Counting simplicial pairs in hypergraphs

TL;DR

It is shown that the simplicial ratio captures the frequency, as well as the rarity, of simplicial interactions in a hypergraph while the simplicial matrix provides more fine-grained details.

Abstract

We present two ways to measure the simplicial nature of a hypergraph: the simplicial ratio and the simplicial matrix. We show that the simplicial ratio captures the frequency, as well as the rarity, of simplicial interactions in a hypergraph while the simplicial matrix provides more fine-grained details. We then compute the simplicial ratio, as well as the simplicial matrix, for 10 real-world hypergraphs and, from the data collected, hypothesize that simplicial interactions are more and more deliberate as edge size increases. We then present a new Chung-Lu model that includes a parameter controlling (in expectation) the frequency of simplicial interactions. We use this new model, as well as the real-world hypergraphs, to show that multiple stochastic processes exhibit different behaviour when performed on simplicial hypergraphs vs. non-simplicial hypergraphs.

Figures (13)

• Figure 1: (left) a graph $G_1$ with 18 vertices, 3 edges of size 6, and 3 edges of size 3, and (right) a graph $G_2$ with 18 vertices, 3 edges of size 6, and 3 edges of size 5. We have $\sigma_{\mathrm{SF}}(G_1) = \sigma_{\mathrm{SF}}(G_2) = 0$, $\sigma_{\mathrm{ES}}(G_1) = \sigma_{\mathrm{ES}}(G_2) = 2/57$, and $\sigma_{\mathrm{FES}}(G_1) = \sigma_{\mathrm{FES}}(G_2) = 2/57$.
• Figure 2: (left) a graph $G_1$ with 6 vertices and 4 edges, and (right) a graph $G_2$ with 10 vertices and 3 edges. We have $\sigma_{\mathrm{SF}}(G_1) = 0$, $\sigma_{\mathrm{ES}}(G_1) \approx 0.07$, $\sigma_{\mathrm{FES}}(G_1) \approx 0.07$, and $\sigma_{\mathrm{SF}}(G_2) = 0$, $\sigma_{\mathrm{ES}}(G_2) \approx 0.07$, $\sigma_{\mathrm{FES}}(G_2) \approx 0.13$.
• Figure 3: The simplicial matrix of 10 real networks, as well as the cell-wise average matrix. For each graph $G$, only non-empty cells of $M_{\mathrm{SR}}\left(G\right)$ are shown, and cells involving edges of size greater than 5 are omitted. The value of a cell is replaced with "$>1\mathrm{k}$" whenever the value is above 1000.
• Figure 4: The temporal simplicial matrix of 7 real networks, as well as the cell-wise average matrix. For each graph $G$, only non-empty cells of $M^{\rightarrow}_{\mathrm{SR}}\left(G\right)$ are shown, and cells involving edges of size greater than 5 are omitted. The value of a cell is replaced with "$>1\mathrm{k}$" whenever the value is above 1000.
• Figure 5: The simplicial matrix of $G_1$ from Example \ref{['ex:short-coming 2']}, presented in a simplified manner.

Theorems & Definitions (13)

• Example 1.1
• Example 1.2
• Lemma 2.1
• proof
• Remark 2.2
• Remark 2.3
• Remark 2.4
• Remark 2.5
• Remark 2.6
• Remark 2.7