Chip-firing Game Research Articles

Finite record sets of chip-firing games

Finite record sets of chip-firing games

On approximating the rank of graph divisors

Baker and Norine initiated the study of graph divisors as a graph-theoretic analogue of the Riemann-Roch theory for Riemann surfaces. One of the key concepts of graph divisor theory is the rank of a divisor on a graph. The importance of the rank is well illustrated by Baker's Specialization lemma, stating that the dimension of a linear system can only go up under specialization from curves to graphs, leading to a fruitful interaction between divisors on graphs and curves.Due to its decisive role, determining the rank is a central problem in graph divisor theory. Kiss and Tóthmérész reformulated the problem using chip-firing games, and showed that computing the rank of a divisor on a graph is NP-hard via reduction from the Minimum Feedback Arc Set problem.In this paper, we strengthen their result by establishing a connection between chip-firing games and the Minimum Target Set Selection problem. As a corollary, we show that the rank is difficult to approximate to within a factor of O(2log1−ε⁡n) for any ε>0 unless P=NP. Furthermore, assuming the Planted Dense Subgraph Conjecture, the rank is difficult to approximate to within a factor of O(n1/4−ε) for any ε>0.

Growth of Replacements

The following game in a similar formulation to Petri nets and chip-firing games is studied: Given a finite collection of baskets, each has an infinite number of balls of the same value. Initially, a ball from some basket is chosen to put on the table. Subsequently, in each step a ball from the table is chosen to be replaced by some $2$ balls from some baskets. Which baskets to take depend only on the ball to be replaced and they are decided in advance. Given some $n$, the object of the game is to find the maximum possible sum of values $g(n)$ for a table of $n$ balls.
 In this article, the sequence $g(n)/n$ for $n=1,2,\dots$ will be shown to converge to a growth rate $\lambda$. Furthermore, this value $\lambda$ is also the rate of a structure called pseudo-loop and the solution of a rather simple linear program. The structure and the linear program are closely related, e.g. a solution of the linear program gives a pseudo-loop with the rate $\lambda$ in linear time of the number of baskets, and vice versa with the pseudo-loop giving a solution to the dual linear program. A method to test in quadratic time whether a given $\lambda_0$ is smaller than $\lambda$ is provided to approximate $\lambda$. When the values of the balls are all rational, we can compute the precise value of $\lambda$ in cubic time, using the quadratic time rate test algorithm and the binary search with a special condition to stop. Four proofs of the limit $\lambda$ are given: one just uses the relation between the baskets, one uses pseudo-loops, one uses the linear program and one uses Fekete's lemma (the latest proof assumes a condition on the rule of replacements).

On the limited increment parallel chip-firing game

In the same paper where Bitar and Goles defined the parallel chip-firing game, they also defined the limited increment parallel chip-firing game to be a chip-firing game on a graph G(V,E) in which each vertex fires as in the parallel chip-firing game but can receive at most one chip in each round. While the parallel chip-firing game has been widely-studied, the limited increment parallel chip-firing game has not been. We hereby prove the analogs of the core results of the parallel chip-firing game for the limited increment parallel chip-firing game: we exactly determine the possible cycle lengths for special classes of G(V,E), show that the cycle length cannot be bounded by a polynomial in the number of chips or vertices, and determine the possible atomic firing sequences.

An exact bound on the number of chips of parallel chip-firing games that stabilize

An exact bound on the number of chips of parallel chip-firing games that stabilize

Tutte short exact sequences of graphs

We associate two modules, the \(G\)-parking critical module and the toppling critical module, to an undirected connected graph \(G\). The \(G\)-parking critical module and the toppling critical module are canonical modules (with suitable twists) of quotient rings of the well-studied \(G\)-parking function ideal and the toppling ideal, respectively. For each critical module, we establish a Tutte-like short exact sequence relating the modules associated to \(G\), an edge contraction \(G/e\) and an edge deletion \(G \setminus e\) (\(e\) is a non-bridge). We obtain purely combinatorial consequences of Tutte short exact sequences. For instance, we reprove a theorem of Merino that the critical polynomial of a graph is an evaluation of its Tutte polynomial, and relate the vanishing of certain combinatorial invariants (the number of acyclic orientations on connected partition graphs satisfying a unique sink property) of \(G/e\) to the equality of the corresponding invariants of \(G\) and \(G \setminus e\).Mathematics Subject Classifications: 13D02, 05E40Keywords: Tutte polynomials, chip firing games, toppling ideals, \(G\)-parking function ideals, canonical modules

Simplicial Dollar Game

The dollar game is a chip-firing game introduced by Baker as a context in which to formulate and prove the Riemann-Roch theorem for graphs. A divisor on a graph is a formal integer sum of vertices. Each determines a dollar game, the goal of which is to transform the given divisor into one that is effective (nonnegative) using chip-firing moves. We use Duval, Klivans, and Martin's theory of chip-firing on simplicial complexes to generalize the dollar game and results related to the Riemann-Roch theorem for graphs to higher dimensions. In particular, we extend the notion of the degree of a divisor on a graph to a (multi)degree of a chain on a simplicial complex and use it to establish two main results. The first of these generalizes the fact that if a divisor on a graph has large enough degree (at least as large as the genus of the graph), it is winnable; and the second generalizes the fact that trees (graphs of genus $0$) are exactly the graphs on which every divisor of degree $0$, interpreted as an instance of the dollar game, is winnable.

Skeleton Ideals of Certain Graphs, Standard Monomials and Spherical Parking Functions

Let $G$ be a graph on the vertex set $V = \{ 0, 1,\ldots,n\}$ with root $0$. Postnikov and Shapiro were the first to consider a monomial ideal $\mathcal{M}_G$, called the $G$-parking function ideal, in the polynomial ring $ R = {\mathbb{K}}[x_1,\ldots,x_n]$ over a field $\mathbb{K}$ and explained its connection to the chip-firing game on graphs. The standard monomials of the Artinian quotient $\frac{R}{\mathcal{M}_G}$ correspond bijectively to $G$-parking functions. Dochtermann introduced and studied skeleton ideals of the graph $G$, which are subideals of the $G$-parking function ideal with an additional parameter $k ~(0\le k \le n-1)$. A $k$-skeleton ideal $\mathcal{M}_G^{(k)}$ of the graph $G$ is generated by monomials corresponding to non-empty subsets of the set of non-root vertices $[n]$ of size at most $k+1$. Dochtermann obtained many interesting homological and combinatorial properties of these skeleton ideals. In this paper, we study the $k$-skeleton ideals of graphs and for certain classes of graphs provide explicit formulas and combinatorial interpretation of standard monomials and the Betti numbers.

Stable Divisorial Gonality is in NP

Divisorial gonality and stable divisorial gonality are graph parameters, which have an origin in algebraic geometry. Divisorial gonality of a connected graph G can be defined with help of a chip firing game on G. The stable divisorial gonality of G is the minimum divisorial gonality over all subdivisions of edges of G. In this paper we prove that deciding whether a given connected graph has stable divisorial gonality at most a given integer k belongs to the class NP. Combined with the result that (stable) divisorial gonality is NP-hard by Gijswijt et al., we obtain that stable divisorial gonality is NP-complete. The proof consists of a partial certificate that can be verified by solving an Integer Linear Programming instance. As a corollary, we have that the total number of subdivisions needed for minimum stable divisorial gonality of a graph with m edges is bounded by mO(mn).

A combinatorial method for computing characteristic polynomials of starlike hypergraphs

By using the Poisson formula for resultants and the variants of chip-firing game on graphs, we provide a combinatorial method for computing a class of of resultants, i.e. the characteristic polynomials of the adjacency tensors of starlike hypergraphs including hyperpaths and hyperstars,which are given recursively and explicitly.

Chip-firing based methods in the Riemann–Roch theory of directed graphs

Baker and Norine proved a Riemann–Roch theorem for divisors on undirected graphs. The notions of graph divisor theory are in duality with the notions of the chip-firing game of Björner, Lovász and Shor. We use this connection to prove Riemann–Roch-type results on directed graphs. We give a simple proof for a Riemann–Roch inequality on Eulerian directed graphs, improving a result of Amini and Manjunath. We also study possibilities and impossibilities of Riemann–Roch-type equalities in strongly connected digraphs and give examples. We intend to make the connections of this theory to graph theoretic notions more explicit via using the chip-firing framework.

Diffusion on graphs is eventually periodic

We study a variant of the chip-firing game called \emph{diffusion}. In diffusion on a graph, each vertex of the graph is initially labelled with an integer interpreted as the number of chips at that vertex, and at each subsequent step, each vertex simultaneously fires one chip to each of its neighbours with fewer chips. Since this firing rule may result in negative labels, diffusion, unlike the parallel chip-firing game, is not obviously periodic. In 2016, Duffy, Lidbetter, Messinger and Nowakowski nevertheless conjectured that diffusion is always eventually periodic, and moreover, that the process eventually has period either 1 or 2. Here, we establish this conjecture.

Woven Nematic Defects, Skyrmions, and the Abelian Sandpile Model.

We show that a fixed set of woven defect lines in a nematic liquid crystal supports a set of nonsingular topological states which can be mapped on to recurrent stable configurations in the Abelian sandpile model or chip-firing game. The physical correspondence between local skyrmion flux and sandpile height is made between the two models. Using a toy model of the elastic energy, we examine the structure of energy minima as a function of topological class and show that the system admits domain wall skyrmion solitons.

A maximizing characteristic for critical configurations of chip-firing games on digraphs

Aval et al. proved in Aval et al. (2016) that starting from a critical configuration of a chip-firing game on an undirected graph, one can never achieve a stable configuration by reverse firing any non-empty subsets of its vertices. In this paper, we generalize the result to digraphs with a global sink where reverse firing subsets of vertices is replaced with reverse firing multi-subsets of vertices. Consequently, a combinatorial proof for the duality between critical configurations and superstable configurations on digraphs is given. Finally, by introducing the concept of energy vector assigned to each configuration, we show that critical and superstable configurations are the unique ones with the greatest and smallest (w.r.t. the containment order), respectively, energy vectors in each of their equivalence classes.

A Chip-Firing Game on the Product of Two Graphs and the Tropical Picard Group

Cartwright (2015) introduced the notion of a weak tropical complex in order to generalize the theory of divisors on graphs from Baker and Norine (2007). A weak tropical complex $\Gamma$ is a $\Delta$-complex equipped with algebraic data that allows it to be viewed as the dual complex to a certain kind of degeneration over a discrete valuation ring. Every graph has a unique tropical complex structure (which is the same structure studied by Baker and Norine) in which divisors correspond to states in the chip-firing game on that graph. Let $G$ and $H$ be graphs, and let $\Gamma$ be a triangulation of $G\times H$ obtained by adding in one diagonal of each resulting square. There is a particular weak tropical complex structure on $\Gamma$ that Cartwright conjectured was closely related to the weak tropical complex structures on $G$ and $H$. The main result of this paper is a proof of Cartwright's conjecture. In preparation, we discuss some basic properties of tropical complexes, along with some properties specific to the product-of-graphs case.

Hall–Littlewood symmetric functions via the chip-firing game

Via the chip-firing game, a class of Schur positive symmetric functions depending on four parameters is introduced for any labeled connected simple graph. Tableaux formulae are stated to expand such symmetric functions in terms of the complete symmetric functions. In the case of a simple path, the resulting symmetric functions reduce to the transformed Hall–Littlewood symmetric functions when the parameters are suitable specialized.

Chip-Firing Game and a Partial Tutte Polynomial for Eulerian Digraphs

The Chip-firing game is a discrete dynamical system played on a graph, in which chips move along edges according to a simple local rule. Properties of the underlying graph are of course useful to the understanding of the game, but since a conjecture of Biggs that was proved by Merino López, we also know that the study of the Chip-firing game can give insights on the graph. In particular, a strong relation between the partial Tutte polynomial $T_G(1,y)$ and the set of recurrent configurations of a Chip-firing game (with a distinguished sink vertex) has been established for undirected graphs. A direct consequence is that the generating function of the set of recurrent configurations is independent of the choice of the sink for the game, as it characterizes the underlying graph itself. In this paper we prove that this property also holds for Eulerian directed graphs (digraphs), a class on the way from undirected graphs to general digraphs. It turns out from this property that the generating function of the set of recurrent configurations of an Eulerian digraph is a natural and convincing candidate for generalizing the partial Tutte polynomial $T_G(1,y)$ to this class. Our work also gives some promising directions of looking for a generalization of the Tutte polynomial to general digraphs.

On Computation of Baker and Norine’s Rank on Complete Graphs

The paper by M. Baker and S. Norine in 2007 introduced a new parameter on configurations of graphs and gave a new result in the theory of graphs which has an algebraic geometry flavor. This result was called Riemann-Roch formula for graphs since it defines a combinatorial version of divisors and their ranks in terms of configurations on graphs. The so called chip firing game on graphs and the sandpile model in physics play a central role in this theory. In this paper we present an algorithm for the determination of the rank of configurations for the complete graph $K_n$. This algorithm has linear arithmetic complexity. The analysis of number of iterations in a less optimized version of this algorithm leads to an apparently new parameter which we call the prerank. This parameter and the parameter dinv provide an alternative description to some well known $q,t$-Catalan numbers. Restricted to a natural subset of configurations, the two natural statistics degree and rank lead to a distribution which is described by a generating function which, up to a change of variables and a rescaling, is a symmetric fraction involving two copies of Carlitz $q$-analogue of the Catalan numbers. In annex, we give an alternative presentation of the theorem of Baker and Norine in purely combinatorial terms.

Complex networks and activity spreading

We give a brief survey of the basic notions of the theory of complex networks and various models of activity spreading in networks. We consider the role of random graph theory for the study of complex networks, describe small world and scale-free networks, activity spreading models in social networks, percolation processes, the concept of self-organized criticality, and two versions of its formalization: cellular automata and the chip firing game. We note the common elements of all considered models.

Divisors on graphs, binomial and monomial ideals, and cellular resolutions

We study various binomial and monomial ideals arising in the theory of divisors, orientations, and matroids on graphs. We use ideas from potential theory on graphs and from the theory of Delaunay decompositions for lattices to describe their minimal polyhedral cellular free resolutions. We show that the resolutions of all these ideals are closely related and that their $\mathbb{Z}$-graded Betti tables coincide. As corollaries, we give conceptual proofs of conjectures and questions posed by Postnikov and Shapiro, by Manjunath and Sturmfels, and by Perkinson, Perlman, and Wilmes. Various other results related to the theory of chip-firing games on graphs also follow from our general techniques and results.