A Unifying Modularity in Networks

We propose a new modularity criterion in complex networks, called the unifying modularity q which is independent of the number of partitions. It is shown that, for a given network, the relationship between the upper limit of Q and the number of the partitions, k, is sup（Qk） = （k - 1）/k. Since the range of Q for each partition number is inconsistent, we try to extend the concept Q to unifying modularity q, which is independent of the number of partitions. Subsequently, we indicate that it is more accurately to determine the number of partitions by using unifying modularity q than Q.