Chapter 14 Spatial Threshold Systems including Self-Organized Criticality

Cellular DEVS Models of Spatial Threshold Systems DEVS abstractions of continuous systems with threshold behavior allow simulations that would otherwise be too complex/time consuming to be feasible An interesting class of such systems are the Self Organized Criticality (SOC) natural systems which depend on threshold properties of continuous systems These properties allow using DEVS to model them with appropriate abstractions We break space into cells and model these with DEVS models; the cells are placed into 1,2,3 or higher dimensional grids and coupled with neighborhoods form a cellular space.

SOC Earthquake expressed as a One Dimensional Cellular Space Due to compression on tectonic plates, each cell accumulates energy at a constant rate until a critical threshold is exceeded. At this point a fraction of its accumulated energy is equally distributed among its neighbors. This reduces the time to achieve their critical levels. If the additional energy causes a cell to exceed its threshold, then it will release energy to its neighbors. The size of an avalanche is the number of such propagated releases that occur until all cells are below the threshold. Releases occur with ta = 0, so the size of an avalanche is the length of a transitory sequences of external transitions. Power law distribution of avalanches (~1/x) Per Bak, “How Nature Works” H. Jensen, “Self Organizing Complexity”

SOC Earthquake expressed as a One Dimensional Cellular Space earthquakeCell inherits from oneDimCell. To insert digraph models as cells they must satisfy the cell interface Neighborhood coupling i-2 i-1 i i+1 public class earthquakeCellSpace extends oneDimCellSpace{ earthquakeCell.numCells = numCells; for (int i = 0;i<numCells;i++) addCell(i,new earthquakeCell(i,energyScale)); hideAll(); //hides only components so far doNeighborCoupling(+1,"outEnergy","in"); //nearest neighbor doNeighborCoupling(-1,"outEnergy","in") coupleAllTo("out",this,"out"); coupleOneToAll(this,"stop","stop"); coupleAllTo("outPair",this,"outPair"); EarthquakeCellSpace inherits from OneDimCellSpace which provides methods for easily creating a one dimentional cellular space by adding addCell(index, oneDimCell) and coupling oneDImCells: doNeighborCoupling(neighborIndex,outPort, inPort) coupleAllTo(sport,devs,dport) coupleOneToAll(sport,devs,dport) Random. earthquakeCellSpace

Self Organizing Critical Earthquake Cell public void deltext(double e,message x){ energy += e*accumulationRate; //updating energy Continue(e); if (somethingOnPort(x,"in")){ energy += getRealValueOnPort(x,"in"); //if you are neighbor get energy if (energy >=energyThresh) holdIn("active",0); else holdIn("active“, (energyThresh – energy)/accumRate); //every cell increases at same rate } else if (somethingOnPort(x,"stop")) passivate();} energyThresh energy released Energy e Updated total ta ta = time to threshold crossing Release cell neigh bor public void deltint(){ super.deltint(); energy = 0;//release all energy holdIn("active",energyThresh /accumRate);) //schedule yourself to reach threshold} public message out(){ message m = super.out(); m.add(makeContent("outEnergy", new doubleEnt( alpha*(energy + sigma*accumRate)))); // alpha small implies non conservation // of accumulated energy return m;} Random. earthquakeCellSpace

output= Time advance = s s’ Make a transition Internal Transition /Output Generation -- a cell reaches its threshold energy and distributes its energy to its neighbors; the cell starts accumulating energy starting from zero output= alpha*(energy + sigma*accumRate) Generate output Time advance = (energyThresh – energy)/accumRate) s Make a transition s’

input Make a transition Time advance, ta Response to External Input – a neighbor cell receives energy and schedules itself to reach the threshold earlier than before (or it may exceed threshold and become critical) input Make a transition energy +=e*accumRate; if (energy >=energyThresh) ta = 0; else ta =(energyThresh –energy)/accumRate); elapsed time, e Time advance, ta

Sand Pile Avalanche model vs Salt Pile model Cells communicate to discover the state (color coded) they are in. The 8 cell Mealy neighborhood is needed. Since signals travel at different speeds, classical CA are not (easily) able to do this. black - falling grain gray -track of falling grain pink - cells with open side blue - cell about to destabilize – open on at least one side and with weight = threshold (here 4) red - cell walled in on all sides Avalanche size – Number of lateral moves in one event – very close to power law = 53/x Salt grain model – grains fill up empty slots, come to rest on top of a red (walled in) cell. Avalanche – sand grains fall down center and pile up to threshold, as determined by weight on bottom cell. This destabilized cell moves and destabilizes those above, which fall with equal probability to the left or right. This process continues with the same rules applying to distributed grains. Large avalanches occur when a number of adjacent columns have cells near the threshold. Sand and Salt Piles show distinct avalanche size distributions

Sand and Salt Piles show distinct avalanche distributions Avalanche size – Number of lateral moves in one event – very close to power law = 53/x Sand pile

Threshold-based watershed rain Discrete flow is injected at the top, middle cell. It randomly diffuses down and to the sides under threshold constraint. That is, it cannot transmit the flow until its accumulated water exceeds a threshold. When it overflows it acts like a burst dam. The distinct increasing-with-depth lateral extents may be related to the threshold value. Avalanches are the propagated series of dam bursts – difficult to measure. depth

Chapter 15 Partial Differential Equation Models Method of Lines One Dimensional Diffusion Example: Triangular Initial Data Distribution Number of Transitions = Activity

Example: One Dimensional Diffusion (PDE) cell tries to move toward average of its neighbors lower than neighbors in between neighbors higher than neighbors

One Dimensional Diffusion (Con’d) higher than neighbors lower than neighbors tend to flatten out u in between neighbors x x over time x x

DEVS Implementation of 1D Diffusion oneDimCellSpace cell 1D DEVS Integrator left right time derivative = k*spatial derivative cell 1D DEVS Integrator left right time derivative = k*spatial derivative cell 1D DEVS Integrator left right time derivative = k*spatial derivative

DEVS Implementation of 1D Diffusion : sloping initial state oneDimCellSpace length and height = 11 and speed = 10, 11 cells

Definition and Measurement of Activity Continuous segment Piecewise Continuous Segment t1 tn ti m1 mi mn T Activity = sum of ranges. Avg Activity ~ A*f for wave with amplitude A and frequency f computational efficiency occurs when number of transitions reflects number of threshold crossings activity in monotonic region reflects avg rate of change

activity distribution 11 5.5 activity distribution activity = 35.5 =(area of space with arrows) length and height = 11 and speed = 10, 11 cells Since there is some oscillartion arround 5.5, I used the first time to hit 5.5 in the table.The oscillation is about 3 times the quantum. quantum activity/quantum for cell 0 Actual number of cell 0 transitions to reach 5.5 .05 110 122 .01 550 561 .005 1100 1101 .001 5500 5501 DEVS transitions = Activity/quantum

q = .01 q = .001 q = .0001 As quantum gets smaller, the #DEVS transitions gets closer to the predicted activity. The reason can be found in the distribution over cells which moves from uniform to the sharp curve predicted. The uniform distribution, at lower quanta is due to increased oscillation as cells adjust to their neighbors in big steps.