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Community detection has drawn significant attention as new media generates big data every day. To provide statistical testing procedures for community detection in social networks, a scanning method has been developed based on the likelihood of Poisson random graph. However, the scan statistics did not consider detecting communities of the attributes with power-law distribution. Power-law distribution, generally followed by network attributes, is conspicuous in many scientific situations. This paper aims at extending the scanning method to analyze a social network in which attributes follow power-law distribution. Besides the theoretical construction, simulation studies are performed to verify the feasibility of the proposed method, and an authorship network is used to demonstrate the proposed method.