In this article, we first propose a graph neural network encoding method for the multiobjective evolutionary algorithm (MOEA) to handle the community detection problem in complex attribute networks. In the graph neural network encoding method, each edge in an attribute network is associated with a continuous variable. Through nonlinear transformation, a continuous valued vector (i.e., a concatenation of the continuous variables associated with the edges) is transferred to a discrete valued community grouping solution. Further, two objective functions for the single-attribute and multiattribute network are proposed to evaluate the attribute homogeneity of the nodes in communities, respectively. Based on the new encoding method and the two objectives, a MOEA based upon NSGA-II, called continuous encoding MOEA, is developed for the transformed community detection problem with continuous decision variables. Experimental results on single-attribute and multiattribute networks with different types show that the developed algorithm performs significantly better than some well-known evolutionary- and nonevolutionary-based algorithms. The fitness landscape analysis verifies that the transformed community detection problems have smoother landscapes than those of the original problems, which justifies the effectiveness of the proposed graph neural network encoding method.

Decomposition-based multi-objective evolutionary algorithms provide a good framework for static multi-objective optimization. Nevertheless, there are few studies on their use in dynamic optimization. To solve dynamic multi-objective optimization problems, this paper integrates the framework into dynamic multi-objective optimization and proposes a memory-enhanced dynamic multi-objective evolutionary algorithm based on $L_p$ decomposition (denoted by dMOEA/D- $L_p$ ). Specifically, dMOEA/D- $L_p$ decomposes a dynamic multi-objective optimization problem into a number of dynamic scalar optimization subproblems and coevolves them simultaneously, where the $L_p$ decomposition method is adopted for decomposition. Meanwhile, a subproblem-based bunchy memory scheme that stores good solutions from old environments and reuses them as necessary is designed to respond to environmental change. Experimental results verify the effectiveness of the $L_p$ decomposition method in dynamic multi-objective optimization. Moreover, the proposed dMOEA/D- $L_p$ achieves better performance than other popular memory-enhanced dynamic multi-objective optimization algorithms.

Detecting overlapping communities of an attribute network is a ubiquitous yet very difficult task, which can be modeled as a discrete optimization problem. Besides the topological structure of the network, node attributes and node overlapping aggravate the difficulty of community detection significantly. In this article, we propose a novel continuous encoding method to convert the discrete-natured detection problem to a continuous one by associating each edge and node attribute in the network with a continuous variable. Based on the encoding, we propose to solve the converted continuous problem by a multiobjective evolutionary algorithm (MOEA) based on decomposition. To find the overlapping nodes, a heuristic based on double-decoding is proposed, which is only with linear complexity. Furthermore, a postprocess community merging method in consideration of node attributes is developed to enhance the homogeneity of nodes in the detected communities. Various synthetic and real-world networks are used to verify the effectiveness of the proposed approach. The experimental results show that the proposed approach performs significantly better than a variety of evolutionary and nonevolutionary methods on most of the benchmark networks.