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Related Experiment Video

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Modeling the Functional Network for Spatial Navigation in the Human Brain

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Published on: October 13, 2023

Structural properties and complexity of a new network class: Collatz step graphs.

Frank Emmert-Streib¹

¹Computational Biology and Machine Learning Laboratory, Center for Cancer Research and Cell Biology, School of Medicine, Dentistry and Biomedical Sciences, Faculty of Medicine, Health and Life Sciences, Queen's University Belfast, Belfast, United Kingdom. v@bio-complexity.com

|February 23, 2013

Summary

This study introduces a novel method for generating complex networks using number theory's Collatz problem. The resulting complex networks (CS graphs) exhibit a unique "smaller" characteristic as they grow in size.

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Area of Science:

Complex networks
Network science
Number theory

Background:

Existing network generation models often lack biological inspiration.
The Collatz problem offers a unique starting point for sequence generation.

Purpose of the Study:

To derive and analyze structural properties of symbol sequences from the Collatz problem (step sequences).
To introduce a novel network construction procedure based on these step sequences.
To investigate the structural properties and scaling behavior of the newly generated complex networks (CS graphs).

Main Methods:

Derivation of step sequences from the Collatz problem.
Development of a network construction algorithm utilizing these step sequences.
Analysis of network properties including complexity, average shortest path lengths, and clustering coefficients.

Main Results:

Characterization of the structural properties of Collatz-derived step sequences.
Successful generation of a new class of complex networks (CS graphs).
Demonstration that CS graphs exhibit decreasing size with increasing network size, contrasting with other models.

Conclusions:

The Collatz problem provides a viable foundation for generating complex networks with unique properties.
CS graphs represent a novel class of networks with distinct scaling behaviors.
This biologically inspired approach offers new perspectives in network science and complexity studies.