Table 1.
Algorithm 1. Main procedure of multi-objective memetic algorithm for community detection.
Fig 1.
Illustration of group based representation.
Left, a network with a community structure. Right, two possible representations corresponding to the community structure.
Table 2.
Algorithm 2. Population initialization procedure.
Table 3.
Algorithm 3. One-way crossover procedure.
Fig 2.
Illustration of one-way crossover procedure.
Left, a toy network with 7 nodes; Right, node 2 is randomly chosen and the set of nodes with the same label as node 2 is {1,2,3} in Xs.
Fig 3.
Illustration of possible designs of fitness evaluation method for local search procedure.
(a) Region I-IV are four regions divided with respect to node A. Individuals in region I dominate A and in region I,II,IV are not dominated by A. Region I is too small to search, while individuals in region II and IV may move to Region III which are dominated by A after several generations; (b) When constant weight vector ω = (0.5,0.5) is applied, individual population will suffer from diversity problem after several searches; (c) Assuming weight vector ω = (0.5,0.5) selects X1 as initial individual according to random weight vector scheme, then the probability to select right side individual is much higher then select left side as there is only one individual on the right side of X1; (d) Pseudoweight vector ωP deviates from normal line vector ωN.
Fig 4.
Illustration of pseudonormal vector.
ωN is the vector of normal line, ωPN is the pseudonormal vector.
Table 4.
Algorithm 4. Local Search Procedure.
Fig 5.
Best NMI values averaged over 10 runs for MMCD.
(a) Different number of generations; (b) Different maximum sizes of dominant population; (c) Different maximum iteration numbers.
Fig 6.
Best NMI and Q values averaged over 10 runs for MMCD and MOA.
(a) Best NMI values with different generation numbers; (b) Best Q values with different generation numbers.
Table 5.
Parameters of algorithms for GN and LFR benchmark datasets.
Fig 7.
Best NMI values averaged over 10 runs for different algorithms on artificial datasets.
(a) On GN benchmark dataset; (b) On LFR benchmark dataset. Meme-Net and MOGA-Net can’t give outputs within a given time (4 hours).
Table 6.
The basic information of the real-world networks used in this paper.
Table 7.
Parameters of algorithms for real-world datasets.
Table 8.
The maximum, average and standard deviation of best modularity values (Qmax, Qavg, Qstd) obtained over 10 runs on fourteen real-word networks.
Table 9.
The maximum, average and standard deviation of best NMI values (NMImax, NMIavg, NMIstd) obtained over 10 runs on five real-word networks with known true partition.
Fig 8.
Results of MMCD on hierarchical GN benchmark.
(a) Nondominated front of final solution population; (b) Relationships between some objective values and the number of communities.
Fig 9.
Two representative community structures obtained by MMCD on hierarchical GN benchmark.
Fig 10.
Results of MMCD on Karate network.
(a) Nondominated front; (b)-(d) correspond to three solutions labeled as I-III in nondominated front, respectively. Squares and circles represent true communities. Different colors denote communities obtained by MMCD.
Fig 11.
Results of MMCD on Journal network.
(a) Nondominated front; (b)-(d) correspond to three solutions labeled as I-III in nondominated front, respectively. Circles, Squares, diamonds and triangles represent physics, chemistry, biology and ecology journals in true partition, respectively. Different colors denote communities obtained by MMCD.
Fig 12.
Results of MMCD on Dolphins network.
(a) Nondominated front; (b)-(d) correspond to three solutions labeled as I-III in nondominated front, respectively. Squares and circles represent true communities. Different colors denote communities obtained by MMCD.
Fig 13.
Results of MMCD on Football network.
(a) Nondominated front; (b)-(d) correspond to three solutions labeled as I-III in nondominated front, respectively. Groups of nodes gathered together represent true communities. Different colors denote communities obtained by MMCD.