Efficient Physical Embedding of Topologically Complex Information Processing Networks in Brains and Computer Circuits
A hierarchical modular network, A(i), is made up of modules, A(ii) green, which are made up of sub-modules, A(ii) yellow, and sub-sub-modules, A(ii) red, which are collectively visualized by a co-classification matrix, A(iii), where hierarchical modularity is evident by layers of color located along the diagonal . To estimate the topological Rent exponent and dimension of a network, B(i), we first cover the network with a single box large enough to cover it entirely; then we recursively partition the box (B(ii) and B(iii)) into halves, quarters, and so on using a partition algorithm that minimizes the number of edges cut by each partition. For each iteration of this process, we count the number of nodes within a partition (), and the number of edges () crossing the partitions; a linear relationship between these two variables plotted on logarithmic axes indicates topological Rentian scaling of network connectivity and provides an estimator of the topological dimension of the network, . To estimate the physical Rent exponent, we randomly place 5000 randomly sized boxes on the physically embedded network, e.g., the human brain network in anatomical space C(i). Then we count the number of nodes and the number of boundary-crossing edges for each box C(ii) and estimate the physical Rent exponent by the linear relationship between these two variables on logarithmic axes, C(iii).