Stochastic Block Models in Logistics

Stochastic Block Models (SBM) originate from the analysis of social networks and have been successfully adapted and extended to processes in physics. Since in logistic and manufacturing many processes can be described as complex network, too, the research question arises whether the modelling with SBMs can be applied to this field and in which advantages this result. The examination of this research question amongst others is part of the DFG founded project “Stochastische komplexe Netzwerke als Vorhersage- und Erklärungsmodell für die dynamische Entwicklung von produktionslogistischen Systemen“ (translated: stochastically complex networks as prediction and analysis model for the dynamic development of production and logistic systems).
 
Therefore, the application of the SBM on real material flow data from different areas is investigated. The first step towards the usage of SBMs in logistics is the transformation of the line by line material flow data into a complex network. Following a suitable SBM has to be extracted from the network representation. This process, the so-called inference of SBM, is already for undirected and unweighted network a non-trivial task and is accordingly challenging in the viewed case of weighted and directed networks.
 
The basic idea of SBM is to partition the nodes of the network into groups in a way that the probability for an edge between two nodes only depends on the group belongings of the nodes. In the process the network can be represented by different SBMs with a varying quality which is explained by the following figure. The aim of the inference of a SBM to a given graph is to find the most probable model and simultaneously determine a suitable number of groups.
 
 

Figure: Two different SBMs for the same undirected network with the partition of the nodes (doted) and the edge probability based on the maximum-likelihood estimation next to it. The maximum-likelihood estimation for the edge probabilities is given by the division of the number of existing edges through the number of possible edges.

 

In this project the following research question amongst others will be examined:
  • How can the modelling with SBM can be applied to complex networks from logistic and manufacturing?
  • Which analysis  and prediction are enabled by the usage of SBM on these complex networks?