Innovation Kinetics

Sometime in Fall 2023, Yanis Yorstos (Dean of USC Viterbi) introduced me to an epidemiology study he had written that used Arrhenius kinetics to model disease transfer. Arrhenius kinetics (used to describe chemical reaction rates) is one of many advances in physics that emerged in the second half of the 19th century thanks to statistical mechanics. Simple yet powerful, Arrhenius kinetics uses the statistical behavior of particles in collision to explain chemical reactions. Only particles that meet in precisely the right orientation and with sufficient momentum react. Others miss entirely or bounce off without consequence. The rare interactions aggregated across trillions upon trillions of molecular collisions powers chemistry as we know it.

Reading texts on innovation, technology, and economics recently, I found myself considering this analogy in a new light. Michael Porter's Competitive Advantage of Nations asks why some regions repeatedly birth innovation and why inefficient firms persist despite external pressure from actors who—in key measures—perform better? Nathan Rosenberg argues in Exploring The Black Box: Technology, Economics, and History that innovation emerges from a confluence of interactions between diverse actors. These channels of information-sharing cross-link ideas in novel ways and spawn new ways of thinking—that is, innovation. Porter emphasizes the logistical advantages of co-located "hubs"—places where convenience and efficient resource-sharing accelerate progress. These observations, taken together, point to a common denominator: interaction is central to the innovative process just as it is for chemistry.

So I spent a few hours expanding on the idea.

In chemical solutions, two reactants must collide. The rate equation is governed by:

\[ k = Ae^{-E_a/RT} \]

where \(k\) is the rate constant, \(A\) is the Arrhenius constant, \(E_a\) is activation energy, \(R\) is the gas constant, and \(T\) is temperature. The reaction rate itself follows:

\[ r = k[A]^a[B]^b \]

Disease transfer can be mapped onto this framework. In the classical SIR (Susceptible-Infected-Recovered) model:

\[ S + I \xrightarrow{k_s} 2I \quad \Rightarrow \quad r_s = -k_s[S][I] \]

\[ I \xrightarrow{\rho} R \quad \Rightarrow \quad r_i = k_s[S][I] - \rho[I] \]

\[ r_R = \rho[I] \]

Now consider innovation. There is no straightforward statistical framework for innovation comparable to disease kinetics. Populations are smaller and the interaciton mechanisms far less straighforward. But the analogy is cool, so let's see what happens.

The variables worth tracking should fall into three groups: independent (population density, physical infrastructure, existing knowledge reserves), dependent (knowledge transfer, action taken), and mediating (what conditions lead to innovation? This is what we need to define).

One crucial observation is that physical infrastructure matters in a way it doesn't for kinetics or SIR models. New cancer treatments come out of labs, not garages. This suggests that collisions between free particles might not be right; innovation might follow logic closer to surface chemistry where reactions happen when a molecule collides with the right part of a surface.

Innovation's location dependency is reinforced by the successes of many (not all) industrial parks and the tendency for people to form domain-specific hubs. In either case, we should expect innovation to depend on both overall population density and the relative density of sub-groups with certain characteristics, say, possessing knowledge of a particular kind. Sector hubs have high concentrations of a key species. If we map sectors to cities:

Finance ↔ NYC  |  Media ↔ LA
Software ↔ Silicon Valley  |  Policy ↔ DC
Auto ↔ Detroit  |  Petroleum ↔ Houston

Then we can write the reaction network where species X collides with site S:

\[ S_i + X_j \xrightarrow{k_{ij}} S_i X_j \]

But this strays from the core thesis that innovation arises from human-human interaction. In that case, I maybe erred in thinking that surface could let us discount free particle collisions entirely.

So a system must depend on both human-human and human-infrastructure collisions. There must also be an attractive force pulling species to their home sector (or towards similar species) and keeping people in a hub that are already there. But it can't be so strong that it will ever draw in all species as time goes to infinity. There are jobs for engineers in DC and for policy wonks in Houston. Could there be some sort of universal baseline population for each species?

The impact of an individual depends on local species densities. An engineer in DC has more impact than a policy worker in SF, because engineers are probably rarer in DC than policy professionals are in SF. This places value on interdisciplinary innovation since new ideas are more likely to come from unfamiliar ways of thinking. Still, same species interactions continuously beget innovation by making up for low probabilities with many interactions.

\[ A + B \to A^* \]

\[ A^* + A \to A^* \]

This lets an initial innovation (A*) catalyze further innovation. [Update: I don't think this is right]