This paper proposes a novel framework that uses a geometric approach — specifically discrete curvature — rather than conventional correlation-based methods to analyze the dynamic changes in brain activity that occur during social interactions. This approach interprets shifts in interbrain connectivity through the evolving geometric structure of neural networks, and presents an analysis pipeline that uses entropy measures derived from curvature distributions to identify critical transition points in network connectivity. This geometric framework substantially enhances the explanatory power of hyperscanning methodology, enabling a deeper understanding of the neural mechanisms underlying interactive social behavior.

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

In social neuroscience research, measures of Interbrain Synchrony (IBS) — particularly metrics such as the Phase Locking Value (PLV) — have been widely used to characterize neural interactions. However, these methods tend to remain at the level of descriptive observation and overlook the dynamic transitions between brain network states that could provide mechanistic insight into social interaction. 🤔

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

In particular, we leverage discrete graph curvature to overcome the limitations of correlation-based measures, aiming to provide richer and more mechanistic insight into how brain networks dynamically reorganize during social interaction. ✨

2. The Graph Geometry Toolkit: Understanding Discrete Curvature 📐

The cornerstone of our proposal is discrete curvature. One such measure, Forman-Ricci curvature (FRC), was originally developed to describe the geometric properties of discrete spaces parameterized as cell complexes. FRC analyzes the connectivity patterns of a network to quantify how much information expands or contracts as it flows through the network.

• Positive curvature values typically identify edges in densely connected regions.
• Negative curvature values highlight edges that bridge highly connected network modules.

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

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

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

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

Hyperscanning — the simultaneous recording of neural signals from multiple interacting individuals — has transformed social neuropsychology and clinical neuroscience. Yet the analysis methods applied to hyperscanning data still rely heavily on purely correlation-based approaches, which fundamentally limits their explanatory power.

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

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

3.1 Interbrain Networks and Their Synchronization 🤝

Interbrain networks represent the joint neural connectivity of two or more individuals within a weighted graph constructed from hyperscanning data. Each node typically corresponds to a neural region, and edge weights are derived by computing IBS measures (e.g., PLV) from the neural activity of those regions. However, such studies have been limited in the mechanistic reasoning they afford researchers. Correlations between brain regions of interacting individuals are interpreted only through the computational-cognitive roles of those regions; the detailed mechanisms by which social behavior unfolds over time and its dynamic evolution remain 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 insight into the information routing strategies that interbrain networks use to perform joint behavioral tasks.

3.2 Capturing Phase Transitions 🔄

If the timing of task-relevant behavioral transitions or events — such as cooperative engagement, misunderstanding, or conflict resolution — aligns with the timing of phase transitions in the interbrain network 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 divergence over time in the differential entropy of the graph curvature distribution 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 illustrates the application of this method to a simulated time-varying brain network with a 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 the FRC distribution diverges between $p=10^{-3}$ and $p=10^{-1}$. This corresponds to the transition of the network from a regular lattice to a random network. Notably, panels E–F show that entropy rises sharply and the curvature distribution broadens (a jump in the 95th percentile) when $p \gtrsim 10^{-2}$, signaling a transition from a segregated lattice-like topology to a more integrated small-world/random regime driven by increasing neighborhood overlap and the formation of shortcuts.

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

3.3 Capturing Information Routing Strategies 🗺️

Theoretical research on information routing in brain networks has used Markov chains to model a range of routing strategies spanning from shortest-path routing to random diffusion. The ORC distribution can therefore be interpreted as identifying the information routing strategy adopted by a network.

Recent deep learning research has shown that FRC can identify information bottlenecks — points where information flow is distorted during message passing in graph neural networks. These findings suggest that FRC could be a valuable tool for evaluating information flow in brain networks, serving as a key component of the mechanistic models proposed by predictive theories of the brain.

4. Toward an Interbrain Geometry 🛣️

Adopting a geometric framework in neuroscience offers methodological and conceptual advances over conventional IBS-based analysis. Geometric Hyperscanning can address the limitations of correlation-based measures 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 distributional entropy marks network reorganization events. While this approach does not inherently resolve all confounds that arise in IBS-based approaches, it provides a complementary network-level account of interbrain interactions that enables additional reasoning necessary for building mechanistic explanations.

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

Conclusion 🎯

This paper proposes an innovative approach that leverages discrete geometry to analyze the dynamic changes in interbrain networks occurring during social interactions. By providing mechanistic insight into network reorganization and information routing strategies that conventional correlation-based analysis cannot offer, it highlights how the explanatory power of hyperscanning research can be substantially enhanced. This contribution is expected to be significant in advancing our deeper understanding of the neural mechanisms underlying complex social behavior.