Community (Panel (b), Fig. 2). The leading eigenvector algorithm slightly overestimates the number of communities of small networks and the prediction worsens with increasing . Moreover, it GW610742 web underestimates the number of communities in large networks and even the behaviour do not change monotonically with (Panel (c), Fig. 2). The Label propagation algorithm is able to deliver the EPZ004777 web correct number of communities with small values of regardless of the network size. However, in the range 0.3 ?0.6, it underestimates the number of communities and the prediction worsens with increasing network size and . For ?0.6, this algorithm fails to detect any community and all nodes are placed into the same community (Panel (d), Fig. 2). It is apparent that the Mutilevel algorithm constantly underestimates the number of communities and such behaviour worsens with increasing network size and (Panel (e), Fig. 2). In Fig. 2, Panel (f), for 0.4, the Walktrap algorithm delivers the correct number of communities regardless of network sizes, although the change of behaviour at which the prediction is correct depends on system size. For 0.4, this algorithm behaves differently depending on network size: it slightly underestimates the number of communities of small networks and significantly overestimates it for large ones. For ?0.6, the Spinglass algorithm constantly overestimates the number of communities, and its prediction worsens with network size. When ?0.6, it fails and tends to put nodes into a few giant communities (Panel (g), Fig. 2). The Edge betweenness algorithm is able to deliver the correct number of communities for ?0.4 regardless of network size. It overestimates C for ?0.4 and the accuracy of the prediction worsens with increasing network size (Panel (h), Fig. 2). Overall, for ?1/2, Infomap, Leading eigenvector, Multilevel, Spinglass, and Edge betweenness algorithms are able to deliver a reasonable estimator of the number of communities for small networks, while the number of communities obtained by Label propagation and Walktrap algorithms are relatively close to the real value regardless of network size. For ?1/2, all the algorithms are much worse at detecting the correct number of communities, and among all the algorithms, Multilevel, Walktrap, and Spinglass algorithms have better outputs when the network sizes are small. Third, we turn to the real computing time of the algorithms. This measure is usually represented in theoretical estimations as a function of the number of nodes and edges. However, the real computing time may be also affected by the structure of the network. Given the number of nodes and a fixed average degree, we illustrate the computing time as a function of the mixing parameter. The results are shown in Fig. 3 on log-linear scale. Each panel presents the computing time of a given community detection algorithm and it is subdivided in two plots: the lower one depicts the average computing time, while the upper sub-panel contains the standard deviation of the computing time when repeated over 100 different network realisations. Some algorithms barely depend on the mixing parameter. This is not the case for Multilevel, Spinglass, and Edge betweenness algorithms (Panel (e,g,h), Fig. 3). There is a slight dependency for Infomap algorithm that cannot be disregarded (Panel (b), Fig. 3). The decrease of computing time for Infomap, Leading eigenvector, and Label propagation algorithms (Panel (b ), Fig. 3) are accompanied with the.Community (Panel (b), Fig. 2). The leading eigenvector algorithm slightly overestimates the number of communities of small networks and the prediction worsens with increasing . Moreover, it underestimates the number of communities in large networks and even the behaviour do not change monotonically with (Panel (c), Fig. 2). The Label propagation algorithm is able to deliver the correct number of communities with small values of regardless of the network size. However, in the range 0.3 ?0.6, it underestimates the number of communities and the prediction worsens with increasing network size and . For ?0.6, this algorithm fails to detect any community and all nodes are placed into the same community (Panel (d), Fig. 2). It is apparent that the Mutilevel algorithm constantly underestimates the number of communities and such behaviour worsens with increasing network size and (Panel (e), Fig. 2). In Fig. 2, Panel (f), for 0.4, the Walktrap algorithm delivers the correct number of communities regardless of network sizes, although the change of behaviour at which the prediction is correct depends on system size. For 0.4, this algorithm behaves differently depending on network size: it slightly underestimates the number of communities of small networks and significantly overestimates it for large ones. For ?0.6, the Spinglass algorithm constantly overestimates the number of communities, and its prediction worsens with network size. When ?0.6, it fails and tends to put nodes into a few giant communities (Panel (g), Fig. 2). The Edge betweenness algorithm is able to deliver the correct number of communities for ?0.4 regardless of network size. It overestimates C for ?0.4 and the accuracy of the prediction worsens with increasing network size (Panel (h), Fig. 2). Overall, for ?1/2, Infomap, Leading eigenvector, Multilevel, Spinglass, and Edge betweenness algorithms are able to deliver a reasonable estimator of the number of communities for small networks, while the number of communities obtained by Label propagation and Walktrap algorithms are relatively close to the real value regardless of network size. For ?1/2, all the algorithms are much worse at detecting the correct number of communities, and among all the algorithms, Multilevel, Walktrap, and Spinglass algorithms have better outputs when the network sizes are small. Third, we turn to the real computing time of the algorithms. This measure is usually represented in theoretical estimations as a function of the number of nodes and edges. However, the real computing time may be also affected by the structure of the network. Given the number of nodes and a fixed average degree, we illustrate the computing time as a function of the mixing parameter. The results are shown in Fig. 3 on log-linear scale. Each panel presents the computing time of a given community detection algorithm and it is subdivided in two plots: the lower one depicts the average computing time, while the upper sub-panel contains the standard deviation of the computing time when repeated over 100 different network realisations. Some algorithms barely depend on the mixing parameter. This is not the case for Multilevel, Spinglass, and Edge betweenness algorithms (Panel (e,g,h), Fig. 3). There is a slight dependency for Infomap algorithm that cannot be disregarded (Panel (b), Fig. 3). The decrease of computing time for Infomap, Leading eigenvector, and Label propagation algorithms (Panel (b ), Fig. 3) are accompanied with the.
