A spatial hypergraph model to smoothly interpolate between pairwise graphs and hypergraphs to study higher-order structures

Abstract

We introduce a spatial graph and hypergraph model that smoothly interpolates between a graph with purely pairwise edges and a graph where all connections are in large hyperedges. The key component is a spatial clustering resolution parameter that varies between assigning all the vertices in a spatial region to individual clusters, resulting in the pairwise case, to assigning all the vertices in a spatial region to a single cluster, which results in the large hyperedge case. An important outcome of this model is that the spatial structure is invariant to the choice of hyperedges. Consequently, this model enables us to study clustering coefficients, graph diffusion, and epidemic spread and how their behavior changes as a function of the higher-order structure in the network with a fixed spatial substrate. We hope that our model will find future uses to distill or explain other behaviors in higher-order networks.

Figures (14)

• Figure 1: How hyperedges are formed in our model. Eight nearest neighbors are computed for node $1$ (leftmost plot) which are then clustered into $3$ clusters (middle plot). Finally, each of the clusters serves as a separate group interaction for node $1$ and they become hyperedges.
• Figure 2: As we vary the number of clusters produced among each node's nearest neighbors, we are able to interpolate between purely pairwise (leftmost plot) and purely higher-order structure (rightmost plot).
• Figure 3: Hyperedges formed around the node $v=1$ as the number of nearest neighbors, $d_v$, increases. We want the neighborhood radius parameter $\varepsilon$ of DBSCAN to scale with $d_v$ and $r_v$ (the maximum distance among the $d_v$ neighbors). To establish a concrete and controllable model, we design a function $\epsilon_\alpha$ that interpolates between individual clusters around each point and a single cluster. This function needs to scale with the parameters of the neighborhood to achieve its aims.
• Figure 4: Hypergraphs generated on $n=250$ nodes in $d=2$ dimensions for $\alpha = 0, 1, 2$ (left to right) for the same spatial embedding $\boldsymbol{X}$ and specified degrees $\boldsymbol{d}$.
• Figure 5: Total number of hyperedges formed using functions for scaling the radius parameter in DBSCAN as we vary dimensions. The functions used for $\epsilon_\alpha$ are: Equation \ref{['eq:dbscan-radius']} (leftmost column), linear scaling with $\epsilon_\alpha=\alpha r_v/2$ (middle column), and Equation \ref{['eq:dbscan-radius-alt']} (rightmost column). Gray to black to gray bands (only visible when zoomed in) in the middle and right columns indicate 10th, 25th, 50th, 75th, and 90th percentiles. This shows that our choice of Equation \ref{['eq:dbscan-radius']} gives the smoothest interpolation between pairwise effects and pure hyperedge effects as $\alpha$ varies from $0$ to $2$.

Theorems & Definitions (2)

• Theorem 1
• proof