Abstract
The Hamiltonian function of a network, derived from the intrinsic distributions of nodes and edges, magnified by resolution parameter has information on the distribution of energy in the network. In brain networks, the Hamiltonian function follows hierarchical features reflecting a power-law behavior which can be a signature of self-organization. Further, the transition of three distinct phases driven by resolution parameter is observed which could correspond to various important brain states. This resolution parameter could thus reflect a key parameter that controls and balances the energy distribution in the brain network.
Highlights
Brain is a complex universe of neurons in which its functional and structural network properties have diverse features, such as, small worldness [1], significant emergence of modules [2], and a distribution of rich club of key hubs [3]
Study of a complex networks within the framework of Potts model was done in order to extract patterns and properties of the networks [13], and the formalism has been used as a powerful method for finding communities specially in hierarchical networks [14]
We have studied the energy distribution in the brain networks of three species [8] using simple but efficient constant Potts model (CPM) approach [12,13,14]
Summary
The energy stored in a complex system represented by a complex network may be interpreted as the energy spent in the distribution and organization of interacting edges (wiring/rewiring of edges), with specified weights and directions, among the constituting nodes in the network. It is trivial from Equations (6) and (7) that |∆N [k+1→1]| ≥ |∆N [k+1→1]| They are equal only when all the nodes in each level are totally distributed among the modules/sub-modules in the other level such that there are no isolated nodes. C1 =c′1 where L[c21] is the number of edges in the c1th module at level-2, constructed from the network at level-1. The number of inter-edges among the connected sub-modules at level-2 cannot be zero. The first terms in these equations are always greater than zero This means the organization of a larger network at a particular level from a smaller network at another level needs significant number of edges for rewiring them to achieve the self-organization at that level. From Equations (6) and (12), we can further prove that |∆H[k+1→1]| > |∆H[k+1→k]|
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