Mathematical Modeling of Social Relationships

Mathematical modeling of social relationships, integrating sociological theories, offers data-driven insights into how social connections form and evolve. By simulating these dynamics, our research unveils how individual agents shape their interactions, revealing patterns of hierarchy and balance in real-world networks.

Challenge

Social networks evolve through complex interactions where individuals form positive and negative relationships. Two key mechanisms influence these connections:

  • Structural balance (e.g., “a friend of my friend is my friend”)
  • Status hierarchies (e.g., “respecting those of higher status”)

Traditional models assume global knowledge, which limits their applicability to real-world networks.

Our Approach

We developed an agent-based model that integrates both structural balance and status mechanisms from an ego perspective—each agent updates only its own relationships based on local information.

Using:

  • Python,
  • NetworkX, and
  • Pandas,

we simulated network dynamics and validated it on multiple real-world signed networks to analyze emerging social structures.

Results

  • The study successfully replicated real-world patterns of hierarchical triads, confirming that social networks tend to favor status-driven structures.
  • When status mechanisms dominate, individuals in the network strive for the top of the hierarchy.
  • Introducing status dynamics can destabilize networks, making the transition to fully positive relations discontinuous and fragile.

Future Plans

Building on this research, there are opportunities to explore:

  • The use of graph-based neural networks for recognizing entities.
  • Detecting influence patterns within social networks.

This approach could enable more advanced predictive modeling, offering valuable insights into organizational and social structures.

Team Expertise

This project resulted from a bilateral collaboration between:

  • Warsaw University of Technology
  • ETH Zurich

It combined expertise in data science and physics.

My Contribution:

  • Applied data science skill set to analyze social network dynamics.
  • Leveraged advanced modeling and analysis techniques.

The team’s strong physics background provided a solid foundation for applying mathematical modeling to complex systems, enabling the development of a comprehensive and robust approach to social relation analysis.