Chapter 123 20.2 Counterintuitive Network Math

The set of mathematics invented by Kaufman, Holland, and others does not yet have a proper name, and I call it "network mathematics" here.Some of these methods have various informal names: parallel distributed processing, Boolean networks, neural networks, spin glasses, cellular automata, classification systems, genetic algorithms, swarm computing, and so on.No matter what kind of network mathematics, the horizontal causality formed by thousands of interacting functions is a common element.They both try to coordinate a large number of simultaneous events—the kind of nonlinear events that are ubiquitous in the real world.Network mathematics is the opposite of classical Newtonian mathematics.Newtonian mathematics is applicable to most physical problems and was once considered the only mathematics serious scientists needed.And network mathematics is useless without a computer.

The wide variety in the mathematics of group systems and networks made Kaufman wonder whether this peculiar logic of groups—which he was sure would generate necessary order—was not a logic more general than specific.Physicists who study magnetic materials, for example, have encountered a vexing problem: the particles that make up ordinary ferromagnets—the kind that stick to refrigerator doors or are used in compasses—will become obsessed with pointing at the same direction, thereby forming a significant magnetic field.In the weakly magnetic "spin glass", the internal particles are more like "grass on the wall", and its orientation will be affected by nearby particles.Nearby particles have a greater influence, while farther particles have less influence.In this network, the magnetic fields that influence each other and link each other form the familiar picture in Kaufman's mind.This nonlinear behavior of spin glasses, which can be modeled using various network mathematics, was later found in other swarm modes as well.Kaufman is convinced that the circuits of genes are similar in architecture.

Network mathematics is not like classical mathematics, and it has properties that often do not conform to people's intuition.In general, in interacting clusters, small changes in the input can cause large changes in the output.This is the butterfly effect - the effect is not proportional to the cause. Even the simplest equation, as long as it feeds intermediate results back to the input, its output is unpredictable.It is difficult to get a glimpse of its properties just by studying the equation itself.The relationship between the various parts is entangled, and trying to describe it clearly with mathematics is tantamount to embarrassing yourself.The only way to know what an equation will do is to run it, or in computer jargon, "execute" it.The same is true for the compression of plant seeds.The chemical pathways underlying it are so intricate that no amount of ingenuity in examining an unknown seed can predict the final plant form.The easiest way to know what a seed will look like is to let it germinate and grow.

The equations are rooted in the computer.Kaufman devised a genetic model that runs on an ordinary computer and contains 10,000 genes, each of which is a tiny piece of code that turns other genes on or off.Associations between genes are set randomly. Kaufmann's point: Whatever the gene's mission, such a complex network topology can generate order—spontaneous order! When Kaufman worked on modeling genes, he realized that what he was doing was building a general genetic model for any population system.His program can model any group of mesons interacting in a massively concurrent domain.They could be cells, genes, businesses, black-box systems, or simple rules—as long as the mesons have inputs and outputs and their outputs serve as inputs to neighboring mesons.

Kaufman randomly connects this large group of nodes to form an interactive network.He made them interact with each other and recorded their behavior.He regards each node in the network as a switch, which can turn on or off certain surrounding nodes.The surrounding nodes can in turn act on this node.Eventually, this chaotic situation of "A triggers B, B triggers A" tends to a stable and measurable state.Then, Kaufman randomly reset the connection relationship of the entire network again, let the nodes interact again, until they all settled down.Repeat this many times until he thinks he has "traveled" every inch of this possible random connection space.From this he can learn about the general behavior of the network, which is independent of the content of the network.If you use real things to do an analogy experiment, you can choose 10,000 companies, connect the employees of each company randomly through the telephone network, and then consider the average effect of the 10,000 networks, regardless of whether people are on the phone or not. What did you say.

After conducting tens of thousands of experiments on these general-purpose interaction networks, Kaufman learned enough about them to paint a rough picture of how such swarm systems behave in specific contexts.In particular, he wanted to understand what types of behavior a typical chromosome would have.To do this he programmed thousands of random combinations of genetic systems and ran them on a computer—the genes changing and influencing each other.He found that they fell into several behavioral "basins." When water comes out of a garden hose at a low velocity, the flow is not steady but continuous.Turn on a large spigot, and the water bursts out in a chaotic but descriptive torrent.Open the faucet fully, and the water will rush out like a river.Carefully adjust the tap so that it is between two speeds, but the water does not stay in the middle mode, but turns quickly to one mode or the other, as if the two modes are attractive to it.Just as a drop of rain falls on the continental divide, it will eventually flow into the Pacific Ocean or the Atlantic Ocean.

Sooner or later, the dynamic process of the system will enter a certain "basin" that can capture the surrounding motion state and make it into a persistent state.According to Kaufman, random combinations of systems will find their way to a certain basin, that is, order emerges from disorder out of chaos. Kaufman ran numerous genetic simulations, and he found that there was a rough ratio between the number of genes in a system (the square root) and the number of "basins" those genes ended up in.The same relationship exists between the number of genes in a biological cell and the number of cell types (liver cells, blood cells, brain cells) that these genes give rise to.This ratio is roughly constant for all organisms.

The fact that this ratio holds for many species, Kaufman claims, suggests that the number of cell types is essentially determined by the cellular structure itself.The number of cell types in the body, then, might have less to do with natural selection and more to do with the mathematics that describe how genes interact.How many other biological phenomena, Kaufman wondered excitedly, have little to do with natural selection? He intuitively believed that the answer to this question could be sought through experimentation.First, though, he needed a way to randomly construct life.He decided to simulate the origin of life.First generate all the "elements" before the birth of life, and then let these elements gather in a virtual "pool" and interact with each other.If this pot of "original soup" can inevitably produce order, then he has an example.The trick is to get the molecules to play a game called "stacking".