Borda Aggregation Dynamics of Preference Orderings on Networks
This addresses the problem of understanding collective opinion formation in social networks, but it is incremental as it builds on existing aggregation methods with a focus on dynamical analysis.
The paper tackles the problem of modeling how preference orderings evolve on networks through local Borda aggregation, and it develops conditions for self-sustained and forced oscillations in such dynamics, with robustness analysis and comparisons between synchronous and asynchronous updating.
We introduce and analyze a discrete-time network process in which each node holds a (weak) preference ordering over a finite set of alternatives and updates by local Borda aggregation. At each step, a node forms a weighted average (row-stochastic random-walk normalization) of its neighbors' Borda score vectors and projects the aggregated score back to a weak order. Updates are bounded: in each round, a node advances by at most one step along a shortest path in the fixed graph of preference orderings, following the direction prescribed by its neighbors' Borda-aggregated preferences. Our emphasis is dynamical: we develop sufficient conditions, stated directly in terms of graph topology, weights, and the bounded step rule, for (i) self-sustained oscillations in the absence of persistent sources, and (ii) forced oscillations under contrarian persistent camps. We also record robustness (structural stability) away from score-tie hyperplanes and contrast synchronous (Variant S) and asynchronous (Variant A) updating.