A JOURNEY DOWN THE RABBIT HOLE: PONDERING PREFERENTIAL ATTACHMENT MODELS WITH LOCATION

We investigate the use of stochastic approximation as a method of identifying conditions necessary to facilitate condensation and coexistence. We did this for a variety of preferential attachment models which are growing by way of some predetermined selection criteria. The main results presented in this thesis concern the choice of r model. This growth method uses preferential attachment to select r vertices from a graph at time n. These r vertices are subsequently ranked according to fixed location assigned at each of their creations and used as an extra level of comparison between vertices. A new vertex is then attached to one of these r selected vertices according to a predetermined vector of probabilities corresponding to this ranking. We have shown that condensation can occur for any of these vectors, if we can find at least two stable fixed points to the corresponding set of stochastic approximation equations. Following this we investigate the degree distribution and complexity associated to the introduction of a higher dimensional location coefficient. Our concluding chapter investigates the coexistence between vertices in preferential attachment networks where vertices posses different types and locations. Using similar methods as in the choice of r model we have shown that coexistence can occur in location type models with phase transitions helping to classify different cases.