Research published in the SIAM Journal on Applied Mathematics describes a new mathematical model for studying influence across social networks. Using tools from the field of topology, Robert Ghrist and Ph.D. graduate Jakob Hansen developed a framework to track how opinions change over time in a wide range of scenarios, including ones where individuals can use deceptive behaviors and propaganda agents can drive a group’s consensus.
With the rise of social media platforms, there has been increased interest in developing different types of models to study behavior over networks; in mathematics, that means studying networks, groups of individuals, known as nodes, and their connections to one another, known as edges. The current challenge, says Ghrist, is developing mathematical frameworks that can incorporate a broader range of features to help model more real-world types of scenarios.
“There are a lot of people putting out models that have one or or two novel features; one allows for multiple opinions, another allows people to selectively lie to their neighbors, and another has the introduction of a propagandist,” he says. “What we were looking to do was come up with a framework that can incorporate all of these different aspects, yet still be able to prove rigorous theorems about how the model behaves.”
To do this, Ghrist and Hansen used topological tools called sheaves, previously used in their group. Sheaves are algebraic data structures, or collections of vector spaces, that are tethered to a network and link information to individual nodes or edges. Using a transportation network as an illustrative example, where train stations are nodes and the tracks are the edges, sheaves are used to carry information about the network, such as passenger counts or the number of on-time departures, not only for specific stations but also on the connections between stations.
“These vector spaces can have different features and dimensions, and they can encode different quantities and types of information,” says Ghrist. “So the sheave consists of collections of vectors over top of each node and each edge with matrices that connect them all together. Collectively, this is a big data structure floating over top of your network.”
One of the core mathematical concepts that enabled this work was the incorporation of Laplacian operators and diffusion dynamics into the model. Laplacians were used in a classic study of opinion dynamics, which found that, for individuals with a scaled opinion on a specific topic, such as their opinion of the president from 1 to 10, interacting with their neighbors in the network would move their opinion towards a local average.
“If that were an accurate model, what that would mean is that the more we talk to each other over social media the more we all come to believe the same thing,” Ghrist says. “That didn't work out so well and leads us to the problem of explaining cleavage or polarization. So what we do in our paper is build this new framework that can accommodate all kinds of interesting twists on the classical situation.”
By incorporating Laplacians into their “discourse shaves,” the researchers were able to create an opinion dynamics model that was incredibly flexible and able to incorporate a wide variety of scenarios, parameters, and features. This includes the ability to have agents who can lie about their feelings on a specific topic or tell different opinions to others depending on how they are connected, all within a rigorous and testable mathematical framework.
“The key mathematical innovation here is a Laplacian for sheaves that allows the system to evolve in such a way that you can prove results about public consensus. What we see when we run certain examples is that you can have systems where people start off being neighbors and very much in disagreement, and the system naturally evolves towards a public agreement while people can maintain their private opinions,” says Ghrist.
Another interesting finding, Ghrist says, is how, using “co-homology,” one can characterize when this model is both observable and controllable, meaning that one can get a social network to evolve to a particular opinion by designating specific agents as inputs, ones that broadcast propaganda, and others as outputs, ones that are observed to track opinion change. “There are conditions under which you can designate a set of target individuals and control their opinions by seeding the network with propaganda and letting the system evolve,” says Ghrist, adding that, while the findings are concerning, there is a gap between using these models to study networks versus controlling how ideas spread in the real world.
The next step for Ghrist and his group is to find ways to work with more complex sheaves, such as ones with logical statements instead of numerical values. “The mathematical challenges associated with this are substantial, and my group and I have been working very hard on trying to lift all the mathematics to incorporate these more complex data types,” he says.
Ghrist also hopes that researchers from a variety of other fields, from economics to neuroscience, will find these tools useful because of their adaptivity and flexibility. “Sheaf theory was developed in the 1950s, and yet it's one of these things that never crossed over into applied math in part because it's very abstract,” he says. “I have been working for about 15 years on adapting ideas from sheaves and sheaf theory into a context that people can use outside of math, and I'm hopeful that this paper really pushes things in that direction.”
Jakob Hansen earned his Ph.D. from Penn’s Applied Mathematics and Computational Science program under the direction of Robert Ghrist in May 2020. He is currently a visiting assistant at The Ohio State University.
This research was supported by the U.S. Department of Defense Office of the Under Secretary of Defense for Research and Engineering Basic Research Office and the U.S. Department of Defense Vannevar Bush Faculty Fellowship.