On Clustering Coefficients in Complex Networks

TL;DR

It is shown that these two definitions of the global clustering coefficients are strongly inequivalent and may significantly impact the accuracy of the outcome.

Abstract

The clustering coefficient is a valuable tool for understanding the structure of complex networks. It is widely used to analyze social networks, biological networks, and other complex systems. While there is generally a single common definition for the local clustering coefficient, there are two different ways to calculate the global clustering coefficient. The first approach takes the average of the local clustering coefficients for each node in the network. The second one is based on the ratio of closed triplets to all triplets. It is shown that these two definitions of the global clustering coefficients are strongly inequivalent and may significantly impact the accuracy of the outcome.

Figures (5)

• Figure 1: The average node degree per node, $\kappa$, as a function of the temperature ($\gamma =2.1$, $T_c =1$). Black dotted line: $\mu =20$, green dashed line: $\mu =10$, red solid line: $\mu =5$. The blue dash-dotted line: $\mu =2$.
• Figure 2: The local clustering coefficient, $c$, as a function of the energy $\varepsilon$ and temperature $T$ ($\gamma =2.1$, $T_c =1$, $\mu =10$).
• Figure 3: The local clustering coefficient, $c(\varepsilon)$, as a function of the energy $\varepsilon$ ($\gamma =2.1$, $T_c =1$, $T =0.1$). Black dotted line: $\mu =20$, green dashed line: $\mu =10$, red solid line: $\mu =5$, blue dash-dotted line: $\mu =2$.
• Figure 4: The global clustering coefficient, $C_1$, as a function of the temperature ($\gamma =2.1$, $T_c =1$). Black dotted line: $\mu =20$, green dashed line: $\mu =10$, red solid line: $\mu =5$, blue dash-dotted line: $\mu =2$. The inset is a zoom of the main figure with the same lines convention. The dash-dotted red lines present the asymptotic value of the clustering coefficient as $T \rightarrow 0$ (see Eq. \ref{['C1']}).
• Figure 5: The global clustering coefficient, $C_2$, as a function of the temperature ($\gamma =2.1$, $T_c =1$). Black dotted line: $\mu =20$, green dashed line: $\mu =10$, red solid line: $\mu =5$, blue dash-dotted line: $\mu =2$. The inset is a zoom of the main figure with the same lines convention. The dash-dotted red lines present the asymptotic value of the clustering coefficient as $T \rightarrow 0$ (see Eq. \ref{['C2']}).