Complex network dimension and path counts

Large complex networks occur in many applications of computer science. The complex network zeta function and the graph surface function have been used to characterize these networks and to define a dimension for complex networks. In this work we present three new results related to the complex network dimension. First, we show the relationship of the concept to Kolmogorov complexity. Second, we show how the definition of complex network dimension can be reformulated by defining the concept for a single node, and then defining the complex network dimension as the supremum over all nodes. This makes the concept work better for formally infinite graphs. Third, we study interesting parallels to zeta dimension, a notion originally from number theory which has found connections to theoretical computer science. These parallels lead to a deeper insight into the complex network dimension, e.g., the formulation in terms of the entropy and a theorem relating dimension to connectivity.