Saturday, November 19, 2005

Thinking Out Loud About Emergent Behavior...Those Power Laws

I’ve addressed the idea of emergence, emergent behavior, network structures, and so on a number of times in the not so distant past, and a new result by a student of mine has me thinking about these topics again. There is an ever-growing list of complex systems in all areas of science, economics, social science, and technology that have been described or categorized as ‘emergent systems.’ Related to these types of systems are things like networks and self-organized systems, where structure/organization arises from what initially is a disorganized system of individual components. In addition, what we commonly call phase transitions can also be grouped in with the mix.

A common problem with all this is understanding what the fundamental organizational principles are that are responsible for the transition from disorganized/chaotic state to an organized state. In other words, we see what the initial state is, we observe what comes out as a final state, but the actual emergent process is typically not well understood. I began wondering if there may be some type of system that is well understood, shows some sort of transition with a signature of emergent behavior, and could be used to give new clues to organizational principles that may be useful in other, different areas of research. One place to look is in the most fundamental system possible, subatomic particles.

A common signature that is linked to emergent behavior, phase transitions, network structure (specifically scale-free networks), across all fields is the presence of power laws relating various quantities relevant to a specific system. Do power laws exist in analyses of subatomic particles that are along the same lines as other emergence studies? Inspired by a study that looked at what ends up being a scale-free network structure in cellular metabolic chemical systems, where power laws result when one counts the number of chemical reactions within the metabolic process particular molecules participate in, we looked at baryons and the number of decay modes they have, as well as the number of times they appear in other particle decays. It was not obvious going into this what to expect, because each type of baryon has specific numbers of allowed decay modes, each with their own branching ratios (i.e. probabilities of occurring), and having a limited number of particles that exist within the quark model. The results were…power laws.

The interesting aspect of this is that the decay of particles is described in detail by the Standard Model (SM). The SM predicts what type of particles may exist (there is a finite set of possibilities because of three families of six quarks, and the quarks only combine in pairs, mesons, or triplet states, baryons), it predicts what they are allowed to decay into, and it predicts the branching ratios. What we observe in experiments fits the predictions beautifully in all cases. The SM bases its predictions on conservation laws and selection rules of certain quantum numbers, similar to the way quantum mechanics predicts allowed electron configurations (i.e. the entire periodic table of the elements) from simple selection rules of electron quantum numbers in bound states.

The questions in my mind now are: Why are power laws found in so many systems that have nothing to do with each other? Why are power laws signatures for emergent behavior? Now that we see power laws in a fundamental system such as baryons, is there a deeper meaning we can gather from the results? That is, are there possible analogs to conservation laws and quantum numbers (specific quantities that can only have certain values) in other systems that haven’t been thought of that could be the organizational principles, and responsible for the observed power laws?

In some sense this is my take on what is being tried in a field such as econophysics, where analysis methods and techniques that have been perfected in describing physical systems is leading to new insights and new ways of thinking in economic models. Could the baryon analysis be used to cue into a new way of thinking about other systems? I have no idea what the answers to these questions are or if these are nonsense questions to ask...I am only thinking out loud at the moment.

2 comments:

Rizalist said...

I think that one of the most interesting challenges for this area of research is in the modelling of scale-free networks like the WWW, because like WWW, human societies are also scale-free networks. Visualizations of WWW's local connectivity structure are easily seen using a tool like TouchGraph and reveal patterns of links among websites that resemble stellar constellations in form and behave as if there was a kind of "link gravity" as you drag sites around, like stars. Now I think as you know power laws are also the signature of the presence of strange attractors, which in WWW would be the equivalent of INSTAPUNDIT. It was after all in the study of WWW as a scale-free network that the Zipf Law Dependence of Blogosphere Popularity was found.

Sorry to be rambling, but i just love all this sexy science talk, this is ..
nobody in particular just passing through...

vonny said...

djb-

You are right that looking at connectivity maps of certain networks does in fact reveal some amazing sructures and interesting pattern formation in their own right, regardles if it is a map of a food web, www, some social network, etc. In terms of sclae-free networks, I think the "link gravity" you mention refers to the "rich get richer" behavior of the network, as Barabasi found with the www. When new nodes enter the www, they are more likely to form links with hubs that have numerous links rather than other nodes with few links, so the Yahoos of the web keep getting bigger and bigger in terms of their links.