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The Continuous Node Degree: a New Measure for Complex Networks

By Sherief Abdallah

Abstract

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). We prove that in general the C-degree reflects how many neighbors are effectively being used (taking interaction disparity into account) and if a node interacts uniformly with its neighbors (no interaction disparity) 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 powerlaw, 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.

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