This article proposes a new framework that uses geometric approaches, specifically discrete curvature, instead of traditional correlation-based methods to analyze the dynamic changes in brain activity during social interactions. This method interprets changes in interbrain connectivity through the evolving geometric structure of neural networks, and presents an analysis pipeline that uses entropy measurements derived from curvature distributions to identify critical transition points in network connectivity. This geometric approach significantly enhances the explanatory power of hyperscanning methodology, enabling a deeper understanding of the neural mechanisms underlying interactive social behavior.

1. Introduction: Limitations of Existing Analytical Methods and the Need for a New Approach

In social neuroscience research, Interbrain Synchrony (IBS) measurements, particularly metrics like Phase Locking Value (PLV), have been widely used to describe neural interactions. However, these methods largely remain at the level of descriptive observation and tend to overlook the dynamic transitions between brain network states that could provide insights into the mechanisms of social interaction.

Recent advances in geometric machine learning have demonstrated the potential of discrete geometric methods as powerful tools for characterizing complex network structures and dynamics. This paper builds on the idea that flexible cognitive processes are modulated by transient connectivity patterns. While previous studies primarily applied geometric tools to intra-brain networks, we propose applying geometric methods to time-varying interbrain networks during social interactions.

In particular, we aim to overcome the limitations of correlation-based measurements by leveraging discrete graph curvature, providing richer, more mechanistic insights into how brain networks dynamically reorganize during social interactions.

2. The Graph Geometry Toolkit: Understanding Discrete Curvature

At the core of our proposal are discrete curvatures. One of these, Forman-Ricci curvature (FRC), was originally developed to describe the geometric properties of discrete spaces parameterized by cell complexes. FRC analyzes the connectivity patterns of networks to quantify how information expands and contracts as it flows through the network.

• Positive curvature values generally identify edges in densely connected regions.
• Negative curvature values highlight edges connecting highly connected network modules.

Another discrete Ricci curvature concept, Ollivier-Ricci curvature (ORC), is defined through Markov chains and can be interpreted differently in the context of interbrain connectivity.

"The curvature of an edge provides a proxy for the tendency of information flow to be attracted, with negative curvature indicating greater attraction."

In other words, regions dense with edges of low (negative) curvature facilitate shortest-path navigation, while regions with high (positive) curvature promote diffusion. The following section examines how this discrete Ricci curvature-based toolkit can be usefully applied in social neuroscience.

3. Hyperscanning Use Case: A New Horizon for Dynamic Network Analysis

Hyperscanning refers to the simultaneous recording of neural signals from interacting individuals, and it has revolutionized the fields of social neuropsychology and clinical neuroscience. However, the analytical methods applied to hyperscanning still rely heavily on purely correlation-dependent approaches, inherently limiting their explanatory power.

We argue that curvature-based analysis of interbrain coupling networks can bring hyperscanning research closer to mechanistic explanations.

"We argue that curvature-based analysis of interbrain coupling networks can bring hyperscanning research closer to mechanistic explanations."

3.1 Interbrain Networks and Their Synchronization

Interbrain networks represent the joint neural connectivity of two or more individuals within weighted graphs constructed through hyperscanning. Each node typically corresponds to a neural region, and edge weights are derived by computing IBS measurements (e.g., PLV) from the neural activity of those regions. However, these studies have been limited in the mechanistic reasoning they provide to researchers. Correlations between brain regions of interacting subjects are interpreted only in terms of the computational-cognitive roles of those regions, while the detailed mechanisms by which social behavior unfolds over time and its dynamic evolution remained speculative.

We propose extending the study of time-varying interbrain networks with graph curvature to detect meaningful phase transitions in interpersonal neural dynamics and to provide insights into the information routing strategies used by interbrain networks to perform joint behavioral tasks.

3.2 Capturing Phase Transitions

If the timing of task-related behavioral transitions or events -- such as cooperative engagement, misunderstanding, or conflict resolution -- synchronizes with the timing of phase transitions in interbrain networks identified by graph curvature, researchers can reason with greater confidence about the neural mechanisms of behavior.

To capture significant dynamic changes in network configuration, we examine the divergence over time in the differential entropy of graph curvature distributions of IBS. This is defined using the probability density function $f_{RC}^t(x)$ of discrete curvature values in the network configuration $G_t$ as follows:

$H_{RC}(G_t) = -\int_{\mathbb{R}}f_{RC}^{t}(x)\log[f_{RC}^{t}(x)],dx$

Figure 1 demonstrates applying this method to a model of time-varying brain networks with small-world topology to detect phase transitions. As the rewiring probability ($p$) used to generate the network evolves from 0 to 1, the differential entropy of FRC distributions diverges between $p=10^{-3}$ and $p=10^{-1}$. This is the phenomenon observed when the network transitions from a regular lattice to a random network. Specifically, panels E-F show that when $p \gtrsim 10^{-2}$, entropy increases sharply and the curvature distribution broadens (95th percentile jump). This signifies a transition from a segregated lattice-like topology to a more integrated small-world/random regime due to increased neighbor overlap and shortcut formation.

Figure 1: Simulation of time-varying brain networks modeled as weighted small-world networks with different rewiring probabilities.

3.3 Capturing Information Routing Strategies

Theoretical research on information routing in brain networks has used Markov chains to model a spectrum of information routing strategies, from shortest-path navigation to random diffusion. Accordingly, ORC distributions can be interpreted as identifying the information routing strategies adopted by the network.

Recent deep learning research has shown that FRC can identify information bottlenecks that distort information flow during message passing in graph neural networks. These findings suggest that FRC can be a useful tool for evaluating information flow in brain networks. This is a key component of the mechanistic models proposed by predictive theories of the brain.

4. Toward Interbrain Geometry

Adopting a geometric framework in neuroscience offers methodological and conceptual advances over existing IBS-based analysis. Geometric Hyperscanning can address the limitations of correlation-based measurements in capturing dynamic network reorganization and characterizing real-time information routing strategies within and between socially interacting brains.

Discrete curvature distributions can summarize constraints on network dynamics, and divergence in distribution entropy indicates network reorganization events. While this does not inherently resolve confounding factors arising from IBS-based approaches, it provides a complementary network-level description of interbrain interactions, enabling the additional reasoning needed to build mechanistic explanations.

This direction aligns with the call by Kulkarni and Bassett (2024) for minimal principled models of brain network complexity, and with the emphasis by Sporns (2010) on meso-scale features (hubs, clusters, bridges), reinforcing the role of discrete curvature as an indicator of structural transitions during social interactions. Curvature-based analysis will allow researchers to explore the information routing implications of IBS and how they dynamically reorganize during real-time interactions, paving the way for a deeper mechanistic understanding of the social brain.

Conclusion

This paper proposes an innovative approach using discrete geometry to analyze the dynamic changes in interbrain networks during social interactions. By providing mechanistic insights into network reorganization and information routing strategies that traditional correlation-based analysis could not offer, it emphasizes the potential to significantly enhance the explanatory power of hyperscanning research. This is expected to make an important contribution to a deeper understanding of the neural mechanisms underlying complex social behavior.