Decomposition-Based Multiobjective Evolutionary Algorithm for Community Detection in Dynamic Social Networks

Algorithm 1

Framework of the MOEA/D-based community detection algorithm.

Input:

 (i)  The target dynamic network 𝐷𝑁={𝐺₁,𝐺₂,…,𝐺₁,…}.

 (ii)  𝑇: the number of time steps.

 (iii)  𝑁: number of sub problems.

 (iv)    𝑆: neighborhood size.

 (v)    Uniform spread of weight vectors (𝜆¹,1−𝜆¹),…, (𝜆^𝑁,1−𝜆^𝑁).

 (vi)   The maximum number of generations, gen_max.

Output: Sequence of network partitions {𝐶𝑅₁,𝐶𝑅₂,…,𝐶𝑅_𝑇}.

Step  1. Generate an initial clustering 𝐶𝑅₁={𝐶¹₁,…,𝐶^𝑘₁} of the network 𝐺₁

    by optimizing only the community quality objective.

Step  2. For 𝑡=2 to 𝑇

 (1)  Set  EP=∅, gen=0, IP_gen=∅.

 (2)  Initialization:

   (a) Generate an initial population IP₀={𝑥₁,…,𝑥_𝑁} randomly. For each individual 𝑥_𝑖,

      each gene 𝑔_𝑗 randomly takes one of its adjacent nodes of node 𝑗.

   (b) Compute the Euclidean distance between each pair of weight vectors. Then, get the

      neighborhood 𝐵(𝑖)={𝑖₁,…,𝑖_𝑆} for 𝑖th sub problem, where 𝜆^𝑖1,𝜆^𝑖2,…,𝜆^𝑖𝑠 are the 𝑆 closest

      weight vectors to 𝜆^𝑖 (including 𝜆^𝑖 itself).

   (c) Initialize the reference point 𝑧^∗=(𝑧₁,𝑧₂), where 𝑧_𝑖 is the best value found so far

      for objective 𝑓_𝑖.

 (3)  Search for new solutions:

   (a) Assign each sub problem a unique representative 𝑥_𝑖∈𝑋.

   (b) Construct a subpopulation 𝑃_𝑖={𝑥_𝑖1,…,𝑥_𝑖𝑟,…,𝑥_𝑖𝑆} for 𝑖th sub problem, where 𝑥_𝑖𝑟=𝑥_𝑖.

   (c) For each sub problem 𝑖=1 to 𝑁 do

      Randomly select a solution 𝑥_𝑗 from the whole population, and two parents

      consist of 𝑥_𝑖 and 𝑥_𝑗.

      Apply the crossover and mutation operators to 𝑥_𝑖 and 𝑥_𝑗 to generate 𝑦_𝑖 for 𝑃_𝑖.

      Local search operator is employed in some subpopulations, which can be seen in

      Section 3.4 in details.

      Update 𝑃_𝑖, 𝑧^∗, IP and EP with 𝑦_𝑖.

 (4)  Stopping Criteria: If gen ≥ gen_max, then go to (5). Otherwise, gen = gen + 1, go to (3).

 (5) Choose an optimal individual from the EP for the user as shown in Section 3.5.

   Decode the optimal individual to generate the partition 𝐶𝑅_𝑡=(𝐶¹_𝑡,…,𝐶^𝑘_𝑡)

   of the graph 𝐺_𝑡 in 𝑘 connected components; Return the solution

    𝐶𝑅_𝑡=(𝐶¹_𝑡,…,𝐶^𝑘_𝑡); 𝑡=𝑡+1

Step  3.  Output the sequence of network partitions (𝐶𝑅₁,𝐶𝑅₁,…,𝐶𝑅_𝑇).