# A Geometrical Interpretation of Disruption and Diffusion

### Spencer Nam, 9/22/2016

*Introduction*

Since the formalization effort of the theory of disruption has begun, debating about a possible relationship between disruption and diffusion has intensified. The theory of the ‘diffusion of innovation,’ first proposed by Everett Rogers in 1962, has been viewed as a fundamental descriptive model for how a new solution in a form of a product or a service is adopted by a consumer population. A myriad of empirical data has supported the diffusion model. In it, a new solution will take time to be adopted, initially consumed by just a small number of early adopters but eventually the adoption expanding across the entire market (Figure 1).

**Figure 1. Diffusion of Innovation**In the diffusion model, we generally characterize the ‘innovators’ and ‘early adopters’ as “most demanding” customers who seek the most “advanced” solutions to problems at hand. Consequently, those who are the late adopters, such as those in the categories of the ‘late majority’ and ‘laggards’ are characterized as “least demanding” customers who wait until the very end to adopt already popular and broadly available solutions. The paradigm of diffusion from the most demanding to the least demanding also seems to be well supported by empirical evidence.

*An anomaly observed*Clayton Christensen’s theory of disruptive innovation (or theory of disruption) proposes a new entrant’s solution upending the established incumbent (Figure 2) by entering a market with a solution that is initially inferior to an incumbent but eventually eclipses and replaces the incumbent. Numerous examples observed in real world examples suggest that the phenomena observed and predicted in the market places follow such disruptive dynamics described by Christensen.

**Figure 2: Theory of Disruption**** **However, when the two models are compared, an apparent contradiction arises. The theory of diffusion predicts every new solution targeting the most demanding customers initially while the theory of disruption predicts successful solutions targeting the least demanding customers first. If the two models are true descriptions of the same market phenomenon, why an apparent contradiction? If it is indeed a contradiction, which model is the more correct description of the dynamics?

This anomaly is described in Figure 3. If we assume the y-axis to be some measure of performance and x-axis to be time, a customer demand on performance can be plotted over time as a continuous non-linear curve that asymptotically approaches an upper limit of performance measure. We can also assume that this curve represents the mean value points of some customer distribution that is normal.

As we have explained above, according to the theory of diffusion, any given vector V

_{diffusion}, representing the adoption path of innovation, targets the most demanding early adopters first. Here V

_{diffusion} is located at the lower left corner, indicating its existence at an early phase of the demand evolution with its performance measure at the low end of the range. On the other hand, any given vector V

_{disruption}, representing an adoption path at a later stage of demand evolution, is found at the upper right corner, and its customer penetration is the opposite of V

_{diffusion}. Can this apparent disconnect be explained?

**Figure 3: Diffusion and Disruption Vectors**If we consider these two vectors from the point of view of the performance curve, we are left with unresolved contradiction between two empirically observed data pointing to a viable market entry point from two different directions. However, if we consider these two vectors from the perspective of a third-party observer, we see a possible relationship between the two vectors.

*The anomaly resolved*We propose that two vectors V

_{diffusion} and V

_{disruption} are just diverging images of a single vector from two different perspectives, one from the point of view of the entrant and the other from the point of view of the incumbent, or a third party observer.

In fact, if we consider the nature of the frameworks in use, it is apparent that the diffusion curve is a description of how an entrant observes its evolution in a competitive market. On the other hand, the disruption framework is the incumbent observing how the disruptor behaves within the established market.

The rationale for this viewpoint is even more strengthened by some additional observations. In particular:

- The late stages of customer adoption for the incumbent is actually the early stages of entrants trying to disrupt the market. There is an overlap of the two phenomena.
- Not every entrant turns out to be a disruptor. The incumbent does not see failed attempts by the entrants. The incumbent’s observation is strictly focused on those entrants who are able to emerge as potential disruptors.
- Most innovations are attempts at improving the existing solutions, or following the sustaining trajectories. Disruption by definition is a rather abnormal behavior exhibited by entrants, often unplanned even by the entrant itself.

Applying these observations into our proposal that we are dealing with a single description from two different viewpoints or frames of references, we arrive at the following theorem.

Theorem: V

_{disruption} is the negative quantity of the transposed vector of V

_{diffusion}:

V

_{disruption} = - (V

_{diffusion})

^{T}Sketch of Proof: Proof by construction

Without loss of generality, we resize the performance-time map such that the customer demand curve is symmetric along some straight line K satisfying the equation F = - kx + f, where f is intercept on F (Figure 4). Then for every V

_{diffusion} below K, there exists V

_{disruption} = - (V

_{diffusion})

^{T}.

**Figure 4: Resolution of V**_{disruption} and V_{diffusion}If V

_{diffusion} is the adoption path observed by the entrant itself, - (V

_{disruption})

^{T} is the adoption path of the entrant observed by an incumbent.

In fact, (V

_{diffusion})

^{T}, an adoption path vector of the entrant observed by a third party observer from its or incumbent’s frame of reference, enters the market from the high demand customers, just as every path of diffusion is described.

However, from an incumbent who only sees those innovations that end up challenging the establishment, it only observes an adoption vector that is a negative quantity of (Vdisruption)T. This distinction is important in that solutions that actually become disruptive cannot emerge straight from attacking the most demanding customers at first. In fact, those innovations are sustaining innovations that cannot overcome the force of resistance from the incumbent (whether the force is the actual competitive force of the incumbent or the force of resistance from the market environment). However, some select innovations that initially attempt to penetrate the market from the most demanding customer base eventually evolve to satisfy the customers with low demand. This reversal of the entrant is the disruption vector, V

_{disruption}, observed from the eyes of the incumbent.

But, for the incumbent in the path of adoption, all it sees is the diffusion curve it is riding along. Unless the entrant deliberately engineers its path to fit the disruption vector, from its own reference frame, it cannot tell the difference between sustaining and disruptive innovations. From the third party observer, however, the only phenomenon it sees from the incumbent is the entrant rising from the bottom, the least demanding customers.

*Which view is the preferred view?*Philosophically, it doesn’t really matter whose frame of reference is the best frame to observe what is unfolding. Whether one is sitting in the frame of reference of the incumbent or the entrant (the disruptor), the ride will be spectacular.

But, if the observer would like to understand which companies fail and which companies succeed in competition, the diffusion curves provide little details on such information. All diffusion curve explains is how a particular solution emerges and is adopted over time. Because every adoption follows an S curve path, at the initial stage it cannot tell the difference between a success or a failure.

On the other hand, the viewpoint from the incumbent (or a third party) provides a powerful distinction between an entrant’s solution that will succeed versus the one that will fail. This is because we can take a particular adoption data at a particular time and see whether the adoption path is following (V

_{diffusion})

^{T} or - (V

_{diffusion})

^{T}. If it’s the former, then the entrant will unlikely to survive unless it evolves onto the path of - (V

_{diffusion})

^{T}. But, if the entrant path is in fact on the - (V

_{diffusion})

^{T} path, it is a guarantee (up to execution) that the entrant will succeed.

Given this extremely instructive characteristics provided by the disruption framework, in which an observer can follow evolution of competitive behaviors, it is time to abandon the diffusion curve and view all innovations from disruption curves. In fact,

**Corollary: An entrant that has successfully follows the adoption path V**_{disruption} will upend the incumbent.