Fig 1.
(a) Its topology, in which the darkness of node color is proportional to its degree. (b) Its degree distribution, where the points represent the frequency distribution of degree, the red line denotes the fitting curve with equation P(k) = 0.1948k−0.6959−0.01029, the goodness of fit is 0.9198.
Fig 2.
(a) Its topology, in which the darkness of node color is proportional to its degree. (b) Its degree distribution, where the points represent the frequency distribution of degree, the red line denotes the fitting curve with equation P(k) = 0.2727k−0.4256−0.06745, the goodness of fit is 0.9434.
Table 1.
The summaries of benchmark networks used in this paper.
Fig 3.
The density of driver nodes nD as a function of removal fraction f for ER network under different node attacks.
The black and blue dashed lines are theoretical results obtained from Eqs (24) and (26), respectively. The numerical results are averaged over 100 independent realizations.
Fig 4.
nD as a function of the removal fraction f under different node attacks for (a) WS network and (b) NW network.
The results are averaged over 100 independent realizations.
Fig 5.
(a) Correlation between the node betweenness CB and the node degree k for WS and NW networks. (b) The degree distribution of WS, NW and ER network.
Fig 6.
nD as a function of removal fraction f under different node attacks for the BA scale-free network.
The simulation results are averaged over 100 independent realizations, the analytical result of RA is obtained by Eq (27).
Fig 7.
(a) Node betweenness-degree correlation of the BA network. (b) The degree distribution of the BA network.
Fig 8.
nD as a function of removal fraction f under different node attacks for the (a) USAir97 and (b) Erdos971 network.
The results are averaged over 100 independent runs.
Fig 9.
The node betweenness-degree correlation for the (a) USAir97 and (b) Erdos971 network.
Fig 10.
nD as a function of the removal fraction fe under different edge attacks for the ER random network.
The results are averaged over 100 independent realizations.
Fig 11.
The degree distribution P(k) of the ER random network under different edge removal fraction fe subject to the ID attack.
fe = 0 denotes the initial degree distribution.
Table 2.
The structural characteristics of ER random network, including average degree 〈k〉, degree heterogeneity H and average betweenness centrality 〈B〉, vary with fe subject to ID and IB attacks.
Fig 12.
nD as a function of removal fraction fe subject to different edge attacks for (a) WS network and (b) NW network.
The results are averaged over 100 independent realizations.
Fig 13.
The degree distribution under different fe subject to the ID attack for (a) WS network and (b) NW network.
Table 3.
The average degree 〈k〉 and degree heterogeneity H under different fe subject to IDA for WS and NW network.
Fig 14.
nD as a function of fe under different edge attacks for the BA scale-free network.
The results are averaged over 100 independent realizations.
Fig 15.
The structural characteristics of BA network vary with the edge removal fraction fe.
The characteristics include (a) the degree heterogeneity H, (b) APL and (c) the average betweenness centrality 〈B〉.
Fig 16.
nD as a function of fe subject to different edge attacks for (a) USAir97 network and (b) Erdos971 network.
The results are averaged over 100 independent realizations.
Table 4.
The node attack vulnerability of the real networks analyzed in this paper.
Table 5.
The edge attack vulnerability of real networks analyzed in this paper.