Agenda

Ioannis Ntzoufras (Athens University of Economics and Business) - Assessing competitive balance in the English Premier League using a stochastic block model

Abstract:

The Stochastic Block Model (SBM) is a foundational tool in network analysis, often extended to address complex problems in various domains. In this work, we develop a Bayesian network model based on an extension of the SBM where the response is categorical and denotes different types of connections between nodes. The data are represented by a large table which is similar to a contingency table but now interest lies in finding similarities in the connections between nodes. The method can be used for either sparse or dense networks without loss of generality. We use the simple multinomial-Dirichlet conjugate Bayesian model for the estimation of the model parameters and the reversible jump algorithm for the identification of blocks/clusters/communities with similar connection properties. The data are represented as a large table analogous to a contingency table, but the inferential objective differs substantially. Rather than estimating marginal frequencies, our goal is to identify latent blocks (clusters or communities) of nodes that share similar interaction profiles. Conditional on block memberships, edges are assumed independent, with their categorical outcomes following a multinomial distribution whose parameters depend on the pair of blocks involved. This captures the notion of stochastic equivalence: nodes within the same block exhibit similar probabilistic behavior in their connections.

We adopt a fully Bayesian specification. Dirichlet priors are placed on both the block interaction probabilities and the block membership proportions, allowing conjugate updating and leading to a collapsed posterior distribution after integrating out nuisance parameters. The number of blocks is treated as unknown and assigned a prior distribution, enabling direct probabilistic inference on the degree of structural heterogeneity in the network. Posterior sampling is performed via a Markov chain Monte Carlo algorithm that updates both the allocation vector and the number of blocks, without requiring reversible jump techniques. The approach applies to both sparse and dense networks without loss of generality.

The proposed methodology can be used to evaluate competitive balance between teams in a sports league. We represent the outcomes of all matches in a football season as a dense network, where nodes correspond to teams and the categorical edges reflect the results of each game—win, draw, or loss. This model is then applied to assess competitive balance, a topic of great interest in sports Economics and of the general public. The primary focus of this application is on the English First Division / Premier League, covering over 40 seasons. Our analysis indicates a structural shift in competitive balance around the early 2000s, transitioning from a reasonably balanced league to a two-tier structure.

Overall, the proposed Bayesian SBM provides a coherent, model-based framework for analysing categorical network data and offers a statistically principled approach to quantifying competitive balance over time.

The seminar can be attended also remotely, connecting to ZOOM.

Link Zoom: bit.ly/insem-2425
ID riunione:  880 2639 9452
Passcode: InSem-2425