Plum Pudding Models for Growing Small-World NetworksAri 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

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

Small-world modelLattice Random

Image from Watts-Strogatz Nature 1998

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

1-Dimensional Thomson Network

Images from Ozik et. al.Physical Review E 2004

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

Path Length is Logarithmic in Plum Pudding Network Model

2D

4D

8D

Clustering Decays with Dimension in Plum Pudding Network Model

High Clustering

Low Clustering

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

With Thanks to J. J. Thomson

Image from Wikimedia Commons