The local structure entropy is a new method which is proposed to identify the influential nodes in the complex networks. In this paper a new form of the local structure entropy of the complex networks is proposed based on the Tsallis entropy. The value of the entropic index $q$ will influence the property of the local structure entropy. When the value of $q$ is equal to 0, the nonextensive local structure entropy is degenerated to a new form of the degree centrality. When the value of $q$ is equal to 1, the nonextensive local structure entropy is degenerated to the existing form of the local structure entropy. We also have find a nonextensive threshold value in the nonextensive local structure entropy. When the value of $q$ is bigger than the nonextensive threshold value, change the value of $q$ will has no influence on the property of the local structure entropy, and different complex networks have different nonextensive threshold value. The results in this paper show that the new nonextensive local structure entropy is a generalised of the local structure entropy. It is more reasonable and useful than the existing one.