The simplicity of planar networks.

Area of Science:

• Graph theory
• Network science
• Spatial analysis

Background:

• Shortest paths are crucial in network analysis but may not be the most navigable.
• Planar networks present unique challenges where path simplicity (fewest turns) differs from path shortness (least distance).

Purpose of the Study:

• To introduce and analyze the 'simplicity index' and 'simplicity profile' for quantifying path simplicity in planar networks.
• To statistically compare shortest and simplest paths to understand spatial organization.
• To differentiate organizational principles between artificial and natural networks.

Main Methods:

• Defining the simplicity index as the average ratio of shortest path length to simplest path length.
• Calculating the simplicity profile to assess simplicity across different scales.
• Applying these metrics to diverse networks: roads, highways, railways, leaves, slime mould, and insect wings.

Main Results:

• Fundamental differences observed between urban and biological network organization.
• Biological networks (leaves, insect wings) display hierarchical organization in straight lines.
• Urban networks (roads) exhibit random lengths and locations for straight lines.
• Simplicity metrics can detect structural changes in evolving networks.

Conclusions:

• The simplicity index and profile offer non-trivial, non-local insights into graph spatial organization.
• Distinct organizational strategies in biological versus urban systems are linked to function and navigation.
• Network simplicity analysis is valuable for understanding both static and dynamic network structures.