Random Walks on Simplicial Complexes and the normalized Hodge 1-Laplacian

Abstract

Graphs modeling pairwise relationships between entities have become a dominant framework to study complex systems and data. Simplicial complexes extend this dyadic model of graphs to polyadic relationships and have emerged as a model for multi-node relationships occurring in many complex systems. For instance, biological interactions occur between sets of molecules, and communication systems include group messages and not only pairwise interactions. While the graph Laplacian and Laplacian dynamics have been intensely studied, corresponding notions of Laplacian dynamics beyond the node-space have so far remained largely unexplored for simplicial complexes. In particular, diffusion processes such as random walks and their relationship to the graph Laplacian, that underpin many methods such as centrality measures, flow-based rankings, community detection, and contagion models, lack a proper correspondence for general simplicial complexes Focusing on the coupling between edges, here we introduce a normalized Laplacian matrix for simplicial complexes and demonstrate its relationship to a random walk model on simplicial complexes as a foundational step towards translating many Laplacian-based analytics from graphs to simplicial complexes. Our key idea is to generalize the relationship between the normalized graph Laplacian and random walks on graphs by devising an appropriate normalization for the Hodge Laplacian, the analog of the graph Laplacian for simplicial complexes. We further discuss how this Hodge Laplacian gives rise to 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 demonstrate how these results can be leveraged for data analytics that extract information about the edge-space of a simplicial complex that complements and extends graph-based analysis. To illustrate the utility of these tools we derive spectral embeddings based on the Hodge Laplacian to examine trajectory data, and exemplify our ideas by an analysis of ocean drifters near Madagascar. We also present a generalization of personalized PageRank for the edge-space of simplicial complexes and apply it for the analysis of a book co-purchasing dataset.

Multibody interactions and nonlinear consensus dynamics on networked systems

“…Such systems may be represented as hypergraphs or simplicial complexes, and a substantial body of work has characterised their structural properties. However, a proper understanding of how multi-body interactions affect spreading dynamics in networked systems is still nascent [3,[8][9][10][11].…”

Configuration Models of Random Hypergraphs

“…In the past decades, the dominant approach to these systems has been to represent these networks dyadically, allowing the analyst to apply standard techniques of dyadic network science, including the configuration model. Recent work, however, has highlighted limitations of the dyadic paradigm in modeling of polyadic systems, both in theory [43] and in application domains including neuroscience [20], ecology [23], computational social science [48,7] among others [6]. The importance of polyadic interactions calls into question the use of the dyadic configuration model in such systems.…”

Consensus dynamics on temporal hypergraphs

“…Recent works have focused on investigating dynamics on higher-order topologies [4], and a variety of dynamical processes such as diffusion [3,24], synchronization [27] or social and opinion dynamics [7,16,23] have been extended to multi-body frameworks. As a result, it was shown that higher-order interactions can significantly modify the dynamical process in comparison to the dynamical process on the underlying reduced network.…”