  A key measure that has been used extensively in analyzing complex networks is
the degree of a node (the number of the node's neighbors). Because of its
discrete nature, when the degree measure was used in analyzing weighted
networks, weights were either ignored or thresholded in order to retain or
disregard an edge. Therefore, despite its popularity, the degree measure fails
to capture the disparity of interaction between a node and its neighbors.
  We introduce in this paper a generalization of the degree measure that
addresses this limitation: the continuous node degree (C-degree). The C-degree
of a node reflects how many neighbors are effectively being used, taking
interaction disparity into account. More importantly, if a node interacts
uniformly with its neighbors (no interaction disparity), we prove that the
C-degree of the node becomes identical to the node's (discrete) degree. We
analyze four real-world weighted networks using the new measure and show that
the C-degree distribution follows the power-law, similar to the traditional
degree distribution, but with steeper decline. We also show that the ratio
between the C-degree and the (discrete) degree follows a pattern that is common
in the four studied networks.

Physics and Society;Data Analysis, Statistics and Probability