Estimating the number of weak balance structures in signed networks

A signed network represents relations of users on social media which can be positive ('like' or 'trust') or negative ('dislike' or 'distrust') in it. The theory of weak balance shows that signed networks tend to cluster to communities. The relations of any two nodes in the same communities are positive while the relations are negative for any two nodes in the different communities. A set of networks are looked as the same structure if they are isomorphic to each other. So, it is a critical problem how many balance structures in signed n-complete networks, because the number of the balance structures and the size of each community can be used to estimate or analyse upper or lower bounds of communities in signed networks. In this paper, we obtain a one-to-one function connecting the number of weak balance structures with the number of solutions of special Diophantine equations. These results are, in turn, used to give a recursive formula for computing the number of weak balance structures and an inductive algorithm for solving n variables of Diophantine equation. Moreover, the complexity of the proposed calculating the number of weak balance structures is a polynomial time of n. In addition, numerical simulations show that the recursive algorithm is an efficient and quick. Finally, we discover the number of communities with the fixed number of nodes displays a normal-like distribution and the average number of communities approximately is nlogn.