The lattice structure of rotor-router model Le Manh Ha Hue University’s College of Education and Institute of Mathematics, 18 Hoang Quoc Viet, Hanoi, Vietnam Email: [email protected]

Abstract—In this paper 1 , we study the rotor router model in the relation with the famous discrete dynamical system - Chip Firing Game. We consider the rotor router model as a discrete dynamical system defined on digraph and we use order theory to show that its state space started from any state is a lattice, which implies strong structural properties. The lattice structure of the state space of a dynamical system is of great interest since it implies convergence (and more) if the state space is finite. Moreover, we also attempt to define the class L(R) of lattices that are state space of a rotor router model, and compare it with the class of distributive lattices and the class of ULD lattices. Index Terms—Chip Firing Game, Discrete dynamical system, state space, rotor router, order and lattice structure.

I. I NTRODUCTION The abelian sandpile or Chip Firing Game (CFG) and rotor-router models were discovered independently in different contexts. The Chip Firing Game (CFG) was introduced by Bjorner, Lov´asz and Shor in 1991 to illustrate the behaviors of distributed jobs in networks [3]. Then this model has became a very famous one which can be used to illustrate many systems in different science domains. For example, in complex systems research, CFG was considered as a paradigm for the so-called self organized criticality [1], [4], [19]; in economy or computer science, CFG was studied as a resource distribution systems [5], [13]. Because of this important role, many approaches to investigate the behavior of CFG were developed, from physics experimental techniques [1], [19] to other methods using algebraic structures [4], formal languages [3], [2] or enumerative combinatorics [17], [20]. Related ideas were explored earlier by Engel [10] in the form of a pedagogical tool (the ”probabilistic abacus”), and by Lorenzini [18] in connection with arithmetic geometry. The rotor-router model was first introduced by Priezzhev et al. [16] (under the name Eulerian walkers model) in connection with self-organized criticality. It was rediscovered several times: by Rabani, Sinclair and Wanka [24] as an approach to load-balancing in multiprocessor systems, by Propp [15] as a way to derandomize models such as internal diffusion-limited aggregation (IDLA) [7], and by Dumitriu, Tetali, and Winkler as part of their analysis of a graph-based game [14]. To define the rotor-router model on a directed graph G, for each vertex of G, fix a cyclic ordering of the outgoing edges. To each vertex v we associate a rotor ρ(v) chosen from among the outgoing edges from v. A chip performs a walk on 1 This work is supported in part by Vietnam’s National Foundation for Science and Technology Development (NAFOSTED).

G according to the rotor-router rule: if the chip is at v, we first increment the rotor ρ(v) to its successor e = (v, w) in the cyclic ordering of outgoing edges from v, and then route the chip along e to w. If the chip ever reaches a sink, i.e. a vertex of G with no outgoing edges, the chip will stop there; otherwise, the chip continues walking forever. In another context, the Conflicting Chip Firing Game (CCFG) model [23], an extension of CFG, has many close connection. In this model, the condition of firing rule is reduced and the transferring of chips similar to the rotorrouter model. The vertex v is firable if it contains at least one chip and its firing is carried out by sending one chip along one edge from v to one of its neighbors. We also denote by CCF G(G, n) the set of all configurations of CCF G(G, n) and call the configuration space of this game. This set is exactly the set of compositions of n into V . It easy to check that the configuration space of CFG is a subset of the configuration space of CCFG on the same support graph and we are going to see that the state space of rotorrouter model is too. In the rotor-route model, at each time a chip at a vertex which is not the sink just can walk to only one of its neighbor, while a chip in the CCFG model can walk to many other neighbor depending on the out-degree of the vertex which it stores. The structure of configuration space of CCFG is not a lattice, but we has shown the order structure of this model on directed acyclic graph by using energy functions [12] and we has constructed the algorithm to determine this order in [8]. We has also studied the reachability of this model in the relation with the flow network problem and we has constructed the polynomial algorithm to determine the reachability of this model on general digraph in [9]. In Section 2, we first represent the definition of rotor-router model in the relation with the discrete dynamical systems Chip Firing Games. Then some important results of the order structure, the lattice structure of CFGs are also represented in this section. In section 3, we investigate the rotor-router models with no closed component and give the strong structural properties of this model. We show the state space of this model has lattice structure by using the notation of shot-vector and some techniques in [17]. In Section 4, we attempt to characterize the class L(R) of lattices that are state space of a rotor-router model.

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II. C HIP F IRING G AMES AND ROTOR - ROUTER MODEL The goal of this section is to introduce some results of Chip Firing Games in the relation with rotor-router models. First of all, we recall here some basic notions in the Order and Lattice Theory. After that, we represent some results of lattice structure of CFG on digraph with no closed component and order structure of CCFG on directed acyclic graph (DAG). An order relation or partial order relation is a binary relation ≤ over a set, such that for all x, y and z in this set, x ≤ x (reflexivity), x ≤ y and y ≤ z implies x ≤ z (transitivity), x ≤ y and y ≤ x implies x = y (antisymmetry). The set is then called a partially ordered set or, for short, a poset. A lattice L is an poset such that for any two elements a and b of L, there exists a unique smallest element which is greater than a and b (the supremum of a and b, denoted by a ∨ b) and there exists a greatest element which is smaller than a and b (the infimum of a and b, denoted by a ∧ b). A lattice L is distributive if it satisfies one of the two following laws of distributivity (which are equivalent):

the way of firings is, after exactly the same time, the system reaches the same fixed point. Moreover, this convergence is very strong in the following sense: for two configurations of a system, there exists a unique first configuration obtain for them, and every configuration which can be obtained from both of them can be obtained from this first one. In [20], the authors proved that the set of configuration spaces of all CFG is strictly included in the set of U LD lattices and they introduced the coloured Chip Firing Game, that generates exactly the class of U LD lattices. Next, we recall one of the most important result about the characterization of the order structure of CCFG on a special class support graph. Theorem 2. [12] The configuration space CCF G(G, n) of conflicting Chip firing game on DAG G is ordered with the reflexive and transitive closure of the successor relation. 2

∀x, y, z ∈ L, x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z)

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∀x, y, z ∈ L, x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z) A lattice is a hypercube of dimension n if it is isomorphic to the set of all subsets of a set of n elements, ordered by inclusion. It is also called a boolean lattice. A lattice is upper locally distributive (denoted by U LD [21]) if the interval between an element and the supremum of all its upper covers is a hypercube. The lower locally distributive lattices (LLD) is dual. Let us present here one of the most important results about the lattice structure of CFG. Theorem 1. [17] The configuration space of a CFG on a directed graph with no closed component and with an arbitrary initial configuration O ordered with the reachability relation is a lower locally distributive lattice.

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The configuration space of a CFG with 9 chips.

This Theorem derives many consequences: first, it proves the convergent property of CFG, then it shows that whatever

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Now, we define the rotor-router model and we will see that on the support digraph G in which every vertex has out-degree at most one, the rotor-router model is nothing but the CCFG model on the same support graph. Given a directed graph G, fix for each vertex v a cyclic ordering of the edges emanating from v. For an edge e with tail v we denote by e+ the next edge after e in the prescribed cyclic ordering of the edges emanating from v. Definition 1. A chip configuration σ on G, is a vector of nonnegative integers indexed by the non-sink vertices of G, where σ(v) represents the number of chips at vertex v. Definition 2. [22] A rotor configuration is a function ρ that assigns to each non-sink vertex v of G an edge ρ(v) emanating from v. If there is a chip at a non-sink vertex v of G, routing the chip at v (for one step) consists of updating the rotor configuration so that ρ(v) is replaced with ρ(v)+ , and then

moving the chip to the head of ρ(v)+ . A single-chip-and-rotor state is a pair consisting of a vertex w (which represents the location of the chip) and a rotor configuration ρ. The rotor router operation is the map that sends a single-chip-and-rotor state (w, ρ) (where w is not a sink) to the state (w+ , ρ+ ) obtained by routing the chip at w for one step. A chip-androtor state (for short, state) is a pair τ = (σ, ρ) consisting of a chip configuration σ and rotor configuration ρ on G. By using the definition of rotor-router in [22], we consider rotor-router as a discrete dynamical system on digraph and redefine it as follows: Definition 3. The rotor-router on digraph G, denoted by R(G), is a dynamical model defined as follows: each state is a chip-and-rotor-state on G; a non-sink vertex is active if it has at least one chip; the evolution rule (firing rule) of this system is the firing of one active vertex v and firing v results in a new state given by replacing the rotor ρ(v) with ρ(v)+ and moving a chip from v to the head of ρ(v)+ (and removing the chip if the head of ρ(v)+ is a sink). We call state space, and denote by R(G), the set of all state on G. 

Definition 4. Given two state τ and τ of a R(G), we say that τ  is reachable from τ , denoted by τ  ≤ τ , if τ  can be obtained from τ by a firing sequence (in the case the firing sequence is empty, τ = τ  ). In particular, we write τ ≺ τ  if τ  is obtained from τ by applying once firing rule and we call τ  a successor of τ . We say that τ is stable if no vertex can fire, i.e., all chips have moved to sinks. Definition 5. Given R(G) and let τ0 be a state of R(G). We denote by R(G, τ0 ) the state space of all reachable states from τ0 .

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The state τ0 = ((2, 1); (1, 3)) (left) and its state space R(G, τ0 )

III. ROTOR -ROUTER WITH NO CLOSED COMPONENT As we will see, there is an important link between chip firing game and rotor-router. A hint at this link comes from a straightforward count of configurations. Recall that a stable chip configuration of classical CFG is a way of assigning some number of chips between 0 and dv − 1 to each nonsink vertex v of G. Thus, the number of stable configurations is exactly v dv , where the product runs over all non-sink vertices. This is also the number of rotor configurations on G. Other connections become apparent when one explores the appropriate notion of recurrent states for the rotor-router model. We will treat the cases of digraphs with no closed component. Definition 6. Let G be a digraph. A closed component of G is a nontrivial (more than one element) proper subset S of the set of the vertices of G such that • there exists a path from any element of S to any other element of S (S is a nontrivial strongly connected component) • there is no outgoing edge from any element of S to any vertex of G \ S (S is closed) It is clear that the (support of the) CFG in Figure 1 has no closed component, since its unique nontrivial strongly connected component is composed of the two topmost vertices, and there is an edge from this component to the third vertex, which is a sink. Let us recall that when a chip arrive to such a sink, it can never go out. Notice that a closed component behaves as a sink, since it also has this property. Now, we will show an important lemma which allows the study of the special case where all the nontrivial strongly connected components of the support of the game have at least one outgoing edge, i.e, an edge from one vertex in the component to a vertex outside the component. This means that the support graph of the game has no closed component. By using the same technique in [17], we can show the order structure of state space of a rotor-router model with no closed component. Lemma 1. Let us consider a non closed strongly connected component C. Starting from a state τ = (σ, ρ) there is no nonempty sequence of firings of vertices in C such that the state τ is reached again. This lemma implies that if we only fire vertices which are not in a closed component, then we can not have a cycle of state. Therefore, we deduce that if the support graph of a rotorrouter model has no closed component then its state space contains no cycle, and so it is a poset. Theorem 3. The state space of a rotor-router with no closed component is partially ordered by the reflexive and transitive closure of the successor relation. Recall that a vertex is a sink if its out-degree is zero. A global sink is a sink s such that from every other vertex there is a directed path leading to s. Note that if there is a global sink,

then it is the unique sink. It easy to see that if G is a digraph with a global sink then G has no closed component. So, by using the similar reasoning in [22], we have the following lemma. Lemma 2. [22][Lemma 3.9] Let G be a digraph with no closed component. Let τ0 , τ1 , . . . , τn be a sequence of chipand-rotor states of R(G), each of which is a successor of the  is another such sequence, and τn one before. If τ0 , τ1 , . . . , τm is stable, then m ≤ n. If in addition τm is stable, then m = n and τn = τn , and for each vertex w, the number of times w fires is the same for both sequences. This lemma shows that starting from a state τ there is at most one stable state which can be reached by a finite sequence of firings. Moreover, the Lemma 1 shows that the state space of a rotor-router with no closed component has no cycle and the number of state is finite, so starting from a state τ there is exactly one stable state. We denote it τ 0 and call it the stabilization of τ . Now, we prove that every path between any two states has the same length, that means the number of applications of firing rule to the vertices v ∈ V during every firing sequence to obtain state τ  from state τ are the same. Given a firing sequence p, we denote by |p|i the number of applications of firing rule to the ith vertex during the sequence p and denote by |p| the number of vertices (may be repeated) in sequence p. By using the Lemma 2, we have the following result: Lemma 3. Given a rotor-router R(G) with no closed component, if starting from the same state, two sequences of firing s and t lead to the same final state, then |s|i = |t|i for each vertex i. This lemma allows us to define the shot vector k(τ, τ  ) of two states τ and τ  if b can be obtained from τ in a rotorrouter R(G) : k(τ, τ  ) = (k1 (τ, τ  ), k2 (τ, τ  ), . . . , kn (τ, τ  )), where ki (τ, τ  ) is the number of firings of vertex i to obtain τ  from τ . If τ and τ  are two states obtained from the same state τ0 , we order k(τ0 , τ ) ≤ k(τ0 , τ  ) if for all i, ki (τ0 , τ ) ≤ ki (τ0 , τ  ). Moreover, if τ  ≥ τ then k(τ0 , τ  ) = k(τ0 , τ ) + k(τ, τ  ). By using the proof technics in [17], we can characterize the order between all the state obtained from the initial one τ0 in a rotor-router by comparing their shot-vectors as follows. Theorem 4. Let τ and ς be two states of R(G, τ0 ). Then τ ≥ ς in R(G, τ0 ) if and only if k(τ0 , τ ) ≤ k(τ0 , ς). By using the same technique in [17], we can show the strong structure, that is the lattice structure of the state space of R(G, τ0 ). Theorem 5. Let R(G) be a rotor-router with no closed component and τ0 be a state of R(G). Then R(G, τ0 ), ordered with the reflexive and transitive closure of the successor relation, is a lattice. Moreover, the infimum of two elements τ and ς is defined as follows: let k be a vector such that for all vertex i, ki = max(ki (τ0 , τ ), ki (τ0 , ς)) then the state κ such

that k(τ0 , κ) = k is the infimum of τ and ς. IV. S TUDY OF THE CLASS L(R) The study of the configuration spaces structure of some discrete dynamical systems has been presented in [20] and [11]. We have seen above that the proof of lattice structure of rotor-router model is similar to the lattice structure of Chip Firing Game. Denote by L(CF G) the class of lattices induced by CFG. In [20] it has been proved that L(CF G) is not the whole ULD class (i.e, there exists a ULD lattice which is the configuration space of no CFG), but contains the class D of distributive lattices. This is an interesting result from the lattice theory point of view, since the distributive and ULD lattices classes are very closed one to another. We have the following relations: D  L(CF G)  U LD. The classify of lattices which induced by some models emphasizes the complexity of the characterization problems in lattice theory and plays an important role in the computation complexity of systems. Given a rotor-router model with no closed component R = R(G, τ0 ), we denote by L(R) its state space which is a lattice. We denote by L(R) the class of lattices that are the state space of a rotor-router model. For a lattice L ∈ L(R), we say a corresponding rotor-router any rotor-router model R = R(G, τ0 ) such that L(R) = L. In this section, we are going to study the class L(R) in the relation with the class D and class ULD. The rotor-router model are quite similar to the CFG, not only at the behavior but also at group structure (sandpile group) (see example [6], [22]) and lattice structure. However, the classes L(R) and L(CF G) are not the same. The class D is not included the class L(R). First of all, we can give an example to show that L(R) is not a subset of D. In Figure 4, we show the state space of a rotor-router model which is not in D. Next, in the following theorem, we show that there exists distributive lattices that is not the state space of any rotorrouter model. Theorem 6. The lattice D5 = 1 ⊕ 22 is not in L(R). Proof: Let us suppose that D5 is in L(R). Then there exists rotor-router model R = R(G, τ0 ) such that L(R) = D5 (see Figure 5), where τf is the stable state. Let a and b be the vertices that are fired in τ2 and τ1 to obtain τ3 . Clearly, a = b then by Lemma 3, we have a and b are vertices that fired at the state τ0 to obtain τ1 and τ2 , respectively. At the state τ0 = (σ0 , ρ0 ), there are only vertices a and b active, so σ0 (a) = 0, σ0 (b) = 0 and σ0 (v) = 0 for all v = a, b. It easy to see that σ0 (a) = σ0 (b) = 1. Similarly, at the state τ3 = (σ3 , ρ3 ), there is only the vertex c active and it is fired exactly once, so σ3 (c) = 1 and σ3 (v) = 0 for all v = c. We can describe the states τ0 and τ3 in the Figure 7. After firing the vertex a at the state τ0 = (σ0 , ρ0 ) we obtain the state τ1 = (σ1 , ρ1 ). So we have ρ0 (b) = ρ1 (b). At the state τ1 = (σ1 , ρ1 ), there is only the vertex b active, so σ1 (b) = 0 and σ1 (v) = 0 for all v = b. Moreover, after firing the

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vertex b at the state τ1 , we obtain the state τ3 , this implies that σ1 (b) = 1 and σ1 (v) = 0 for all v = b. Thus, after firing the vertex b at the state τ1 chip must be transferred from b + to c, that mean ρ+ 1 (b) = (b, c) and therefore ρ0 (b) = (b, c) (here we denote by (u, v) for the arc in digraph which u is the head and v is the tail). By using similar argument we have + ρ+ 0 (a) = ρ2 (a) = (a, c). Now, at the state τ0 , consider the firing sequence a

initial state. A subset X ⊆ V is a valid shot-set if there exists a state τ reachable from the initial state such that s(τ ) = X. It is easy to check that the lattice of the state space of a CFG is isomorphic to the lattice of the shot-sets of its state space ordered by inclusion. The join is given by the following formula: Lemma 4. Let R = R(G, τ0 ) be a rotor-router model, L(R) be its state space and τ, τ  be two states. The join of τ and τ  in L(R) is determined by:

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The classes of lattices induced by various systems.

The state space lattice of a rotor-router is not distributive.

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→ τ1 − → τ3 . τ0 − + Since ρ+ 0 (a) = (a, c) and ρ1 (b) = (b, c) then σ3 (c) = 2. This contraction implies that there is no rotor-router models that its state space is the lattice D5 = 1 ⊕ 22 . Now, we will show that the class L(R) is included in the class of ULD by using the notation of shot-set of a state presented in [20]. By Lemma 3, we can define the shot-set s(τ ) of a state τ the set of rules fired to reach τ from the

s(τ ∨ τ  ) = s(τ ) ∪ s(τ  ). As we see, the behavior of rotor-router models is similar to the behavior of CFG in the sense the firing of a vertex v does not prevent the firing of any vertex v  , v  = v because when v is fired, the number of chips in v  stays the same or increases. So the following result is immediate: Theorem 7. The lattice of the state space of a rotor-router model with no closed component is ULD. From this Theorem and Theorem 6, we conclude that the class L(R) of lattices induced by rotor-router models is strictly included in the class of ULD. Now, fix a chip configuration σ on digraph G by considering different rotor configurations, we obtain different states. The corresponding lattices induced from these states are also different. They are eventually very different from one another at the time to reach fixed points and their complexities. These issues are particular of interest in learning about the relationship between the dynamical systems. Firstly, in this paper, we see that if two rotor configuration are acyclic then two corresponding induced lattices are disjoint (see Figure 3,

V. C ONCLUSION

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Figure 9 and Figure 8, lattices are pairwise disjoint) otherwise they contain each other (see Figure 9 lattice L(R(G, (21, 23))) is included in lattice L(R(G, (21, 14)))). Let ρi , i = 1,2, be two rotor configurations on G and let τi = (σ, ρi ) be two states. Let Ri = R(G, τi ) and let Li = L(Ri ), for i = 1, 2. Then we have the following result: Theorem 8. If ρ1 and ρ2 are acyclic then L1 ∩ L2 = ∅ otherwise L1 ⊆ L2 or L2 ⊆ L1 . This theorem implies that if there is a state τ obtained from both τ1 and τ2 then there exists the path between τ1 and τ2 . Proof: Suppose that L1 ∩ L2 = ∅, that means there exists state τ such that τ ≤ τ1 and τ ≤ τ2 . Without loss of generality we may assume that τ is a fixed point and the rotor configuration of state τ is acyclic. Notice that the chip addition operator is a permutation on the set of acyclic rotor configuration (for more detail see [22]), so either ρ1 or ρ2 is not acyclic. Let ci be the number of directed cycles in rotor configuration ρi , (i = 1, 2) and assume that c1 ≤ c2 . Let s(τi ) be the set of rules fired to reach τ from τi . Because of σ1 = σ2 and τ1 differ from τ2 at some cycles of walking of chips and c1 ≤ c2 implies that s(τ2 ) ⊆ s(τ1 ). Therefore, at the state τ1 , we may apply the firing rules at vertices to routing chip to the cycles of ρ2 and we have τ2 ≤ τ1 , that mean L2 ⊆ L1 .

We study the rotor-router model as a discrete dynamical system. We show the lattice structure of this model. We have also compared the class L(R) of lattices that are the state space of rotor-router models with class D and class ULD. We predict that the class L(R) is included in the class L(CF G). In future works, we will study the relation between lattices which have the common initial chip configurations but differ at rotor configurations and the problem determining the shortest path and study the complexity of lattices. Acknowledgements: The author would like to thank PHAN Thi Ha Duong for her invaluable advising, and for providing the initial motivation of this work. R EFERENCES [1] P. Bak, C. Tang, and K. Wiesenfeld. Self-organized criticality: An explanation of 1/f noise. Physics Rewiew Latters, (59):381, 1987. [2] A. Bjorner and L. Lov´asz. Chip firing games on directed graphes. Journal of Algebraic Combinatorics, 1:305–328, 1992. [3] A. Bjorner, L. Lov´asz, and W. Shor. Chip-firing games on graphes. E.J. Combinatorics, 12:283–291, 1991. [4] E. M. Coven and A. Meyerowitz. Tiling the intergers with translates of one finite set. Journal of Algebra, 212:161–174, 1999. [5] J. Desel, E. Kindler, T. Vesper, and R. Walter. A simplified proof for the self-stabilizing protocol: A game of cards. Information Processing Letters, (54):327–328, 1995. [6] D. Dhar. Self-organized critical state of sandpile automaton models. Phys. rev. Lett., (64):1613–1616, 1990. [7] P. Diaconis and W. Fulton. A growth model, a game, an algebra, lagrange inversion, and characteristic classes. Rend. Sem. Mat. Univ. Politec. Torino, 49(1):95119, 1991. [8] LE Manh Ha; NGUYEN Anh Tam; PHAN Thi Ha Duong. Algorithmic aspects of the reachability of conflicting chip firing game. Studies in Computational Intelligence, Springer, 283:359–370, 2010. [9] LE Manh Ha; PHAM Van Trung; PHAN Thi Ha Duong. Reachability of conflicting chip firing game and flow network. (submited), 2010. [10] A. Engel. The probabilistics abacus. Ed. Stud. Math., 6(1):1–22, 1975. [11] E. Goles, M. Latapy, C. Magnien, M. Morvan, and H. D. Phan. Sandpile models and lattices: a comprehensive survey. Theoret. Comput. Sci., 322:383–407, 2004. [12] Le Manh Ha and Phan Thi Ha Duong. Order structure and energy of conflicting chip firing game. (to appear in Acta Math. Vietnam.), 2008. [13] S.-T. Huang. Leader election in uniform rings. ACM Trans. Programming Languages Systems, 15 (3):563–573, 1993. [14] P. Tetali I. Dumitriu and P. Winkler. On playing golf with two balls. SIAM J. Discrete Math., 16(4):604615 (electronic), 2003. [15] J.Propp. Correspondence with david griffeath. 2001. [16] V. B. Priezzhev; D. Dhar; S. Krishnamurthy. Eulerian walkers as a model of self-organised criticality. Phys. Rev. Lett., 77:5079–5082, 1996. [17] M. Latapy and H.D. Phan. The lattice structure of chip firing games. Physica D, 115:69–82, 2001. [18] D. J. Lorenzini. Arithmetical graphs. Math. Ann, 285(3):481–501, 1989. [19] M. Morvan M. Latapy, R. Mataci and H.D. Phan. Structure of some sand piles model. Theoret. Comput. Sci, (262):525556, 2001. [20] C. Magnien, H. D. Phan, and L. Vuillon. Characterization of lattices induced by (extended) chip firing games. Discrete Math. Theoret. Comput. Sci., AA:229–244, 2001. [21] Bernard Monjardet. The consequences of dilworths work on lattices with unique irreductible decompositions. The Dilworth theorems Selected papers of Robert P. Dilworth, Birkhauser, Boston:192–201, 1990. [22] A. E. Holroyd; L. Levine; K. Meszaros; Y. Peres; J. Propp and D. B. Wilson. Chip-firing and rotor-routing on directed graphs. In and out of equilibrium 2., Progr. Probab., 60(3):331–364, 2008. [23] Thi Ha Duong PHAN Tra An PHAM and Thi Thu Huong TRAN. Conflicting chip firing games on directed graphs and on treese. VNU Journal of Science. Natural Sciences and Technology, 24:103–109, 2007. [24] A. Sinclair Y. Rabani and R. Wanka. Local divergence of markov chains and the analysis of iterative load-balancing schemes. In IEEE Symp. on Foundations of Computer Science, 77:694705, 1998.