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Modeling the Functional Network for Spatial Navigation in the Human Brain

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Distance distribution in random graphs and application to network exploration.

Vincent D Blondel¹, Jean-Loup Guillaume, Julien M Hendrickx

¹Department of Mathematical Engineering, Université catholique de Louvain, 4 avenue Georges Lemaitre, B-1348 Louvain-la-Neuve, Belgium. vincent.blondel@uclouvain.be

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

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Summary

We studied edge discovery in Erdos-Rényi graphs using shortest paths. A new method reveals oscillatory patterns and phase transitions in graph connectivity as edge probability changes.

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Area of Science:

Graph theory
Network analysis
Probability theory

Background:

Erdos-Rényi graphs are fundamental models in network science.
Understanding edge discovery is crucial for network analysis.
Shortest path computations are key to network structure analysis.

Purpose of the Study:

To determine the proportion of discovered edges in Erdos-Rényi graphs via shortest paths.
To analyze the oscillatory behavior of edge discovery with varying edge probabilities.
To introduce a novel method for computing node distance distributions.

Main Methods:

Developing a new approach to compute distance distributions from a source node.
Analyzing the proportion of edges connecting nodes at equal distances.
Investigating the impact of edge probability on graph connectivity.

Main Results:

Identified intriguing oscillatory behavior in edge discovery as edge probability changes.
The novel distance distribution method shows superior performance compared to prior analyses.
Estimates from the new method align exceptionally well with numerical simulations.

Conclusions:

The developed method accurately characterizes phase transitions in graph connectivity.
The findings provide new insights into the structure and properties of Erdos-Rényi graphs.
This work offers a more efficient way to analyze network properties related to shortest paths.