The Mathematics of Memory and Growth: Exploring Common Patterns

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Aug 12, 2023
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The Mathematics of Memory and Growth: Exploring Common Patterns
In a world where a person's true demise only occurs when they are forgotten, the importance of continued and shared attention becomes evident. This attention not only shapes our identity as individuals but also influences the structures and priorities of society through the choices we make in what to commemorate and remember. Surprisingly, the pattern of decay in collective memory follows a mathematical law, as revealed by a recent study (C. Candia et al. Nature Hum. Behav. http://doi.org/cxq2; 2018). By analyzing online views of Wikipedia profiles, citations of scientific papers, patents, and online play counts of songs and film trailers, researchers found that collective memory follows a biexponential function.
The biexponential function, characterized by two phases, shows a rapid initial decline in attention followed by a much gentler slope. This decline in attention can be seen across various domains, with music experiencing the shortest and sharpest decline (taking around 6 years) and the online biographies of sports stars enduring the longest (20-30 years). The first phase of decline is dominated by communicative memory, where information is transferred through word-of-mouth. However, the second phase relies more on cultural memory, which is sustained by the physical recording of information. This highlights the importance of recording and accessibility in preserving collective memory.
Interestingly, this pattern of decay in attention can be likened to the words of Chilean writer Pablo Neruda, who wrote, "Love is so short, forgetting is so long." Just as love fades over time, so does our collective memory, but it lingers on in a more enduring and slower manner.
Shifting our focus to the myth of exponential hypergrowth in companies, it becomes clear that growth is not always exponential, but rather quadratic. This phenomenon, known as Growth Decay or Growth Persistence, highlights the natural decline in growth as a company scales, even when everything is going well. As Benjamin Brewster once said, "In theory, theory and practice are the same. In practice, they're not."
While viral products may experience exponential growth due to the enforced need for user invites, word-of-mouth products encourage sharing without such restrictions. However, even if a product experiences exponential growth initially, it cannot sustain this growth indefinitely. The logistic curve, often used to model the growth of a product, shows exponential growth in the early stages when it is far from reaching its natural limit. But as the product approaches around 25% market penetration, the curve flattens into linear growth. This is due to the tension between the exponential force of growth and the diminishing pool of remaining targets. Eventually, the product reaches its "carrying capacity," the fully-saturated market.
To overcome this plateau, companies are willing to invest billions of dollars in expanding the market. This strategy becomes crucial in creating growth beyond raising prices. The concept of Elephant Curves becomes more apparent when growth is plotted as market share, taking into account the varying carrying capacity of the underlying market. In the early stages, the focus should be on winning market share in a specific space, creating the first Elephant Curve. However, as the product matures, it becomes necessary to introduce wholly new products or significant updates to address new markets.
In the pursuit of growth, word-of-mouth-driven growth proves to be more effective and cost-efficient than traditional marketing methods. As John Wanamaker famously said, "Half my advertising is wasted. I just don't know which half." Building word-of-mouth into the product itself becomes a valuable strategy, as it grows automatically alongside the company's expansion.
In light of these insights, here are three actionable pieces of advice:
- 1. Prioritize the recording and accessibility of information to preserve collective memory. This applies not only to historical events but also to personal stories and experiences.
- 2. Understand that growth in companies follows a quadratic pattern, not an exponential one. Plan for the eventual plateau and invest in expanding the market to create sustained growth.
- 3. Focus on incorporating word-of-mouth-driven growth into your product design. Create a product that naturally encourages sharing and referrals, reducing reliance on traditional marketing methods.
In conclusion, the mathematics of memory and growth reveal common patterns that shape our lives and the success of companies. By understanding these patterns, we can better navigate the complexities of collective memory and business growth, ensuring a lasting impact in the world around us.
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