Games on deBruijn Graphs and Cycle Means

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.

💡 Research Summary

The paper investigates a novel game‑theoretic method for assigning edge weights to doubly weighted de Bruijn graphs so that every directed cycle has the same arithmetic mean weight. The setting begins with a de Bruijn graph G defined on an alphabet of size n and word length d, yielding N = n^d vertices. Each vertex m carries a prescribed real weight c(m). The authors introduce a repeated two‑person zero‑sum game played on G between players Paul and Carol. At each discrete time step t (0 ≤ t ≤ T) Paul chooses real numbers f(t,(m,m|ℓ)) for the n outgoing edges from the current vertex m, subject to the zero‑sum constraint Σ_ℓ f(t,(m,m|ℓ)) = 0. After Paul’s assignment, Carol selects (deterministically or probabilistically) one of the successors m|ℓ, moving the token accordingly. Paul’s objective is to minimize the total accumulated cost (the sum of vertex and edge weights along the path), while Carol aims to maximize it.

The authors formulate a value function v(t,m) representing the optimal cost-to‑go from state (t,m). Using the Dynamic Programming Principle they derive the recursion

 v(t,m) = c(m) + (1/n) ∑_{ℓ=0}^{n‑1} v(t+1,m|ℓ) for t < T,

with terminal condition v(T,m) = c(m). This recursion implies that, under optimal play, Paul’s edge weights must equalize the quantities c(m)+f(t,(m,m|ℓ))+v(t+1,m|ℓ) across all ℓ. Solving the linear system together with the zero‑sum constraint yields an explicit formula

 f(t,(m,m|ℓ)) = (1/n) ∑_{ℓ′} v(t+1,m|ℓ′) − v(t+1,m|ℓ).

Crucially, for all steps earlier than T − d the optimal edge weights become independent of the time index; they depend only on the structure of the de Bruijn tree rooted at m. Consequently, the value function admits a closed‑form expression as a repeated averaging of vertex weights along all possible length‑(T‑t) extensions of m. When T − t exceeds the word length d, the expression stabilizes:

 v(t,m) = c(m) + (1/n) ∑{ℓ₁}c(m|ℓ₁) + … + (1/n^d) ∑{ℓ₁,…,ℓ_d}c(m|ℓ₁|…|ℓ_d) + (T‑t‑d+1)·(1/n^d) ∑_{x∈M}c(x).

From this, the authors prove that any cycle whose traversal finishes before time T − d has total weight equal to its length multiplied by the global average of the vertex weights, i.e.,

 average weight of any such cycle = (1/n^d) ∑_{x∈M}c(x).

Thus, despite the exponential number of cycles in a de Bruijn graph, a single edge‑weight assignment (given by the formulas above) forces all cycles to share the same mean weight. This result is striking because it solves an apparently over‑determined linear system: the number of cycle‑mean equations vastly exceeds the number of edge variables, yet a solution exists and is unique under the game’s optimality conditions.

Moreover, the value function v satisfies a discrete Poisson equation on the graph. Defining the discrete Laplacian Δh(m) = h(m) − (1/n) ∑_{ℓ}h(m|ℓ), the authors show

 Δv(t,m) = c(m) − (1/n^d) ∑_{x∈M}c(x) for t < T − d.

Hence the optimal game value is precisely a solution of an inhomogeneous Poisson problem with source term equal to the deviation of the vertex weight from its global average.

The paper also discusses two related games, showing that they lead to the same edge‑weight formulas and the same discrete Poisson solution, thereby reinforcing the robustness of the approach.

In summary, the contribution of the work is threefold: (1) it establishes the existence of edge‑weight assignments that equalize cycle means for any prescribed vertex weights on de Bruijn graphs; (2) it provides explicit constructive formulas derived from minimax optimal strategies of a repeated two‑person game; and (3) it reveals a deep connection between this game‑theoretic construction, over‑determined linear systems, and discrete Poisson equations on directed regular graphs. The results open new avenues for applying game theory to graph weighting problems, with potential implications for network design, distributed systems, and combinatorial optimization.