Generalized Percolation Processes UsingPretopology Theory

Soufian Ben Amor, Vincent Levorato and Ivan LavalleeComplex Systems Modeling and CognitionEurocontrol and EPHE Joint Research Lab

41 rue G. Lussac, F75005 ParisEmail :{sofiane.benamor, vincent.levorato, ivan.lavallee}@ephe.sorbonne.fr

Abstract- We propose in this paper a generalization of perco- section 5 and 6, a percolation based modeling of forest fire,lation processes in Z2 using the pretopology theory. We formalize expressed on a pretopological space, and its simulation. Wethe notion of neighborhood by extending it to the concept of conclude, in section 7, with a discussion of the results andproximity, expressing different types of connections that may future work.exist between the elements of a population. A modeling andsimulation of forest fire using this approach shows the efficiencyof this formalism to provide a realistic and powerful modeling II. PERCOLATION THEORYof complex systems. The first percolation model was introduced by Simon Broad-

I. INTRODUCTION bent et John M. Hammersley in 1957, using the example of a

Percolation Theory studies the deterministic propagation porous stone immersed in a bucket of water. This fundamentalfluid' on a random medium [4]. As phase transition question was asked: What is the probability that the center

phenof enon, an abrupt change in the behavior of the systemof the stone is wetted? Equivalently, what is the probabilityphenomenon, an abrupt change in the behavior of the system, tha an inint siz peclto 'lse fprseit.Ti

observed~~~~~~~~~~~~~~~inpeclto'rcse sacaatrsi rprthat an infinite size percolation cluster of pores exists. This

y probability depends on the porosity of the stone (i.e. theof complex systems, percolation theory was successfully ap-plied to model and simulate complex phenomena in statistical densityo pores [4]Physical problems are mathematically modeled as a networkphysics, economy and recently in social networks. However, of points (or vertices) and the connections (or edges) betweenpercolation processes are based on the neighborhood concept each two neighbors may be open (allowing the liquid to passwhich iS not well formalized. It iS difficult to express a ..

through) with probability p, or closed with probability (1-p),certain kind of relations between the components of theand we assume they are independent. For a given p, what issystem such as remote connections, the dynamical evolution ofthe probability that an open path exists from the top to thethe neighborhood basis, the heterogeneity of the interactions bottom? Generall the interest concerns the behavior for large

and the hierarchy between the components. For example, toy'hc wm n. As is quite typical, it is actually easier to examine infinitemodel the spreading of oil in water, of fire through forests networks than just large ones. In this case the correspondingor infectious diseases among a population, it is difficult to question is: does there exist an infinite open cluster ? That is,express the qualitative properties of the system. is there a path of connected points of infinite length "through"In this work, we provide a generalized formulation of perco-lation processes in z2 using pretopology theory[2]. This can be tentok nti aew a s omgrvszr-nlachievproed sesinusingthe pretopological conceptsof p hiseoclue law to see that, for any given p, the probability that an infiniteachieved using the pretopological concepts of pseudoclosure clse exst is eihe zeoo n .Sic thspoaiiyiand closed subsets. It is a general formalism that expresses.crexissi ther zero orione. Sincebthisyprobabilityaiincreasing, there must be a critical probability PC such thatdifferent types of connections that may exist between the (figure 1)components of a system. It provides also theoretical tools toexpress specific phenomena in distributed systems such as the p() =O if P<Pcalliance phenomena, acceptability processes and emergence of =1 if p>Pccollective behavior. where P is the percolation probability which indicates the

The rest of the paper is organized as follows: in section 2, probability of appearance of the giant cluster in the system.we present an overview of percolation theory. The section 3 A model where we open and close vertices rather thanconcerns a general presentation of pretopology theory. The edges, is called site percolation (figure 2 a) while the modeldynamical aspect of the structural transformation and the Ib ao imrpo l cl b percoatio

neighborhood~ ~ ~~ ~ ~ ~~ esrenotionareforalzeinpelsection4.nToeillustratneigboroodnoton re ormlize insecion4 T ilustate (figure 2 b). The model where the uncertainty concerns boththe efficiency of our approach we present, respectively in sieanbodiscldmxdproatn(fge2c)

1The word fluid is considered in the widest sense of the term(water, If we consider the percolation processes on 22, for example,information, disease, etc.). we have different models: depending on the chosen neighbor-

1-4244-0695-1/07/$25.00 ©2007 IEEE. 130

r 1 al 7 I I I I L T 10

I( L,t- Ili, I, ii!] I L

Fig. 4. Moore neighborhood (eight neighbors) and Von Neumann neighbor-hood (four neighbors).

0 t. ip] 0 _ lg~~ti

Fig. 1. Phase transition around the critical probability. Source in [8] Fig. 5. Pretopological space.

(a) (b) (c)

phenomena, tolerance and acceptability processes and emer-gence of collective behavior. Even if closure operators havebeen widely studied in algebra, topology and computer sciencetheory the axiomatics that define them are too limited to

Fig. 2. Basic percolation models express concrete problems.[6]

A. Pretopology formalismLet us consider a non-empty finite set E, and 'P(E) desig-

hood, on the nature of the studied aspect (dynamic or static ....nates all of the subsets of E.aspect of a system) and on the nature of the studied object (a 1) Pretopological space:specific phenomenon or a property of a system). The most used Definition 1: A pretopological space is a pair (E, a) (figurekind of neighborhoods are the Von Neumann neighborhood 5) where a is a map a(.) : 19(E) - 'P(E) called pseudo-and the Moore neighborhood (figure 4). The values of the closure and defined as follows: VA, A C E the pseudo-closurecritical parameters depend on the used neighborhood. We can of A, a(A) C E such thatsee in the figure 3 that an infinite cluster (according to aVon Neumann neighborhood) appears abruptly at a critical 0a(0) 0 (PF)threshold Pc - 0, 6. This value is different if we consider * A C a(A) (2)a Moore neighborhood. In fact, with a Moore neighborhood The pseudo-closure is associated to the dilation process (figurewe obtain a giant connected cluster with less density (i.e. 6).probability of active sites. We can also define a pretopological space (E, i) using the

interior map (figure 7). VA, A C E we define the interiorIII. PRETOPOLOGY THEORY i(A) C E such that:

Pretopology theory expresses the structural transformation i(A)= [a(Ac)]cof sets composed of interacting elements. It allows the repre-sentation of discrete structures, thanks to a general notion of VA, A C E,i(A) c Aneighborhood. The pretopological concepts of pseudoclosure The interior map is associated to the erosion process (figureand closed subsets developed in [2] allows a mathematical 8).representation of various aspects and phenomena such asconstitution of decisive coalition among a population, alliance

(a0- o14 (b - 6 (Xp- U a

Fig. 3. Appearance of a giant cluster. Fig. 6. Dilation process.

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E ~~ ~ ~ ~ ~ ~ ~

Fig. 9. Examples of neighborhood basis.

a) b) L

Fig. 7. Interior map. ,5 A I

Fig. 10. The dilation process with a user-defined neighborhood (in red) andthe pseudo-closure (red & blue).

of connections and take into account their evolution overtime by changing their neighborhood basis. An example ofthe efficiency of this approach is presented in the followingparagraph and concerns the forest fire modeling. Our choice

D was motivated by the fact that this phenomenon exhibits aA C E: phase transition behavior [3], which is a common property of

. A is a closed subset if a(A) A complex systems, and the fact that several percolation based

. A is an open subset if (A) A models were developed to simulate it. We can also show2) V Pretopological spaceE: through it, the difficulty to express a certain kind of qualitativeDefinition 3: A V pretopological space (E, i, a) is defined factors using the traditional neighborhood definition.

by:. VA, B, A C E B C E and A c B then a(A) c a(B) V. EXAMPLE OF APPLICATION: FOREST FIRE MODELING. equivalently i(A) c i(B) A. Percolation andforestfire modeling

IV. EXPRESSING A PERCOLATION PROCESS ON A In the case of the forest fire modeling, the density of treesPRETOPOLOGICAL SPACE is the most important factor. In fact, fire spreads by ignition of

nearby trees, and the density of trees decides how far the fireUsing pretopology we can define a general percolation spreads. If the trees are sparse, the fire will burn out before it

process including the classical ones and allowing new ones. gets very far. If the forest density exceeds a certain threshold,The advantage of this approach is that it allows us to formulate the fire will burn straight through the forest.and treat structural dynamical aspects of complex systems in Several percolation based models were presented to simulatea unified manner. Percolation processes on Z2 depend on the the propagation of fire through forests. In these models,neighborhood. Using a neighborhood basis (set of objects of generally, trees are arranged in a rectangular grid where theV(x)) expressed in a pretopological space, any neighborhood spread of fire is propagated from a tree to its neighbors,V(x) can be constructed. The diversity of the connections defined in an ad-hoc manner. The basic difference betweenbetween the elements can be expressed by adapting the choice these model is the neighborhood definition (figure 11). Forof the pseudoclosure to the studied problem. example, some models using a Von Neumann neighborhood

A. Neighborhood sets are not realistic because they do not take into account the windLet (E, i, a) be a V pretopological space. effect. For this reason models using the anisotropic percolation

Vxc C E :or oriented percolation were developed in order to provideV(z) {V c E/ E i(V)} a more realistic modeling [7]. All these models could be

considered as particular cases of a general percolation modelV(x) is a family of neighborhoods of x. if the the neighborhood concept was well formalized. With

B. Pretopological transformation in Z2 the mathematical definition of the neighborhood expressed ina pretopological space we are able not only to unify perco-

V1A, A C E, a(A) = {x C E/\1VV V C V(z), VOA #t 0} If lation processes formalism but also provide a more efficientwe consider, for example, three different neighborhoods as in modeling.figure 9 and construct the associated pseudoclosure the dilationprocess have different propagation effects (figure 10a, 10b, B. Percolation based model of forest fire expressed in a10c). pretopological spaceWith the concepts of pseudoclosure, interior and neighbor- Using the pseudo-closure, we express the propagation pro-

hood sets we can express in a general way different types cess of the fire according to the direction of the wind, defined

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Moore

Fig. 14. By applying the interior map we model the consumed trees (inactiveFig. 11. The neighborhood is the basic difference between various percolation sites in the propagation process).processes.

Ms

|~~~~~~~~~~~~~~~~~~i ,.!*XXik d ft =e=

I |

Fig. 12. Propagation process using the specified neighborhood basis. 11...........using the proximity bases determining the neighborhood V(x)(figure 12).By changing the neighborhood basis we can model the e

wind direction changes (figure 13). We can also model theconsumed trees using the interior mapping (figure 14).

The interior map is defined as follows TreeI r

VA, A C E, i(A) = etE E/3V,V E V(x), V C A} = G.id =

VI. SIMULATION Fig. 15. The propagation process using the successive computation of theVI.SIMULATION pseudoclosure and interior maps.

A preliminary simulation and implementation of the conceptof neighborhood and in a pretopological space was performedusing Java (figure 15 and 16). The user can define until fiveneighborhood bases (to express the direction of the wind or ercoFrethe presence of an obstacle) and specify the number of theiterations. Trees are generated randomly according to fixedprobability (i.e. density) using a slider. We assumed thatburning trees will be consumed after three iterations. The Ne4ihboue6rhd 3results of the simulation are reported in the plot representing a S: othe evolution of the consumed trees proportion over time. Set

VII. CONCLUSIONWe showed that interesting concepts developed in pretopol- a

ogy theory may be applied to formalize the neighborhood no- Spd

b)

wIiXt.tblg N-N W > I; A j -B vtXgmf ________________c__ Tee lDensia iaz rwPoi : 1 S X F i W ; ;7S% Grid size, is

Fig. 16. The effect of the canges of the wind direction on the propagation

Fig. 13. By changing the neighborhood basis we model the wind direction process.changes.

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tion, to generalize percolation processes in Z2 and to enhancethe modeling of complex phenomena and distributed systems.In a future work, we aim to provide a general formalismunifying the mathematical description of percolation processesand cellular automata in order to express the heterogeneity ofneighborhoods and local dynamics in asynchronous distributedsystems.

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