Cascades and connectivity

Michael Feldstein gave us a nice description of the cascade problem in networks in his last opinion column. In cases of serial decision-making—one person decides to adopt Plan A, then another, and so on—people tend to rely on the decisions of those made before. Thus the result is that every person in a network has decided to adopt Plan A, based on the opinions of their predecessors, even though Plan B may be the optimum plan.

There are numerous instances where a cascade phenomenon is undesirable, and not simply in cases where Plan A is not the best plan. In many instances, following the leader is not the most viable strategy, for example, in cases when being the leader confers significant advantages. By being ahead of the pack, Amazon.com was able to create a sustainable business. But businesses that followed faced a problem Amazon did not—competition from the established leader in the field.

According to Feldstein, the problem of cascades in networks is caused by the nature of the network itself. Because a person relies on the opinions of someone else, their own knowledge is not taken into account, thus causing an “information loss.” Communication from other people in the network overwhelms the information that a person might rely upon on his or her own, and that information therefore never informs the group as a whole.

Not surprisingly, Feldstein’s response is to limit the information flow. “You can do that by simply not giving the participants the chance to hear other people’s answers before they respond to a question.” This prevents one person’s opinion from influencing another, and hence forces the other to rely on local information, thus ensuring that it is entered into the network in the form of a decision to adopt Plan B.

Though Feldstein’s solution would certainly solve the cascade problem, it does so at the cost of adding substantial overhead. “Informational cascades can be prevented but generally only with deliberate and specific intervention,” he writes. But the cost of such intervention impairs the functioning of the network. For example, Feldstein suggests the employment of “active moderators who have the authority to direct the group’s information-sharing activities.” People would be, for example, stepped through a polling process such that they would decide simultaneously whether to adopt Plan A or Plan B, thus ensuring that no person is influenced by the choice of another.

The problem of coordination this raises is staggering. Suppose four people are ready to choose a plan but the fifth is not. Are the first four retarded in their progress, or is a hasty decision forced on the fifth? Moreover, it is not even clear that communications between the people can be managed in such a way—what prevents their use of backchannels (such as telephone calls or after-hours meetings) to circumvent the limitations imposed in the communications network? Further still, some activities are inherently serial. How could we conduct an ongoing activity such as stock-market purchases were all transactions required to be conducted at the same time?

There is a tendency when a network produces less-than-desirable results to want to suggest that the solution may be found in imposing some sort of control or organization over the network as a whole. The presumption is that a centralized authority will be able to manage what are perceived to be coordination problems within the network, such as the timing of decisions made by individuals in the network. But beyond a very simple network, the difficulties involved in controlling the network become greater than the problems being addressed by the network. The likelihood of error is thus increased to the point where the benefits of the network are completely negated.

Though cascade phenomena are usually represented as ‘groupthink’ or ‘herd mind’ (decisions made by individuals based on the influence of other individuals), cascade phenomena are generally better represented as the likelihood of the majority of entities in a network entering into a certain state. Cascade phenomena in electricity networks, for example, have nothing to do with decisions or opinions—they are simply the case where one power station entering an “overload” state as a result of connected stations being in overload. Epidemics of disease are also cascade phenomena, where the cascade is defined as the majority of the entities in the network entering the state of ‘being diseased’ as a result of contact with another, contagious, diseased entity.

When viewed in this manner, the futility of central-state administration becomes apparent. It is simply not possible to direct all power stations to decide to go into overload (or not) at the same time. It is unreasonable to require that all people be exposed to a disease (or not) at the same time. No amount of central control can dictate the cost of wheat, the flow of power, the spread of disease—were it possible it would have been accomplished long ago (certainly, we have had enough authoritarian regimes that have tried, as they say, to make the trains run on time).

Ironically, the employment of a centralized management function exaggerates this, because it decreases the degree of connectedness between the members. Communication between the members is magnified, reinforced, made more direct. The existence of a centralized and controlling agent makes a cascade phenomenon more likely, because any intervention by the central authority is immediately broadcast to every entity and has a disproportionate influence on that entity. If the mechanism deployed in any way favors Plan A over Plan B, it becomes indistinguishable from a directive that Plan A, rather than Plan B, be employed. The presumption is that the central agent is neutral in such matters; such a presumption assumes a complete separation between mechanism and output that is impossible to attain.

If you have no friends, your choices will not be influenced by your friends. But if you have one friend then your friend will have a disproportionate influence on you (the centralized authority model). If you have 100 friends, however, the influence of one friend is once again reduced to the point where that one opinion, by itself, is unlikely to sway your decision. Cascade phenomena, therefore, are caused not simply because a network of connections exists, but because the network that exists is not connected enough.

As Duncan Watts said in “A simple model of global cascades on random networks,” says, “When the network of interpersonal influences is sufficiently sparse, the propagation of cascades is limited by the global connectivity of the network; and when it is sufficiently dense, cascade propagation is limited by the stability of the individual nodes.” Cascade phenomena occur, if you will, in a “sweet spot” where there is enough connectivity to permit influence and the propagation of an idea, but not enough connectivity to provide the stabilizing influence of dissenting opinions.

To return to the practical example set out by Feldstein, let’s look at the case of various managers opting for Plan A or Plan B. In the example, where there is a small number of managers, the problem isn’t simply that one manager is being influenced by the other, the problem is that the influence of the one has a disproportionate influence on the other. But instead of cutting off communication with the other manager—Feldstein’s solution—a more robust response would be to increase the number of managers with whom the first interacts. Thus, when one manager opts for Plan A, it will not automatically cause the other manager to opt for Plan A; the other managers’ inertia (or varied choices) counsels caution, and this allows for the influence of local knowledge to be felt.

When we look at phenomena like the Kerry nomination, we see that the structure of the communication network that conveyed voter intentions was more like the manager model and less like a densely connected network. Voters did not typically obtain information from each other; they obtained information from centralized sources, such as broadcast agencies. These broadcasters, themselves sharply limited in the number of sources of information they could receive (and receiving it mostly from each other) were very quick to exhibit cascade properties, and when transmitted to the population at large, exhibited a disproportionate influence. Were the broadcasters removed from the picture, however, and were voters made aware of each others’ intentions directly, through bilateral rather than mediated communications, the influence of any one voice on the eventual vote would be minimized.

In a similar manner, when people complain about reading the same item over and over on the Web, it is because of the disproportionate influence of a small group of writers who, in essence, propagate ideas that are then replicated on numerous other sites. These influential bloggers are riding the top of what is called the “power curve” of connectivity; they are in the same position as the manager who opted for Plan A. By virtue of being first into the market they attracted the most readers, and their position of having the most readers only made it more likely that other people (all other things being equal) would read them. [All other things are not equal, of course—a power blogger can vault into this position by bringing reputation from other spheres, such as television (Wil Wheaton) or journalism (Andrew Sullivan) or connections (Ann Marie Cox).

Networks that develop dynamically tend to evolve into this formation naturally; power laws are typically limited only by physical constraints. Thus, although the hub airports of the United States have benefited from the tendency of flights to gravitate toward airports already used by other flights, the physical limitations of airport management have ensured that there is an upper limit to airline growth. Similarly, though some proteins exhibit hub behavior in the function of a cell, physical constraints create an upper limit on the number of interactions a protein molecule can undertake. To a certain degree, no such limits exist on the Web; hence a hub like Google exists that is connected to every other Web site, and blogs like Instapundit can have massive numbers of readers. Thus, while the connected nature of the web demonstrates a lesser tendency to cascade phenomena than the centralized model of mass media, the power law ultimately prevails even in this environment.

In my view, this will remain the case so long as access to content on the web is organized by Web site authors. Because of this, it remains difficult to find content on a particular topic, and readers will gravitate to a few sites that tend to cover topics in which they are interested rather than expend the time and effort to find items more precisely matching their interests. By drawing content from a wide variety of sites and organizing these contents into customized content feeds, the range of sites made available to a reader is much greater, decreasing the power law and reducing the probability of cascade phenomena. The shift from Web sites to blogs was, in effect, this sort of transition; the development of specialized RSS feeds will be a significant move in this direction.