Backup: Notes on Aldana's 'Boolean dynamics of networks with scale-free topology'

This post was written around 7 months ago.

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After having learned about Kauffman's attractors and then the various amazing features of scale free networks, I was trying to see how these two areas of network theories meet each other. Then I found this interesting paper written by Maximino Aldana in 2003 (DOI:10.1016/S0167-2789(03)00174-X). Through reading this paper I have also learned more basics of dynamic systems.

It is great to know that attractors also exist in a scale-free network, although in the model discussed there only the input connections follow the power-law distribution while the output connections are still Poissonian distributed. Roughly speaking, like in the random NK network, less connections per node also render a more 'ordered' system, while more connections lead the system towards chaos in the scale-free network. The difference is that the scale-free exponent, gamma, is used as the index instead of the average K.

One misunderstanding I used to have is now corrected. Once I thought a system in 'chaotic phase' has no attractors. Now it appears to me that it is only the number of the attractors are highly variable in a chaotic system. A chaotic system just contains more uncertainties which are measured by a set of parameters.

Understood is understood. After all, there are so many parameters that are proposed to measure the characterictics of a dynamic system. The most fascinating discovery here may be the quantum-like distribution of 'ns' (the sizes of attraction basins) in ordered or critical phases. At this moment I don't know what it means, but could only guess it must have implied something really important.

While Kauffman assumed life systems to be 'at the edge of chaos' because they must have both stability at steady states and evolvability to change, the author of this paper didn't stick to that. As Aldana believed, the power-law distribution of connections sufficiently allows a system to possess evolvability even under the ordered phase, because those 'super nodes' with the most connections are susceptible to disturbance and easy to change the state of the entire system.

However, that's far from explaining the behaviour of a system moving from one attractor to another in such predictable manners as in cell differentiation. Such moves are not necessarily in tight control, I belive, but are neatly tamed by the network. The challenge now is to find out how it works.

Finally, one thing the author didn't show is what happens when both input and output connections are of scale-free topology. It matters if the connections in both directions are scale-free distributed in realistic biological networks, e.g. those composed of transcription factors and miRNAs.