Mathematicians are exploring the intricate dynamics of opinion formation within large groups, a subject that transcends sociology and intersects with mathematics. On November 19, 2025, Ph.D. candidate Federico Capannoli defended his thesis, which focuses on understanding how opinions change over time within complex networks.
Understanding Opinion Dynamics through Mathematical Models
Capannoli emphasizes that many real-life processes are too complex for direct analysis. Instead, they can be represented through mathematical models. He cites examples beyond social opinions, such as the movements of cells in biological materials or the spread of diseases, all of which can be modeled similarly.
The research took place at Leiden University, where Capannoli pursued his Ph.D. after completing his Master’s in Padova. He describes the university’s probabilistic research group as a hub of expertise, stating, “There is real expertise in this field here and I always felt like I was welcome.”
Modeling Elections as Complex Networks
Central to Capannoli’s research is the idea of representing individuals as dots within a network, where each dot’s color signifies their opinion. Connections between these dots represent friendships. He explains that during elections, individuals tend to influence each other, altering the colors of their dots over time.
In his theoretical model, Capannoli examines how long it takes for a consensus to emerge. He finds that the time required for this consensus is proportional to the number of people involved and the number of connections they have. “If only a handful of people are very well-connected to most of the other individuals while the vast majority has few connections, then you still reach consensus much faster,” he notes, highlighting the significant impact of social media influencers.
Capannoli’s work also delves into the effects of bias in opinion formation, particularly during political campaigns. He explains that if two individuals converse, one may adopt a biased opinion based on propaganda. “If this bias is high, the time to reach consensus is cut drastically,” he states. Conversely, a low bias does not significantly affect the consensus time.
The Complexity of Friendships and Polarization
Capannoli acknowledges the complexities of real-life interactions. Friendships may dissolve if individuals express differing opinions, or new connections may form. This dynamic co-evolution of opinions and friendships complicates the study of opinion dynamics.
At Leiden University, researchers like Frank den Hollander and Rajat Subhra Hazra are at the forefront of exploring these intricate relationships. Capannoli warns of the dangers of polarization, where groups become isolated and fail to communicate with one another. “It’s quite scary if you think about it,” he reflects, urging the importance of engaging with diverse viewpoints to counteract the effects of social media bubbles.
Capannoli’s research sheds light on the mathematical underpinnings of how opinions form and shift within society, demonstrating the critical role of connections and biases in shaping collective beliefs. His findings offer valuable insights into the dynamics of opinion formation, particularly in our increasingly interconnected world.
