Outer Median and Probabilistic Cellular Automata on Network Topologies · 2019-03-20 · Outer Median and Probabilistic Cellular Automata on Network Topologies Abigail Nussey Wolfram
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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.
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.
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
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
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r = 3
rule 2150870686
r = 4
rule 2150870686
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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.
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
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of cells
density of k = 3 states for rule 11455
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of cells
density of k = 3 states for rule 11455
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density of k = 3 states for rule 11455
20 40 60 80 100steps
10
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Figure 7.
Outer Median and Probabilistic CAs on Network Topologies 463
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
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 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
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.
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
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
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of cells
50 100 150 200steps
50
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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.
Outer Median and Probabilistic CAs on Network Topologies 473
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.