摘要：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.
其他摘要：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.
