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Table 3 Calculation of the temporal correlation coefficient C for time series with identical unconnected components of equal size and isolated node

From: Adaption of the temporal correlation coefficient calculation for temporal networks (applied to a real-world pig trade network)

Snapshots 1st calculation step 2nd calculation step 3rd calculation step
\(t_{m} , t_{m + 1}\) \(C_{i = 1} \left( {t_{m} , t_{m + 1} } \right) = \frac{1}{\sqrt 3 }\) Method 1: \(C_{m} = \frac{1}{N}\mathop \sum \nolimits_{i = 1}^{N} C_{i} \left( {t_{m} , t_{m + 1} } \right) \approx 0.32\) Method 2: \(C_{m} = \frac{1}{{\text{max} \left[ {N\left( {t_{m} } \right), N\left( {t_{m + 1} } \right)} \right]}}\mathop \sum \nolimits_{i = 1}^{N} C_{i} \left( {t_{m} , t_{m + 1} } \right) \approx 0.39\) Method 3: \(C_{m} = \frac{1}{{\text{max} \left[ {A\left( {t_{m} } \right), A\left( {t_{m + 1} } \right)} \right]}}\mathop \sum \nolimits_{i = 1}^{N} C_{i} \left( {t_{m} , t_{m + 1} } \right) \approx 0.39\) Method 1: \(C = \frac{1}{M - 1}\mathop \sum \nolimits_{m}^{M - 1} C_{m} \approx 0.56\) Method 2: \(C = \frac{1}{M - 1}\mathop \sum \nolimits_{m}^{M - 1} C_{m} \approx 1.20\) Method 3: \(C = \frac{1}{M - 1}\mathop \sum \nolimits_{m}^{M - 1} C_{m} \approx 0.70\)
\(C_{i = 2} \left( {t_{m} , t_{m + 1} } \right) = 1\)
\(C_{i = 3} \left( {t_{m} , t_{m + 1} } \right) = 0\)
\(C_{i = 4} \left( {t_{m} , t_{m + 1} } \right) = 0\)
\(C_{i = 5} \left( {t_{m} , t_{m + 1} } \right) = 0\)
\(t_{m + 1} , t_{m + 2}\) \(C_{i = 1} \left( {t_{m + 1} , t_{m + 2} } \right) = 1\) Method 1: \(C_{m + 1} = \frac{1}{N}\mathop \sum \nolimits_{i = 1}^{N} C_{i} \left( {t_{m + 1} , t_{m + 2} } \right) = 0.80\) Method 2: \(C_{m + 1} = \frac{1}{{\text{max} \left[ {N\left( {t_{m + 1} } \right), N\left( {t_{m + 2} } \right)} \right]}}\mathop \sum \nolimits_{i = 1}^{N} C_{i} \left( {t_{m + 1} , t_{m + 2} } \right) = 2\) Method 3: \(C_{m + 1} = \frac{1}{{\text{max} \left[ {A\left( {t_{m + 1} } \right), A\left( {t_{m + 2} } \right)} \right]}}\mathop \sum \nolimits_{i = 1}^{N} C_{i} \left( {t_{m + 1} , t_{m + 2} } \right) = 1\)
\(C_{i = 2} \left( {t_{m + 1} , t_{m + 2} } \right) = 1\)
\(C_{i = 3} \left( {t_{m + 1} , t_{m + 2} } \right) = 1\)
\(C_{i = 4} \left( {t_{m + 1} , t_{m + 2} } \right) = 1\)
\(C_{i = 5} \left( {t_{m + 1} , t_{m + 2} } \right) = 0\)