Table 1 Vulnerability, critical nodes (CN) and driver nodes (\(N_D\)) characteristics of natural RNs. C denotes controllability, \(N_{DJ}\) is the number of driver nodes on junctions of a network, N is the total number of nodes in a network, \(N_{CN{\cap N}_{DJ}}\) is the number of common driver and critical nodes, and k is the total number of identified critical nodes in a network.

From: Controllability and heterogeneity of river networks using spectral graph theory approach

Basin’s CAI	\(C=N_D/N\) (%)	Vulnerability	c-value	\(N_{DJ}/N\) (%)	\(N_{CN\cap N_{DJ}}\)(%) \(k=5\)	\(N_{CN\cap N_{DJ}}\)(%) \(k=10\)	\(N_{CN\cap N_{DJ}}\)(%) \(k=N_D\)
0.30	17.07	0.81	3.01	0.61	0	0	0.61
0.34	17.28	0.84	3.26	0	0	0	0
0.48	17.69	0.79	2.07	0.68	0.34	0.34	0.68
0.57	20.83	0.72	4.62	0	0	0	0
0.61	18.03	0.72	2.55	0.41	0	0	0
0.61	17.42	0.74	2.51	0.43	0	0.43	0.43
0.62	16.81	0.88	2.39	0.57	0	0.28	0.57
0.64	16.42	0.79	3.04	0.86	0.43	0.43	0.86
0.66	16.04	0.67	3.02	0	0	0	0
0.69	15.54	0.88	1.81	1.01	0	0	1.01
0.69	14.78	0.76	2.59	0.82	0	0	0.41
0.72	13.04	0.90	3.21	1.52	0	0.43	1.09
0.95	16.81	0.84	3.20	0.85	0	0	0.28
0.96	13.80	0.86	3.19	0.61	0	0	0.61
1.13	14.08	0.79	3.73	0.14	0	0	0.14
1.13	13.04	0.86	2.27	0	0	0	0
1.15	14.37	0.67	1.36	1.45	0.15	0.15	1.45
1.15	12.35	0.82	2.93	0	0	0	0
1.15	17.24	0.92	1.56	1.72	1.72	1.72	1.72
1.16	15.84	0.84	2.53	0.78	0.16	0.31	0.62
1.34	14.79	0.77	1.51	0.97	0	0.19	0.78
1.35	15.20	0.74	2.03	0.49	0.25	0.25	0.25
1.36	16.81	0.85	2.82	0	0	0	0
1.36	14.19	0.81	2.56	0.83	0	0	0.67
1.36	15.38	0.84	2.67	0	0	0	0
1.40	15.60	0.70	2.78	0.46	0	0	0.46
1.42	16.67	0.93	2.93	2.27	0.76	0.76	1.52
1.44	14.69	0.85	3.40	0	0	0.00	0
1.54	12.50	0.94	2.53	2.08	1.04	2.08	2.08
1.56	15.63	0.83	2.33	0.78	0	0.78	0.78
1.64	14.56	0.85	2.12	0	0	0	0
1.67	14.29	0.89	2.70	1.07	0	0.36	1.07
1.68	15.44	0.87	1.38	0.67	0	0	0.67
1.76	17.09	0.85	1.76	0.43	0	0	0
1.84	16.44	0.82	1.11	2.05	0.68	1.37	0.68
1.93	15.14	0.75	2.29	0.40	0	0	0.40
2.03	16.05	0.71	2.42	1.85	0.62	1.23	1.23
2.06	14.96	0.88	2.55	0.88	0	0.59	0.88
2.20	14.49	0.85	3.52	0.72	0	0	0.72
2.20	14.58	0.69	2.49	1.04	0	1.04	1.04
2.20	17.78	0.90	1.81	1.11	0	1.11	0
2.32	13.23	0.78	1.13	0.53	0	0.53	0.53
3.47	19.23	0.72	2.71	0	0	0	0
3.57	10.53	0.92	1.16	0.67	1.32	1.32	0.67
4.82	12.02	0.85	2.53	0	0	0	0
4.84	13.51	0.88	1.30	0	0	0	0
5.42	13.79	0.74	1.60	0.34	0	0.34	0.34
5.73	16.10	0.93	1.92	0	0	0	0
5.79	15.57	0.77	1.91	0	0	0	0
5.80	16.46	0.89	1.90	0.42	0	0	0.42
5.82	13.51	0.81	1.62	0	0	0	0
8.31	10.00	0.90	2.53	3.33	3.33	3.33	3.33
9.77	15.12	0.84	1.77	0.58	0	0.58	0.58
10.11	14.70	0.99	1.48	0	0	0	0
11.19	14.29	0.94	1.57	1.26	0	0	1.26
12.45	13.87	0.99	1.24	1.54	0.77	1.54	1.54

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