Skip to main content

Higher-order interactions and Laplacian dynamics in complex networks: structure, dynamics and control

Periodic Reporting for period 2 - HIntNets (Higher-order interactions and Laplacian dynamics in complex networks: structure, dynamics and control)

Reporting period: 2019-07-01 to 2020-06-30

Complex networks underpin integral parts of our biological, physical and socio-economic universe. Understanding such networks is thus a core problem arising in many different applications. Thus far, such networks have been mainly represented as graphs. However, while graphs can capture pairwise interactions between nodes, fundamental interactions in networks often take place between multiple nodes. For example, in socio-economic networks, the joint coordinated activity of several agents (e.g. buyer, seller, broker); the formation and interactions of coalitions; the emergence of peer pressure; and the existence of triadic closure are all prevalent.

The objective of this interdisciplinary project was to investigate such non-binary, group-based relationships in complex networks and their dynamical implications. Specifically, we examined how such interactions can be accounted for in the modelling, analysis and design of complex networks, and will investigate generalized network models. While data about group interactions is readily available in some cases, often the groupings or even the network of interactions might be unknown. We thus investigated techniques to infer (generalized) networks and group structures from available data.

To provide an analysis framework for non-binary interactions, we extended the geometrical framework of simplicial complexes to account systematically for non-binary couplings of nodes, node-pairs, triplets, etc., allowing us to assess and design higher-order interactions. We focused on Laplacian dynamics such as consensus and random walks as prototypical examples of a range of phenomena. By combining tools from Network Science and Control Theory, we were able to develop new tools for (higher order) network analysis.

In conclusion, the project developed new tools to analyze, design and utilize higher-order network interactions that arise in applications and real-world data.
"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]"
The project enabled the fellow to obtain a unique academic training and ultimately secure him a tenure track position in the EU.

This project made progress on the modelling of complex systems via the use of simplicial complexes, thus going beyond the current practice of modelling complex systems via network. We developed a first suite of tools for analysis of simplicial complexes, focusing especially on the spectral analysis of simplicial complexes. In particular, we advanced the analysis of higher-order networks by so-called Hodge-Laplacian. This Hodge Laplacian generalizes the standard graph Laplacian, which is currently employed in a large number of network analysis tasks. Our results thus have the potential to gain new insights into to augment or replace standard network models in a wide range of application areas.

Another exciting development came from the realization that our tools could be fruitfully applied in the context of (higher-order) graph signal processing, e.g. for the processing of signals defined in the edge-space of graphs. As the analysis of such kind of signals defined on higher-order graphs is an emerging area in the context of complex systems analysis, we believe that this research will have an important long-term impact in the field of network and data science.