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

Büttner, Kathrin; Salau, Jennifer; Krieter, Joachim

doi:10.1186/s40064-016-1811-7

SpringerPlus

Table 4 Calculation of the temporal correlation coefficient C for time series with identical unconnected components of different sizes and isolated nodes

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: \(C_{m} = \frac{1}{N}\mathop \sum \nolimits_{i = 1}^{N} C_{i} \left( {t_{m} , t_{m + 1} } \right) \approx 0.23\) 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: \(C_{m} = \frac{1}{{\text{max} \left[ {A\left( {t_{m} } \right), A\left( {t_{m + 1} } \right)} \right]}}\mathop \sum \nolimits_{i = 1}^{N} C_{i} \left( {t_{m} , t_{m + 1} } \right) \approx 0.32\)	Method 1: \(C = \frac{1}{M - 1}\mathop \sum \nolimits_{m}^{M - 1} C_{m} \approx 0.47\) Method 2: \(C = \frac{1}{M - 1}\mathop \sum \nolimits_{m}^{M - 1} C_{m} \approx 1.03\) Method 3: \(C = \frac{1}{M - 1}\mathop \sum \nolimits_{m}^{M - 1} C_{m} \approx 0.66\)
	\(C_{i = 2} \left( {t_{m} , t_{m + 1} } \right) = 1\)
	\(C_{i = 3} \left( {t_{m} , t_{m + 1} } \right) = 0\)
	\(C_{i = 4} \left( {t_{m} , t_{m + 1} } \right) = 0\)
	\(C_{i = 5} \left( {t_{m} , t_{m + 1} } \right) = 0\)
	\(C_{i = 6} \left( {t_{m} , t_{m + 1} } \right) = 0\)
	\(C_{i = 7} \left( {t_{m} , t_{m + 1} } \right) = 0\)
\(t_{m + 1} , t_{m + 2}\)	\(C_{i = 1} \left( {t_{m + 1} , t_{m + 2} } \right) = 1\)	Method 1: \(C_{m + 1} = \frac{1}{N}\mathop \sum \nolimits_{i = 1}^{N} C_{i} \left( {t_{m + 1} , t_{m + 2} } \right) \approx 0.71\) Method 2: \(C_{m + 1} = \frac{1}{{\text{max} \left[ {N\left( {t_{m + 1} } \right), N\left( {t_{m + 2} } \right)} \right]}}\mathop \sum \nolimits_{i = 1}^{N} C_{i} \left( {t_{m + 1} , t_{m + 2} } \right) \approx 1.67\) Method 3: \(C_{m + 1} = \frac{1}{{\text{max} \left[ {A\left( {t_{m + 1} } \right), A\left( {t_{m + 2} } \right)} \right]}}\mathop \sum \nolimits_{i = 1}^{N} C_{i} \left( {t_{m + 1} , t_{m + 2} } \right) = 1\)
	\(C_{i = 2} \left( {t_{m + 1} , t_{m + 2} } \right) = 1\)
	\(C_{i = 3} \left( {t_{m + 1} , t_{m + 2} } \right) = 0\)
	\(C_{i = 4} \left( {t_{m + 1} , t_{m + 2} } \right) = 0\)
	\(C_{i = 5} \left( {t_{m + 1} , t_{m + 2} } \right) = 1\)
	\(C_{i = 6} \left( {t_{m + 1} , t_{m + 2} } \right) = 1\)
	\(C_{i = 7} \left( {t_{m + 1} , t_{m + 2} } \right) = 1\)

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