Abstract
The measurement error of the network topology caused by missing network data during the collection process is a major concern in analyzing collected network data. It is essential to clarify the error between the properties of an original network and the collected network to provide an accurate analysis of the entire topology. However, the measurement error of the clustering coefficient, which is a fundamental network property, has not been well understood particularly from an analytical perspective. Here we analytically and numerically investigate the measurement error of two types of clustering coefficients, namely, the global clustering coefficient and the network average clustering coefficient, of a network that is randomly missing some proportion of the nodes. First, we derive the expected error of the clustering coefficients of an incomplete network given a set of randomly missing nodes. We analytically show that (i) the global clustering coefficient of the incomplete network has little expected error and that (ii) conversely, the network average clustering coefficient of the incomplete network is underestimated with an expected error that is dependent on a property that is specific to the graph. Then, we verify the analytical claims through numerical simulations using three typical network models, i.e., the Erdős–Rényi model, the Watts–Strogatz model, and the Barabási–Albert model, and the 15 real-world network datasets consisting of five network types. Although the simulation results on the three typical network models suggest that the measurement error of the clustering coefficients on graphs with considerably small clustering coefficients may not behave like the analytical claims, we demonstrate that the simulation results on real-world networks that typically have enough high clustering coefficients sufficiently support our analytical claims. This study facilitates an analytical understanding of the measurement error in network properties due to missing graph data.
Highlights
Network are a major concern in analyzing collected networks
We verify the analytical claims through numerical simulations using the three typical network models, i.e., the Erdős–Rényi m odel[32], the Watts–Strogatz m odel[3], and the Barabási–Albert model[33], and the 15 real-world network datasets consisting of five network types
The simulation results on the Erdős–Rényi model and the Barabási–Albert model suggest that the measurement errors of the clustering coefficients on graphs with considerably low clustering coefficients may not behave as shown in the analytical results, we demonstrate that our analytical claims sufficiently hold for real-world networks that typically have high clustering coefficients
Summary
When researchers discuss the relative magnitude of the clustering coefficients of a collected network, underestimation and overestimation of the measured values can seriously affect the claims of the research If such concerns are present, the qualitative effects of missing data, including overestimation or underestimation, can typically be predicted based on numerical simulations using certain real-world network data. Analytical results for general networks have not been obtained; it is not clear whether little measurement error against randomly missing nodes is the characteristic of the global clustering coefficient or results from the specific type and topology of real-world networks. The second analytical result claims that the network average clustering coefficient of an incomplete network is underestimated with an expected relative error that depends on a property that is specific to the graph. The simulation results on the Erdős–Rényi model and the Barabási–Albert model suggest that the measurement errors of the clustering coefficients on graphs with considerably low clustering coefficients may not behave as shown in the analytical results, we demonstrate that our analytical claims sufficiently hold for real-world networks that typically have high clustering coefficients
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