During the outgoing phase of the project we have investigated group-based interactions in networks from different perspectives. Our main results so far are as follows.
1) We examined the use of simplicial complexes as a framework for modelling non-dyadic interactions and introduced a higher-order link prediction task to evaluate models for such interaction data. In this context we also investigated the phenomenon of simplicial closure as a higher-order analog of the well known process of triadic closure in networks.
This work has been consolidated in the publication:
Benson, Austin R., Rediet Abebe, Michael T. Schaub, Ali Jadbabaie, and Jon Kleinberg. "Simplicial closure and higher-order link prediction." Proceedings of the National Academy of Sciences 115, no. 48 (2018): E11221-E11230.
2) We investigated how diffusion processes on simplicial complexes can be defined and in how far such processes can be used to generalize the toolkit of network science to simplicial complexes, specifically focusing on extensions of PageRank and spectral embeddings for simplicial complexes. To this end we introduced a normalized 1-Hodge Laplacian operator, an extension of the normalized graph Laplacian for simplicial complexes.
Schaub, M.T.; Benson, A.R.; Horn, P.; Lippner, G. & Jadbabaie, A. (2020), "Random walks on simplicial complexes and the normalized Hodge 1-Laplacian", SIAM Review., May, 2020. Vol. 62(2), pp. 353-391.
3) In many problems modeled using graphs, the data of interest is located on the edges (as opposed to the nodes). A typical scenario of practical interest is a flow on the edges – signal, mass, energy, information – of a graph that is measured and has to be analyzed further, such as traffic flow associated with the edges of a traffic network. To analyze these types of signal we have developed techniques for analyzing the edge-space of graphs and simplicial complexes in more detail.
An important tool in this context is the Hodge decomposition, a decomposition of edge flows into intuitively interpretable components that are analogous to notions such as gradient flows or rotational flows from vector calculus. We have demonstrated how this decomposition can be leveraged for data analytics that extract information about the edge-space that complements and extends typical graph-based analysis.
Schaub M. T.; Segarra, S. “Flow smoothing and denoising: graph signal processing in the edge-space”. 2018 IEEE Global Conference on Signal and Information Processing (GlobalSIP), Anaheim, CA, USA, 2018, pp. 735-739.
Jia, J.; Segarra, S.; Schaub, M. T. & Benson, A. R. “Graph-based Semi-Supervised & Active Learning for Edge Flows”. Proceedings of the 25th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD 2019), ACM, 2019
4) Extending our previous work in this direction, we further investigated the detection of group structures in networks from a dynamical perspective. In contrast to many standard techniques for the detection of groups within networks, our work focuses on group structures that are defined in a dynamical manner.
5) In addition, we investigated the inference of group-based generative network models from dynamical networks only, thereby generalizing the problem of network inference.
[A full list of all publications is provided on the project homepage]
