# Detect Overlapping Community Based on the Combination of Local Expansion and Label Propagation
## 1 Introduction  
In real-world social networks, the individual behavior is not simply obedient to the collective. Restricted by diverse social relations, they often naturally gather into some social structure, named a community in network research [1]. The community structure is considered to be the functional unit in a network system, as it often accounts for the functionality. Although there is no unified definition of community, many technologies have been developed to detect community, such as modularity maximization [2], random walks [3], particle swarm optimization [4], spectral clustering [5], etc. In the early studies of community detection, much of the focus was on disjoint communities, where individuals have only one membership [6]. However, it is obvious that people's social memberships are not singular. For example, a person could be a member of family while also belonging to other social groups, such as those of friends and colleagues. Moreover, in large-scale social networks, the membership of an individual is more complex.  
In general, the characteristic of nodes being able to belong to multiple communities is called overlap. There are extensive works on detecting overlapping communities [7] for which we provide a more detailed introduction in the next section. However, these studies are still insufficient in some ways. First, the static local optimization goal for detecting overlapping communities is adopted generally, such as fitness function in local expansion [8], optimization for modularity [9] and local Nash equation [10] in the game algorithm. The density, scale and edge distribution of different communities in real social networks are diverse. A static optimization goal can just be applied to detect some communities, andnot to the whole. Second, there is a lot of randomness in the detection. For example, the integration order of a clique has no rules in most algorithms based on clique percolation. The selection of the start position in local expansion is always random. Some updated rules of labels in label propagation are not strict when nodes face the same degree of label influence, which may lead to unstable results. The existence of randomness often means that the algorithm performance can be optimized by some beneficial rules.  
In this paper, an algorithm that combines local expansion and label propagation (\(LELP\)) is proposed. We use an extended local expansion to obtain the immature communities instead of the final communities so that the disadvantages of static optimization are avoided. The seed of local expansion is selected based on the node pair with the highest _closeness_, an index expanded from the Jaccard coefficient [11], which governs the direction of expansion. The immature communities detected by local expansion are pruned to be dense regions in order to conduct label propagation, a speaker-listener-based information propagation process. As the source of propagation, these dense regions can reduce the process of reaching local stability in the early stage of label propagation, making this dynamic process more accurate. We conduct experiments on both synthetic and real-world networks to compare our algorithm with others. The experimental results show that our algorithm generally performs well and is stable in different conditions.