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This month's topics:

Taming Complexity

"The mathematics of network control--from cell biology to cellphones" is the cover story for the May 12, 2011 Nature. The authors, Yang-Yu Liu, Jean-Jacques Slotine and Albert-László Barabási, present new theoretical results with surprising practical consequences.

small network

The image on the left shows a small but typical network. There are nodes numbered $i = 1, 2, 3, 4$, each in a state measured by the function $x_i(t)$; so the state of the entire network is represented by the time-dependent vector ${\bf x}(t) = (x_1(t), x_2(t), x_3(t), x_4(t))$. The nodes interact: the coefficient $a_{ij}$ measures the strength of the effect of node $j$ on node $i$; $a_{ij}=0$ if the nodes do not interact. Additionally the node $i$ may have input ($b_i = 0,1$) of strength $u_i(t)$ from outside the network. In this model, the interactions are linear: the state $x_1$ obeys the differential equation $dx_1/dt = a_{12}x_2 + a_{13}x_3 + a_{14}x_4 + u_1$, and similarly for the other nodes. In vector form, $$ \frac{\displaystyle d{\bf x}}{\displaystyle dt} = A{\bf x} + B{\bf u}$$ where $A$ is the matrix of the $a_{ij}$, and $B$ is the matrix specifying which nodes have inputs. For our network, $$ A = \left ( \begin{array}{cccc} 0&0&0&0\\ a_{21}&0&0&0\\ a_{31}&0&0&a_{34}\\ a_{41}&0&0&0 \end{array}\right ); ~~~~~B = \left ( \begin{array}{cc} b_1&0\\ 0&b_2\\ 0&0\\ 0&0 \end{array}\right );$$ The network is said to be controllable if the functions $u_i(t)$ can drive the network from any one state ${\bf x} = (x_1, x_2, x_3, x_4)$ to any other. The authors begin their exposition with the statement of Kalman's controllability rank condition: a linear system like ours with $N$ nodes is controllable if the rank of the matrix $C = (B, AB, A^2B, \dots, A^{N-1}B)$ is maximum, i.e. $N$. In our case $$ C = \left ( \begin{array}{cccccccc} b_1&0 & 0 & 0 & 0 &0 &0 &0\\ 0 & b_2 & a_{21}b_1 & 0 & 0 &0 &0 &0\\ 0&0& a_{31}b_1 & 0 & a_{34}a_{41}b_1 &0&0&0\\ 0&0 & a_{41}b_1 &0 & 0 &0 &0 &0 \end{array}\right ).$$ clearly has rank 4, so the system is controllable. Note that without the nonzero $a_{34}$ connection this would not be the case.

The article first describes an efficient algorithm for determining, in an arbitrary network, the minimum number $N_D$ of driver nodes: a set of nodes whose inputs control the entire network. Then they show (they consider this their most important finding) that $N_D$ is determined mainly by the degree distribution, a feature of the topology of the network: the probabilistic analysis of the number of ingoing and outgoing links per node.

A salient feature of the article (picked up in a "News & Views" piece by Maurice Egerstedt in the same issue of Nature), is its analysis of a large collection of real-world networks, including the counter-intuitive finding that "Social networks are much easier to control than biological regulatory networks, in the sense that fewer driver nodes are necessary to fully control them." Furthermore, "engineered networks such as power grids and electronic circuits are overall much easier to control than social networks and those involving gene regulation." Which, as Egerstedt remarks, "may or may not be a good thing, depending on who is trying to control the network."

"Capturing Rogue Waves"

"Rogue Wave Observation in a Water Wave Tank" by A. Chabchoub, N. P. Hoffmann (Hamburg University of Technology) and N. Akhmediev (Australian National University, Canberra) appeared in Physical Review Letters for May 20, 2011. Rogue waves are anomalous, giant waves (typically three times the height of the ambient wave pattern) that seem to appear out of nowhere and to vanish equally mysteriously. The authors give evidence that a rogue wave is in fact an example of Peregrine's soliton, a solution to the nonlinear Schrödinger equation discovered by the mathematician D. Howell Peregrine in 1983; it is localized both in space and in time. Peregrine's solitons are also called "breather waves" because they temporarily suck up energy from their neighbors. The authors managed to program their wave-maker to reproduce the initial conditions for a breather wave in a water tank.

Peregrine soliton

Breather wave (detail). Read from the bottom: as the wave packet progresses down the tank (each record of height/time is taken 1 m. beyond the one below it) the soliton develops, up to its maximum. The almost symmetrical decay of the anomaly would have required a tank more than twice as long to work itself out. Tick-marks are 10 sec. apart. Click for larger and more complete image. Images courtesy of Amin Chabchoub.

theory vs experiment P. soliton

The waveform at its maximum almost exactly matches the shape predicted by Peregrine: the measured surface height at the position of maximum amplitude (solid blue line) compared with the theoretical Peregrine solution (dashed red line).

This work was picked up by Devin Powell for Science News, under the title "Rogue Waves Captured: Re-creating monster swells in a tank helps explain their origin." He quotes Chabchoub: "It's possible that the wind could generate a similar ... perturbation in the open sea."

Tony Phillips
Stony Brook University
tony at

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