The maximum, average and standard deviation of best <i>NMI</i> values (<i>NMI</i>_<i>max</i>, <i>NMI</i>_<i>avg</i>, <i>NMI</i>_<i>std</i>) obtained over 10 runs on five real-word networks with known true partition.

<p>— denotes the value is removed. Bold number in each row denotes the best value in corresponding item.</p><p>The maximum, average and standard deviation of best <i>NMI</i> values (<i>NMI</i>_<i>max</i>, <i>NMI</i>_<i>avg</i>, <i>NMI</i>_<i>std</i>) obtained over 10 runs on five real-word networks with known true partition.</p

Illustration of pseudonormal vector.

<p><i>ω</i>^<i>N</i> is the vector of normal line, <i>ω</i>^<i>PN</i> is the pseudonormal vector.</p

Results of MMCD on Dolphins network.

<p>(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.</p

Results of MMCD on Karate network.

<p>(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.</p

Algorithm 2. Population initialization procedure.

<p>Algorithm 2. Population initialization procedure.</p

Algorithm 4. Local Search Procedure.

<p>Algorithm 4. Local Search Procedure.</p

Illustration of possible designs of fitness evaluation method for local search procedure.

<p>(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 <i>ω</i> = (0.5,0.5) is applied, individual population will suffer from diversity problem after several searches; (c) Assuming weight vector <i>ω</i> = (0.5,0.5) selects <i>X</i>₁ 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 <i>X</i>₁; (d) Pseudoweight vector <i>ω</i>^<i>P</i> deviates from normal line vector <i>ω</i>^<i>N</i>.</p

Two representative community structures obtained by MMCD on hierarchical GN benchmark.

<p>Two representative community structures obtained by MMCD on hierarchical GN benchmark.</p

Algorithm 3. One-way crossover procedure.

<p>Algorithm 3. One-way crossover procedure.</p

The maximum, average and standard deviation of best modularity values (<i>Q</i>_<i>max</i>, <i>Q</i>_<i>avg</i>, <i>Q</i>_<i>std</i>) obtained over 10 runs on fourteen real-word networks.

<p>— denotes the value is removed. \ denotes the corresponding algorithms can’t give outputs within a given time (3 hours). Bold number in each row denotes the best value in corresponding item.</p><p>The maximum, average and standard deviation of best modularity values (<i>Q</i>_<i>max</i>, <i>Q</i>_<i>avg</i>, <i>Q</i>_<i>std</i>) obtained over 10 runs on fourteen real-word networks.</p