Dynamics on expanding spaces: modeling the emergence of novelties

Abstract: Novelties are part of our daily lives. We constantly adopt new technologies, conceive new ideas, meet new people, experiment with new situations. Occasionally, we as individuals, in a complicated cognitive and sometimes fortuitous process, come up with something that is not only new to us, but to our entire society so that what is a personal novelty can turn into an innovation at a global level. Innovations occur throughout social, biological and technological systems and, though we perceive them as a very natural ingredient of our human experience, little is known about the processes determining their emergence. Still the statistical occurrence of innovations shows striking regularities that represent a starting point to get a deeper insight in the whole phenomenology. This paper represents a small step in that direction, focusing on reviewing the scientific attempts to effectively model the emergence of the new and its regularities, with an emphasis on more recent contributions: from the plain Simon's model tracing back to the 1950s, to the newest model of Polya's urn with triggering of one novelty by another. What seems to be key in the successful modelling schemes proposed so far is the idea of looking at evolution as a path in a complex space, physical, conceptual, biological, technological, whose structure and topology get continuously reshaped and expanded by the occurrence of the new. Mathematically it is very interesting to look at the consequences of the interplay between the "actual" and the "possible" and this is the aim of this short review.

Dynamics on Expanding Spaces: Modeling the Emergence of Novelties

The paper "Dynamics on expanding spaces: modeling the emergence of novelties" by Vittorio Loreto, Vito D. P. Servedio, Steven H. Strogatz, and Francesca Tria, provides a comprehensive review of models that aim to explain the emergence of novelties and innovations in complex systems. Novelties, which can turn into societal innovations, emerge throughout social, biological, and technological systems. Despite being a critical element of human progress, the processes leading to their emergence remain obscure. The paper reviews various modeling efforts, emphasizing the transition from early Simon’s models to advanced Polya urn models that incorporate the adjacent possible concept.

Review of Modeling Approaches

The paper explores different models, highlighting their ability to capture the statistical regularities associated with novelties. A key commonality among successful models is the treatment of evolution as a path through an expanding and reshaping complex space. This section summarizes the major models discussed:

1. Simon's Model: Simon's model of text generation from the 1950s serves as the foundational framework for modeling frequency distributions. It employs a rich-gets-richer mechanism that implies a linear rate of innovation, but is restricted by its assumption of constant novelty creation rates, which does not align well with empirical observations.
2. Zanette-Montemurro (ZM) Model: This model introduces a time-dependent probability to better approximate the sub-linear growth of new entries, which is more consistent with real data. By incorporating a variable invention rate, this model addresses some of Simon's limitations.
3. Dorogovtsev-Mendes (DM) and Cattuto-Loreto-Pietronero (CLP) Models: These models extend Simon’s framework by including aging effects to account for the relevance of memory. The DM model incorporates a preferential attachment mechanism sensitive to the age of elements while the CLP model includes a temporal kernel impacting the selection bias towards older elements.
4. Sample-space Reducing Model: This hypothetical framework challenges the need for rich-gets-richer dynamics by proposing that possibilities reduce over time; however, it fails to reproduce observed empirical phenomena like sub-linear growth or Zipf’s law slopes greater than one.
5. Hoppe's Urn Model: An extension to Polya urn models that introduces the concept of innovations consistent with Ewens' sampling formula, but it predicts a logarithmic, rather than power-law, novelty growth.
6. Urn Model with Triggering: This recent and most promising model builds on the idea of the adjacent possible where novelties can trigger further innovations. By systematically linking the emergence of novelties to previous events, this model closely mirrors empirical observations from varied domains, displaying sub-linear growth rates consistent with real-world data.

Implications and Future Developments

The paper points out the profound theoretical and practical implications of understanding novelty emergence through these models. Practically, accurate models can enhance prediction and management across ecological, social, and technological landscapes. Theoretically, they foster insights into the structural and dynamic nature of complex systems.

By modeling the interplay between the actual and the possible, insights into innovation processes can lead to more refined predictions about the evolution of different systems. Future research directions might explore the granularity of triggering mechanisms, different scales of individual versus collective innovation processes, and empirical validations across more diverse datasets.

Conclusion

The paper effectively synthesizes decades of research into a coherent narrative that emphasizes how complex systems evolve through the continuous emergence of novelties. The reviewed models, especially the advanced Polya urn systems, provide a quantitative foundation for understanding how innovations emerge, correlate, and propagate, offering valuable avenues for future scientific and practical endeavors in understanding and harnessing the unpredictable nature of innovation.