Plum Pudding Models for Growing Small-World Networks

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

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 Random Lattice Small-world model 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 2D 4D 8D

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