How to forecast trends amid uncertainty

Who needs a crystal ball when you can use mathematics to calculate whether a product or idea will catch on with the masses?

Featured faculty: Philip Zhang Renyu and Lin Yunduan
Written by Putro Harnowo

Not everything needs to carry meaning, especially on social media. Take 67, a nonsensical expression Gen Alpha uses to confuse adults, for example. While digital platforms can forecast trends by analysing users’ behaviour, humans are inherently unpredictable and easily swayed by others. Algorithms may struggle to keep pace.

Digital anthropologist Brian Solis said, “Social media is about sociology and psychology, not technology.” This can explain many inconsequential trends exploding online. Within their social network, people randomly influence and are influenced by others, even when they don’t actually know each other.

“You may have heard of the six degrees of separation, where everyone in the world is connected through a chain of no more than six acquaintances. It means that everyone is actually more connected than they realise through social networks,” says Lin Yunduan, Assistant Professor of the Department of Decisions, Operations and Technology at the Chinese University of Hong Kong (CUHK) Business School.

Social networks lie at the intersection of many disciplines, from sociology and marketing to even politics. Understanding how ideas spread within communities or whether a new product thrives in the market amid unpredictable human behaviour becomes critical.

Given that people respond to one another in messy, often unpredictable ways, it can be hard to pin down why an idea catches on, or why a product takes off in one community but not another. To cut through that complexity, Professor Lin introduces the fixed-point approximation, a method that distils the back-and-forth of social influence into a clear picture of how these behaviours ultimately settle across the network.

“Imagine it as if a group of people want to schedule a gathering. It starts as an unspecified plan, as anyone may still change their minds. When someone confirms they can make it, their friends may become more likely to join, but when someone who confirmed later cancelled due to a sudden change, this can also ripple through the group,” she says.

“The fixed point refers to a certain level where, after these influences play out, each person’s likelihood of adopting an idea becomes steady and doesn’t change anymore.”

Predicting the trends with a mathematical formula

In a paper titled Nonprogressive diffusion on social networks: Approximation and applications, Professor Lin and Associate Professor Philip Zhang Renyu from the same department collaborate with Zhang Heng of Arizona State University and Max Shen of the University of Hong Kong to develop a deterministic approach to decode interactions within the unpredictable social network.

The fixed-point approximation starts from a few interpretable ingredients, including network structure to know who is connected to whom, intrinsic value or how much each person likes or dislikes the new idea before influences from others, noise distribution or the unpredictable whims that sway a person’s mind, and network effect intensity or how sensitive a person is to being influenced by their connections.

“Basically, we try to find a middle ground to estimate how people will behave under the influence of a network structure.”

Professor Lin Yunduan

This framework offers a way to capture social influence at scale without tracking every possible chain reaction in the network. Instead, it estimates each person’s likelihood of adoption under peer influence. For example, in a small neighbourhood, A has a 90 per cent chance of buying and B has a 30 per cent chance.

The approach is most reliable for people embedded in large, well-connected communities, since no single contact can easily dominate the outcome, and the influence of many peers creates a more stable signal. By contrast, for individuals with very few connections, the prediction can be harder, since one friend’s decision can meaningfully tilt the result, and random factors play a larger role.

To address those outliers, the paper proposes a small add-on step. After producing the main estimate, it focuses on low-connection individuals and generates many plausible scenarios for what their close contacts might do, then averages the results to refine that person’s adoption likelihood.  Professor Lin provides the formula of the framework in a GitHub repository here.

“Basically, we try to find a middle ground to estimate how people will behave under the influence of a network structure,” she says. “The approach does not require simulating every possible ripple through the network, yet it still captures the influence patterns accurately, turning messy, shifting interactions into a clear picture of each person’s likelihood of adopting a new trend.”

The researchers have examined their framework using five actual Facebook networks available in an open-access digital archive. The results show that the framework can accurately measure the likelihood of a new idea adoption, with an average error of less than 3.5 per cent. The graph below illustrates the framework’s efficiency in a small network compared with real-world results.

The researchers also compare it with other models that examine interactions within a network and find that their framework is 70 to 230 times faster than the basic simulations and 23 to 30 times faster than the advanced simulations.

Wider adoption in businesses and communities

A strong suit of fixed-point approximation is its ability to quickly pinpoint the key actors to maximise the adoption of a trend or idea, while accounting for unpredictable factors within a social network. This framework can be used in any practical setting where one’s behaviour impacts others.

“For instance, in a product launch, someone will purchase the new product, and these first purchasers may influence others to follow,” Professor Lin says. “Our framework can help to identify which first purchasers have a high downstream impact more quickly. These purchasers don’t necessarily have large numbers of followers, but are those positioned to spread adoption efficiently through their networks.”

While the framework can operate offline, digital platforms have advantages due to their infrastructure, connectivity and ability to utilise data in real time. Therefore, the framework would enable platforms to respond more quickly to market changes and stay ahead of competitors, while also adjusting their strategies over time to sustain momentum.

Another strong point of fixed-point approximation is its ability to help firms set pricing strategies by accounting for network influence. For retailers, this means the framework can help measure how many customers are likely to purchase a new product at different prices and set realistic sales goals.

Beyond profits, government or community leaders trying to spread an important message or encourage a new behaviour can use this framework to identify key community members whose participation will most effectively encourage others. The framework can also help understand how to seed these messages within the community to achieve widespread adoption.