insight review articles
268 NATURE | VOL 410 | 8 MARCH 2001 | www.nature.com
Networks are on our minds nowadays.Sometimes we fear their power — and withgood reason. On 10 August 1996, a fault intwo power lines in Oregon led, through acascading series of failures, to blackouts in 11
US states and two Canadian provinces, leaving about 7
million customers without power for up to 16 hours1. The
Love Bug worm, the worst computer attack to date, spread
over the Internet on 4 May 2000 and inflicted billions of
dollars of damage worldwide.
In our lighter moments we play parlour games about
connectivity. ‘Six degrees of Marlon Brando’ broke out as a
nationwide fad in Germany, as readers of Die Zeit tried to
connect a falafel vendor in Berlin with his favourite actor
through the shortest possible chain of acquaintances2. And
during the height of the Lewinsky scandal, the New York
Times printed a diagram3 of the famous people within ‘six
degrees of Monica’.
Meanwhile scientists have been thinking about
networks too. Empirical studies have shed light on the
topology of food webs4,5, electrical power grids, cellular and
metabolic networks6–9, the World-Wide Web10, the Internet
backbone11, the neural network of the nematode worm
Caenorhabditis elegans12, telephone call graphs13, coauthor-
ship and citation networks of scientists14–16, and the
quintessential ‘old-boy’ network, the overlapping boards of
directors of the largest companies in the United States17
(Fig. 1). These databases are now easily accessible, courtesy
of the Internet. Moreover, the availability of powerful
computers has made it feasible to probe their structure;
until recently, computations involving million-node
networks would have been impossible without specialized
facilities.
Why is network anatomy so important to characterize?
Because structure always affects function. For instance, the
topology of social networks affects the spread of informa-
tion and disease, and the topology of the power grid affects
the robustness and stability of power transmission.
From this perspective, the current interest in networks is
part of a broader movement towards research on complex
systems. In the words of E. O. Wilson18, “The greatest
challenge today, not just in cell biology and ecology but in all
of science, is the accurate and complete description of
complex systems. Scientists have broken down many kinds
of systems. They think they know most of the elements
and forces. The next task is to reassemble them, at least
in mathematical models that capture the key properties of
the entire ensembles.”
But networks are inherently difficult to understand, as
the following list of possible complications illustrates.
1. Structural complexity: the wiring diagram could be an
intricate tangle (Fig. 1).
2. Network evolution: the wiring diagram could change
over time. On the World-Wide Web, pages and links are
created and lost every minute.
3. Connection diversity: the links between nodes could
have different weights, directions and signs. Synapses in
Exploring complex networks
Steven H. Strogatz
Department of Theoretical and Applied Mechanics and Center for Applied Mathematics, 212 Kimball Hall, Cornell University, Ithaca,
New York 14853-1503, USA (e-mail: strogatz@cornell.edu)
The study of networks pervades all of science, from neurobiology to statistical physics. The most basic
issues are structural: how does one characterize the wiring diagram of a food web or the Internet or the
metabolic network of the bacterium Escherichia coli? Are there any unifying principles underlying their
topology? From the perspective of nonlinear dynamics, we would also like to understand how an enormous
network of interacting dynamical systems — be they neurons, power stations or lasers — will behave
collectively, given their individual dynamics and coupling architecture. Researchers are only now beginning
to unravel the structure and dynamics of complex networks.
Dynamical systems can often be modelled by differential
equations dx/dt4v(x), where x(t)4(x1(t), …, xn(t)) is a
vector of state variables, t is time, and v(x)4(v1(x), …,
vn(x)) is a vector of functions that encode the dynamics.
For example, in a chemical reaction, the state variables
represent concentrations. The differential equations
represent the kinetic rate laws, which usually involve
nonlinear functions of the concentrations.
Such nonlinear equations are typically impossible to
solve analytically, but one can gain qualitative insight by
imagining an abstract n-dimensional state space with
axes x1, …, xn. As the system evolves, x(t) flows through
state space, guided by the ‘velocity’ field dx/dt4v(x) like
a speck carried along in a steady, viscous fluid.
Suppose x(t) eventually comes to rest at some point
x*. Then the velocity must be zero there, so we call x* a
fixed point. It corresponds to an equilibrium state of the
physical system being modelled. If all small disturbances
away from x* damp out, x* is called a stable fixed point
— it acts as an attractor for states in its vicinity.
Another long-term possibility is that x(t) flows
towards a closed loop and eventually circulates around it
forever. Such a loop is called a limit cycle. It represents a
self-sustained oscillation of the physical system.
A third possibility is that x(t) might settle onto a
strange attractor, a set of states on which it wanders
forever, never stopping or repeating. Such erratic,
aperiodic motion is considered chaotic if two nearby
states flow away from each other exponentially fast.
Long-term prediction is impossible in a real chaotic
system because of this exponential amplification of small
uncertainties or measurement errors
Box 1
Nonlinear dynamics:
terminology and concepts97
© 2001 Macmillan Magazines Ltd

the nervous system can be strong or weak, inhibitory or
excitatory.
4. Dynamical complexity: the nodes could be nonlinear dynamical
systems. In a gene network or a Josephson junction array, the state
of each node can vary in time in complicated ways.
5. Node diversity: there could be many different kinds of nodes. The
biochemical network that controls cell division in mammals
consists of a bewildering variety of substrates and enzymes6, only
a few of which are shown in Fig. 1c.
6. Meta-complication: the various complications can influence
each other. For example, the present layout of a power grid
depends on how it has grown over the years — a case where
network evolution (2) affects topology (1). When coupled
neurons fire together repeatedly, the connection between them is
strengthened; this is the basis of memory and learning. Here
nodal dynamics (4) affect connection weights (3).
To make progress, different fields have suppressed certain
complications while highlighting others. For instance, in nonlinear
dynamics we have tended to favour simple, nearly identical
dynamical systems coupled together in simple, geometrically regular
ways. Furthermore we usually assume that the network architecture
is static. These simplifications allow us to sidestep any issues of
structural complexity and to concentrate instead on the system’s
potentially formidable dynamics.
Laser arrays provide a concrete example19–24. In the single-mode
approximation, each laser is characterized by its time-dependent
gain, polarization, and the phase and amplitude of its electric field.
These evolve according to four coupled, nonlinear differential
equations. We usually hope the laser will settle down to a stable state,
corresponding to steady emission of light, but periodic pulsations
and even chaotic intensity fluctuations can occur in some cases19.
Now suppose that many identical lasers are arranged side by side in a
regular chain20 or ring21, interacting with their neighbours by evanes-
cent coupling or by overlap of their electric fields22. Will the lasers
lock their phases together spontaneously, or break up into a standing
wave pattern, or beat each other into incoherence? From a technolog-
ical standpoint, self-synchronization would be the most desirable
outcome, because a perfectly coherent array of N lasers would
insight review articles
NATURE | VOL 410 | 8 MARCH 2001 | www.nature.com 269
Replication
Skp1
CBF3
primase
p68
Pol α
P
PCNA
RPA
c y c E
p 2 7P
c y c D
PT286
c d k 4 / 6
P P
p 1 6
c y c A
c y c B
cdk1
PT14Y15
p 2 1
c d c 2 5 A
P
PT380
cdk2
P
cdc25C
PS216
C-TAK1
Chk1
Wee1
P
PT161
Myt1
PY15
Skp2
14-3-3
Cyclin box
DMP1
P
Cks1
P
C1
C2
C2
C3
C4
C5
p 5 7
C7
C6
C8
C9
C10
C11
C12
C13
C15
C16
C17
C18
C19
C20
C21
C22
C23
C24
C25
PS76
C28
C29
S9
Gadd45
C30
CycH Cdk7
P
p 5 3
M d m 2
P
R2
P
C26
R1
C14
p53 box
P
C32
C34
C35 Myc:Max
C36
SL1
P
SL1
C37 C38
C39C40
APC P
Plk1
C37
C41
C41
C41
P26
R11
R 6
C11a
C11b
p19ARF
P42
cyclins, pol α
p19ARF
D
M
P1

E
2F
p53

Fo
s
E20
C31
PCNA
C27
Transcription
C42
C42
?
C14
c
dk
cdk1
p68
p68
R
PA
CycD
p21/Gadd45
C26
CycE
CycA
R2
R2
R11
R10
S9 CycA
CycA
pR
b
E
2F
D
P
1
C11b
p16
C43
CAK
R2
P 4 4
P 4 3
P48
a
b c
Figure 1 Wiring diagrams for complex networks. a, Food web of Little Rock Lake,
Wisconsin, currently the largest food web in the primary literature5. Nodes are
functionally distinct ‘trophic species’ containing all taxa that share the same set of
predators and prey. Height indicates trophic level with mostly phytoplankton at the
bottom and fishes at the top. Cannibalism is shown with self-loops, and omnivory
(feeding on more than one trophic level) is shown by different coloured links to
consumers. (Figure provided by N. D. Martinez). b, New York State electric power grid.
Generators and substations are shown as small blue bars. The lines connecting them
are transmission lines and transformers. Line thickness and colour indicate the
voltage level: red, 765 kV and 500 kV; brown, 345 kV; green, 230 kV; grey, 138 kV
and below. Pink dashed lines are transformers. (Figure provided by J. Thorp and
H. Wang). c, A portion of the molecular interaction map for the regulatory network
that controls the mammalian cell cycle6. Colours indicate different types of
interactions: black, binding interactions and stoichiometric conversions; red,
covalent modifications and gene expression; green, enzyme actions; blue,
stimulations and inhibitions. (Reproduced from Fig. 6a in ref. 6, with permission.
Figure provided by K. Kohn.)
© 2001 Macmillan Magazines Ltd