Hi, I'm Steve Lansing, Co-Director of The Complexity Institute at Nanyang Technological University in Singapore. Welcome to the second of our video lectures, our MOOC, on complexity science. Today we'll be talking about three core concepts in complexity science: robustness , resilience, and sustainability. There are many misunderstandings about those concepts. It's often assumed that they mean the same thing, but they do not. They refer to different aspects of the behavior of complex systems. Today, we'll probe the differences and in the process gain a better understanding of complex systems. Dictionary.com defines robust as strong and effective in all our most situations and conditions. Merriam-Webster is pretty similar. Robust means capable of performing without failure under a wide range of conditions. The Oxford Dictionary is a little different. It defines robust as able to withstand or overcome adverse conditions, which misses the breadth of conditions aspect in the other two definitions. For resilience, we find more consistency in the dictionary definitions. Dictionary.com says resilient means returning to the original form or position after being kept compressed or stretched. Merriam-Webster defines resilient as able to become strong, healthy, or successful again after something bad happens. Finally, the Oxford Dictionaries defined resilient as able to withstand or recover quickly from difficult conditions. These three definitions are quite similar. All three definitions involve a drastic change to the complex system in question and describe the ability to recover from that change. Therefore, we can already distinguish between robustness, which has to do with a complex system retaining its function over a broad range of conditions, and resilience, which instead has to do with a complex system recovering after experiencing a sudden change. Finally, let us look at the dictionary definitions of sustainability. Here we find considerable differences between the three dictionary definitions. For example, Dictionary.com defines sustainable to mean the ability to be maintained or kept going as an action or process. Dictionary.com's focus is more on process and a sustainable process is one that can be kept going for a long time. Finally, Oxford Dictionaries define sustainable as the quality of being possible to maintain at a certain rate or level. This definition can apply to both processes and resources, but focuses on the existence of thresholds. Beyond those thresholds, the processes and resources will no longer be sustainable. These differences in definitions reflect different groups of experts who want to use robust, resilient, and sustainable to describe the different things they're interested in. We'll not say whether they're right or wrong to do so, but for the purpose of this course, we'll use a set of definitions that's consistent with our basic understanding of complex systems. First, let's consider how a simple system versus a complex system respond to change. We can think about those interactions using a graph or a network. Here's a network showing a simple system. The red dots are variables and the lines between them indicate that there is some connection between them, there is some interaction effect. This network is sparsely connected. That means that not much is going on and something like a doubling of some force here can lead to a doubling in the response of this system. The change is linear, it is proportional to the cause. Doubling the cause doubles the effect. On the other hand over here, we have a complex network. Here there are more variables indicated by more nodes with more edges, and notice that in some cases they connect as a feedback loop. In other words, there is a relationship that continues linking all three variables together. Under those conditions, this system can undergo nonlinear change. It means that effects may not be proportionate to their causes. Here's a loop connecting three variables into a feedback system in a complex system. It's the existence of loops like this that are largely responsible for the non-trivial conflicts behavior of densely connected complex systems. They force the system to settle down into a small number of stable states. Those stable states can be fixed points where the variables remain at a fixed unchanging value or limit cycles where some of the variables undergo oscillations, or more complex attractor sets. These feedback loops also change how the complex system responds to changes and its variables. When it is already in one stable state, a 10 percent change in some of its variables may lead to imperceptible change to the complex system. In fact, we must make some large changes to a few variables before the complex system undergoes an observable change to a different stable state. We call the change of the complex system from one stable state to another a regime shift. Now, we can give a definition of robustness. Here's a complex system that is already robust which means it's mostly in a stable state and most of the time small changes in the variables will not cause a major change. In fact, there are only a small number of variables whose change can cause this system to undergo a change from one state to the other. The breadth of this condition, how persistent that condition is defines the robustness of that complex system. Small changes have very little effect. As we discussed in the previous slide, the complex system starts out in a stable state and stays in that stable state even as we tune the control parameter over a range of variables. If this range of values is large we can say that this stable state is robust. But eventually, when we increase the control parameter beyond its critical value the complex system undergoes a forward regime shift to a different stable state which could be more robust, less robust or just as robust as the original stable state. Now, suppose this new stable state is undesirable and we'd like to restore the old stable state. We can do this by reducing the value of the control parameter. However and this is the key point, this reverse regime shift typically does not occur when the forward regime shift occurs. In other words, we may have to reduce the value of the control parameter well below the first critical value, passed a second critical value before the complex system returns to its old stable state. This phenomenon is called hysteresis or path dependence. Because of the presence of hysteresis, a complex system that's kicked out of its stable state by a sudden shock will need time to recover to return to its original stable state. Often very large shocks don't just change the control parameters for a system they actually remove some of the links which means the system as a whole is changed. For example, an earthquake can change the connections in a power grid and the engineers will have to go back and repair those broken links. Once that happens, the whole behavior of the system is going to change. A natural or a social complex system is also equipped with some repair mechanisms. However, since a complex system is the product or the result of a self-organizing process, not design, there's no blueprint for the complex system to refer to after the shock, to identify the broken links. Instead, new links will just form until the graph of variables self organizes into a functionally equivalent new network. This new network will not have the same links as the old network before the shock but it can serve the complex system in very much the same ways. Now, imagine tracking the recovery of the complex system. Let's say, that after the shock the recovery is not complete but the complex system nevertheless regained most of its vital functions. We can then say that the complex system is resilient. This is to be contrasted against a different system which collapses right after the shock. Such a system we would call fragile because it's incapable of any kind of recovery. Now, suppose we compare two complex systems which recover to the same level after a shock and find that one of them recovers faster than the other. We can then say that the complex system that recovers faster is more resilient. Alternatively, if we compare two complex systems and find that one recovers to a level closer to the initial state than any other then we can also say, that complex system that recovers more completely is more resilient even if the recovery is not faster. Therefore, from the complex systems point of view there are two qualities of recovery from shock that we can establish, speed and completeness. All of this suggests that we should exercise a lot of care when we talk about resilience. Finally, based on the discussions we've just had about robustness and resilience we can conclude that from a complexity perspective, a complex system is sustainable if it's in a robust regime, has mechanisms that allow it to recover from shocks and in its day-to-day functions stays far away from regime shifts. Let's move on now to sustainability. Sustainability became an analytical question in two lectures which were published by the British Economist William Foster Lloyd in 1833 about the tragedy of the commons. That concept was later picked up and popularized by an American Ecologist, Garrett Hardin in 1968. In a nutshell, we start out with a common-pool resource available to all agents. In Lloyd's 1833 lecture, that was a common pasture that was not owned by any of the herders. With access to sunlight, carbon dioxide and water the grassland renews at a rate Alpha and is being extracted at a rate Beta. Clearly, if the extraction rate is larger than the renewal rate the grassland would become barren in a finite amount of time and the situation would become unsustainable. On the other hand, if the extraction rate is smaller than the renewal rate the grassland would remain close to the carrying capacity and the cows belonging to the herders could graze there forever. This represents a sustainable solution. We can count how many seeds each grass plant produces and how many of them sprout into plants. We can also count how many of the grass plants are eaten by the cows. But those two rates, Alpha and Beta, use different units so they can't be directly compared. To analyze the sustainability of the grasslands with the cows, we'll need to write down a model. There are several points we need to consider in modeling a problem like this. First, we must acknowledge that we can't consider all the variables like individual cows, different species of grass, intensity of sunlight, cloud cover, rain, temperature, length of day over the seasons, and so on. That would make the model practically intractable, as we would not have enough data to represent all the variables. Therefore, we model only those variables that we assume are the most important ones, like the grass population density and the cow population density. Next, we need to make a choice of the outset, whether to make the model time-independent, that is to say, an equilibrium model or time-dependent, in other words, a dynamical model. We'd also need to decide whether our model will be deterministic involving no random factors, or probabilistic involving some random factors. Finally, we need to decide whether the model will be continuous or discrete. For continuous models, we do not track individual grass plants or individual cows. We'll write the model down using the mathematical language of differential equations. For a discrete model, we'll track the individual cows, though well, perhaps not the individual grass plants. We'll have to write them all down in the form of algorithms that can be simulated using a computer. Let's say we've decided to model the question using a continuous model, which means differential equations. Okay? First, we've got S, that's the resource which is just the grasslands. That's the grasslands. We also have L, which is the exploitation, that's the cows. Here we have the cows. First of all, let's grow the grass. Here we have change in S, which is change in the resource, which means the grass is a function of Alpha, that's the growth parameter, times the amount of grass that S, times 1 minus S, again, that's the grasslands, over K, which is the carrying capacity. That simple equation is going to grow the grass at the rate Alpha, which depends upon how fast grass grows. However, we also got to put in the cows. Here are the cows, that's consumption by the cows. That depends upon Beta, which is the rate at which cows eat grass, times L, which is the population of cows at time t. We grow the grass with a logistic growth curve. Logistic means it's a continuous growth. Then we remove some grass with the consumption term. In this toy model, we assume that the exploitation rate is proportional to the population density of exploiters, meaning cows, L of t. In addition, we assume that the cows can only be efficient in exploiting the grassland if the grassland is full of grass. If for some reason grass occurs only in patches within the grassland, then the cows would need to spend part of their time searching for these grass patches, leaving them less time to eat the grass. The further apart the grass patches are, the larger the fraction of time the cows must commit to searching for them. Therefore, for simplicity's sake, we assume that the exploitation rate is also proportionate to the common pool resource level, in other words, S of t. Combining these two dependencies, we write the exploitation rate as Beta LS. For the exploiters, the cows, we assume that the rate of growth of its population density is proportional to the population density itself. Basically, we argue that if there are more cows, there will be more calves born per unit time. If we doubled the number of cows, we'll double the number of calves being born per unit time. In this toy model, we see that the rate of change of exploiter population density depends upon a complicated-looking effective growth rate per capita. There are two parts to this effective growth rate. First, Delta is the difference between the intrinsic growth rate and the intrinsic death rate. By intrinsic, we mean the number of calves born per unit time, if the calves have nothing to eat, and the number of cows that will die if they have nothing to eat. Clearly, if the cows have nothing to eat, there probably won't be any calves born. Therefore, we can assume that the intrinsic growth rate is zero. Also, if the cows are well-fed all along, but then suddenly the grass vanishes, then the cows will keep digging into their reserves. Every now and then a cow will starve to death, and eventually, all the cows will die of starvation. This tells us that Delta is effectively just the intrinsic death rate, which is a negative number. Next, if the cows do get to eat, they will not die of starvation. If they get to eat enough, they would have enough energy reserves to also give birth to calves. Therefore, resource consumption can lead to a positive rate of growth in the exploiter population. This added rate of positive growth should be proportional to the rate of exploitation, which is Beta LS. However, a cow may have to eat a lot of grass in order to give birth to a calf. We therefore replace Beta with Phi, a conversion factor, in the positive growth rate of exploiters. Now that we've got a model, let's analyze it. Suppose we start with different levels of cows and rates of grass growth, then we let the model go forward in time, so we allow Alpha, Beta, Delta, Phi, all of these parameters to evolve. What will be the total levels of S and L, in other words, grasslands and cows, when they are no longer changing? What happens when the model moves to a fixed point or an equilibrium? To get these answers, we perform a fixed-point analysis on the model. This is done by first setting Delta S over Delta T equals 0, and Delta L over Delta T equals 0. We then solve the resulting algebraic equations simultaneously to find the solutions S and L, and we find three such solution. The first solution is S equals zero and L equals zero, which corresponds to the situation when there's no grass and no cows. The second solution is S equals K. In other words, the grasslands are at carrying capacity, and L equals zero, which means no cows, and so the grassland is at carrying capacity. Finally, the third solution is S equals minus Delta over 5 times Beta, and L equals Alpha over Beta times 1 plus Delta over Phi Beta K or carrying capacity. That corresponds to a situation where a non-zero level L of exploiters coexists with a non-zero level of common-pool resource, in other words, grass, so that's the interesting solution. In the real world, the common-pool resource and the exploiter levels do not remain constant. There are always fluctuations. In the presence of these fluctuations, some of the solutions will be stable while others will be unstable. A proper stability analysis would be too lengthy for this video lecture, so I invite you to read up on this technique in any introductory texts on nonlinear dynamics. In the meantime, we'll just quote the results to tell you that the first two fixed points are unstable. Well, the third fixed point is stable. Intuitively, we can understand why this is so. In the first fixed point, we have S equals zero, no grasslands, and L equals zero, no cows. If we sprinkle some grass seeds without bringing in any cows, then after some time, the grassland will be restored. In other words, a small patch of grass representing S of T, only slightly larger than zero, will bring us from the first fixed point to the second fixed point. In the second fixed point, where S equals K, which means grasslands are at carrying capacity and L equals zero, if we bring in some cows, the cow population will start to grow exponentially, eating the grassland barren, and thereafter starving themselves to death. Therefore, introducing sufficiently many cows will bring us from the second fixed point back to the first fixed point, but in the third fixed point, there's a balance between the growing grass population and the cow populations in such a way that if the grass population increases, the cows will reproduce faster and eat more grass, bringing the grass population down. Conversely, if the grass population decreases, the cows will reproduce slower and eat less grass bringing the grass population up. We've just created a nice, simple model that has three fixed points and tells us something about the tragedy of the commons. Next step is to add a little bit more realism, which takes us to a different kind of model. Suppose that the herders have noticed that too many cows leads to the disappearance of the pasture. In that case, we have a different kind of dynamic as they decide whether to cooperate with one another and maintain the herd at a sustainable level, so then each herder or each player can make a decision. They can either decide to put as many cows on the field as can be sustained, or they could decide to go ahead and put as many cows as they've got on the pasture, in which case that resource will disappear. A different kind of model has been developed to analyze the dynamics in that kind of situation, and it generally goes by the name of the prisoner's dilemma. Imagine we've got this homogeneous group of extractors. Those are people who are extracting the resource. In other words, the cow owners. What happens if the number of cooperators, who are restraining themselves and not putting too many cows on, suppose that small and the number of defectors, meaning those who are going to go ahead and put in as many cows and they can on the field, suppose that's large. In that case, more defectors are going to result in a much smaller equilibrium solution as the grasslands diminish as a result of overgrazing by the cows. The relationship between cooperators and defectors is going to determine the stability of this system. The problem of cooperation and defection, which is modeled by the prisoner's dilemma, it has been extensively studied in the branch of applied mathematics called game theory. The problem itself is commonly known as the prisoner's dilemma, and in the next video, we'll hear an analysis of the prisoner's dilemma by Brian Arthur, a distinguished complexity economist.
