The Continuous Node Degree: a New Measure for Complex Networks

TL;DR

The paper introduces the continuous node degree (C-degree), a new measure for complex networks that accounts for interaction disparity, providing a more nuanced understanding of node connectivity in weighted networks.

Contribution

It proposes the C-degree as a generalization of the traditional degree, capturing interaction disparity and aligning with the degree in uniform cases, with validation on real-world networks.

Findings

C-degree distribution follows a power-law with a steeper decline.

The ratio of C-degree to degree shows a consistent pattern across networks.

C-degree provides a more detailed view of node interactions in weighted networks.

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). 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.