insight review articles 268 NATURE | VOL 410 | 8 MARCH 2001 | www.nature.com N etworks are on our minds nowadays. Sometimes we fear their power — and with good reason. On 10 August 1996, a fault in two power lines in Oregon led, through a cascading series of failures, to blackouts in 11 US states and two Canadian provinces, leaving about 7 million customers without power for up to 16 hours 1 . 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 acquaintances 2 . And during the height of the Lewinsky scandal, the New York Times printed a diagram 3 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 webs 4,5 , electrical power grids, cellular and metabolic networks 6–9 , the World-Wide Web 10 , the Internet backbone 11 , the neural network of the nematode worm Caenorhabditis elegans 12 , telephone call graphs 13 , coauthor- ship and citation networks of scientists 14–16 , and the quintessential ‘old-boy’ network, the overlapping boards of directors of the largest companies in the United States 17 (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. Wilson 18 , “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/dtǃv(x), where x(t)ǃ(x 1 (t), …, x n (t)) is a vector of state variables, t is time, and v(x)ǃ(v 1 (x), …, v n (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 x 1 , …, x n . As the system evolves, x(t) flows through state space, guided by the ‘velocity’ field dx/dtǃv(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 concepts 97 © 2001 Macmillan Magazines Ltd