An in-person Sunbelt conference at Paris. This time I’ve fortunately got a prime time slot to present on the soon-to-be-wrapped-up work on “evidencing preferential attachment in dependency network evolution”; slides are in pdf and html.
This is built on the following two pieces of work:
- Last year we incorporated extreme value theory & mixture modelling when fitting to degree distributions. What we’ve found is that most networks are partially scale-free, meaning that the degrees follow the power law up to a certain threshold $u$, and the integer generalised Pareto distribution above $u$. This led to the paper in Statistica Neerlandica.
- My PhD student Thomas Boughen also presented this time, based on this preprint. What he has shown is that the preference function of preferential attachment model can be modified to accommodate both a flexible heavy-tail behaviour and the partial scale-freeness of the degree distribution. The degree is raised to a power (which can be greater or smaller than 1) up to a threshold $v$, followed by a straight line above $v$. Note that $v$ has nothing to do with $u$ above. This will be called the piecewise preference function, and denoted by $g(d)$ where $d$ is the degree.
Utilising the above results, this project does something different, in the presence of network evolution data i.e. granular increments of the network over time:
- We model the increments directly in a statistical way in order to evidence preferential attachment;
- At the same time, we use the piecewise preference function and infer its parameters.
One might wonder, how is this different to the existing approaches such as ERGM or REM, as they can also evidence preferential attachment? I had a little think about this on the train to the airport, and this is a rough comparison assuming directed networks:
- Under traditional models, $Y_{ij}$ is a Bernoulli random variable with probability $p_{ij}$ where $\log(p_{ij}/(1-p_{ij})) = \beta_0+\beta_1~d_j^{\alpha}$ or some more general linear form involving $d_j$ ($j$’s total or in-degree).
- Under our approach, the collection ${Y_{ij}}_j$ follows the multinomial distribution with probabilities ${g(d_j)/\sum_k~g(d_k)}_j$. This allows us to put a Poisson prior on the mean of the random variable of the number of new edges and integrate this variable out using the multinomial splitting property of Poisson.
The traditional models are more flexible in the sense that other features or properties such as homophily and reciprocity can be incorporated by additional terms in the linear predictor. This is going to be slightly trickier for our approach - I haven’t figured out how to apply our framework to other properties yet.