Games on deBruijn Graphs and Cycle Means

TL;DR

This paper addresses balancing directed cycles in a doubly weighted deBruijn graph by assigning edge weights, given fixed vertex weights, so that every cycle shares the same average weight $W(C)/|C|$ (equal to the vertex-average $\bar c=\frac{1}{n^{d}}\sum_{ar m} c(\bar m)$). It introduces a repeated two-player zero-sum game and derives explicit minimax strategies that produce edge weights with zero-sum outgoing edges and equal cycle means, with the value function $v(t,m)$ connected to a discrete Poisson equation via $\Delta v= c - \bar c$. The framework yields a closed-form expression for $v$ and edge-weights, and reduces the problem to solving an overdetermined linear system with $(n-1)n^{d}$ degrees of freedom; these results extend to related games and to general directed graphs. The findings provide a constructive approach to cycle-mean balancing in graphs, with potential implications for cycle analysis, network timing, and graph-based control problems.

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.

Figures (1)

• Figure 1: The de Bruijn Graph, $n=2, d=3$

Theorems & Definitions (25)

• Lemma 1
• proof
• Lemma 2
• proof
• Theorem 3
• proof
• Lemma 4
• proof
• Corollary 5
• proof