Phase transitions in small-world systems: application to functional brain networks

Inhomogeneous Coherent States in Small-World Networks: Application to the Brain Networks

“…The network generated using the algorithm presented in this paper is a system with weak disorder. The regular substructure of this system possesses the symmetry of the discrete subgroup of the Galilean group [10]. To describe a concrete process in such system, it is necessary to construct the Rapp diagram, which allows determining the number of components of the order parameter [2,10].…”

“…The regular substructure of this system possesses the symmetry of the discrete subgroup of the Galilean group [10]. To describe a concrete process in such system, it is necessary to construct the Rapp diagram, which allows determining the number of components of the order parameter [2,10]. Thus, it is possible to define the integer basis of invariants of an irreducible representation on which depends the free energy functional [10].…”

“…To describe a concrete process in such system, it is necessary to construct the Rapp diagram, which allows determining the number of components of the order parameter [2,10]. Thus, it is possible to define the integer basis of invariants of an irreducible representation on which depends the free energy functional [10]. The fractional differential equation of motion for the order parameter takes into account the effects of spatial and temporal memory.…”

“…Using these transformations (10) we reduce the fractional differential equation to an integer nonlinear differential equation…”

“…It intuitively makes sense that in case of small enough, but finite 𝑝 the obtained small-world network could be considered as a lattice with a weak disorder. Then knowing the number of the components of the order parameter we could determine its transformational properties [10].…”