Table 3 Results justifying the optimal window size (in days) of nine window sizes as determined by the proposed approach applied to four longitudinal networks.

UCI Network
Validation tests	Window Size (days)
Validation tests	1	2	3	4	5	6	7	15	30
1. Best Fit ARIMA	(0,1,1)	(0,1,1)	(0,1,1)	(0,1,0)	(0,1,0)	(0,1,0)	(0,1,0)	(0,1,0)	(0,1,0)
2. Time Series Anomaly (%)	0	0	0	0	0	0	0	3.15	3.57
3. k-means clustering
3.1 Optimal number of clusters	8	8	8	8	9	8	7	8	8
3.2 Minimum Total Within-cluster Variance	0.084	0.127	0.155	0.151	0.121	0.163	0.226	0.170	0.150
MIT Network
1. Best Fit ARIMA	(1,0,0)	(2,0,2)	(0,1,1)	(0,1,1)	(0,1,1)	(0,1,0)	(0,1,0)	(0,1,0)	(0,1,0)
2. Time Series Anomaly (%)	0.85	0.85	1.28	0	0	0	0	0	0
3. k-means clustering
3.1 Optimal number of clusters	1	1	1	2	1	2	2	2	2
3.2 Minimum Total Within-cluster Variance	0.969	1.092	1.119	0.284	1.082	0.256	0.255	0.252	0.225
Email Network
1. Best Fit ARIMA	(1,0,0)	(1,1,1)	(2,1,3)	(1,1,1)	(0,1,1)	(0,1,1)	(0,1,1)	(0,1,0)	(0,1,0)
2. Time Series Anomaly (%)	4.62	4.51	4.40	4.41	3.64	4.35	2.56	0	0
3. k-means clustering
3.1 Optimal number of clusters	6	4	4	4	3	3	3	3	3
3.2 Minimum Total Within-cluster Variance	0.059	0.196	0.236	0.222	0.400	0.357	0.344	0.260	0.214
Facebook Network
1. Best Fit ARIMA	(3,1,3)	(0,1,1)	(4,1,4)	(2,1,2)	(0,1,1)	(2,1,2)	(4,1,0)	(0,1,0)	(0,1,0)
2. Time Series Anomaly (%)	0	0.87	0.85	0.56	0.68	0.83	0.96	0.96	4.00
3. k-means clustering
3.1 Optimal number of clusters	8	8	9	9	9	9	9	8	9
3.2 Minimum Total Within-cluster Variance	0.026	0.081	0.115	0.172	0.232	0.301	0.363	1.032	1.442

The evaluations involved three types of validation tests: the best-fit ARIMA model, the percentage of time series anomalies in time series of the positional dynamicity of SINs of nine different lengths and the total within-cluster variance or total within-cluster sum of squares of the optimal number of clusters determined from k-means clustering on a distribution of actor positional dynamicity values.

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