The literature is flooded with centrality indices and new ones are introduced on a regular basis. Although there exist several theoretical and empirical guidelines on when to use certain indices, there still exists plenty of ambiguity in the concept of network centrality. To date, network centrality is nothing more than applying indices to a network:
The only degree of freedom is the choice of index. The package comes
with an Rstudio addin (index_builder()), which allows to
build or choose from more than 20 different indices. Blindly (ab)using
this function is highly discouraged!
The netrankr package is based on the idea that
centrality is more than a conglomeration of indices. Decomposing them in
a series of microsteps offers the posibility to gradually add ideas
about centrality, without succumbing to trial-and-error approaches.
Further, it allows for alternative assessment methods which can be more
general than the index-driven approach:
The new approach is centered around the concept of positions, which are defined as the relations and potential attributes of a node in a network. The aggregation of the relations leads to the definition of indices. However, positions can also be compared via positional dominance, leading to partial centrality rankings and the option to calculate probabilistic centrality rankings.
For a more detailed theoretical background, consult the Literature at the end of this page.
To install from CRAN:
install.packages("netrankr")To install the developer version from github:
# install.packages("remotes")
remotes::install_github("schochastics/netrankr")This example briefly explains some of the functionality of the package and the difference to an index driven approach. For a more realistic application see the use case vignette.
We work with the following small graph.
library(igraph)
library(netrankr)
data("dbces11")
g <- dbces11
Say we are interested in the most central node of the graph and
simply compute some standard centrality scores with the
igraph package. Defining centrality indices in the
netrankr package is explained in the centrality indices
vignette.
cent_scores <- data.frame(
degree = degree(g),
betweenness = round(betweenness(g), 4),
closeness = round(closeness(g), 4),
eigenvector = round(eigen_centrality(g)$vector, 4),
subgraph = round(subgraph_centrality(g), 4)
)
# What are the most central nodes for each index?
apply(cent_scores, 2, which.max)
#> degree betweenness closeness eigenvector subgraph
#> 11 8 6 7 10
As you can see, each index assigns the highest value to a different vertex.
A more general assessment starts by calculating the neighborhood inclusion preorder.
P <- neighborhood_inclusion(g)
P
#> 1 2 3 4 5 6 7 8 9 10 11
#> 1 0 0 1 0 1 1 1 0 0 0 1
#> 2 0 0 0 1 0 0 0 1 0 0 0
#> 3 0 0 0 0 1 0 0 0 0 0 1
#> 4 0 0 0 0 0 0 0 0 0 0 0
#> 5 0 0 0 0 0 0 0 0 0 0 0
#> 6 0 0 0 0 0 0 0 0 0 0 0
#> 7 0 0 0 0 0 0 0 0 0 0 0
#> 8 0 0 0 0 0 0 0 0 0 0 0
#> 9 0 0 0 0 0 0 0 0 0 0 0
#> 10 0 0 0 0 0 0 0 0 0 0 0
#> 11 0 0 0 0 0 0 0 0 0 0 0Schoch &
Brandes (2016) showed that P[u,v]=1 implies that u is
less central than v for centrality indices which are defined via
specific path algebras. These include many of the well-known measures
like closeness (and variants), betweenness (and variants) as well as
many walk-based indices (eigenvector and subgraph centrality, total
communicability,…).
Neighborhood-inclusion defines a partial ranking on the set of nodes. Each ranking that is in accordance with this partial ranking yields a proper centrality ranking. Each of these ranking can thus potentially be the outcome of a centrality index.
Using rank intervals, we can examine the minimal and maximal possible rank of each node. The bigger the intervals are, the more freedom exists for indices to rank nodes differently.
plot(rank_intervals(P), cent_scores = cent_scores, ties.method = "average")
The potential ranks of nodes are not uniformly distributed in the
intervals. To get the exact probabilities, the function
exact_rank_prob() can be used.
res <- exact_rank_prob(P)
res
#> Number of possible centrality rankings: 739200
#> Equivalence Classes (max. possible): 11 (11)
#> - - - - - - - - - -
#> Rank Probabilities (rows:nodes/cols:ranks)
#> 1 2 3 4 5 6
#> 1 0.54545455 0.27272727 0.12121212 0.04545455 0.01298701 0.002164502
#> 2 0.27272727 0.21818182 0.16969697 0.12727273 0.09090909 0.060606061
#> 3 0.00000000 0.16363636 0.21818182 0.20909091 0.16883117 0.119047619
#> 4 0.00000000 0.02727273 0.05151515 0.07272727 0.09090909 0.106060606
#> 5 0.00000000 0.00000000 0.01818182 0.04545455 0.07532468 0.103463203
#> 6 0.00000000 0.05454545 0.08484848 0.10000000 0.10649351 0.108658009
#> 7 0.00000000 0.05454545 0.08484848 0.10000000 0.10649351 0.108658009
#> 8 0.00000000 0.02727273 0.05151515 0.07272727 0.09090909 0.106060606
#> 9 0.09090909 0.09090909 0.09090909 0.09090909 0.09090909 0.090909091
#> 10 0.09090909 0.09090909 0.09090909 0.09090909 0.09090909 0.090909091
#> 11 0.00000000 0.00000000 0.01818182 0.04545455 0.07532468 0.103463203
#> 7 8 9 10 11
#> 1 0.00000000 0.00000000 0.000000000 0.00000000 0.00000000
#> 2 0.03636364 0.01818182 0.006060606 0.00000000 0.00000000
#> 3 0.07272727 0.03636364 0.012121212 0.00000000 0.00000000
#> 4 0.11818182 0.12727273 0.133333333 0.13636364 0.13636364
#> 5 0.12727273 0.14545455 0.157575758 0.16363636 0.16363636
#> 6 0.10909091 0.10909091 0.109090909 0.10909091 0.10909091
#> 7 0.10909091 0.10909091 0.109090909 0.10909091 0.10909091
#> 8 0.11818182 0.12727273 0.133333333 0.13636364 0.13636364
#> 9 0.09090909 0.09090909 0.090909091 0.09090909 0.09090909
#> 10 0.09090909 0.09090909 0.090909091 0.09090909 0.09090909
#> 11 0.12727273 0.14545455 0.157575758 0.16363636 0.16363636
#> - - - - - - - - - -
#> Relative Rank Probabilities (row ranked lower than col)
#> 1 2 3 4 5 6 7
#> 1 0.00000000 0.66666667 1.0000000 0.9523810 1.0000000 1.0000000 1.0000000
#> 2 0.33333333 0.00000000 0.6666667 1.0000000 0.9166667 0.8333333 0.8333333
#> 3 0.00000000 0.33333333 0.0000000 0.7976190 1.0000000 0.7500000 0.7500000
#> 4 0.04761905 0.00000000 0.2023810 0.0000000 0.5595238 0.4404762 0.4404762
#> 5 0.00000000 0.08333333 0.0000000 0.4404762 0.0000000 0.3750000 0.3750000
#> 6 0.00000000 0.16666667 0.2500000 0.5595238 0.6250000 0.0000000 0.5000000
#> 7 0.00000000 0.16666667 0.2500000 0.5595238 0.6250000 0.5000000 0.0000000
#> 8 0.04761905 0.00000000 0.2023810 0.5000000 0.5595238 0.4404762 0.4404762
#> 9 0.14285714 0.25000000 0.3571429 0.6250000 0.6785714 0.5714286 0.5714286
#> 10 0.14285714 0.25000000 0.3571429 0.6250000 0.6785714 0.5714286 0.5714286
#> 11 0.00000000 0.08333333 0.0000000 0.4404762 0.5000000 0.3750000 0.3750000
#> 8 9 10 11
#> 1 0.9523810 0.8571429 0.8571429 1.0000000
#> 2 1.0000000 0.7500000 0.7500000 0.9166667
#> 3 0.7976190 0.6428571 0.6428571 1.0000000
#> 4 0.5000000 0.3750000 0.3750000 0.5595238
#> 5 0.4404762 0.3214286 0.3214286 0.5000000
#> 6 0.5595238 0.4285714 0.4285714 0.6250000
#> 7 0.5595238 0.4285714 0.4285714 0.6250000
#> 8 0.0000000 0.3750000 0.3750000 0.5595238
#> 9 0.6250000 0.0000000 0.5000000 0.6785714
#> 10 0.6250000 0.5000000 0.0000000 0.6785714
#> 11 0.4404762 0.3214286 0.3214286 0.0000000
#> - - - - - - - - - -
#> Expected Ranks (higher values are better)
#> 1 2 3 4 5 6 7 8
#> 1.714286 3.000000 4.285714 7.500000 8.142857 6.857143 6.857143 7.500000
#> 9 10 11
#> 6.000000 6.000000 8.142857
#> - - - - - - - - - -
#> SD of Rank Probabilities
#> 1 2 3 4 5 6 7 8
#> 0.9583148 1.8973666 1.7249667 2.5396850 2.1599320 2.7217941 2.7217941 2.5396850
#> 9 10 11
#> 3.1622777 3.1622777 2.1599320
#> - - - - - - - - - -For the graph g we can therefore come up with 739,200
indices that would rank the nodes differently.
rank.prob contains the probabilities for each node to
occupy a certain rank. For instance, the probability for each node to be
the most central one is as follows.
round(res$rank.prob[, 11], 2)
#> 1 2 3 4 5 6 7 8 9 10 11
#> 0.00 0.00 0.00 0.14 0.16 0.11 0.11 0.14 0.09 0.09 0.16relative.rank contains the relative rank probabilities.
An entry relative.rank[u,v] indicates how likely it is that
v is more central than u.
# How likely is it, that 6 is more central than 3?
round(res$relative.rank[3, 6], 2)
#> [1] 0.75expected.ranks contains the expected centrality ranks
for all nodes. They are derived on the basis of
rank.prob.
round(res$expected.rank, 2)
#> 1 2 3 4 5 6 7 8 9 10 11
#> 1.71 3.00 4.29 7.50 8.14 6.86 6.86 7.50 6.00 6.00 8.14The higher the value, the more central a node is expected to be.
Note: The set of rankings grows exponentially in the number of nodes and the exact calculation becomes infeasible quite quickly and approximations need to be used. Check the benchmark results for guidelines.
netrankr is based on a series of papers that appeared in
recent years. If you want to learn more about the theoretical background
of the package, consult the following literature:
Schoch, David. (2018). Centrality without Indices: Partial rankings and rank Probabilities in networks. Social Networks, 54, 50-60.(link)
Schoch, David & Valente, Thomas W., & Brandes, Ulrik. (2017). Correlations among centrality indices and a class of uniquely ranked graphs. Social Networks, 50, 46-54.(link)
Schoch, David & Brandes, Ulrik. (2016). Re-conceptualizing centrality in social networks. European Journal of Appplied Mathematics, 27(6), 971–985. (link)
Brandes, Ulrik. (2016). Network Positions. Methodological Innovations, 9, 2059799116630650. (link)
Please note that the netrankr project is released with a Contributor Code of Conduct. By contributing to this project, you agree to abide by its terms.