Minimal partition coverings and generalized dimensions of a complex network

Computing the generalized dimensions Dq of a complex network requires covering the network by a minimal number of “boxes” of size s. We show that the current definition of Dq is ambiguous, since there are in general multiple minimal coverings of size s. We resolve the ambiguity by first computing, for each s, the minimal covering that is summarized by the lexicographically minimal vector x(s). We show that x(s) is unique and easily obtained from any box counting method. The x(s) vectors can then be used to unambiguously compute Dq. Moreover, x(s) is related to the partition function, and the first component of x(s) can be used to compute D∞ without any partition function evaluations. We compare the box counting dimension and D∞ for three networks.