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Modeling the Functional Network for Spatial Navigation in the Human Brain

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Published on: October 13, 2023

Ising model on the scale-free network with a Cayley-tree-like structure.

Takehisa Hasegawa¹, Koji Nemoto

¹Division of Physics, Hokkaido University, Sapporo, Japan. hase@statphys.sci.hokudai.ac.jp

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|March 16, 2007

Summary

The Ising model on a leaf-bound random graph shows no magnetization at finite temperatures. Its susceptibility diverges below a critical temperature, especially for gamma <= 4, unlike models without boundaries.

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Area of Science:

Statistical mechanics
Network science
Condensed matter physics

Background:

The Ising model is a fundamental tool for studying magnetism and phase transitions.
Random graphs are used to model complex networks found in nature and technology.
Understanding boundary effects in network models is crucial for predicting system behavior.

Purpose of the Study:

To derive exact expressions for magnetization and susceptibility in a leaf-bound Ising model on a random graph.
To investigate the influence of network boundaries (leaves) on magnetic properties.
To compare results with models lacking boundaries.

Main Methods:

Derivation of exact analytical expressions for thermodynamic properties.
Analysis of the Ising model on random graphs with a specific degree distribution (P(k) proportional to k^-gamma).
Inclusion of a boundary condition defined by leaf nodes (degree 1 vertices).

Main Results:

The system exhibits no magnetization at any finite temperature.
Zero-field susceptibility diverges below a critical temperature (Ts) dependent on the exponent gamma.
For gamma <= 4, the critical temperature Ts tends to infinity.
These findings differ significantly from random graph models without boundaries.

Conclusions:

Leaf nodes play a nontrivial role in the magnetic properties of random graph systems.
The presence of a boundary fundamentally alters the phase transition behavior compared to unbounded networks.
The study highlights the importance of network topology, specifically boundary structure, in statistical physics models.