When group level is different from the population level: an adaptive network with the Deffuant model

We propose a model coupling the classical opinion dynamics of the bounded confidence model, proposed by Deffuant et al., with an adaptive network forming a community or group structure. At each step, an individual can decide if it changes groups or interact on its opinion with one of its internal or external neighbour. If it decides to look at the group level, it changes groups if its opinion is far from the average of its group from more than a threshold. If it is the case, it joins the group which has proportionally the closest average opinion from its. If it decides to interact with one of its neighbour, it becomes closer in opinion to it when its opinion and the one of the selected-to-interact neighbour are less distant from the threshold. From the study of this coupled model, we discover some surprising behaviours compared to the known behaviour of the Deffuant bounded confidence model(BC): The coupled model exhibits a total consensus for an threshold value lower than the BC model; the distribution of sizes of the groups changes: some groups become larger while other decrease in size, sometimes until containing only one individual; from the point of view of the groups, the consensus remains for a large set of threshold values while, looking at the population level, there are a lot of opinion clusters.

💡 Research Summary

The paper introduces an adaptive‑network extension of the classic Deffuant bounded‑confidence opinion model by coupling it with a community (group) structure. In the baseline Deffuant model, two agents interact only if the absolute difference between their continuous opinions is smaller than a confidence threshold ε; when interaction occurs, each agent moves a fraction μ toward the other’s opinion. This mechanism alone yields consensus only for relatively large ε, while for smaller ε the population fragments into several opinion clusters.

To explore how group affiliation can reshape this dynamics, the authors endow each of N agents with a group label among G groups. At every discrete time step an agent randomly decides between two actions: (1) “group‑level” assessment or (2) “neighbor‑level” interaction. In the group‑level action the agent compares its opinion to the average opinion of its current group. If the distance exceeds ε, the agent may leave its group and join another one whose average opinion is the closest (and within ε). The probability of choosing the group‑level action is denoted p; the complementary probability (1‑p) leads to the neighbor‑level action, where the agent selects a randomly chosen neighbor (either from within its own group or from outside) and, if the opinion distance is ≤ ε, updates both opinions according to the standard Deffuant rule.

The authors systematically vary ε, μ, and p in Monte‑Carlo simulations and monitor three observables: (i) the number of opinion clusters at the population level, (ii) the distribution of group sizes, and (iii) the degree of consensus within each group. Several striking phenomena emerge.

1. Consensus at lower ε – Compared with the original Deffuant model, the coupled system reaches global consensus for substantially smaller confidence thresholds (e.g., ε≈0.3 versus ε≈0.5 in the classic case). The ability of agents to relocate to more compatible groups acts as a “re‑allocation” mechanism that rapidly reduces opinion heterogeneity, thereby lowering the ε required for full agreement.

2. Heterogeneous group size evolution – Starting from an egalitarian allocation, groups diverge in size over time. Groups whose average opinion is close to the prevailing consensus attract members and grow, while groups far from the consensus lose members, sometimes shrinking to a single individual or disappearing altogether. This creates a highly skewed size distribution reminiscent of real‑world communities where a few large factions dominate and many small or singleton factions persist.

3. Decoupling of group‑level and population‑level dynamics – Within most surviving groups, opinions become nearly homogeneous, indicating strong intra‑group consensus. However, at the population level multiple opinion clusters can coexist, especially when p is moderate. Thus, the system exhibits a dual‑scale structure: high agreement inside groups but persistent diversity across groups.

4. Influence of μ and p – Larger μ accelerates opinion convergence, promoting earlier consensus but also intensifying the size disparity among groups because agents switch groups more frequently. Higher p (more frequent group‑level checks) further encourages agents to seek compatible groups, which speeds up global consensus and amplifies the “rich‑get‑richer” effect on group sizes. Conversely, low p reduces group mobility, making the dynamics resemble the original Deffuant model with its characteristic fragmentation.

5. Model limitations and extensions – The current network topology is a simple two‑layer structure (intra‑group links and inter‑group random links). Real social networks display multi‑scale, weighted, and directed connections, as well as overlapping community memberships. Moreover, the decision rule based solely on distance to the group mean neglects factors such as identity, reputation, or external information sources (e.g., media). Future work could incorporate weighted migration probabilities, dynamic creation and dissolution of groups, and exogenous signals to better capture realistic opinion‑formation processes.

Overall, the study demonstrates that allowing agents to adaptively re‑assign themselves to more compatible groups dramatically alters the macroscopic outcome of bounded‑confidence dynamics. It suggests that policies or organizational interventions that facilitate flexible group membership could lower the confidence threshold needed for societal consensus, offering a novel lever for managing polarization.