A rendering of a discourse sheet, showing vector spaces (depicted in rectangles), connected to a network (shown with a series or circles or nodes and connecting lines or edges). Credit: Robert Ghrist
A new mathematical model has been developed to study influence across social networks, according to research published in SIAM Journal on Applied Mathematics. Robert Ghrist, a Ph.D. student, and Jakob Hansen, a topologist, created a framework that tracks how opinions change over time in many scenarios. This includes situations where individuals can deceive others and propaganda agents can be used to drive consensus.

There has been an increase in interest in the development of different models to study behavior over networks. In mathematics, this means studying networks, groups or individuals (known as nodes) and their connections with one another (known as edges). Ghrist says that the current challenge is to develop mathematical frameworks that can include a wider range of features in order to model real-world scenarios.

He says, "There are many people who put out models that have one or two new features. One allows for multiple opinions. Another allows people to selectively lie about their neighbors. And another introduces a propagandist." "We wanted to create a framework that could incorporate all these aspects and still prove theorems about the model's behavior.

Ghrist, Hansen, and others used sheaves, which are topological tools that were previously used in their group. Sheaves, which are algebraic data structures or collections of vector space, are tied to a network and link information with individual edges or nodes. As an example, consider a transportation network. Train stations are nodes, and tracks are the edges. Sheaves can be used to transport information about the network such as passenger counts, on-time departures, etc., but also the connections between stations.

Ghrist says that vector spaces can be different in their dimensions and features, as well as encoding different types and quantities of information. The sheave is made up of vectors placed over each node and edge, with matrices connecting them all. This is collectively a big data structure that floats over your network.

This work was possible because of the inclusion of Laplacian operators as well as diffusion dynamics in the model. In a classic study on opinion dynamics, Laplacians were employed. This showed that individuals with scaled opinions, such as those about the president, would interact with their network neighbors to move their opinion towards a local average.

Ghrist states that if that were a true model, it would mean that we all believe the same things the more we communicate over social media. "But that didn't work and we now have to solve the problem of explaining cleavage and polarization. Our paper proposes a new framework that allows for many interesting twists to the classic situation.

Researchers were able create an opinion dynamics model which could be used to include a variety of parameters and scenarios by incorporating Laplacians. This allows agents to lie about their feelings or give different opinions depending on how they are connected. All this is possible within a mathematical framework that can be tested and proven rigorous.

"The Laplacian for sheaves is the key mathematical innovation that allows the system evolve in a way that can be proven to have reached public consensus. We see that systems can be built from people starting out as neighbors but being in disagreement. The system evolves naturally towards public consensus while people can keep their private opinions.

Ghrist also mentions another interesting finding: one can use "co-homology" to determine when a model is both observable/controllable. This means that it is possible to get a social media network to develop to a specific opinion by designating agents that broadcast propaganda as inputs and other outputs as outputs. These agents can be used to control opinion change and can be observed to track the evolution of opinion. Ghrist says that there are certain conditions where you can target people and control their opinions. He also adds that while these findings are alarming, there is a difference between using these models for studying networks and controlling how ideas spread in real life.

Ghrist's next task is to develop new ways of working with complex sheaves. For example, those that use logical statements rather than numerical values. He says that the mathematical problems associated with this are significant and that his group and he have worked hard to find ways to remove all the mathematics required to include these more complex data types.

Ghrist hopes these tools will be useful for researchers in a range of other fields such as economics and neuroscience due to their flexibility and adaptability. He says that although sheaf theory was first developed in 1950, it has remained largely abstract. "I've been trying for 15 years to adapt ideas from sheaves, sheaf theory into a context that people could use outside of mathematics. I hope this paper pushes the envelope."

Further information: Jakob Hansen et. al., Opinion Dynamics on discourse sheaves, SIAM Journal on Applied Mathematics (2021). Information about the Journal: SIAM Journal on Applied Mathematics Jakob Hansen et. al, Opinion Dynamics on Discourse Sheaves. (2021). DOI: 10.137/20M1341088