Two species coagulation approach to consensus by group level interactions

We explore the self-organization dynamics of a set of entities by considering the interactions that affect the different subgroups conforming the whole. To this end, we employ the widespread example of coagulation kinetics, and characterize which interaction types lead to consensus formation and which do not, as well as the corresponding different macroscopic patterns. The crucial technical point is extending the usual one species coagulation dynamics to the two species one. This is achieved by means of introducing explicitly solvable kernels which have a clear physical meaning. The corresponding solutions are calculated in the long time limit, in which consensus may or may not be reached. The lack of consensus is characterized by means of scaling limits of the solutions. The possible applications of our results to some topics in which consensus reaching is fundamental, like collective animal motion and opinion spreading dynamics, are also outlined.

💡 Research Summary

The paper introduces a novel extension of classical Smoluchowski coagulation theory by incorporating two distinct particle species, denoted A and B, to model the dynamics of sub‑groups interacting within a larger population. The authors argue that many real‑world consensus‑forming processes—such as flocking, opinion spreading, or leader‑follower coordination—cannot be captured by a single‑species framework because the interaction rules between different sub‑populations often differ qualitatively. To address this, they formulate a set of coupled kinetic equations for the concentrations (c_k^{(A)}(t)) and (c_k^{(B)}(t)) of clusters of size (k) belonging to each species.

A central technical contribution is the explicit construction of analytically tractable coagulation kernels (K_{XY}(i,j)) (with (X,Y\in{A,B})). Three families of kernels are examined: (i) homogeneous‑homogeneous (AA, BB) kernels that are linear in the product of cluster sizes, representing symmetric intra‑group merging; (ii) heterogeneous‑homogeneous (AB, BA) kernels that allow one species to absorb the other with a rate proportional to a single size factor, capturing leader‑follower or opinion‑dominance mechanisms; and (iii) heterogeneous‑heterogeneous (AB*) kernels that are nonlinear and encode mutual mixing of attributes, reflecting situations where two different groups combine to create a hybrid consensus. Each kernel contains tunable parameters (e.g., (\alpha,\beta)) that control the relative strength of intra‑ versus inter‑species collisions.

Using generating‑function techniques, the authors reduce the coupled Smoluchowski equations to a system of nonlinear ordinary differential equations for the generating functions (G_A(z,t)) and (G_B(z,t)). For the linear kernels the solution reproduces the well‑known result that the average cluster size grows linearly with time, leading to a single macroscopic cluster that contains essentially all mass—a state the authors identify as “consensus”. In contrast, when the nonlinear heterogeneous‑heterogeneous kernel dominates, the system exhibits self‑similar scaling. The asymptotic cluster‑size distribution assumes the form
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