Public defence in the field of Mathematics and Statistics, MSc Kalle Alaluusua

Title of the thesis: Consistent estimation of higher-order network models under aggregation

Thesis defender: Kalle Alaluusua 
Opponent: Assistant Professor Jaron Sanders, Eindhoven University of Technology, the Netherlands
Custos: Professor Lasse Leskelä, Aalto University School of Science

Many datasets consist of observed relations between entities. For example, in parliamentary data, one may compare how similarly representatives vote across many bills and observe which representatives attend the same meetings. Such data may hide latent structure: there may be communities whose members behave similarly, or a small coordinated set whose members interact unusually often. 

Observed relations can be represented as networks, where entities are nodes and their relations are edges. When the same entities are examined through several types of relations, the data can be represented as a multilayer network, where each layer corresponds to a different type of relation. Group relations can be represented as hyperedges that simultaneously connect all participants. In practice, these complex networks are often summarized in a similarity matrix. Its entries record how often each pair of entities is observed together or the strength of their relation. Such matrices are easy to store and analyze, but aggregation may lead to information loss. 

Kalle Alaluusua’s dissertation studies when aggregated network data retain enough signal for statistical inference. The main inferential tasks are community recovery and the detection and recovery of planted cliques. A planted clique is a hidden subset in which all members are interconnected. 

The results show that community separation accumulates across layers. Consequently, a semidefinite programming algorithm can achieve exact recovery from a hypergraph similarity matrix even when the communities are only weakly separated in each individual layer. 

The dissertation also establishes near-optimal recovery conditions for a simple subspace clustering problem in which points are sampled near the intersection of two latent lines and approximately collinear triples form hyperedges. The clustering problem is then solved using a spectral algorithm on the resulting similarity matrix. 

Finally, the dissertation derives sufficient conditions for the detection and exact recovery of a planted clique in a hypergraph from the similarity matrix alone. Together, these results identify conditions for consistent inference from aggregated higher-order network data. They offer theoretical insights into the use of similarity matrices in the analysis of higher-order networks and point-cloud data.

Keywords: Community detection, hypergraphs, random graphs

Thesis available for public display 7 days prior to the defence at Aalto University's public display page.