Generating consensus and dissent on massive discussion platforms with an $O(N)$ semantic-vector model

TL;DR

This work tackles consensus formation on massive discussion platforms by replacing frequency-based adoption with a local energy landscape built from semantic vectors. It introduces a 2D lattice of agents with $N$-component semantic vectors $\phi(i_n)$ and a Hamiltonian $H=-\sum_{n,\mu} \phi(i_n)\cdot\phi(i_{n+\mu})$ under $Z=\int_{O(N)} e^{-\beta H}$, enabling a Markov process with $β$ controlling the balance between cohesion and diversity. By testing both synthetic and real CI embedding distributions and several annealing schedules, the work demonstrates ferromagnetic consensus for $β>0$, antiferromagnetic dissent for $β<0$, and faster convergence under a consensus-dissent protocol. The framework offers a scalable, locality-based method to steer online collective intelligence toward desired diversity-then-cohesion outcomes, with practical implications for large-scale CI platforms.

Abstract

Reaching consensus on massive discussion networks is critical for reducing noise and achieving optimal collective outcomes. However, the natural tendency of humans to preserve their initial ideas constrains the emergence of global solutions. To address this, Collective Intelligence (CI) platforms facilitate the discovery of globally superior solutions. We introduce a dynamical system based on the standard $O(N)$ model to drive the aggregation of semantically similar ideas. The system consists of users represented as nodes in a $d=2$ lattice with nearest-neighbor interactions, where their ideas are represented by semantic vectors computed with a pretrained embedding model. We analyze the system's equilibrium states as a function of the coupling parameter $β$. Our results show that $β> 0$ drives the system toward a ferromagnetic-like phase (global consensus), while $β< 0$ induces an antiferromagnetic-like state (maximum dissent), where users maximize semantic distance from their neighbors. This framework offers a controllable method for managing the tradeoff between cohesion and diversity in CI platforms.

Figures (8)

• Figure 1: Distributions used in the experiments. The graph of the function $-\ln(s)$ represents the probability density function of $P(X \cdot Y)$ where $X$ and $Y$ are independent random variables with a uniform distribution on $[0,1]$. The CI is the histogram of the distribution of the similarity matrix between the embeddings of the responses from a real CI experiment with more than 5000 different responses.
• Figure 2: Example of evolution for the Standard annealing process with $\beta\in[1,8]$. Results for the CI distribution are represented with points, and results for the $P(X \cdot Y)$ distribution are represented with lines.
• Figure 3: Snapshots of the response configuration in the ferromagnetic regime. Nodes are colored according to the response (idea) they represent. Arrows indicate the direction of time evolution. The corresponding local energy distribution for each frame is shown in \ref{['fig:ferro_energy_conf']}.
• Figure 4: Evolution of the local energy configuration in the ferromagnetic state. Nodes are colored according to their local energy, as defined in \ref{['eq:Elocalsem']}. Each panel corresponds to the respective time step shown in \ref{['fig:ferro_conf']}.
• Figure 5: Example of evolution for the Negative Standard annealing process with $\beta\in[-8,-1]$. Results for the CI distribution are represented with points, and results for the $P(X \cdot Y)$ distribution are represented with lines.