A New Statistical Model to Infer Complex Structures in Multilayer Networks

Imagine a network consisting of links between entities (nodes) that interact. For example, people linked by friendships, brain regions connected by white matter fibers, or even criminals interacting for coordination purposes. Now add another source of complexity: these nodes are divided into various subgroups that generate different “layers” within the network, such as lobes in the brain or operational cells within a criminal organization.

How can we study such an intricate network, where nodes are not all equal but belong to distinct layers? How can we rigorously exploit this layering information to unveil unexplored modular structures underlying the connections among nodes of the same layer and across different layers?

This is precisely the challenge addressed by Daniele Durante, Antonio Lijoi, and Igor Prünster (all from Bocconi’s Department of Decision Sciences) and Francesco Gaffi (BIDSA researcher and former postdoc at the University of Maryland), who have recently published in the prestigious Journal of the American Statistical Association an innovative model that answers these questions. The article is part of Francesco Gaffi’s doctoral thesis, carried out during his PhD in Statistics and Computer Science at Bocconi University. With the same work, Francesco Gaffi also won an award in the American Statistical Association’s Student Paper Competition.

Why is a new model needed?

In recent years, network science has witnessed important advancements, but most developments are designed for simple cases where nodes are not allocated to different layers of the network. When this does happen—for example, brain regions divided into lobes, citizens of different social classes, or criminals affiliated with various operational cells—traditional methods fail to account for this information and thus produce limited findings. So far, there has been no approach capable of exploiting the peculiarities of nodes’ layering to guide the modeling and prediction of modular connectivity patterns within complex multilayer networks, without sacrificing mathematical rigor and probabilistic coherence.

An innovative perspective

A key concept underlying the proposed model is partial exchangeability. In Statistics, saying that nodes are “exchangeable” essentially means that the order in which we observe them does not matter. But in multilayer networks, not all nodes are equivalent: a region in the frontal lobe cannot be swapped with one in the temporal lobe, and the same applies to two affiliates in different criminal cells. The new model therefore assumes that nodes are exchangeable only within each layer, but not across different layers. This informs the model on the fact that it is more likely to expect modular patterns among nodes of the same layer, while still allowing for complex macro-structures involving nodes in different layers. In other words, this assumption offers a rigorous perspective to account for differences between layers, while still allowing for the possibility to infer recurring cross-layer patterns.

From theory to practice: criminal networks

To demonstrate its effectiveness, the authors applied the model to a real criminal network reconstructed from judicial records of a large investigation conducted some years ago in Italy. In this network, each node represents a member of the criminal organization, and edges indicate co-attendances to summits. The layers (the “colors” of the nodes) correspond to different operational cells.

The model revealed dynamics that could not have been detected by other formulations:

• Core–periphery structures, in which leaders of different territorial cells form complex central cores, separated from modular peripheral structures among affiliates;
• Hidden roles, such as affiliates who act as genuine coordinators, even mediating between leaders of different cells;
• Internal fragmentations, for example that of a cell destabilized by prior power instabilities within the criminal organization.

Why is this important?

This article not only makes a major methodological contribution. The proposed model is a key for understanding the complexity of multilayer connectivity structures across several fields: neuroscience, sociology, political science, criminology. Thanks to its rigorous construction, it allows not only the study of observed structures, but also prediction of future ones: what will happen when new nodes enter the network? Which ties are likely to form?

Looking ahead

The article paves the way for a broader vision, highlighting the importance of associating the most suitable notion of “exchangeability” with the various types of modern networks (multilayer, multilevel, multiplex). In the case of multilayer networks, partial exchangeability is the key. For other types of networks—for example, those where the same nodes interact through various types of relationships—different concepts will be needed.

Daniele Durante, Francesco Gaffi, Antonio Lijoi and Igor Prünster, “Partially Exchangeable Stochastic Block Models for (Node-Colored) Multilayer Networks”, Journal of the American Statistical Association, 1–32. https://doi.org/10.1080/01621459.2025.2507825

Source: Bocconi news