CHIN.PHYS.LETT. Vol. 25, No. 10 (2008) 3822
Cascading Failures of Complex Networks Based on Two-Step Degree
∗
WU Zhi-Hai(
Ç
£
°
)
∗∗
, FANG Hua-Jing(
u
®
)
Department of Control Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074
(Received 7 March 2008)
We propose a new concept, two-step degree. Defining it as the capacity of a node of complex networks, we
establish a novel capacity–load model of cascading failures of complex networks where the capacity of nodes
decreases during the process of cascading failures. For scale-free networks, we find that the average two-step
degree increases with the increase of the heterogeneity of the degree distribution, showing that the average two-
step degree can be used for measuring the heterogeneity of the degree distribution of complex networks. In
addition, under the condition that the average degree of a node is given, we can design a scale-free network with
the optimal robustness to random failures by maximizing the average two-step degree.
PACS: 89. 75. Hc, 89. 75. Fb
Complex networks such as the Internet, the elec-
trical power grid, and the transportation network, are
an essential part of a modern society. The security of
such a network under random failures or intentional
attacks is of a great concern.
[1−10]
A capacity–load
model of cascading failures of complex networks, due
to the ability to describe the electric power system
blackout and failure phenomena of other complex sys-
tems, has been quite studied.
[11−15]
However, in this
model, researchers assume that the capacity of a node
is invariant during the process of cascading failures.
For example, Motter et al.
[12]
assume that the capac-
ity C
i
of node i is proportional to its initial load L
i
,
C
i
= (1 + δ)L
i
, i = 1, 2, · · · , N, (1)
where the constant δ > 0 is the tolerance parameter,
and N is the initial number of nodes. This assump-
tion does not agree with the fact that in many complex
networks the capacity of a node is time dependent. In
this Letter, we propose a new concept of complex net-
works, two-step degree of a node, and using it as the
capacity of a node we establish a novel capacity–load
model where the capacity of partial or all nodes de-
creases with the transmission of cascading failures.
The heterogeneity of complex networks, i.e., the
diversity of the degree distribution leads to many im-
portant and unique properties, such as the robustness
to node failures, synchronization ability and so on.
[1]
A complex network is thought homogeneous, if its
nodes have approximately the same degree equivalent
to the average degree. In contrast, a complex network
is thought heterogeneous, if its numerous nodes have
small degree but a few nodes have large degree. Con-
sequently, the measure of the heterogeneity is impor-
tant to research the topology and functions of complex
networks. Wang et al.
[10]
provide an average measure
of the heterogeneity of complex networks using the
entropy of the degree distribution. However, the mea-
sure only describes the diversity of the degree distri-
bution but does not contain the information about the
network topology.
With the help of the two-step degree of a node, we
can compute the average two-step degree of complex
networks that will be defined later. Especially, for
scale-free networks,
[16]
we can gain the average two-
step degree using the continuous approximation. We
find that the average two-step degree increases with
the increase of the heterogeneity of the degree distri-
bution, showing that the average two-step degree can
be used for measuring the heterogeneity of the degree
distribution of scale-free networks. Lastly, based on
the relationship among the average two-step degree,
the heterogeneity of the degree distribution and the
robustness of complex networks against random fail-
ures, a scale-free network with the optimal robustness
to random failures can be designed when the average
degree of a node is given.
For a complex network G = (V, E, A) with the set
of nodes V = (v
1
, · · · , v
N
), the set of edges E ⊆ V ×V
and the adjacency matrix A = (a
i,j
)
N×N
defined by
a
i,j
=
1, (v
i
, v
j
) ∈ E,
0, (v
i
, v
j
) /∈ E,
(2)
we define the two-step degree D
i
of node v
i
by
D
i
=
X
v
j
∈N
i
d
j
, (3)
where the set of neighbours N
i
of node v
i
is defined
by N
i
=
v
w
∈ V : v
i
∈ V, (v
w
, v
i
) ∈ E
and d
j
is
the degree of node v
j
, i.e., |N
j
|. We can also compute
D
i
, i = 1, · · · , N by
Ad = D, (4)
∗
Supported by the National Natural Science Foundation of China under Grant Nos 60574088 and 60874053.
∗∗
Email: wuzhihai@smail.hust.edu.cn
c
2008 Chinese Physical Society and IOP Publishing Ltd
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