[Paper Review] Games on deBruijn Graphs and Cycle Means

Nadejda Drenska|arXiv (Cornell University)|Jan 30, 2026

Game Theory and Voting Systems0 citations

TL;DR

The paper shows that for a de Bruijn graph with fixed vertex weights, one can assign edge weights via a two-player repeated game so that every cycle has the same average weight; the resulting value function solves a discrete Poisson equation on the graph.

ABSTRACT

deBruijn graphs are widely used in genomics and computer science. In this paper we present a novel approach to finding weights on edges of doubly weighted deBruijn graphs. Given any fixed set of weights on vertices, we use a repeated two-person zero-sum game to find weights on edges so that every cycle on the deBruijn graph has the same average weight, providing explicit formulas. This approach uses minimax optimal strategies of the players. Once the weights on the edges are determined, we observe that they correspond to solving a set of linear equations with as many equations as there are cycles. This is very surprising, because there are many more cycles than unknowns. Moreover we analyze other, related games on graphs.

Motivation & Objective

• Motivates analyzing cycle means on deBruijn graphs due to applications in genomics and computer science.
• Introduces a repeated two-player zero-sum game to determine edge weights given fixed vertex weights.
• Proves the existence of edge weights that equalize the average cycle weight across all cycles.
• Shows the edge-weight solution corresponds to solving a discrete Poisson equation on the graph.

Proposed method

• Define a deBruijn graph with vertex weights and edge weights subject to zero-sum outgoing edge constraint at each vertex.
• Model a repeated two-person zero-sum game where Paul selects edge weights and Carol selects the next vertex.
• Use dynamic programming to derive the value function v(t,m) and show it satisfies a minimax recurrence.
• Derive explicit edge weights f(t,(m,m|ℓ)) from v via a linear relation, independent of time before a cutoff, leading to equal cycle means.
• Show that cycles ending before time T−d have the same average weight equal to the mean of vertex weights over all graphs vertices.
• Demonstrate that the value function solves a discrete Poisson equation on the graph.

Experimental results

Research questions

• RQ1 Can edge weights be chosen so that every cycle in a deBruijn graph has the same average weight given fixed vertex weights?
• RQ2 What is the structure of optimal strategies in the induced two-player game, and how do these relate to solving linear systems or Poisson equations on the graph?
• RQ3 Do analogous results hold for related games and for general directed graphs without sinks?

Key findings

• There exists an edge-weight assignment making all cycles share the same average weight, equal to the arithmetic mean of all vertex weights.
• The common cycle mean equals (1/n^d) times the sum of vertex weights over all vertices.
• The value function v(t,m) satisfies a closed-form expression and, for t < T−d, solves a discrete Poisson equation Δv = c − mean(c).
• The edge weights used by Paul become time-invariant for times t ≤ T−d.
• Carol’s optimal path choice can be modeled as deterministic or probabilistic without changing the value of the game.