The literature is flooded with centrality indices and new ones areintroduced on a regular basis. Although there exist several theoreticaland empirical guidelines on when to use certain indices, there stillexists plenty of ambiguity in the concept of network centrality. Todate, network centrality is nothing more than applying indices to anetwork:
The only degree of freedom is the choice of index. The package comeswith an Rstudio addin (index_builder()), which allows tobuild or choose from more than 20 different indices. Blindly (ab)usingthis function is highly discouraged!
Thenetrankr package is based on the idea thatcentrality is more than a conglomeration of indices. Decomposing them ina series of microsteps offers the posibility to gradually add ideasabout centrality, without succumbing to trial-and-error approaches.Further, it allows for alternative assessment methods which can be moregeneral than the index-driven approach:
The new approach is centered around the concept ofpositions, which are defined as the relations and potentialattributes of a node in a network. The aggregation of the relationsleads to the definition of indices. However, positions can also becompared viapositional dominance, leading to partialcentrality rankings and the option to calculate probabilistic centralityrankings.
For a more detailed theoretical background, consult theLiterature 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 thepackage and the difference to an index driven approach. For a morerealistic 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 andsimply compute some standard centrality scores with theigraph package. Defining centrality indices in thenetrankr package is explained in the centrality indicesvignette.
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 differentvertex.
A more general assessment starts by calculating the neighborhoodinclusion 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 thatP[u,v]=1 implies that u isless central than v for centrality indices which are defined viaspecific path algebras. These include many of the well-known measureslike closeness (and variants), betweenness (and variants) as well asmany walk-based indices (eigenvector and subgraph centrality, totalcommunicability,…).
Neighborhood-inclusion defines a partial ranking on the set of nodes.Each ranking that is in accordance with this partial ranking yields aproper centrality ranking. Each of these ranking can thus potentially bethe outcome of a centrality index.
Using rank intervals, we can examine the minimal and maximal possiblerank of each node. The bigger the intervals are, the more freedom existsfor 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 theintervals. To get the exact probabilities, the functionexact_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 graphg we can therefore come up with 739,200indices that would rank the nodes differently.
rank.prob contains the probabilities for each node tooccupy a certain rank. For instance, the probability for each node to bethe 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 entryrelative.rank[u,v] indicates how likely it is thatv is more central thanu.
# How likely is it, that 6 is more central than 3?round(res$relative.rank[3,6],2)#> [1] 0.75expected.ranks contains the expected centrality ranksfor all nodes. They are derived on the basis ofrank.prob.
round(res$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 thenumber of nodes and the exact calculation becomes infeasible quitequickly and approximations need to be used. Check the benchmark resultsfor guidelines.
netrankr is based on a series of papers that appeared inrecent years. If you want to learn more about the theoretical backgroundof the package, consult the following literature:
Schoch, David. (2018). Centrality without Indices: Partial rankingsand 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 rankedgraphs.Social Networks,50, 46-54.(link)
Schoch, David & Brandes, Ulrik. (2016). Re-conceptualizingcentrality in social networks.European Journal of ApppliedMathematics,27(6), 971–985. (link)
Brandes, Ulrik. (2016). Network Positions.MethodologicalInnovations,9, 2059799116630650. (link)
Please note that the netrankr project is released with aContributorCode of Conduct. By contributing to this project, you agree to abideby its terms.