Fig 1.
The communities (red),
(blue) form the strategy profile s = (1, 1, 1, 2, 2, 2, 2).
Fig 2.
The communities (red),
(blue) form the strategy profile q = (1, 1, 2, 2, 2, 2, 1).
Fig 3.
Distance to Nash equilibrium for the Cournot oligopoly and different p values.
The null hypothesis that there is no statistical difference between means of results could not be rejected by using a Wilcoxon sum-rank test (0.05 significance level).
Fig 4.
Evolution of the distance to NE in all cases: again no significant diference between the four p values is observed.
Fig 5.
Duration of the runs (in sec.).
The null hypothesis that differences between mean values are not significant was rejected by using a Wilcoxon sum-rank test with a significance level of 0.05.
Fig 6.
GN zout = 6, 7, 8; boxplots of NMI values for all methods considered (left).
On the right, the black-white matrix represents Wilcoxon h values for each pair of methods considered, numbered in the order they appear in the boxplot. A black square indicates a statistical difference between results. For zout = 8 results obtained with all values of p are significantly better than all other methods considered.
Fig 7.
LFR small, 1000 nodes; boxplots of NMI values for all methods considered (left).
On the right, the black-white matrix represents Wilcoxon h values for each pair of methods considered, numbered in the order they appear in the boxplot. A black square indicates a statistical difference between results.
Fig 8.
Evolution of the mean NMI values over time for the GN zout = 8 set and different p values.
Fig 9.
Real-world networks; boxplots of NMI values for all methods considered (left).
On the right, the black-white matrix represents Wilcoxon h values for each pair of methods considered, numbered in the order they appear in the boxplot. A black square indicates a statistical difference between results.
Table 1.
Fraction of nodes that can improve their payoffs by unilateral deviation in solutions reported by pMNEO for networks with known community structure.
A zero value indicates that reported solutions are indeed Nash equilibria of game . We report average values ± standard deviations and confidence limits of the mean.
Table 2.
Fraction of nodes that can improve their payoffs by unilateral deviation in solutions reported by pMNEO for real world networks with unknown community structure.
A zero value indicates that reported solutions are indeed Nash equilibria of game . We report average values ± standard deviations and confidence limits of the mean also for modularity Q.
Fig 10.
Political books dataset: The ‘real’ community structure is represented.
We can see yellow nodes that have more colored links than yellow, indicating more links in the other two communities.
Fig 11.
Political books dataset: A solution reported by pMNEO with NMI = 0.52 and verified to be a Nash equilibrium; some of the ‘difficult’ nodes are placed in a different community.
Fig 12.
Differences in running times for the four tested p values as percentage of the running time for the p = 100% case for each network set.
Fig 13.
Number of payoff function calls in Alg. 2, 10 individuals.
Notches indicate confidence limits of the mean.
Table 3.
Wilcoxon sum-rank results regarding differences in payoff function calls for the tested p values. An * indicates significant difference.