It can be advocated that a graph is a powerful representation for several kinds of systems. Does anyone use them to study self-organizations? It think it has been argued that the study of the degree power law of a graph could give clues for the evolution of the system.
Any further idea?
Hi, you could check out work by Pacheco and collegues and Fehl et al 2011 on evolution of cooperation on graphs / networks. Or Cordero and Hogeweg and similar papers on evolution of protein networks.
See book of A.N. Sharkovsky, V. Fedorenko et. al. Of course, the graph theory can be applied to the one-dimensional dynamical systems. For example, for unimodal maps.
Quite a number of papers have tackled this question but the area remains somewhat poorly investigated. Erdos-Renyi, Bollobas, and Kaufman explored criticality and phase transition in random graphs. For example, the N-K model. Evolution of power law graphs and criticality in power law graphs were studied by a number of people, including Fan Chung Graham, and Ashish Bhan. In late 1999 or in 2000-2001 (don't remember exactly) there were a couple of papers, one in Nature, on statistical mechanical theory of graphs, where they explored phase transition behavior. I think Barabasi has also explored this question. If you search with these names you will find enough to lead you to the literature.
There is the famous Sharkovsky theorem for the one- dimension systems. The question is equivalent to the question: Is there an analog of the Sharkovsky theorem in the two-dimension case?
Hi Matthias,
If you are looking for a book "Networks, An Introduction" by Mark Newman might be of interest. Especially part V Processes on networks.
Dear Matthias, I think the most promising application of graphs to self-organization was initiated by Rutledge et al. (J. Theor. Biol. 57:355-371) when they compared the input distributions to the output distributions of the *same* network using the decomposition of the network's functional entropy into the mutual information (representing constraint [organization]) and the conditional entropy (reflecting flexibility). The power of this decomposition is enormous, and I devoted most of my professional career to demonstrating its practical ramifications. The best, Bob
Evolution of complex networks is one of the areas supporting answer your question.
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