A Simple Model of Fads and Cascading Failures

Maximizing the Spread of Influence through a Social Network (click on “Full Text: PDF” to view article)

Our class discussion of information cascades has mostly centered around a scenario in which individuals receive information about a particular choice and make a decision based on their information as well as other individuals’ choices. While this approach allows for reasonable predictions regarding whether a cascade will occur, it makes the severely limiting assumption that the population is entirely homogeneous. Realistically, there is nearly always some variety among a population, and this variety can influence the possibility and structure of an information cascade.

Duncan J. Watts’ paper “A Simple Model of Fads and Cascading Failures” examines a few ways in which a diverse population might influence the probability of a cascade, while “Maximizing the Spread of Influence through a Social Network” by Kempe, Kleinberg, and Tardos explores various ways in which this diversity might be modeled in order to build analysis algorithms. (Please note: my decision to reference a paper written by Professor Kleinberg and two other Cornell professors was purely incidental; my search criteria did not include any reference to either of the course professors, nor did they include Cornell University.) Both papers analyze the population in terms of an undirected connected graph in which a node represents an individual and an edge connects any two nodes between which information is exchanged. Watts postulates that each node has a (randomly assigned) threshold such that the node does not become part of the cascade unless at least a certain percentage of its neighbors have done so already. Nodes with a relatively low threshold value are termed “vulnerable,” and Watts notes that the success of a cascade can depend largely on the prevalence of such nodes, since a large percentage of “vulnerable” nodes increases the likelihood of overcoming nodes with high threshold values. He also points out that the problem is significantly exacerbated by clustering individuals into groups in which there is frequent interaction, while inter-group interaction occurs with lower frequency. This creates a situation in which a trend will likely propagate quickly within a group, but inter-group propagation may be much slower.

The paper written by Kempe, Kleinberg, and Tardos also suggests the notion of node threshold values and suggests several other useful schemes. These include examining the “influence” of an initially affected set of nodes in terms of the number of affected nodes at the end of the cascade sequence, assigning a “weight” to each node in the initial set based on its effect on the final outcome, or assigning a probability value to each node in the graph and using this value to determine which neighbors of a particular node may be affected. The latter idea is particularly interesting. In the authors’ words, “When node v first becomes active, […] it is given a single chance to activate each currently inactive neighbor w; it succeeds with a probability [p …] independently of the history thus far” (2). This lends itself to a model in which some function could be created to assign values of p to each node based on the particular dynamics of the population in question. For example, one might propose a function where the value of p for a particular node is directly related to the number of that node’s neighbors, reasoning that a node with many connections is a naturally outgoing individual who is likely to influence others (perhaps a “Maven,” “Connector,” or “Salesman,” as described in Gladwell’s The Tipping Point). By deriving an appropriate function to assign the p-values over the network, one can create a model for an extremely large set of scenarios.