Robustness of a community detection algorithm
Null model
robin offers two choices for the null model:
it can be generated by using the function random
it can be built externally and passed directly to the argument graphRandom of the robinRobust function.
The function random creates a random graph with the same degree distribution of the original graph, but with completely random edges. The graph argument must be the same returned by prepGraph function.
graphRandom <- random(graph=graph)
graphRandom## IGRAPH 32769a3 U--- 115 613 --
## + attr: id (v/n), label (v/c), value (v/n)
## + edges from 32769a3:
## [1] 1-- 65 1-- 88 10--114 1-- 41 1-- 6 1-- 34 1-- 36 42-- 91
## [9] 52-- 66 52-- 88 34-- 94 54-- 60 2-- 26 75-- 92 2-- 34 39-- 42
## [17] 2-- 75 2-- 58 2-- 94 2--105 82--104 87--106 105--110 61-- 87
## [25] 58-- 84 3-- 63 3-- 15 30-- 53 3-- 51 3-- 61 3-- 42 3-- 73
## [33] 3-- 26 56-- 97 3-- 91 4-- 79 42--109 4--112 4-- 41 4-- 83
## [41] 55-- 59 85--110 3-- 4 4--115 85--101 4--103 5--108 5-- 8
## [49] 5--102 5-- 16 25-- 29 5-- 76 5-- 38 40-- 71 5-- 59 5-- 73
## [57] 6-- 11 12-- 32 70--112 10-- 49 6-- 82 6--107 91-- 99 6-- 98
## + ... omitted several edgesrobinRobust
robinRobust function implements the validation of the community robustness. In this example we used “vi” distance as stability measure, “independent” type procedure and “louvain” as community detection algorithm.
Users can choose also different measures as: “nmi”,“split.join”, “adjusted.rand”.
The graph argument must be the one returned by prepGraph function. The graphRandom must be the one returned by random function or own random graph.
proc <- robinRobust(graph=graph, graphRandom=graphRandom, measure="vi",
method="louvain", type="independent")## Detected robin method independent type
## Perturbed 31 edges
## Perturbed 61 edges
## Perturbed 92 edges
## Perturbed 123 edges
## Perturbed 153 edges
## Perturbed 184 edges
## Perturbed 215 edges
## Perturbed 245 edges
## Perturbed 276 edges
## Perturbed 306 edges
## Perturbed 337 edges
## Perturbed 368 edgesAs output robinRobust will give all the measures at different level of perturbation from 0% to 60% for the real and random graph.
plotRobin function plots the curves. The (x,y)-axes represents the percentage of perturbation and the average of the stability measure, respectively. The arguments of model1 and model2 must be the measures for the real graph and the random graph that are the outputs of the robinRobust function.
We will expect that with a robust algorithm the behavior of the two curves is different. We expect that the curve of the real graph vary less than the curve of the random graph, this visually means that the curve of the real graph is lower than the one of the random graph, so it is more stable than a random graph.
plotRobin(graph=graph, model1=proc$Mean, model2=proc$MeanRandom,
legend=c("real data", "null model"))
The procedure implemented depends on the network of interest. In this example, the Louvain algorithm fits good the network of interest,as the curve of the stability measure assumes lower values than the one obtained by the null model.
Statistical tests
The differences between the stability measure curves are tested using:
Functional Data Analysis (FDA);
Gaussian Process (GP).
Moreover to quantify the differences between the curves when they are very close the Area Under the Curves (AUC) are evaluated.
robinFDATest function implements a test giving a p-value for different intervals of the curves. It tests in which interval the two curves are different.
robinFDATest(graph=graph, model1=proc$Mean, model2=proc$MeanRandom,
legend=c("real graph", "random graph"))## [1] "First step: basis expansion"
## Swapping 'y' and 'argvals', because 'y' is simpler,
## and 'argvals' should be; now dim(argvals) = 13 ; dim(y) = 13 x 20
## [1] "Second step: joint univariate tests"
## [1] "Third step: interval-wise combination and correction"
## [1] "creating the p-value matrix: end of row 2 out of 9"
## [1] "creating the p-value matrix: end of row 3 out of 9"
## [1] "creating the p-value matrix: end of row 4 out of 9"
## [1] "creating the p-value matrix: end of row 5 out of 9"
## [1] "creating the p-value matrix: end of row 6 out of 9"
## [1] "creating the p-value matrix: end of row 7 out of 9"
## [1] "creating the p-value matrix: end of row 8 out of 9"
## [1] "creating the p-value matrix: end of row 9 out of 9"
## [1] "Interval Testing Procedure completed"
## TableGrob (1 x 2) "arrange": 2 grobs
## z cells name grob
## 1 1 (1-1,1-1) arrange gtable[layout]
## 2 2 (1-1,2-2) arrange gtable[layout]## $adj.pvalue
## [1] 0.2293 0.0080 0.0048 0.0022 0.0021 0.0026 0.0031 0.0040 0.0055
##
## $pvalues
## [1] 0.2293 0.0020 0.0020 0.0020 0.0020 0.0020 0.0031 0.0040 0.0020The first figure represents the stability measure plot using Louvain algorithm for detecting communities. The second one represents the corresponding p-values and adjusted p-values of the Interval Testing procedure. Horizontal red line corresponds to the critical value 0.05.
robinGPTest function implements the GP testing.
robinGPTest(model1=proc$Mean, model2=proc$MeanRandom)## Profile 1
## Profile 2## [1] 139.2966It tests the two curves globally. The null hypothesis claims that the two curves come from the same process, the alternative hypothesis that they come from two different processes. The output is the Bayes Factor. One of the most common interpretations is the one proposed by Harold Jeffereys (1961) and slightly modified by Lee and Wagenmakers in 2013:
IF B10 IS… THEN YOU HAVE…
° > 100 Extreme evidence for H1
° 30 – 100 Very strong evidence for H1
° 10 – 30 Strong evidence for H1
° 3 – 10 Moderate evidence for H1
° 1 – 3 Anecdotal evidence for H0
° 1 No evidence
° 1/3 – 1 Anecdotal evidence for H0
° 1/3 – 1/10 Moderate evidence for H0
° 1/10 – 1/30 Strong evidence for H0
° 1/30 – 1/100 Very strong evidence for H0
° < 1/100 Extreme evidence for H0
robinAUC function implements the AUC.
robinAUC(graph=graph, model1=proc$Mean, model2=proc$MeanRandom)## $area1
## [1] 0.1210938
##
## $area2
## [1] 0.2602397The outputs are the area under the two curves.
