# Distances in Higher-Order Networks and the Metric Structure of Hypergraphs

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## Abstract

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## 1. Introduction

- (1)
- If one assigns weights to hyperedges as proper functions of their sizes, then the distance between a couple of nodes $s,\phantom{\rule{0.166667em}{0ex}}t$ is the sum of the weights of all hyperedges in the shortest path, i.e.,$$d(s,t)=\underset{P\in {\mathbf{P}}_{st}}{min}\left(\right)open="("\; close=")">\sum _{e\in P}f\left(\right|e\left|\right)$$
- (2)

## 2. Methods

#### 2.1. Weighted Linegraphs

- The distance between the nodes obtained from Formula (2) must be the same as the classic distance in a graph at all times that one considers a (dyadic) complex network $\mathcal{G}=(V,E)$ as a hypergraph.
- The bigger the intersection of the hyperedges, the smaller the distance should be. For instance, if one considers the case illustrated in Figure 3, then the weighted distance between i and j in panel (a) should be smaller than that in panel (b). Therefore, the distance should be inversely proportional to the intersection size.
- The bigger the hyperedges involved, the larger the weighted distance should be. In particular, taking as an illustrative example the case of Figure 4, the weighted distance in panel (b) should be larger than that in panel (a) since the sizes of edges are bigger in case (b), while the size of the intersection is the same.
- Finally, the larger is the number of hyperedges involved in the path, the larger weighed distances one should obtain. In other words, the new metric structure should be sensitive to the number of hyperedges in the paths considered. In particular, if one takes two paths, one with only one hyperedge and another with two hyperedges but with the same number of nodes involved in both cases, then the path length should be smaller in the first case (see the illustration in Figure 5, where panel (b) must give a larger distance with respect to the case of panel (a)).

- As the Jaccard index [23] between ${e}_{i}$ and ${e}_{j}$ is given by $\mathcal{J}({e}_{i},{e}_{j})=\frac{|{e}_{i}\cap {e}_{j}|}{|{e}_{i}\cup {e}_{j}|}$, then$${w}_{ij}=\frac{1}{3}\left(\right)open="("\; close=")">|{e}_{i}\cup {e}_{j}|+\frac{1}{\mathcal{J}({e}_{i},{e}_{j})}$$
- If one takes ${e}_{i}={e}_{j}$, then ${w}_{ii}=\frac{1}{3}\left(\right)open="("\; close=")">|{e}_{i}|+1$. Furthermore, if one starts from a (dyadic) network $\mathcal{G}=(V,E)$, then ${w}_{ii}=0$ for every ${e}_{i}\in E$, and if ${e}_{i}\ne {e}_{j}$ (but ${e}_{i}$ connected with ${e}_{j}$ in L), then $|{e}_{i}\cup {e}_{j}|=3$ and $|{e}_{i}\cap {e}_{j}|=1$, which make that ${w}_{ij}=1$. Hence, one has that for this choice of weight function, the distance between the nodes obtained from formula (2) is the same as the classic distance in a graph (first desired property).
- Properties 2–4 also hold for this choice of weight function.

#### 2.2. Some Structural Measures

## 3. Results

#### 3.1. Synthetic Examples

#### 3.2. Real World Examples

- 1.
- In the Contact High School hypergraph, the difference in rankings is significantly less considerable than that occurring in the other two hypergraphs, especially for the case of closeness centrality;
- 2.
- The sets of the first 50 top nodes for the cases of the Email Enron and Senate Committees hypergraphs are significantly different when one uses the traditional distance measure and our measure for both betweenness and closeness centralities (see Figure 9b,d);
- 3.
- The KRC coefficient is low in the case of betweenness centralities, even for the case of the Contact High School hypergraph (see Figure 9a);
- 4.
- The KRC coefficient for the case of closeness centrality is low for the Email Enron and Senate Committees hypergraphs (see Figure 9c), being even negative for the case of the Email Enron graph.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Illustrative sketch of two hypergraphs made of two communities of nodes, which are bridged by a series of hyperedges. Hypergraph in the panel (

**a**) is used to provide explanation of the hyperedges’ cardinality importance for the path length calculation, while the one in the panel (

**b**) illustrates the impact of hyperedges intersection size (see explanation in the text).

**Figure 2.**A 11-node and 3-hyperedge hypergraph $\mathcal{H}$ and its weighted line graph with self-loops $L\left(\mathcal{H}\right)$.

**Figure 3.**Two paths from nodes i to j in a hypergraph, each of them with 2 hyperedges of the same size but with different intersection size. The weighted path length in panel (

**a**) should be smaller than that in panel (

**b**).

**Figure 4.**Two paths from nodes i to j in a hypergraph, each of them with two hyperedges with the same intersection size, but with different number of nodes in each hyperedge. The weighted path length in panel (

**a**) should be smaller than that in panel (

**b**).

**Figure 5.**Two paths from nodes i to j in a hypergraph, one involving only one hyperedge and another with two hyperedges. Both paths include the same number of nodes. The weighted path length in panel (

**a**) should be smaller than that in panel (

**b**).

**Figure 6.**An illustrative example of a ring-like hypernetwork with an additional hyperedge grouping all nodes. If the central hyperedge is removed, the hypernetwork is transformed into a ring structure.

**Figure 7.**Illustrative example of a hypergraph, where information transfer occurs from a source node i to a target node j. The hypothesis is made that in each intermediate node, information is copied and errors may appear. In the example, there are two possible paths from node i to node j: $({e}_{1},{e}_{2})$ and $({e}_{3},{e}_{4})$. $a,b,c$ are nodes forming the intersection between different hyperedges.

**Figure 8.**Distributions of differences between the proposed weighted hypergraph distance measure and the one calculated in the clique projection approach, for the real world hypergraphs analyzed in our study. In the first histogram, the number of pair of nodes is reported for which the difference between the two calculated distances takes the values specified in the horizontal axis. In the second and third histograms, we report instead the number of pairs of nodes, for which the difference between the two distances lies within the intervals indicated in the horizontal axis.

**Figure 9.**Correlations between the rankings based on betweenness (panels (

**a**,

**b**)) and closeness (panels (

**c**,

**d**)) centralities, calculated using the traditional distance measure and the weighted hypergraph one. (

**a**,

**c**) Kendall rank correlation (KRC) coefficients between the top nodes rankings. (

**b**,

**d**) The $\mu $ measure values (see Equation (10) for the definition of the $\mu $ measure). In all panels, the horizontal axis reports the number of top nodes considered in the rankings. The color code for identifying the curves plotted in all panels is reported in the horizontal bar at the bottom of the figure.

**Table 1.**Efficiency values of ${\mathcal{H}}_{1}$ and ${\mathcal{H}}_{2}$ calculated using different distance definitions. $k=20$.

${\mathit{E}}^{\mathit{w}}(\xb7)$ | $\mathit{E}(\xb7)$ | |
---|---|---|

${\mathcal{H}}_{1}$ | 0.311 | 1 |

${\mathcal{H}}_{2}$ | 0.303 | 0.303 |

High School | Email Enron | Senate Committees | |
---|---|---|---|

Number of nodes | 327 | 143 | 282 |

Number of unique hyperedges | 7818 | 1457 | 315 |

Maximal hyperedge size | 5 | 18 | 31 |

Minimal hyperedge size | 2 | 2 | 4 |

Mean hyperedge size | 2.3 | 3.1 | 17.2 |

Median hyperedge size | 2 | 2 | 19 |

**Table 3.**Efficiency of the real-world hypergraphs computed using the proposed weighted hypergraph distance measure (${E}^{w}(\xb7)$) and the traditional distance measure ($E(\xb7)$).

Contact High School | Email Enron | Senate Committees | |
---|---|---|---|

${E}^{w}(\xb7)$ | 0.505 | 0.443 | 0.106 |

$E(\xb7)$ | 0.510 | 0.546 | 0.670 |

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**MDPI and ACS Style**

Vasilyeva, E.; Romance, M.; Samoylenko, I.; Kovalenko, K.; Musatov, D.; Raigorodskii, A.M.; Boccaletti, S.
Distances in Higher-Order Networks and the Metric Structure of Hypergraphs. *Entropy* **2023**, *25*, 923.
https://doi.org/10.3390/e25060923

**AMA Style**

Vasilyeva E, Romance M, Samoylenko I, Kovalenko K, Musatov D, Raigorodskii AM, Boccaletti S.
Distances in Higher-Order Networks and the Metric Structure of Hypergraphs. *Entropy*. 2023; 25(6):923.
https://doi.org/10.3390/e25060923

**Chicago/Turabian Style**

Vasilyeva, Ekaterina, Miguel Romance, Ivan Samoylenko, Kirill Kovalenko, Daniil Musatov, Andrey Mihailovich Raigorodskii, and Stefano Boccaletti.
2023. "Distances in Higher-Order Networks and the Metric Structure of Hypergraphs" *Entropy* 25, no. 6: 923.
https://doi.org/10.3390/e25060923