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Outer Median and Probabilistic Cellular Automata on Network Topologies
Abigail Nussey
Wolfram Research, Inc.100 Trade Center DriveChampaign, IL [email protected]
One- and two-dimensional cellular automata have traditionally mod-eled systems with an underlying regular lattice well. However, for othersystems, such as those with nonlocal phenomena, running cellular au-tomata over different topologies could result in a simpler, more naturalmodel. The creation of so-called “outer median” rules, and the use ofmodified probabilistic rules, make it possible to run cellular automataover network topologies. In this way, the rules themselves remain sim-ple while more complicated nonlocal information is encoded in thetopology. In this manner we can model systems with nonlocal elements,and briefly consider two examples of such systems: country border rela-tionships and forest fire spread.
1. Introduction
One- and two-dimensional cellular automata have traditionally mod-eled particular phenomena well. A simple method for biological muta-tion without resorting to evolutionary processes can be modeled withone-dimensional, three-color, nearest-neighbor cellular automata. Thegrowth of snowflakes can be modeled with two-dimensional, two-color cellular automata on a hexagonal grid as given in [1].
The topologies of noncyclic one- and two-dimensional cellular au-tomata can be expressed as networks, namely, a one-dimensional lin-ear network and a two-dimensional lattice network as shown in Fig-ure 1. Cellular automata on these particular network topologies havebeen studied well relative to other topologies.
The two network topologies in Figure 1 are not necessarily themost intuitive topologies for certain systems. It would stand to reasonthat some systems would benefit from a topological structure thattakes nonlocality into account, or global influential sources, or “deadzones” that allow no geographical connections, or other topologicalstructures.
Complex Systems, 18 © 2010 Complex Systems Publications, Inc.
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1D Linear Network
2D Lattice Network
Figure 1.
For instance, in socioeconomic models we may wish to encodedifferent degrees of influence. The first three structures in Figure 2could be relevant to encoding influence in a socioeconomic networkmodel. That way one would not have to write complicated rulesabout how particular agents interact more frequently with a particu-lar nonlocal agent on a grid: the information is already there, in thetopology itself.
25-node
complete graph
single global source
for 25 nodes
multiple global
sources for 25 nodes
100-node lattice
network with hole
100-node lattice
network with tunnel
100-node lattice
network with tunnel
and hole
Figure 2.
Similarly, in models of epidemic or fire spread, sometimes there arecorridors where fire/disease/information travels in some preferentialdirection in a nonlocal way. One example is wind in a forest firemodel, which has the potential to spread fire from one tall tree to atall tree downwind from the first, but not necessarily near to the first.This kind of nonlocal behavior is easily encoded by creating a tunnelof connections between one cluster of points on a grid to another, asshown in Figure 2. Additionally, dead zones or holes could representstanding bodies of water, or other impassable natural barriers.
458 A. Nussey
Complex Systems, 18 © 2010 Complex Systems Publications, Inc. https://doi.org/10.25088/ComplexSystems.18.4.457
Similarly, in models of epidemic or fire spread, sometimes there arecorridors where fire/disease/information travels in some preferentialdirection in a nonlocal way. One example is wind in a forest firemodel, which has the potential to spread fire from one tall tree to atall tree downwind from the first, but not necessarily near to the first.This kind of nonlocal behavior is easily encoded by creating a tunnelof connections between one cluster of points on a grid to another, asshown in Figure 2. Additionally, dead zones or holes could representstanding bodies of water, or other impassable natural barriers.
Certainly, there is a way to encode nonlocality and other propertiesinto the rules of the cellular automaton itself. But to do so often re-quires unintuitive manipulations that can get complicated quickly. Itis much more intuitive to keep the rules as simple as possible and en-code as much information in the topology as is technically feasible.
In Section 2, one way of running cellular automata over networksis considered using so-called “outer median” rules. A second way ofrunning cellular automata over networks using a probabilistic modelis considered in Section 3.
2. Outer Median Rules on Networks
2.1 General Setup
One of the key features of a network topology is that each node has awell-defined neighborhood that generally differs in neighbor numberand composition compared to other nodes. To contrast, one- and two-dimensional cellular automata have user-defined neighborhoods thatare the same for each cell.
Outer median rules are constructed to be a simple and intuitiveway to run a cellular automaton on a topology where the nodes,which correspond to the active cells in the one- and two-dimensionalcases, generally have different numbers of neighbors from node tonode.
The outer median rules operate as follows: take the median of thestates of the neighbors of the active cell, represented by numbers inthe range 80, k - 1<, with k being the number of possible states. Thenfind the floor value of this median and use the result to make a pairwith the current active state: {state of current active cell, floor of themedian of neighbor states}. The usual cellular automaton rules of in-teraction for a one-dimensional, k-state, 1/2-radius cellular automa-ton are then used.
2.2 Outer Median Rules in One Dimension
The pure function OuterMedianCA1D is built to work as describedwhen operating on a list of rules.
Outer Median and Probabilistic CAs on Network Topologies 459
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OuterMedianCA1D@rules_D :=8Ò@@Ceiling@Length@ÒDê2DDD, Floor@Median@
Delete@Ò, Ceiling@Length@ÒDê2DDDD< ê. rules &
The simplest case, k = 2, r = 1, is enumerated in Figure 3.
rule 1 rule 2 rule 3 rule 4
rule 5 rule 6 rule 7 rule 8
rule 9 rule 10 rule 11 rule 12
rule 13 rule 14 rule 15 rule 16
Figure 3.
Figure 4 shows a sample of k = 3 rules that are enumeratedthrough r = 5.
460 A. Nussey
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r = 1
rule 946
r = 2
rule 946
r = 3
rule 946
r = 4
rule 946
r = 5
rule 946
r = 1
rule 2058
r = 2
rule 2058
r = 3
rule 2058
r = 4
rule 2058
r = 5
rule 2058
r = 1
rule 2503
r = 2
rule 2503
r = 3
rule 2503
r = 4
rule 2503
r = 5
rule 2503
r = 1
rule 5060
r = 2
rule 5060
r = 3
rule 5060
r = 4
rule 5060
r = 5
rule 5060
r = 1
rule 5172
r = 2
rule 5172
r = 3
rule 5172
r = 4
rule 5172
r = 5
rule 5172
r = 1
rule 6284
r = 2
rule 6284
r = 3
rule 6284
r = 4
rule 6284
r = 5
rule 6284
r = 1
rule 11732
r = 2
rule 11732
r = 3
rule 11732
r = 4
rule 11732
r = 5
rule 11732
r = 1
rule 13956
r = 2
rule 13956
r = 3
rule 13956
r = 4
rule 13956
r = 5
rule 13956
r = 1
rule 15513
r = 2
rule 15513
r = 3
rule 15513
r = 4
rule 15513
r = 5
rule 15513
Figure 4.
Outer Median and Probabilistic CAs on Network Topologies 461
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Figure 5 shows a sample of k = 4 rules that are enumeratedthrough r = 5.
r = 1
rule 1738358822
r = 2
rule 1738358822
r = 3
rule 1738358822
r = 4
rule 1738358822
r = 5
rule 1738358822
r = 1
rule 2150870686
r = 2
rule 2150870686
r = 3
rule 2150870686
r = 4
rule 2150870686
r = 5
rule 2150870686
r = 1
rule 2184775770
r = 2
rule 2184775770
r = 3
rule 2184775770
r = 4
rule 2184775770
r = 5
rule 2184775770
r = 1
rule 2484070686
r = 2
rule 2484070686
r = 3
rule 2484070686
r = 4
rule 2484070686
r = 5
rule 2484070686
r = 1
rule 2845724924
r = 2
rule 2845724924
r = 3
rule 2845724924
r = 4
rule 2845724924
r = 5
rule 2845724924
Figure 5.
2.3 Outer Median Rules in Two Dimensions
In two dimensions, the outer median rules run on a cyclic lattice topol-ogy. Unlike the simple noncyclic lattice topology in the introduction,the cyclic lattice topology connects the outer edges of the lattice to-gether.
A few rules are chosen to illustrate the behavior of the outer me-dian rules on this cyclic lattice topology. A sample of the evolution isrun on a grid, then the density of the k states of the rule is plotted ona longer time scale.
OuterMedianCA2D@rules_D :=8Part@Flatten@ÒD, Ceiling@Length@Flatten@ÒDDê2DD,
Floor@Median@Delete@Flatten@ÒD,Ceiling@Length@Flatten@ÒDDê2DDDD< ê. rules &
First, the simple case of rule 9, k = 2, r = 1 is shown in Figure 6.
462 A. Nussey
Complex Systems, 18 © 2010 Complex Systems Publications, Inc. https://doi.org/10.25088/ComplexSystems.18.4.457
rule 9, k = 2, r = 1
step 1 step 2 step 3 step 4
step 5 step 6 step 7 step 8
step 9 step 10 step 11 step 12
step 13 step 14 step 15 step 16
density of k = 2 states for rule 9
5 10 15 20steps
30
40
50
number of cells
Figure 6.
Next, Figure 7 shows four rules with complex evolutions chosenfor the case of k = 3, r = 1. The density of states is plotted against thenumber of steps, for 64 cells and 100 steps.
density of k = 3 states for rule 11455
20 40 60 80 100steps
10
15
20
25
30
35
number
of cells
density of k = 3 states for rule 11455
20 40 60 80 100steps
15
20
25
30
35
number
of cells
density of k = 3 states for rule 11455
20 40 60 80 100steps
15
20
25
30
35
number
of cells
density of k = 3 states for rule 11455
20 40 60 80 100steps
10
20
30
40
number
of cells
Figure 7.
Outer Median and Probabilistic CAs on Network Topologies 463
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2.4 Outer Median Rules on Other Topologies
Figure 8 shows six different topologies run over four different rules,for 25 nodes and 50 nodes, respectively.
Topology rule 3748829411 rule 3748865004 rule 3748855964 rule 3748801727
Gear 12
Firecracker 85,5<
JohnsonSkeleton 32
KnightsTour 85,5<
Paulus 825,7<
HeawoodFourColorGraph
Figure 8.
Figure 9 shows six different 50-node topologies with k = 4, r = 1.
464 A. Nussey
Complex Systems, 18 © 2010 Complex Systems Publications, Inc. https://doi.org/10.25088/ComplexSystems.18.4.457
Topology rule 3748815569 rule 3748790710 rule 3748858507 rule 3748824608
CubicNonhamiltonian 850,1<
SzekeresSnark
Grid 82,5,5<
FiveleapersTour 85,10<
GeneralizedPetersen 825,9<
HoffmanSingletonGraph
Figure 9.
2.5 Application: Country Borders
Figure 10 shows a graph of bordering countries created with eachnode a country as assigned in Table 1. Each connection indicates aborder relationship.
Outer Median and Probabilistic CAs on Network Topologies 465
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China Ø 1 HongKong Ø 2 India Ø 3 Kazakhstan Ø 4
Kyrgyzstan Ø 5 Laos Ø 6 Macau Ø 7 Mongolia Ø 8
Myanmar Ø 9 Nepal Ø 10 NorthKorea Ø 11 Pakistan Ø 12
Russia Ø 13 Tajikistan Ø 14 Vietnam Ø 15 Austria Ø 16
CzechRepublic Ø 17 Germany Ø 18 Hungary Ø 19 Italy Ø 20
Liechtenstein Ø 21 Slovakia Ø 22 Slovenia Ø 23 Switzerland Ø 24
Algeria Ø 25 Libya Ø 26 Mali Ø 27 Mauritania Ø 28
Morocco Ø 29 Niger Ø 30 Tunisia Ø 31 WesternSahara Ø 32
Guinea Ø 33 GuineaBissau Ø 34 IvoryCoast Ø 35 Liberia Ø 36
Senegal Ø 37 SierraLeone Ø 38 France Ø 39 Luxembourg Ø 40
Monaco Ø 41 Spain Ø 42 DemocraticRepublicCongo Ø 43 RepublicCongo Ø 44
Rwanda Ø 45 Sudan Ø 46 Tanzania Ø 47 Uganda Ø 48
Zambia Ø 49 Cameroon Ø 50 CentralAfricanRepublic Ø 51 Chad Ø 52
EquatorialGuinea Ø 53 Gabon Ø 54 Nigeria Ø 55 Afghanistan Ø 56
Iran Ø 57 Turkmenistan Ø 58 Uzbekistan Ø 59 Mozambique Ø 60
SouthAfrica Ø 61 Swaziland Ø 62 Zimbabwe Ø 63 Iraq Ø 64
Jordan Ø 65 Kuwait Ø 66 SaudiArabia Ø 67 Syria Ø 68
Turkey Ø 69 Romania Ø 70 Serbia Ø 71 Ukraine Ø 72
BurkinaFaso Ø 73 Ghana Ø 74 Togo Ø 75 Bulgaria Ø 76
Greece Ø 77 Macedonia Ø 78 Belarus Ø 79 Latvia Ø 80
Lithuania Ø 81 Poland Ø 82 Kenya Ø 83 Somalia Ø 84
SanMarino Ø 85 VaticanCity Ø 86 Israel Ø 87 Lebanon Ø 88
WestBank Ø 89 Netherlands Ø 90 Egypt Ø 91 GazaStrip Ø 92
Croatia Ø 93 Montenegro Ø 94 Benin Ø 95 Belgium Ø 96
Azerbaijan Ø 97 Georgia Ø 98 Armenia Ø 99 Angola Ø 100
Namibia Ø 101 Albania Ø 102 Kosovo Ø 103 Oman Ø 104
UnitedArabEmirates Ø 105 Yemen Ø 106 Malawi Ø 107 Thailand Ø 108
Finland Ø 109 Norway Ø 110 Sweden Ø 111 Ethiopia Ø 112
Djibouti Ø 113 Eritrea Ø 114 Cambodia Ø 115 Burundi Ø 116
Botswana Ø 117 BosniaHerzegovina Ø 118 SouthKorea Ø 119 Moldova Ø 120
Indonesia Ø 121 Malaysia Ø 122 PapuaNewGuinea Ø 123 Estonia Ø 124
Bhutan Ø 125 Bangladesh Ø 126 Andorra Ø 127 Qatar Ø 128
Portugal Ø 129 Lesotho Ø 130 Gibraltar Ø 131 Gambia Ø 132
EastTimor Ø 133 Denmark Ø 134 Brunei Ø 135
Table 1. Node assignments by country.
466 A. Nussey
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56
1
5712
14
58
59
102
77
103
78
9425
26
27
28
2930
31
32
127
39
42
100
43
101
44
49
99
97
98
69
16 17
18
19
20
21
22
23
24
13
1263
9
798081
82
72
9640
90
957355
75
125
11893
71
117
61
63
135
122
76
7074
35
11645
47
115
6
108
15
50
51
52
5354
46
2
4
5781011
48
134
113
114112
84
133
121
91
92
87
124
83
109110
111
41
132
37
131
33
34
36
38
123
64
656667
68
8889
85
86130
107
60
120
62
119
104105106
129
128
Figure 10. Plot of country border graph for Asia, Africa, and Europe.
Figure 11 shows the evolution of the border graph network fork = 4 and 50 steps, for a certain number of interesting rules. Nodesare evolved from left to right, with node 1 (China) being the leftmostnode, and node 135 (Brunei) being the rightmost node. Rule icons forsampled rules are included below each rule’s evolution.
Outer Median and Probabilistic CAs on Network Topologies 467
Complex Systems, 18 © 2010 Complex Systems Publications, Inc.
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rule 3748882801
rule 3748791840
rule 3748831394
rule 3748794665
rule 3748798620
rule 3748813309
rule 3748826868
rule 3748833083
rule 3748864156
rule 3748882801
Figure 11.
3. Probabilistic Rules on Networks
3.1 General Setup
In creating the outer median rules on networks, we endeavored totake the evolution of a cellular automaton “off the lattice”, as it were.We conjectured that in fact a system could be built to encode some ofthe specialized, nonlocal phenomena found in many natural andsocioeconomic systems.
468 A. Nussey
Complex Systems, 18 © 2010 Complex Systems Publications, Inc. https://doi.org/10.25088/ComplexSystems.18.4.457
In creating the outer median rules on networks, we endeavored totake the evolution of a cellular automaton “off the lattice”, as it were.We conjectured that in fact a system could be built to encode some ofthe specialized, nonlocal phenomena found in many natural andsocioeconomic systems.
In order to take into account nonlocal phenomena for some proba-bilistic cellular automata models on the lattice, complicated and/or un-intuitive means are often devised for active cells communicating overa distance. But it would seem more intuitive to merely encode nonlo-cal behavior into the topology on which the cellular automata rulesare being evolved, like we did earlier with the outer median rules. Itturns out that probabilistic rules on networks are very easy to define,and only differ slightly from their typical two-dimensional-latticecousins.
In order to clearly introduce probabilistic rules on networks, we setup a simple forest fire model.
3.2 Example: A Forest Fire Model
Forest fire spread has been modeled using two-dimensional cellular au-tomata [2|4] and small-world networks [5]. To set up the model, afew simple examples of the kinds of nonlocal behavior are enumer-ated that might be of interest when modeling forest fire spread.
In Figure 12 we first show a regular 400-node lattice, where theneighborhood of each node consists of itself and its nine nearest neigh-bors. Then we have a similar lattice with a hole punctured in it,through which no information may pass. The third network has a tun-nel connecting the simple neighborhood of a point with the neighbor-hood of another point. Information may in this way jump from oneside of the network to the other, where before it did not have a directmethod of communication. The fourth network shown has both ahole and a tunnel.
Setting up probabilistic rules on network topologies is straightfor-ward: for the simple lattice, it takes a well-studied form where the evo-lution of each active cell usually depends on the density of certainstates in a nine-nearest-neighbor neighborhood [2].
Outer Median and Probabilistic CAs on Network Topologies 469
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400-node lattice network 400-node lattice network
with hole
400-node lattice network
with tunnel
400-node lattice network
with tunnel and hole
Figure 12.
Here are the states in the forest fire model:
State Cell Color Number
Empty ground Brown 0
Trees Green 1
Fire Red 2
The rules we set up are similar to the two-dimensional case givenin [2|4], in that at an empty or 0-valued site tree grows with probabil-ity = pGrowth. At a 1-valued site with an existing tree, that tree burns(becomes 2-valued) if either at least one of its neighbors is burning, orwith a small spontaneous probability = pBurn. After a tree hasburned, it always becomes an empty site (2-valued sites go to 0-val-ued sites with probability = 1).
The function neighbors outputs the neighborhood of each nodeon whatever topology the rules are being run on, nextstepvaluesFFNtakes an initial distribution of states and applies the described rulesfor the next step, and evolveFFN iterates the rules for a certain num-ber of steps.
470 A. Nussey
Complex Systems, 18 © 2010 Complex Systems Publications, Inc. https://doi.org/10.25088/ComplexSystems.18.4.457
neighbors@node_, graphrules_D :=
Flatten@node ê. Ò & êü graphrules ê. node Ø 8<D
nextstepvaluesFFN@init_,graphrules_, pGrowth_ ê; 0 § pGrowth § 1,
pBurn_ ê; 0 § pBurn § 1D := Switch@init@@ÒDD,0, If@RandomReal@D § pGrowth, 1, 0D,1, Which@Count@Part@init, neighbors@Ò, graphrulesDD, 2D >
0, 2, RandomReal@D § 0.001, 2, True, 1D,2, 0, 3, 3D & êü Range@Length@initDD
evolveFFN@init_, graphrules_, steps_D := NestList@
nextstepvaluesFFN@Ò, graphrulesD &, init, stepsD
Since it is our intention to show that it is possible to use a proba-bilistic cellular automaton on a network to model forest fire spread,we pick two reasonable burn and growth probabilities, and then it-erate the evolution over the four different topologies (i.e., lattice, lat-tice with hole, lattice with tunnel, and lattice with tunnel and hole)for three different initial values. Of course, to get any sense for thereal behavior of the model over different topologies, we should care-fully construct topologies to encode certain nonlocal behaviors, makethe networks dynamic as nonlocal behavior in this case can often be,and sample over thousands of runs in order to determine the overallbehavior of the model.
Figure 13 shows the density plots for 100 nodes, sampled overthree different initial conditions, and run for 200 steps withpGrowth = 0.02 and pBurn = 0.001.
Outer Median and Probabilistic CAs on Network Topologies 471
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density plot for 100-node lattice
0 50 100 150 200steps
20
40
60
80
100
number
of cells
50 100 150 200steps
20
40
60
80
number
of cells
0 50 100 150 200steps
20
40
60
80
number
of cells
density plot for 100-node lattice with hole
50 100 150 200steps
20
40
60
80
number
of cells
50 100 150 200steps
20
40
60
80
number
of cells
0 50 100 150 200steps
20
40
60
80
number
of cells
density plot for 100-node lattice with tunnel
50 100 150 200steps
20
40
60
80
number
of cells
50 100 150 200steps
20
40
60
80
number
of cells
0 50 100 150 200steps
20
40
60
80
number
of cells
density plot for 100-node lattice with tunnel and hole
50 100 150 200steps
20
40
60
80
number
of cells
50 100 150 200steps
20
40
60
80
number
of cells
0 50 100 150 200steps
20
40
60
80
number
of cells
Figure 13.
Figure 14 shows the density plots for 400 nodes, sampled overthree different initial conditions, and run for 200 steps.
472 A. Nussey
Complex Systems, 18 © 2010 Complex Systems Publications, Inc. https://doi.org/10.25088/ComplexSystems.18.4.457
density plot for 400-node lattice
50 100 150 200steps
50
150
250
350
number
of cells
50 100 150 200steps
50
150
250
350
number
of cells
50 100 150 200steps
50
150
250
350
number
of cells
density plot for 400-node lattice with hole
50 100 150 200steps
50
150
250
number
of cells
50 100 150 200steps
50
150
250
number
of cells
50 100 150 200steps
50
150
250
number
of cells
density plot for 400-node lattice with tunnel
50 100 150 200steps
50
150
250
350
number
of cells
50 100 150 200steps
150
250
350
number
of cells
50 100 150 200steps
50
150
250
350
number
of cells
density plot for 400-node lattice with tunnel and hole
50 100 150 200steps
50
150
250
number
of cells
50 100 150 200steps
50
150
250
number
of cells
50 100 150 200steps
50
150
250
number
of cells
Figure 14.
4. Conclusion and Future Directions
There are several commonly studied problems that could yield to themethods described in this paper. One could develop a model based onour approach to probabilistic cellular automata on networks in orderto study epidemic spread. While epidemic spread on networks hasbeen studied to some extent [6, 7], it has yet to be modeled in thisway.
One could also apply the methods described to socioeconomics, forinstance, running games on network topologies. Again, while runninggames on network topologies has been and is currently being studied[8, 9], it has typically been for special cases and limited numbers ofgames. The methods we described could provide a way to more ex-haustively enumerate game theories and strategies on network topolo-gies.
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Complex Systems, 18 © 2010 Complex Systems Publications, Inc.
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One could also apply the methods described to socioeconomics, forinstance, running games on network topologies. Again, while runninggames on network topologies has been and is currently being studied[8, 9], it has typically been for special cases and limited numbers ofgames. The methods we described could provide a way to more ex-haustively enumerate game theories and strategies on network topolo-gies.
Finally, our methods could themselves be improved by allowing thetopologies to change dynamically throughout time. For instance, if weencode wind in the topology of a model of forest fire spread, wewould also want to be able to change the direction of the wind, whichwould require the ability to change the topology of the model in a dy-namic way, as the model is being run.
References
[1] S. Wolfram, A New Kind of Science, Champaign, IL: Wolfram Media,Inc., 2002.
[2] B. Beckage and J. Cawley. “Cellular Automaton Model of Pine SavannaDynamics in Response to Fire and Hurricanes” from The WolframDemonstrations Project~A Wolfram Web Resource.http://demonstrations.wolfram.com/CellularAutomatonModelOfPineÖSavannaDynamicsInResponseToFireA.
[3] A. Hernández Encinas, L. Hernández Encinas, S. Hoya White,A. Martín del Rey, and G. Rodríguez Sánchez, “Simulation of ForestFire Fronts Using Cellular Automata,” Advances in Engineering Soft-ware, 38(6), 2007 pp. 372-378.
[4] L. Hernández Encinas, S. Hoya White, A. Martín del Rey, andG. Rodríguez Sánchez, “Modelling Forest Fire Spread Using HexagonalCellular Automata,” Applied Mathematical Modeling, 31(6), 2007pp. 1213-1227.
[5] I. Graham and C. Matthai, “Investigation of the Forest-Fire Model on aSmall-World Network,” Physical Review E, 68(3), 2003, 036109.
[6] A. Ganesh, L. Massoulie, and D. Towsley, “The Effect of NetworkTopology on the Spread of Epidemics,” in Proceedings IEEE INFO-COM Vol. 2 (24th Annual Joint Conference of the IEEE Computer andCommunications Societies 2005), Miami, FL, 2005 pp. 1455-1466.
[7] M. Newman, “The Spread of Epidemic Disease on Networks,” PhysicalReview E, 66, 2002, 016128.
[8] M. Kearns, M. Littman, and S. Singh, “Graphical Models for Game The-ory,” in Proceedings of the 17th Conference in Uncertainty in ArtificialIntelligence (UAI 2001), Seattle, WA (J. Breese and D. Koller, eds.), SanFrancisco, CA: Morgan Kaufmann Publisher Inc., 2001 pp. 253-260.
[9] F. C. Santos, J. F. Rodrigues, and J. M. Pacheco, “Graph TopologyPlays a Determinant Role in the Evolution of Cooperation,” Proceed-ings of the Royal Society B, 273(1582), 2006 pp. 51-55.
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