Proceedings of a Workshop on Statistics on Networks (CD-ROM) (2007) / Chapter Skim
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Keynote Address, Day 1 Network Complexity and Robustness--John Doyle, California Institute of Technology
Keynote Address, Day 1 Network Complexity and Robustness--John Doyle, California Institute of Technology
Pages 1-61
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Keynote Address, Day 1 1
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I will talk a little bit about both of these. One of the common themes I am going to talk about is the fact that we see power laws all over.
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The significance of this is that the worst event is orders of magnitude worse than the median event, which we are learning much more about this last year. It also means that demonstrations of these behaviors (events of the type represented in the upper left corner of Figures 1 and 2)
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Why is that and what do probability theory and statistics tell us about power laws? Well, if I were to ask you why are Gaussians so ubiquitous you would say it is central limit theorem.
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We should not call Gaussians normal; we should call power laws normal. They arise for no reason other than the way you measure data, and the way you look at the world will make them appear all over the place, provided there is high variability.
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. I generated exponentially distributed data, and on Figure 3's semi-log plot you see it is nearly a straight line.
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There are a bunch of networks that have been claimed to have power laws in them, and the data that was presented actually didn't. Figure 8 comes from a paper-on-protein interaction power law, an article from Nature.
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So, once you get into power laws, the slopes are really important -- again the point is, it doesn't exactly look like a power law, the real data. That is not the big deal.
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variability is everywhere.
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Figure 14 is a chart of Abilene, which is the Internet 2, and Figure 15 is actually a weather map taken on November 8. There are a lot of Internet experts here; if you look at the Abilene weather map, you might find this picture quite peculiar.
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Yes, you can create a network that has a power law degree distribution, and the next slide shows such a power law. But following that, Figure 16 is another graph with exactly the same degree distribution.
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fact, there is high variability everywhere.
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Ironically, the weird thing is, it is like those with low variability have been found and then presented as having power laws -- very strange. Anyway, there are enormous sources of high variability, and it is connected with this issue of robust yet fragile, and I want to talk about that.
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FIGURE 18 In Figure 19, I zoom in on the center part of this, which is glycolysis. Experts in the room will notice that I am doing a little weird thing like an abstract version of what bacteria do, and I am going to stick to bacteria since we know the most about them.
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FIGURE 19 What is left out on a cartoon like this? First of all, it is autocatalytic loops, which are fueling loops.
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At the minimum, you could think about stoichiometry plus regulation.
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The enzyme levels themselves, which are the enzymatic reactions, are controlled by transcription, translation, and degradation.
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The rates of the enzymes themselves are controlled also. There are two layers of control, and you might talk about how those two layers of control relate to the two layers of control on the Internet, which is TCP and IP, why they are separated, and what they do.
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Precursors are those things connected by the blue lines, the basic building blocks for the cell, and the carriers are things like ATP, NADH, carriers of energy, redox and small moieties, again, standard terminology. FIGURE 27 Figures 28-32 illustrate how to map catabolism into this framework.
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FIGURE 28 FIGURE 29 That core metabolism then feeds into the polymerization and complex assembly process by which all of these things are made into the parts of the cell, as shown in Figure 30 below.
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FIGURE 30 Of course, if you are a bacterium you have to do taxis and transport to get those nutrients into catabolism, and then there are autocatalytic loops, because not only are those little autocatalytic loops inside for the carriers but, of course, you must make all the proteins that are the enzymes to drive metabolism, and then there are lots of regulations and controls on top of that. So, schematically, we get the nested bow ties in Figure 31.
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They eat uranium, which is a cool thing to do, too. There is a huge variety in what they will eat.
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What you get is this nest of bow ties. We have this big bow tie and inside it you have little bow ties, and you have also regulation and control on top of it.
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It turns out we build everything this way, so if we look at manufacturing, this is how we build everything. We collect and import raw materials, you make common currencies and building blocks, and you undergo complex assembly.
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If you look at the core you get highly efficient, very special purpose enzymes that are controlled by competitive inhibition and allostery, and small metabolites. It is like this machine is sitting there and the metabolites are running through it -- a big machine, little metabolites.
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It is very much similar to core metabolism versus complex molecular assembly. Again, there are these universal arcs.
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If you take a modern car, a modern high end Lexus, Mercedes or so on, essentially all the complexity is in these elaborate control systems. If you knock one of those out what you lose is robustness, not minimal functionality.
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The explanation is straightforward, straight out of standard undergraduate biochemistry. This supplies materials and energy; this supplies robustness.
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That is why we call it robust yet fragile. They are extremely robust most of the time but when they fail they fail big time.
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It is obesity and diabetes. At a physiological level we have a bow tie architecture with glucose and oxygen sitting in the middle, as shown in Figure 40 below.
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Figure 42 shows a transcriptional regulatory motif for heat shock. What you have is the gene which codes for the alternative signal factor that then regulates a bunch of operons.
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What is happening is you have chaperons that both fold nascent proteins and refold unfolded proteins, and you also have proteases that degrade particular aggregates and then they can be reused. This is a system and if this is all you had, you could survive heat shock provided you made enough of these things.
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Hopefully, you are familiar with the Internet hourglass shown below. You have a huge variety of applications and a huge variety of linked technologies, but they all run IP under everything, IP on everything.
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If you are going to build architecture you have got to deal with enormous uncertainty at both edges. Again, it's similar to the bow tie, but now we are talking about it a little bit differently: it is the control system and not the flow of material and energy.
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You have got the hardware at the bottom: routers, links, servers, or DNA, genes, enzymes. You pick a raw physical network or you pick a raw genome.
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Each level of decentralized and asynchronous, so there is no centralized control. It all runs based on these protocols, so you get this bow tie picture.
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If you are thinking about this, how does this relate to the Internet? The cell metabolism is an application that runs on this bow tie architecture -- it is the bow tie architecture that runs on this hourglass.
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Those are the attacks that we need to worry about most and on the Internet file transfers you have got navigation, you have got congestion control, and you have ack-based re-transmission. On a little broader time scale, if routers fail you can reroute around failures, you can do hop swaps and rebooting of hardware.
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Also, they are fragile on time scales. The fluctuations of demand and supply can actually exceed regulatory capabilities so you have heard of glycolytic oscillations.
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This is actually how Shannon first thought about it. He said the entropy reduction is limited by the channel capacity, standard story, and the other big result is that, if the delay of a disturbance is long enough -- if the delay goes to infinity -- and the entropy of the disturbance is less than the channel capacity, then you can make the error zero.
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So, everybody takes their undergraduate course in controls. How many people took an undergraduate course in controls?
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If you think of the other thing as a conservation law based on channel capacity, you can't exceed channel capacity here. You have another conservation law that says that the total sensitivity is conserved.
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FIGURE 53 These two things have been sitting staring at us for 60 years, which is a total embarrassment. What happens if you stick them together?
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The way you connect these things is it is really a constrained optimization problem. It starts out as constrained optimization, but then you really have to redo optimization theory to get it to work in this layer decomposition way.
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I show you the raw data. Then you can do a little bit of statistics by just sort of eyeballing the straight line.
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DR. HANDCOCK: I would like to pick up on that question a little bit, about the identification and characterization of power laws.
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What we can prove now is that some of these protocols are optimal in the sense that they are as good as they can be. For example, TCP properly run achieves a global utility sharing, that is fair sharing among all the users that use it.
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2004. "Evidence for Dynamically Organized Modularity in the Yeast ProteinProtein Interaction Network." Nature 430.
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Network Models 61
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