A new adaptive two-layer model for opinion spread in hypergraphs: parameter sensitivity and estimation

Abstract In studies of opinion spread, peer pressure is often modeled through interactions of more than two individuals (higher-order interactions). We introduce a two-layer random hypergraph model, in which households and workplaces form the layers and hyperedges represent individual households and workplaces. Within this overlapping adaptive structure, individuals may react when their opinion is in the minority within their groups. The process evolves through stochastic steps: individuals can either change their opinion, or quit their workplace and join another one in which their opinion belongs to the majority. Our first goal is to describe, via computer simulations, the effects of the parameters governing opinion change and workplace switching on homophily, speed of polarization, and number of components. We also analyze the model as a Markov chain, and study the absorbing states. We then quantitatively compare how different statistical and machine learning methods—namely linear regression, XGBoost, convolutional neural network (CNN) and long short-term memory network (LSTM)—perform in estimating these parameters from partial observations of the process, such as the distribution of opinion configurations within households and workplaces. Among other observations, we conclude that in most of the cases, especially for shorter observation periods LSTM is the best, while in many cases XGBoost is also a strong contender. For longer time horizons, CNN is also competitive. It is also an important observation that the information required for accurate estimation depends on the strength of the peer pressure effect. Similar content being viewed by others Materials availability Not applicable. Acknowledgment The authors are grateful to Edit Bognár for useful discussions on the model and the measurement of polarization. Funding This work was supported by the National Research, Development and Innovation Office within the framework of the Thematic Excellence Program 2021 – National Research Subprogramme: “Artificial intelligence, large networks, data security: mathematical foundation and applications” (grant number: TKP2021-NKTA-62). Author information Authors and Affiliations Corresponding author Ethics declarations Ethics approval and consent to participate Not applicable. Consent for publication All authors have given consent for publication. Competing interests The authors declare no competing interests. Additional information Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Appendices Appendix A. Parameter sensitivity In this section, we state simple lemmas about the model described in section “The model” and in section “Absorbing states and absorption time”. We investigate the effects of the parameters on the absorbing states. In addition, we present simulation based observations on hypergraphs of size \(n=1000\), which provide further insight into the main effects of parameter changes in the model. In these parameter sensitivity discoveries, we preferred to simulate longer runs of the process. Hence, we increased the maximal number of logging steps to 5000 (we have logged the measurements per 1000 steps taken by the process); on the other hand, we decreased the number of simulation samples to 100 due to computational limitations. In this way we could measure the absorption time for more cases, i.e. the first logging step where the simulation hits an absorbing state. An absorption time close to 5000 means that many processes still did not reach any absorbing state. All measurements presented are means of the 100 samples in the absorbing state or the maximal 5000th final logging step. A.1 The initial size of workplace hyperedges Varying the initial workplace size does not seem to change the qualitative structure of the absorbing states. In the linear model, the absorbing states are unchanged in the sense that just homogeneous hyperedges are allowed, while in the nonlinear model, the same types of absorbing configurations remain possible. The main effect of the initial workplace size is therefore quantitative: it influences the transient dynamics, the absorption time, and the probabilities of the different final states. A.2 Workplace weights \(\lambda\) In the linear model \(r_1=r_2=1\), for \(\beta > 0\) and \(\lambda > 0\) every absorbing state consists only of homogeneous households and homogeneous workplaces. Hence, both homophily indices converge almost surely to \(1\). The absorbing set itself is independent of \(\lambda\), but the transient dynamics is not, so the expected absorption time may depend on \(\lambda\) in a non-monotone way. Moreover, there is no general monotone relationship between the expected absorption time and the expected number of connected components in the absorbing state. Still, in our simulations we observe that parameter regimes with fewer final components often exhibit larger average absorption times. Furthermore, in some cases, increasing \(\lambda\) increases the probability of full consensus, see Fig. A3, showing that the number of components decreases with \(\lambda\). In the nonlinear model, we can state more: increasing \(\lambda\) may enlarge the absorbing state set. Lemma 1 Assume that \(r_1=r_2=\tfrac12\) and \(q > 0\). Let \(\mathcal A_\lambda\) denote the set of absorbing states for the parameter value \(\lambda\). Then for \(\lambda < \lambda^{\prime}\), Proof Let \(x\) be an absorbing state for \(\lambda\). Since \(r_2=\tfrac12\) and \(q > 0\), no vertex can satisfy \(w(v) < \tfrac12\), hence every workplace is either homogeneous or exactly balanced. Therefore for every vertex \(v\), we have \(w(v)=1\) or \(w(v)=\tfrac12\). If \(w(v)=\tfrac12\), then so this condition is independent of \(\lambda\). If \(w(v)=1\), then Since \((1-\lambda)/2\) decreases as \(\lambda\) increases, any vertex that is stable for \(\lambda\) is also stable for every \(\lambda^{\prime} > \lambda\). Thus every vertex that is stable in \(x\) for \(\lambda\) is also stable for \(\lambda^{\prime}\), and the workplace condition is unchanged. Hence \(x\) is also absorbing for \(\lambda^{\prime}\). Therefore □ The consequence of the above lemma can be seen on the \(H_{\mathrm{hh}}\) measures in Fig. A4: \(H_{\mathrm{hh}}\) decreases when we increase \(\lambda\). This suggests that the process ends in absorption states consisting of less homogeneous households as \(\lambda\) increases, where mixed households are more frequent. A.3 Threshold parameters \(r_1, r_2\) When \(r_1=1\) and \(\lambda=0.5\) fixed every absorbing state consists only of homogeneous households and homogeneous workplaces. The change of \(r_2\) effects only the absortion time and the number of components. When \(r_1=1/2\) and \(\lambda\) are fixed, the absorbing state space depends monotonically on \(r_2\). Decreasing \(r_2\) weakens the workplace-stability condition \(w(v)\ge r_2\), while the opinion-stability condition remains unchanged. Hence, the absorbing set enlarges as \(r_2\) decreases. A workplace hyperedge in an absorbing state may be mixed only if the minority opinion occupies at least an \(r_2\)-fraction of the workplace. In particular, for \(r_2 > 1/2\), all absorbing workplaces are homogeneous, for \(r_2=1/2\), they are either homogeneous or exactly balanced, and for \(r_2 < 1/2\), mixed workplaces with sufficiently large minorities are also possible. Lemma 2 Assume that \(\lambda=0.5\), \(q > 0\), and \(r_2 > \tfrac12, r_1 > 0.6\). Then every absorbing state consists only of homogeneous households and homogeneous workplaces. In particular, every connected component of the absorbing state is monochromatic. Consequently, if both opinions survive with positive probability in the absorbing state, then Proof The condition \(r_2 > \tfrac12\) forces every workplace in an absorbing state to be homogeneous. The condition \(r_1 > 0.6\) then forces every household in an absorbing state to be homogeneous as well. Therefore, each connected component is homogenized. If both opinions survive in the final state, then there must be at least two connected components, one carrying opinion \(A\) and one carrying opinion \(B\). □ The consequences of the above lemma can be seen in the simulation results in Fig. A7c. 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To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/. About this article Cite this article Backhausz, Á., Csiszár, V., Kolok, C.B. et al. A new adaptive two-layer model for opinion spread in hypergraphs: parameter sensitivity and estimation. Appl Netw Sci (2026). https://doi.org/10.1007/s41109-026-00829-9 Received: Accepted: Published: DOI: https://doi.org/10.1007/s41109-026-00829-9