These findings reflect a simplicity that lurks behind complexity, and they hold fundamental importance for modern science. For a century or more, physics has focused principally on systems in “equilibrium”; indeed, virtually everything we know about the properties of ordinary substances, from metals to liquid crystals, from semiconductors to superfluids, rests on equilibrium theories. So do many of the more “exotic” applications of physics to such areas as quantum computation. In sharp contrast, power laws emerge naturally in systems that are decidedly not in equilibrium, such as the Earth’s crust or the Internet, which evolve perpetually and never settle into an unchanging state.
Complexity science has grown out of physicists’ attempts to build theories for this huge and largely unexplored area of “nonequilibrium” systems, with applications in physics, chemistry, and biology, but also in other settings, such as economics.
Firms, for example, also show power-law organization. In the United States, one might naturally assume that the diversity among more than 5 million commercial entities — publicly traded and privately held, as varied as oil companies, auto body shops, and dog-walking services — would make it nearly impossible to generalize about the scope of business. Yet sociologist Robert Axtell of the Brookings Institution, who has studied the empirical distribution of U.S. firms according to their total sales, has found a strikingly simple pattern: The number of firms having total sales S is proportional to 1/S2. In words, firms with sales of $1 million are four times as numerous as those with $2 million, which are four times as numerous as those with $4 million, and so on, right across the board for firms ranging from tiny news agents up to vast multinationals.
Distribution of wealth also conforms to a power law and proves empirically similar across many nations, despite different political orientations and economic foundations. The distribution of cities by population within any country also follows a power law, with the number of cities having N inhabitants being proportional to 1/N2. Or consider the financial markets. Exhaustive statistical studies have recently shown that the likelihood of a fractional change in price of amount f falls off as 1/f4, with this simple regularity holding for virtually all financial markets of all kinds and at all times. This means that for Microsoft, or General Electric, or any other stock, a 1 percent change (up or down) is precisely 16 times as likely as a 2 percent change, which itself is 16 times as likely as a 4 percent change, and so on, right across the board. Price changes may be largely random, yet they also reveal a surprising order.
What Is Normal?
Power laws reflect a pattern of organization and change that is typical for complex systems. Hence, familiarity with their properties offers some clues to the expected character of any complex system — including the modern business environment. Systems that follow power laws defy our intuitive expectations in surprising ways.
If you set 500 salespeople to work independently on the telephones, then, according to the “bell curve” of mathematics, their total weekly sales will almost always fall within a narrow range around some average; large deviations are exceedingly rare. The bell curve reflects a baseline theory for what happens when many independent events contribute to some outcome. It is what makes past averages of everything from corporate earnings to a baseball player’s batting percentage useful as guides to the future.
But the Achilles’ heel of the bell curve is the word independent. When one event influences another, we enter the world of interdependence, where the bell curve does not apply. Consider our salespeople again. In real companies, members of the sales force rarely act completely independently. They are likely to be members of a product group, or a regional body; they meet to agree on goals and compare tactics, and they compete for bonuses. In short, what one person does has the potential to influence the behavior of others, leading to collective swings in sales effectiveness.