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Nonbacktracking expansion of finite graphs
G Timár1, R A da Costa1, S N Dorogovtsev1,2
1Departamento de Física da Universidade de Aveiro & I3N, Campus Universitário de Santiago, 3810-193 Aveiro, Portugal.
Physical Review. E
|May 17, 2017
Summary
Message passing equations provide exact solutions for cooperative models on infinite trees, generalizing finite graphs. These solutions approximate large, uncorrelated graphs, with critical behavior linked to nonbacktracking matrix eigenvectors.
Area of Science:
- Graph theory
- Statistical physics
- Network science
Background:
- Message passing equations approximate behavior on finite graphs.
- Locally treelike approximations can introduce artifacts, like sharp transitions.
Purpose of the Study:
- To develop a method for obtaining exact solutions for cooperative models on finite graphs.
- To generalize the Bethe lattice for analyzing local graph structures.
Main Methods:
- Constructing an infinite tree with identical local properties to a finite graph for a nonbacktracking walker.
- Utilizing message passing equations on this infinite tree construction.
- Expressing critical region solutions using eigenvectors of the nonbacktracking matrix.
Main Results:
- The infinite tree construction provides exact solutions for cooperative models.
- These solutions serve as accurate approximations for large, uncorrelated finite graphs.
- Analysis of message passing algorithm limitations and accuracy across network types.
Conclusions:
- The nonbacktracking expansion offers exact solutions and approximations for graph models.
- Critical phenomena can be analyzed via nonbacktracking matrix eigenvectors.
- Direct simulations can be more computationally efficient than message passing for certain networks.
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