Home > Author > Steven H. Strogatz
101 " From a purely mathematical perspective, a power law signifies nothing in particular—it’s just one of many possible kinds of algebraic relationship. But when a physicist sees a power law, his eyes light up. For power laws hint that a system may be organizing itself. They arise at phase transitions, when a system is poised at the brink, teetering between order and chaos. They arise in fractals, when an arbitrarily small piece of a complex shape is a microcosm of the whole. "
― Steven H. Strogatz , Sync: The Emerging Science of Spontaneous Order
102 " With further increases in connectivity, the cascades at first become even larger and more likely, as one might expect, but then—paradoxically—they become larger yet rarer, suddenly vanishing when the network exceeds a critical density of connections. This second tipping point arises because of a dilution effect: When a node has too many neighbors, each of them has too little influence to trigger a toppling on its own. (Remember that each node compares its threshold to the fraction of its neighbors that have tipped, not the absolute number. The more neighbors there are, the less impact any one of them has, in a fractional sense.) "
103 " The challenge now is to decode the underlying meaning of small-world and scale-free architecture, if there is any. In one recent attempt, Solé has observed that electronic circuits tend to be wired in a small-world fashion, and he thinks he knows why. Whether he was analyzing the latest digital microchips or the clunky circuits found in old televisions, he found that all the components were just a few electrical steps from one another, yet they were much more clustered than they would have been in an equivalent random circuit, thanks to the modular design favored by engineering practice. "
104 " Here’s what’s going on, intuitively. Lurking within the network is a connected subset of nodes that Duncan calls the vulnerable cluster. The geometric structure of this cluster—the way it percolates through the rest of the network—is what matters. In marketing language the vulnerable cluster is composed of “early adopters”: not innovators themselves but nodes that are poised and ready to tip, if just one of their neighbors has already toppled. Close to the second tipping point, the vulnerable cluster is spindly and almost invisible—it occupies a very small percentage of the whole network—so the odds of igniting it with a random seed are small. But once ignited, it spreads a slow-burning fire to its neighbors, enough of which pass it on to their neighbors, continuing inexorably until the entire giant component (the vast, interconnected meshwork of nodes that dominates the system) is engulfed in flame. "
105 " Here, it seems to me, is where sync has been uniquely successful. As one of the oldest and most elementary parts of nonlinear science (dealing, as it does, with purely rhythmic units), sync has offered penetrating insights into everything from cardiac arrhythmias to superconductivity, from sleep cycles to the stability of the power grid. It is grounded in rigorous mathematical ideas; it has passed the test of experiment; and it describes and unifies a remarkably wide range of cooperative behavior in living and nonliving matter, at every scale of length from the subatomic to the cosmic. Aside from its importance and intrinsic fascination, I believe that sync also provides a crucial first step for what’s coming next in the study of complex nonlinear systems, where the oscillators are eventually going to be replaced by genes and cells, companies and people. "
106 " Barabási and his team pointed out that scale-free networks also embody a compromise bearing the stamp of natural selection: They are inherently resistant to random failures, yet vulnerable to deliberate attack against their hubs. Given that mutations occur at random, natural selection favors designs that can tolerate haphazard insults. By their very geometry, scale-free networks are robust with respect to random failures, because the vast majority of nodes have few links and are therefore expendable. Unfortunately, this evolutionary design has a downside. When hubs are selectively targeted (something that random mutation could never do), the integrity of the network degrades rapidly—the size of the giant component collapses and the average path length swells, as nodes become isolated, cast adrift on their own little islands. "