The Continuous Node Degree: a New Measure for Complex Networks

arXiv:0903.5025v3 [physics.soc-ph] 15 Apr 2009 The Continuous Node Degree: a New Measure for Complex Networks Sherief Abdallah1,2,∗ 1Faculty of Informatics, the British University in Dubai, UAE 2School of Informatics, University of Edinburgh, UK ∗To whom correspondence should be addressed; E-mail: shario@ieee.org. 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 dis- crete 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 ad- dresses 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 identi- cal 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. 1 Introduction Network analysis is an interdisciplinary field of research that spans over biology, chem- istry, computer science, sociology, and others. A key measurement that has been used extensively in analyzing networks is the degree of a node. A node’s degree is the number of edges incident to that node. Intuitively, the degree of a node reflects how connected the node is. This simple measure (along with other network measures) al- lowed the discovery of universal patterns in networks, such as the power law of the degree distribution [3, 8]. One of the limitations of the degree measure is that it ignores any disparity in the interaction between a node and its neighbors. In other words, the degree measure assumes uniform interaction across each node’s neighbors. This can result in giving an incorrect perception of the effective node degree. For example, a person may have 10 or more acquaintances but mainly interacts with only two of them (friends). Should that 1 person be considered 2 times more connected than a person with only 5 acquaintances but also interacting primarily with two of them? Several network measures were proposed to analyze weighted networks [2, 4, 5, 10], where an edge’s weight quantifies the amount of interaction over the edge. How- ever, none of the previously developed measures is a proper generalization of the de- gree measure. A proper generalization of the degree measure that captures the dis- parity of interactions needs to satisfy three properties. The first property is preserving the maximum traditional degree: if all weights incident to a node are equal (maximum utilization of neighbors), then the generalized degree is maximum and should be equal to the traditional (discrete) degree. The second property is preserving the minimum traditional degree: if all edges incident to a node have weights that are almost zero except one edge that has a weight much larger than zero (the node interacts primarily with one neighbor) then the generalized degree should be very close to 1. The third and final property is the consistent handling of disparity: the partial order imposed by the generalized degree on any two nodes needs to be consistent with the previous two properties. Intuitively, this means the more equal the weights are, the higher their gen- eralized degree should be. We formalize these properties into axioms in the following section. A generalization of the degree measure is significant because it bridges the gap be- tween the extensive research made using the degree (which ignored weights) and the research on weighted networks. Furthermore, it allows more accurate analysis of the networks that were previously analyzed using the degree measure. For example, it is known that the degree distribution of the Internet follows the power law [8]. How- ever, if one takes the disparity of interactions into account, does the effective degree distribution of the Internet still follow a power law? We introduce in this paper a new measure for analyzing weighted networks: the continuous degree (C-degree). What sets our measure apart from previous work is that it is a continuous generalization of the degree measure that captures the disparity of interaction. In particular, we prove that if every node interacts with all its neighbors equally, then the C-degree becomes identical to traditional (discrete) degree measure of the same node. However, if there is a disparity in

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