In our class, we have discussed a simplistic model to analyze to the diffusion of new technologies through a network of people. Our model operates in the framework of a coordination game, where we consider the payoffs to a pair of adjacent nodes if they adopt either technology. From these payoffs, a threshold value can be calculated and if the fraction of your neighbors that adopt the new technology exceeds this value, then you must make the switch. This model has a number of shortcomings, a number of which have been addressed by Frederic Deroian in his article, “Formation of Social Networks and Diffusion of Innovations”. This article can be downloaded from ScienceDirect. (To get it for free, go through library.cornell.edu)

Deroian’s model accounts for:

1. Global network activity - If your circle of friends is stubbornly resistant to change, but the majority of the rest of the world has made the change, then you might still be inclined to change.
2. Node influence - Depending on the amount of influence and popularity a person has, his neighbors may be either drawn to or repulsed by his opinion.
3. Node receptiveness - Indicates whether a person has a greater tendency to follow the crowd or follow his/her own gut.
4. Network evolution strength - As more and more people adopt a new technology, how much pressure is there on the remaining individuals to also make the change? How long does this pressure take to build up?

Based on a node’s influence, its evaluation of a technology lends itself to the formation of “influence links” to people that either share the same opinion or are receptive to the opinion of others. Both our model and Deroian’s model use the concept of a threshold. However, Deroian’s claim is that “critical mass” sets in not when a critical number of people have adopted the technology, but rather when people achieve a certain level of interpersonal influence. At this point, pitchfork bifurcation arises. More specifically, this is an occurrence of a supercritical pitchfork, which is characterized by one stable point (when the number of people with technology A and B are constant) turning into an unstable point (a “tipping point”) and two new stable points arise (nearly everyone ending up with A or B). A mechanical analogy is the buckling of a beam. You can apply a force to the top of the beam, and the beam will be stable until some threshold. After this threshold is surpassed, then the beam will buckle either to the left or to the right (assuming that it has been constrained to only go in those 2 directions). For a more detailed discussion, consult pp.55-57 in “Nonlinear Dynamics and Chaos” by Steven Strogatz.

Deroian’s model, complicated as it is, produces intuitive results. For example, the average time before there is a noticable change in network opinion decreases with increased node receptivity and decreases with increased network evolution strength(see Table 1).  One of the strengths of the model is its ability to capture the transient effects of innovation diffusion. In addition, the model offers insight into how the dynamic phenomenon occurs–interaction between nodes not only conveys information (”what are you using?”), but also influences their relationship (”should I use what you are using?”).