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Borda Aggregation Dynamics of Preference Orderings on Networks

Moses Boudourides

Abstract

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.

Borda Aggregation Dynamics of Preference Orderings on Networks

Abstract

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.

Paper Structure

This paper contains 28 sections, 18 theorems, 23 equations, 3 figures.

Table of Contents

  1. Introduction and social-scientific motivation
  2. Model
  3. Social graph and weights
  4. State space and move graph
  5. Borda scores and projection
  6. Local aggregation and the bounded step
  7. Variants and persistent nodes
  8. Fixed points and eventual periodicity
  9. Eventual periodicity (Variant S)
  10. Consensus fixed points
  11. Network dependence of convergence
  12. Self-oscillations without persistence
  13. Oscillations on directed cycles
  14. Even-period lifting
  15. Forced oscillations with contrarian camps
  16. ...and 13 more sections

Key Result

Proposition 3.1

Every trajectory of Variant S (with fixed $P$ and deterministic $\textsf{Step}$) is eventually periodic: there exist $\mu\ge 0$ and $p\ge 1$ such that $\sigma(t+p)=\sigma(t)$ for all $t\ge \mu$. $\blacktriangleleft$$\blacktriangleleft$

Figures (3)

  • Figure 1: Hasse diagrams (cover graphs) of the weak-order lattices $\Omega(3)$ and $\Omega(4)$. For total preorders (ties allowed), $|\Omega(3)|=F(3)=13$ and $|\Omega(4)|=F(4)=75$, where $F(m)$ is the ordered Bell (Fubini) number.
  • Figure 2: A self-oscillation as in Theorem \ref{['thm:self_traveling_wave']}.
  • Figure 3: Minimal forcing gadget underlying Corollary \ref{['cor:two_node_gadget']}

Theorems & Definitions (46)

  • Remark 2.1: Move graph versus Kemeny/Kendall distances
  • Remark 2.2
  • Definition 2.1: Bounded step
  • Definition 2.2: Persistent (boundary) nodes
  • Definition 2.3: Variant S: synchronous bounded Borda dynamics
  • Definition 2.4: Variant A: asynchronous bounded Borda dynamics (contrast)
  • Proposition 3.1: Eventual periodicity
  • proof
  • Proposition 3.2: Local Borda equilibria as fixed points
  • proof
  • ...and 36 more