In a recent paper, Aletti et al. (2023) attempted to address this question.

The central concept is based on urns. Indeed, in an effort to model innovative processes, the authors drew on statistical literature that links innovations to Poisson-Dirichlet processes.

This model begins with an urn filled with balls of various colors, representing different innovations. When a ball of a new color is drawn for the first time, a set of balls in new colors is added to the urn. This simulates the stimulation of new ideas triggered by an original innovation. Specifically as stated in the text:

If the color of the drawn ball is new (meaning the innovation is a breakthrough or unrelated to previous innovations), then the "ball" is replaced with additional balls of the same color as well as balls of different, new colors. This illustrates that a breakthrough not only stimulates innovations that are, to varying degrees, based on the same breakthrough but also can inspire completely new ideas. For example, machine learning algorithms have not only spurred the development of generative AI but also the advancement of self-driving cars;

If the color of the extracted ball is old, then a bunch of balls of te old color are added simulating the popularity of that innovation.

Now, let's delve into why these adjustments are essential. Innovation, contrary to what some might think, is far from a haphazard process. Without incorporating the strategies outlined in the bullet points above, the selection of either new or existing colors would be purely random, failing to accurately mirror the genuine dynamics of innovation. These adjustments (see bullet points), serve as mechanisms to skew the likelihood of drawing balls in a manner that reflects the frequency at which an innovation is revisited (e.g., patented) or a novel one is introduced. This approach ensures that the process aligns more closely with the real-world patterns of innovation.

### The novelty of the work (overview)

The paper slightly modifies the model described above introducing a system of interacting urns with triggering. In particular it introduced N urns.

The described model is a simulation of a system with several urns (agents/countries/CPC classes) that interact with one another through a process involving colored balls. The urns represent different entities within the system, and each urn contains a unique set of colored balls. The simulation proceeds in discrete steps, with the following rules applied at each step:

If a ball is drawn from an urn and the color is new to the entire system (first appearance), then it is replaced in the same urn with a number of new balls of the same color (this number is defined by a specific parameter) plus one additional ball of a new color.

If a ball drawn from an urn is of a color that has been seen before in the system (an old color), then it is replaced in the same urn with more balls of that same color (the exact number is again defined by a specific parameter).

When a ball is drawn from one urn and it is new, other urns will also receive a number of new balls of that same color (the number is defined by a parameter).

If a ball drawn from one urn is old, other urns will receive additional balls of that same color (the number is determined by a parameter).

The terms 'new' and 'old' are relative to the system's history of ball colors. A 'new' color means it has not been drawn from any urn before, while an 'old' color means it has been drawn at least once from any urn, although it may still be new to some individual urns.

The authors explain via a metaphor how the model works and may reproduce the process of interacting innovation in real life. They specifically uses the metaphor of the "chinese restaurant with an infinite number of tables".

Each time a customer comes in, there's a chance they'll sit at an empty table or join others at an occupied one. This chance is determined by specific probabilities. The way customers are distributed across the tables at a particular moment represents the arrangement of the system up to that point.

In a more detailed model, every customer is categorized (from 1 to N), and every time a group of customers enters the restaurant, they decide where to sit based on their category.

The probability that a new customer from a certain category will sit at an empty table, or join a table with customers from the same or different categories, depends on previous seating arrangements and certain parameters. These parameters control how likely it is that customers from different categories will share a table and are driven basically by the logic of how balls are reintroduced in urns described in 1., 2., 3. and 4. above. Essentially, the model takes into account not only the number of tables that customers from a category have occupied but also how tables are shared among different categories. Each category influences the others to a degree determined by these parameters.

#### How does this metaphor relates to urns and innovation?!

The urn model and the Chinese restaurant metaphor, at first glance distinct, are deeply intertwined in illustrating the proliferation of innovation. Imagine each urn as a domain of expertise and each colored ball as an idea; when a new idea is introduced, it's like a customer choosing a table in the restaurant—either joining an existing hub of thought (table) or establishing a new one. As ideas are exchanged and transferred between urns, akin to customers moving between tables, they evolve and combine, fostering a rich ecosystem of innovation. This constant flux of ideas, mirrored by the fluidity of patrons in a restaurant, encapsulates the essence of creativity and progress—where the birth of a novel concept can influence and transform the entire landscape of knowledge and practice.

###### In more details...

Imagine the restaurant represents a market or a community where new ideas are being introduced. Each table represents a different type of innovation or a domain of ideas. The customers are individuals or organizations within this market.

Sitting at an Empty Table (Novel Innovation):

When a customer (innovator) sits at an empty table, it's like introducing a brand-new idea into the market that has never been seen before. The probability of them doing this could relate to the market's openness to new concepts or the innovator's tendency to create novel ideas.

Joining an Occupied Table (Adopting or Building on Existing Innovations):

If a customer sits at a table that's already occupied, it's akin to someone adopting or improving upon an existing idea. This could reflect how individuals or companies are influenced by existing trends or successful innovations.

Category of Customers (Specific Sectors or Groups within the Market):

The different categories of customers represent various sectors or groups that are more or less likely to innovate. For instance, tech companies might be more inclined to innovate than companies in more traditional industries.

Probability Parameters (Influence and Interaction Between Sectors):

The parameters that affect the probabilities of where customers sit symbolize the factors that influence how ideas spread within and between sectors. Some sectors might be more influential, causing other sectors to follow their lead. These represent 1.-4. above (i.e. probabilities of sitting in tables by certain category of customers represent, out of metaphor, the probabilities of extracting balls of certain colors based on how balls are introduced to urns following rules 1.-4. above).

Shared Tables (Cross-sectoral Innovation):

Tables shared by customers from multiple categories represent cross-sectoral or interdisciplinary innovations, where an idea from one domain is applied to another, often leading to breakthrough innovations.

By considering how customers from different categories choose their tables, we can understand the dynamics of how innovative ideas might spread through a community or market. Some ideas may remain within one sector, while others cross-pollinate and lead to new developments in different areas. The parameters of interaction reflect the complex network of influences, such as collaboration, competition, and knowledge transfer, that affect the innovation landscape.

### Results (a glimpse)

Results relate to the estimation of the parameters guiding power law guiding the process and can be accessed via the paper directly (I cannot share them here due to copyright).

In short:

Uniform Growth of Innovations: If the interaction matrix among agents is irreducible (meaning that the network is fully connected), then the number of distinct innovations (items) grows proportionally to a power-law function for all agents. This uniform growth is characterized by Heaps' exponent γ∗γ∗, which lies between 0 and 1. Tis means that, at the steady state, all the agents of the network produce innovations for the system at the same rate.

Estimation of the Relative Centrality Scores: A consistent estimation method is introduced for the relative centrality scores of the nodes in the network, using the ratio of growth rates of distinct items (ideas/innovations) between any two agents.

Authors point out that for an irreducible matrix, the growth rate of distinct items (ideas/innovations) for each agent is proportional to the right eigenvector entries of the matrix, and the proportionality involves a finite strictly positive random variable. This relationship also holds pairwise for any two agents.

Total Number of Distinct Items: It is shown that the total number of distinct items (ideas/innovations) observed in the system until a certain time grows as a power of time with the exponent γ∗γ∗.

Discovery Process: For each agent, the number of distinct items (ideas/innovations) discovered by time tt is bounded, and the reciprocal of this number grows as a power of time with the exponent −γ∗−γ∗.

Asymptotic Power Law Behavior: When the number of distinct items (ideas/innovations) discovered by an agent exhibits power law behavior, it is indicative of the Heaps' exponent being equal to γ∗γ∗.

Theorem 3.2: It suggests that if the matrix representing interactions is irreducible and the network is strongly connected, then as time progresses, for each observed item, the number of times it has been adopted by an agent grows linearly and the distribution of items among agents becomes uniform. This is akin to having a uniform composition of items across different categories, comparable to customers being uniformly distributed across tables in the Chinese restaurant metaphor.

In summary, the results indicate that in a strongly connected system, the spread of innovations follows a predictable pattern, where the growth of distinct innovations is uniform across all agents and the distribution of items (ideas/innovations) becomes even over time. The centrality of agents within the network can be inferred from the growth rates of their innovations, and overall, the system tends towards a state where each innovation is equally likely to be adopted by any agent.

Recently, the authors reached out to me to replicate their results in a real-world scenario using patent data. I am excited to embark on this new project, which aims to reveal the underlying mechanisms that drive the flourishing of innovations and ideas, and to determine if the developed theory accurately reflects our data. Stay tuned!

References

Aletti, Giacomo, Irene Crimaldi, and Andrea Ghiglietti. "Interacting innovation processes." Scientific Reports 13, no. 1 (2023): 17187.

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