Dynamics of Innovation: From Particles to Ideas*

Innovation dynamics is a widespread phenomenon found across various seemingly unrelated domains, including viral evolution, corporate market share, social networks, and the propagation of ideas. In this study, we propose that innovation processes share analogous characteristics, all regulated by a common mechanism related to the creation and annihilation of particles through collisions.

Our model portrays individuals engaging with and disengaging from innovation trends through a diffusive process encapsulated by a time-dependent Markov chain. 

By calculating the expected number of participants in an innovation trend, we formulate a function that captures the temporal evolution of these trends, transcending specific contextual nuances.

We validate our results using a Recurrent Neural Network (RNN). To achieve this, we introduce a novel training approach that fits ordinary differential equations (ODE) with a loss function that considers both data and its derivative. The RNN fitting process yields confidence values around 90% that substantiate the suitability of our proposed model for describing innovation trends.

Our model also exhibits predictive capabilities, enabling the anticipation of future tendencies with limited observations, particularly beneficial in contexts necessitating trend management. Additionally, our observations reveal that deviations from the model's analytical solution can be attributed to unforeseeable external 

factors influencing innovation dynamics.