From September 2005 until June 2007 I looked for the base of a logarithmic function to model information distribution in a network. In June 2007 I found a formula C log (n) where the base of the log was the network’s mean path length, in steps. That seemed to work though it was a little mystifying for a while. When I began to puzzle through it seemed that the mean path length was proportional to the average energy connecting network nodes. Instead of a mean path length one can substitute a constant path length between nodes (equal to the mean path length), and then the network would be isotropic and homogeneously scaled. By Jensen’s inequality, the log formula with a mean as the base of the log would give optimum entropy. Because the mean path length was both scale factor and length, that implied that the scale factor for the network is the natural logarithm. In other words, the natural logarithm is the scale factor of an isotropic network. That would explain the ubiquity of the natural logarithm in modeling physical systems. I liked this result because I had not looked for it and because I had not even imagined that the natural logarithm arose out of a physical attribute in the distribution of energy. Three of the four articles I had on arxiv up to 2015 (on lexical scaling, allometry, and isotropy which are also on RG) mention this idea. I have not seen it elsewhere. Is it well-known?
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