Of course, since any mathematical model excludes some features and oversimplifies others, we must be careful not to draw overly broad conclusions. History is littered with utopian ideas that looked great on paper but collapsed in practice. Still, mathematical modeling can be quite effective in separating promising ideas from those that are conceptually flawed.

Recently, I led a team of investigators to mathematically model how the structure of a society can encourage or suppress the evolution of cooperative behavior. We represented structure as a network, in which every individual is linked to a certain set of “neighbors.” Links can be strong, as in the case of a close friend or family member, or weak, as for a rarely seen acquaintance.

Individuals can cooperate, helping their neighbors at a cost to themselves, or not. This choice is an example of what game theory calls the “prisoner’s dilemma.” Each individual, if acting in pure self-interest, would choose not to cooperate. Yet cooperation by everyone leads to greater prosperity for all.

The two strategies, cooperation and noncooperation, spread through the network as individuals imitate, or learn from, their neighbors. Individuals are more likely to imitate neighbors who do better in the prisoner’s dilemma. Over time, one strategy will win out: Society will converge to a state where either everyone cooperates or no one does.

An earlier study had examined a simple case of this model, in which each individual has the same number of neighbors. They found that, for cooperation to flourish, the benefit-cost ratio of cooperation must be greater than the number of neighbors per individual. For example, if everyone has exactly five neighbors, cooperation succeeds if it provides at least five times as much benefit as the cost a cooperator pays. But while this is a beautiful result, its applicability is limited: In typical real-world networks, individuals differ widely in their number of neighbors, with some having a great many neighbors and others having very few.

We found a way to calculate whether cooperation is favored on *any *network. The key quantity is the *critical benefit-cost ratio*. If this ratio is three, for example, then any cooperative behavior providing a threefold or greater benefit is favored. We showed how to calculate the critical cost-benefit ratio of any given network by solving a system of linear equations (a mathematically straightforward task). The smaller this ratio, the easier cooperation is to achieve. But for some networks, this ratio is negative, which means that cooperation is *never* favored to evolve.

So which networks are best for promoting cooperation? Cooperation flourishes best when each individual has strong, reciprocated connections to a small number of others. In this case, cooperation spreads locally, along these connections, leading to clusters of cooperators who share benefits with each other. In contrast, if all individuals are equally connected to all others, the benefits of cooperation become diluted in the sea of noncooperators, and the behavior cannot spread. Thus, for cooperation to thrive, a few strong ties are better than a myriad of weak ones.