchange experiments [8] that initially led to the development of BC [6], and that has been used to validate BC and GC [6, 11]. Then I compare them in four additional networks used by Fronzetti Colladon and Naldi in their original analysis of D1: a weighted toy network (see Fig 1B), the unweighted Florentine marriage network [14], the weighted Zachary karate club network [15], and 1000 weighted scale-free networks. The scale-free networks were generated via preferential attachment. They contain 50 nodes with 2 edges added in each step, and the edges’ weights are randomly assigned from the uniform distribution ranging from 1 to 20.

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Fig 1. (A) Unweighted [8] and (B) weighted [1] toy networks.

https://doi.org/10.1371/journal.pone.0298666.g001

Analysis

Within each network, I compute the vector of centrality scores using each measure for tuning meta-parameters in the range −1 < p < 1. I use the distinctiveness package for R to compute D1 [10], and use the R code shown above to compute BC and GC. Following [1], I then compute the Spearman correlation between D1 and BC, and between D1 and GC, for each p to measure the similarity in node rankings that these measures yield. The code and data necessary to reproduce these results is available at https://osf.io/s6pqg/.