A Model of Communities Multi-Layered Networks John Stevens

1. Introduction This poster describes a multi-layered model of a community. This model builds on work by Kleinburg (1999) which holds friendships and acquaintances between individuals as the key elements of a community. To assess the validity of the new multi-layered model, this poster compares it with the output from an implementation of the Kleinberg model for a 10000 node 50 degree simulation averaged over 3 trials. This poster specifically compares results with the small world number, the distribution of out degrees of each node and the number and distribution of cliques. The Model This work represents in the simulation of multiple tiers of lattices representing acquaintance/friendship links. Each individual is represented by a point that has a home location (position in the tier 1 lattice), a work location (tier 2), followed by social and family location, so that the non-home locations are derived from the home location based on a distribution. The number and distribution of cliques within the networks produced offers a more clear comparison of the differences between the models. The Kleinberg model has a large number of cliques of size 5 and produces a odd distribution of cliques. The multi-layered model produces a more realistic distribution of both the number and size of cliques. Discussion The multi-layered simulation produces a better approximation of community structures and acquaintance graph than a Kleinberg simulation tacking into account the average number of acquaintances of each individual (mean degree of each node), the distribution of these acquaintances, the small world number (mean path length between to nodes) of the resultant graph and number and size of cliques in the final graph. Reference Jon Kleinberg, "The Small-World Phenomenon: An Algorithmic Perspective", Cornell Computer Science Technical Report 99-1776 (1999) About the author: John Stevens is a Research Student (PhD) in the Department of Sociology, University of Essex, UK Email: jstevens@essex.ac.uk Some basic features of the Multi-Tiered Model, with two tiers for simplicity The green box represents the lattice (town), the red dots represent the points (people) in their tier 1 (home) locations, the green dots represent the tier 2 (work) locations, with the arrows showing the relationship between tiers 1 and 2. D D C • Consider the links of A: • A is linked to B (ie A knows B) because B lives close to A (within the circle around A). • A knows C because C works close to A (within the circle around A’s work location). • A does not know D because they neither live nor work close enough to each other. A C A B B A knows B and C, but not D A Model of CommunitiesMulti-Layered NetworksJohn Stevens Results I use Kleinberg's method to compare models in this table. Very little difference separates the models for either the small world number or the clustering coefficient (though the similarity is lower in the latter case). Alternatively a greater difference can be discovered by comparing out degree distribution. These two graphs show a differing distribution of number of friends known (out degree) for the two simulations. The output of Kleinberg simulation looks less natural with thee majority of nodes having 10 links to other nodes. The multi-layered model, in contrast, gives a more natural distribution for each node (individual) with a average of 21.93 friends per person (standard deviation= 3.31).