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Robustness and Fragility



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Robustness and Fragility 317

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Robustness and Fragility Jean M. Carlson, University of California at Santa Barbara DR. CARLSON: I’m in the in the area of complex systems in the physics department. Over the last 17-18 years I have been looking at trying to come up with simple models and fundamental kinds of guidelines in thinking about complex systems. That has first led me into earthquakes and more recently into forest fires. Part of the goal has been to try to make connections between simple theory, detailed models and practical kinds of issues so I have benefited a lot from collaborations. In light of everything that dominates the news these days my aim here is to step back and talk about natural disasters. I’m pulling from a lot of things that for me are less directly what I’m working on, but what are the impacts, what are the consequences of robust yet fragile behavior in terms of dynamics on the planet, and then also in terms of people. This is also an invitation and a question for all the people who work on problems in sociology, what can we do from the point of view of passing information from the scale of the modeling and the geophysical phenomena up to where it has a real impact, which is on sociological issues such as response as well as policy and planning and so on. I think we’ve got to do that better. Clearly, that is responsible for an enormous amount of the impact of these events, and the part that I have worked on is just the very beginnings of the geophysical phenomenon themselves. Stepping back, I want to provide an overview of natural disasters. The question at the end is whether or not there are some interesting ways we can think about this. First of all, all these natural disasters are a natural part of the evolution of the planet. If we didn’t have this sort of stuff we wouldn’t have life, so there are many good things about natural disasters. When you go around and start collecting data though, you start to see that there are a lot of trends in terms of natural disasters such as costs as well as loss of life. Figure 1 shows some statistics that have been drawn, and you can see that things are on the increase. Economic losses are on the increase and insurance on these losses is not keeping up with the actual losses themselves. 318

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FIGURE 1 Figure 2 updates that picture through 2004. It doesn’t include 2005, but it does reflect the tsunami in the Indian Ocean. You can see another big spike in terms of economic losses associated with the Kobe earthquake in Japan. And Figure 3 displays the trend in economic costs. Even correcting for inflation, you see that there is an overall increase in terms of global economic losses associated with natural disasters. This goes hand-in-hand with increased population in the world, and a lot of the population increase is associated with less developed countries. 319

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FIGURE 2 FIGURE 3 320

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Natural disasters are also occurring increasingly in urban areas. The world is now approaching the point where the fraction of population in urban areas will equal the fraction in rural areas, and the total world rural population is expected to remain relatively flat or even decline in the 2020s. Figure 4 drives this home. Each set of bars represents a different large city’s expected population growth over coming decades. In looking at some of the large cities you can see most of them are on the increase. Particularly, places like Nigeria and India are showing huge population increases in the large cities. FIGURE 4 . 321

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FIGURE 5 Figure 5 shows the class of cities that have more than 10 million people, called mega cities. People look at these cities, which are often on coastlines, and classify them in terms of their hazard (yellow on the graphic), vulnerability (red), and exposed value (blue) in terms of hazard. People are evaluating these kinds of things. You can see that Tokyo is considered to be at high risk because of all of the geophysical kinds of phenomenon such as tsunamis, and so are the coastal cities in the United States. This is a natural hazard risk index. 322

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FIGURE 6 There are two things that people look at when they try to measure and plot this kind of data: fatalities and damage. Interestingly, the trends are opposite one another, as shown in Figures 6 and 7. The poorest nations dominate in terms of deaths from natural disasters, but industrialized nations dominate the economic costs. What are you trying to protect, how is a natural disaster measured if you try to think about where you are going to put your resources? You see that there are these two different measures to consider, loss of life and dollar impact. FIGURE 7 323

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Returning to this issue of robust yet fragile and the talk that John Doyle gave yesterday, if you look at the distribution of sizes of events themselves (Figure 8) what you see are these power law distributions. Yesterday John talked about plotting the size statistics for natural disasters here. This figure is in terms of dollars for the natural disasters. They follow a power law distribution. That means that on a log-log scale they are very broadly distributed. Each dot represents one increase in the cumulative number, so does the largest event, second largest event and so on. FIGURE 8 This is a power law distribution, and what that means in that the worst event, the events that dominate the losses, are much worse, orders of magnitude worse than a typical event. This is shown in Figure 9. Large events that we see, like Hurricane Katrina, are completely consistent with these statistical distributions, so they’re not outliers or anything like that. They are consistent with the statistics and they are to be expected. 324

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FIGURE 9 One of the things that John Doyle and I have been looking at in recent years has been trying to think about complex systems, cascading failure events, in the context of power law distributions in a fundamental way. This is a very quick summary of the work that we have been doing in that area. We try to bring in insights from biological and technological systems; systems which are highly optimized or have evolved through years and years of Darwinian mechanisms to be behave optimally in some sense, not like generic random systems but in an optimal way. We found that this demands a new theoretical framework on many different levels. We have used some of the simple kinds of models that arise in physics in order to try to show how robustness issues change these models, and robustness trade-offs lead to new fragilities and sensitivities. One hallmark of this is these kinds of power laws. Another statement is that sometimes heterogeneity and high variability can create opportunities for increased performance in some systems. This HOT framework is one that talks about optimization or evolution of systems. It also fits into a broader framework for describing large network systems that have robust, yet fragile behaviors. Not necessarily all systems are obvious solutions of some kind of optimization problem like earthquakes, but they still have this kind of robust yet fragile behavior. We are finding that feedback plays an important role. Interesting issues arise in terms of multi-scale and multi-resolution modeling and analysis. This really is a demand for a new approach to complex systems theory that goes all the way through. My hope is that we’ll be able 325

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to learn from what we study. Social systems will be helpful to understanding complex systems and natural systems. It doesn’t mean they are the same, but it does mean that we are sharing tools. More and more we need to share those tools and we also need to be able to talk to each other, because our cascading failures pass from natural disasters into social systems and our network communication systems as well. In any case, this HOT fits within this general picture and a big, primary message from the HOT systems is to imagine having some sorts of resources that you are allocating to a spectrum of events. If you become, through design or evolution, robust to center kinds of perturbations, you introduce new fragilities as well, as a consequence of this architecture. One of the kinds of pictures that John and I have worked on to describe this is a generalization of Shannon Coding Theory where you imagine that you have some kind of resources in your systems. We want to compress our data, which calls for optimizing d-1 dimensional cuts in d dimensional spaces. Fires, as shown in Figure 10, provide an example, where there is some template of trees in a forest and the resources might be associated with fire breaks or means to suppress fire. Given a constraint on the resources, you want to optimally assign the resources, given some spectrum of possible fires. We have also looked at this generalization of Shannon Coding Theory in the context of Web design. FIGURE 10 326

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I’m not going to tell you the specifics of the model behind the curves in Figure 11, because I want to focus more on the robust and fragile features. We compared it with data, and a very simple model gave rise to very accurate fits that was much more accurate than we were expecting for statistics of forest fire sizes and Web file downloads. We then based it on data compression, so it has to fit there. FIGURE 11 In this graph, “rank” is just frequency of possibility as a function of size, measured obviously differently in these two different cases, but the straight-line parts of the slope are really signatures of the power laws. In this particular case they have different exponents, because there is a different underlying dimensionality of the 2-dimensional forest and chopping 1-dimensional documents into files. 327

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FIGURE 16 How do earthquakes happen? What’s the physics? Figure 16 presents a cartoon-like picture. The idea is you’ve got plate tectonics, and you’ve got the crust of the earth that is like an egg shell on top of the mantle that is slowly convecting, which drives the relative motion of these plates. There are weak interfaces in between tectonic plates where stresses accumulate that create a slipping event when the material along the interface fails, and that sets off waves that radiate through the ground. Figure 16 shows a lateral fault, which is like the San Andreas Fault, but there are different kinds of faults. Again, this is a cartoon; the slip does not occur homo- geneously. There are different things you can look at such as the dynamics, the complexity of the slip itself, and the fact that it’s complicated. You can also look at the dynamic complexity of the radiation as shown via simulation in Figure 17. It doesn’t have much interesting dynamics in the slip itself, but it is showing you what would happen in Los Angeles if you had a very simple slip pulse propagating down the San Andreas Fault, which is cartooned by that line. 332

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FIGURE 17 Figure 17 shows Los Angeles and its freeways. You can see that the ground motion is complicated, and the reason is because the hard rock basin underneath Los Angeles is complicated. What it looks like is known because of oil exploration. You can find that there are some places where you don’t want to be, which has to do with such things as resonance effects. These kinds of models are the things that people use to try to set building codes in different parts of the city. I’ll come back to earthquakes a little bit at the end, which is the area in which I have worked the most. But first I’ll talk a bit about the Sumatran case and the tsunami. What caused the Sumatran earthquake? A simple answer is 200 million years of continental drift, as the Indian plate slides under the Asian plate. Figure 18 shows that collision, with the red area being the portion that slid and caused the Sumatran earthquake. 333

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FIGURE 18 If you superimpose the region that slid on a map of California (as shown in Figure 19), it’s an enormous magnitude 9 earthquake. This just shows how significant the Sumatran quake was. FIGURE 19 This particular earthquake is not the lateral kind, the kind of earthquake that makes tsunamis happen at subduction zones. Subduction zones are where you have two plates that, rather than slipping side-by-side, one is going under the other. They are out in the ocean where things are spreading up; the sea floor is spreading and there is a divergent zone. Material comes out and travels across the ocean very quietly. When it hits another plate it goes down, as shown 334

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in Figure 20, and when it goes down something goes up. This happens underwater, so you have this water that doesn’t want to sit like this and it has to cope with that. That is how you get a tsunami; waves are set off in both directions, as shown in Figure 21. FIGURE 20 FIGURE 21 When the tsunami is a wave in the ocean, it is about one meter high—less than a meter high so it’s nothing; it just goes along in the ocean. When it comes up against the coast, the slope amplifies the wave, as shown in Figure 22, and it can typically go to 10 meters, but it can also be 335

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hundreds of meters. There are some known examples where it is higher. High does not necessarily mean high damage, but there is a wide range of heights of these waves. You know that they are coming, and you can estimate how long it’s going to be. But tsunamis travel at hundreds of miles an hour across the ocean. The question is whether or not people know one is coming. FIGURE 22 There are all kinds of simulations of the Sumatran tsunami. We have a simulation of the disturbance traveling across the ocean in these simulations, and you can estimate how long it will be until it hits various coastal places. Figure 23 shows the tsunami’s progress in hours. It took about two hours to get to Thailand. It was Sumatra that didn’t have very much warning. It’s a lot more warning than you would ever have for an earthquake, but that might not be enough if you’re out on the beach. So the question is whether or not we really have effective warning systems for some of these things. 336

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FIGURE 23 There are lots of tsunamis, and Figure 24 shows statistics for a bunch of recent ones. A magnitude 9 earthquake is a big earthquake, so there was a lot of damage. But not all tsunamis are damaging. The red ones in Figure 24 are the damaging ones and the white ones aren’t. Along the Pacific Coast in the United States we are always at risk for tsunamis. The biggest chance for tsunamis is most likely up in Alaska and in the Chile fault. We would have a lot of warning so we sort of sat around and said, well, look, what would you do? I guess what happens is that the show is on the TV, you probably get an e-mail hours in advance, but you might be out on the beach on vacation, and you might not know, so the policemen and fire departments go out and tell you and try to clear people off the beach. Some people are afraid to tell you because they fear people will go to the beach to see it. That’s also taken into account. 337

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FIGURE 24 Hurricanes are another category of natural disasters that can be extremely costly. Scientists are not surprised that a large one like Katrina will happen, but it doesn’t happen every day so we get used to thinking that it’s not going to happen. There is a question of policy and where you allocate your resources. There is lots of research going on in predicting the intensity, how the intensity will change, and predicting the tracks of a tsunami or a tropical storm once it starts. There is also a lot of effort going on to predict what’s vulnerable by modeling so there’s a lot of interesting work in progress going on out there with the urban area topography. The thing that you can do here regarding something like earthquakes is couple with field research; go out and measure the hot spots in the ocean, and you can tell what is going to spin up the tsunami rate. 338

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A lot of the vulnerabilities that we see are things that have to do with finite resource allocations and their impact over all kinds of scales on our sort of social network structure, as indicated by Figure 25. There is the geophysics and hydrodynamics, which has its impact on homes and families, infrastructure and energy. It puts stress on our hospitals. If they are already full or they are closing because of other stresses like insurance, you won’t have Emergency Rooms for people to receive treatment. We won’t have the military as much on the homeland to come and respond to our disasters if they are already allocated abroad. This impacts our transportation and communications systems. Fuel prices rise and it impacts airlines. Airlines are already going bankrupt; therefore, there is a huge stress on the system. It comes all the way up to global economic issues, politics and natural resources on large scales. I think the thing that is so striking is how a shock, like a hurricane, to a about robust-yet-fragile system can lead to cascading failures all the way up the chain. FIGURE 25 339

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FIGURE 26 In Figure 26 I tried to cartoon the whole issue of scale, going up in time/space. Since most things happen on the diagonal, you might collapse that into a vertical time and space connecting these different scales, and horizontal issues associated with modeling the physics or geophysics on any one of these scales. I would say for natural disasters we focus enormously on the horizontal aspect and not very much on how we can transmit information from one scale to the other. I think it’s a huge issue, and my plea is to the people here that are involved in sociology is how to think better about this problem. I have been thinking about how to address the risk of earthquakes in that multi-scale way. One can model fine-scale geophysics, the impact of that on friction laws, the impact of friction on faults and networks, and all the way up to things that have to do with hazard evaluation policy and building codes. Part of the problem is that in this case, there is the issue of modeling on the horizontal scales of Figure 26 and then trying to connect the scales. When you get to the point of understanding how to set insurance rates and policies in terms of reaction to disasters, the vertical challenges dominate the issues. So, dealing with uncertainty in seismic hazard analysis requires addressing the horizontal challenges—identifying the range of physical behaviors that are plausible—and also addressing the vertical challenges, such as uncertainty management and how to pass information between scales in a useful way. You might think of seismic hazard analysis as being represented by the elements in Figure 27, and to some extent they are there in the background. In the end, though, a lot of economics and policies are based on a single number, which in this particular case is a 62 340

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percent chance of a magnitude greater than 6.7 happening in a 20-year period or something like that in the Bay area. There are all kinds of statistical problems associated with what this number is, and that’s what my student Morgan Page and I have been looking at. FIGURE 27 There needs to be more rigorous statistical methodology for combining data in these uncertain worlds, and also to incorporate physical constraints that come from modeling and simulation and ground motion estimates and so on. In this issue of dealing with uncertainty, vertical challenges really dominate our ability to estimate things like losses and risk to human life. If we could address these in a more systematic way maybe we would have a stronger impact on policy. QUESTIONS AND ANSWERS DR. GROSS: I’m Shula Gross from the City University of New York. My question is you showed a plot that suggested a power law for fire damages. It’s funny, because statisticians and econometricians usually look at the tails, and we care more about the tails than about the center. You somehow did the reverse. 341

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DR. CARLSON: I think there are cut offs partly because of physical sampling, and in some cases, if you look at data for a particular region, like Los Padres National Forest, there is a larger size that is a constraint in that particular forest based on terrain, and where we have got urban areas, or where you hit desert or where you hit rivers and so on. But the tails are really important. There is again, this very broad span. There is a cut off at the low end, too, of fires we just don’t bother to measure, and so it’s the fact that there is this broad span and natural cut offs at the two ends. DR. BANKS: This goes a bit outside the purview of the conference, but I wonder if you have any comments on the following. When you have a country like the United States, which basically is self-insuring against natural disasters, one can usually look at the historical record, and one of your early slides did that, to sort of give you a forecast of what the total costs are going to be in any given year. That sets the level at which money must be collected in order to maintain a balance on that. But then one might very well use some of that money to invest in efforts to harden areas against disaster. I just don’t know about the economic theory that drives self-insured agencies. Do you know if anybody is looking at that type of thing? DR. CARLSON: I don’t know about that, but I think it falls within this category of 100- to-1 reactive spending, where we don’t invest very much in research, and we don’t invest as much as we should in building stronger barriers against these kinds of events. I think that’s huge. We know in many cases that we are operating at or near capacity. So, with things like the power grid, we know that we are operating at or near capacity. If we put more resources in, we would be okay, but instead we have power failures. It's going to get worse instead of better because of increased population, increased demand. DR. SZEWCZYK: So, why do you live in California? DR. CARLSON: Yes, I think that’s a really good question. I grew up in Indiana, and I was afraid of tornados in Indiana. So, in Indiana you would watch the news, and there would be these tornados that come through, then I went to school on the East Coast, and I went to California. I think the first earthquake that occurred after I moved to California was the one in Canada that people felt in New York. California is a little bit crazy but it’s beautiful though. 342