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

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

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

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