1. Plum Pudding Models for Growing Small-World Networks
Ari Zitin (University of North Carolina), Alex Gorowara (Worcester Polytechnic Institute)
S. Squires, M. Herrera, T. Antonsen, M. Girvan, E. Ott (University of Maryland)
Image Credit to transductions.net

2. Motivation and Background
Small-World Networks
Path lengths are short (grow logarithmically or slower with the number of nodes N)
Clustering (probability that a node's neighbors are connected to each other) is high
Real networks grow in spatial dimensions
Neurological and cellular networks exist, expand, and connect in three dimensions of space
The formation of new connections between nodes is limited by proximity
Lattice
Small-world model
Random
Image from Watts-Strogatz
Nature 1998

3. Our Models
We place new nodes in a ball (the Plum Pudding Network Model) or on a sphere (the Thomson Network Model) of d dimensions
Each new node connects to its m nearest neighbors
Nodes repel each other until they achieve a roughly uniform spatial distribution
Image from Wikimedia Commons

4. 1-Dimensional Thomson Network
Images from Ozik et. al.
Physical Review E 2004

5. Addition of a New Node to the 2-D Plum Pudding Network

6. Path Length is Logarithmic in Plum Pudding Network Model

7. Clustering Decays with Dimension in Plum Pudding Network Model
High Clustering
Low Clustering

8. General Results
Different models (Plum vs. Thomson) of the same dimension have similar characteristics
Some contribution due to edge effects
Consistent small-world characteristics
Logarithmic path length, asymptotic clustering
Substantial differences due to dimension
Approaches “dimensionless” behavior as the dimension grows large
Applications to neuronal networks

9. With Thanks to J. J. Thomson
Image from Wikimedia Commons