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One of the most popular topics in analyzing complex networks is the detection of its community structure. In this paper, we introduce a new criterion for community detection, called the E‐quality function. The quality of an individual community is defined as a difference between its benefit and its cost, where both are exponential functions of the number of internal edges and the number of external edges, respectively. The obtained optimization problem, maximization of the E‐quality function over all possible partitions of a network, is solved by the variable neighborhood search (VNS)‐based heuristic. Comparison of the new criterion and modularity is performed on the usual test instances from the literature. Experimental results obtained both on artificial and real networks show that the proposed E‐quality function allows detection of the communities existing in the network.

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One of the most important properties of graphs that represents real complex systems is community structure, or clustering, i.e., organizing vertices in cohesive groups with high concentration of edges within individual groups and low concentration of edges between vertices in different groups. In this paper, we analyze Exponential Quality function for network clustering. We consider different classes of artificial networks from literature and analyze whether the maximization of Exponential Quality function tends to merge or split clusters in optimal partition even if they are unambiguously defined. Our theoretical results show that Exponential Quality function detects the expected and reasonable clusters in all classes of instances and the Modularity function does not.

This paper considers the Balanced Multi-Weighted Attribute Set Partitioning (BMWASP) problem which requires finding a partition of a given set of objects with multiple weighted attributes into a certain number of groups so that each attribute is evenly distributed amongst the groups. Our approach is to define an appropriate criterion allowing to compare the degree of deviation from the "perfect balance" for different partitions and then produce the partition that minimizes this criterion. We have proposed a mathematical model for the BMWASP and its mixed-integer linear reformulation. We evaluated its efficiency through a set of computational experiments. To solve instances of larger problem dimensions, we have developed a heuristic method based on a Variable Neighborhood Search (VNS). A local search procedure with efficient fast swap-based local search is implemented in the proposed VNS-based approach. Presented computational results show that the proposed VNS is computationally efficient and quickly reaches all optimal solutions for smaller dimension instances obtained by exact solver and provide high-quality solutions on large-scale problem instances in short CPU times.