# (Hyper)Graph Embedding and Classification via Simplicial Complexes

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## Abstract

**:**

## 1. Introduction

- feature generation and feature engineering, where numerical features are ad-hoc extracted from the input patterns
- embedding via information granulation.

## 2. Information Granulation and Classification Systems

#### 2.1. An Introduction to Simplicial Complexes

**Čech complex:**- for each subset $S\subset \mathcal{P}$ of points, form an $\u03f5$-ball (A ball with radius $\u03f5$) centred at each point in S, and include S as a simplex if there is a common point contained in all of the balls created so far.
**Alpha complex:**- for each point $x\in \mathcal{P}$, evaluate its Voronoi region $V\left(x\right)$ (i.e., the set of points closest to it). The set of Voronoi regions forms the widely-known Voronoi diagram and the nerve of the latter is usually referred to as Delaunay complex. By considering an $\u03f5$-ball around each point $x\in \mathcal{P}$, it is possible to intersect said ball with $V\left(x\right)$, leading to a restricted Voronoi region and the nerve of the set of restricted Voronoi regions for all points in $\mathcal{P}$ is the Alpha complex.
**Vietoris-Rips complex:**- for each subset $S\subset \mathcal{P}$ of points, check whether all of their pairwise distances are below $\u03f5$. If so, S is a valid simplex to be included in the Vietoris-Rips complex.

- build the Vietoris-Rips neighbourhood graph ${\mathcal{G}}_{\mathrm{VR}}(\mathcal{V},\mathcal{E})$ where $\mathcal{V}$ is the set of vertices and $\mathcal{E}$ is the set of edges, hence $\mathcal{V}\equiv \mathcal{P}$ and $e({v}_{i},{v}_{j})\in \mathcal{E}$ if $d({v}_{i},{v}_{j})\le \u03f5$ for any two nodes ${v}_{i},{v}_{j}\in \mathcal{V}$ with $i\ne j$
- evaluate all maximal cliques in ${\mathcal{G}}_{\mathrm{VR}}$.

**Clique complex:**- for a given underlying graph $\mathcal{G}$, the Clique complex is the simplicial complex formed by the set of vertices of its (maximal) cliques. In other words, a clique of k vertices is represented by a simplex of order $(k-1)$.

#### 2.2. Proposed Approach

#### 2.2.1. Embedding

- the match between two simplices (possibly belonging to different simplicial complexes) can be done in an exact manner: two simplices are equal if they have the same order and they share the same set of node labels
- simplicial complexes become multi-sets: two simplices (also within the same simplicial complex) can have the same order and can share the same set of node labels
- the enumeration of different (unique) simplices is straightforward.

#### 2.2.2. Classification

- there is no guarantee that all symbols in $\mathcal{A}$ are indeed useful for the classification problem at hand
- as introduced in Section 1, it is preferable to have a small, yet informative, alphabet in order to eventually ease an a-posteriori knowledge discovery phase (less symbols to be analysed by field-experts).

- trains the $\nu $-SVM with regularisation parameter $\nu $

## 3. Results

#### 3.1. On Benchmark Data

**The Weighted Jaccard Kernel**. Originally proposed in Ref. [78], the Weighted Jaccard Kernel (WJK) is an hypergraph kernel working on the top of the simplicial complexes from the underlying graphs. As a proper kernel function, WJK performs an implicit embedding procedure towards a possibly infinite-dimensional Hilbert space. In synthesis, the WJK between two simplicial complexes, say $\mathcal{S}$ and $\mathcal{R}$, is evaluated as follows: after considering the ’simplices-of-node-labels’ rather than the ’simplices-of-nodes’ as described in Section 2.2.1, the set of unique simplices belonging to either $\mathcal{S}$ or $\mathcal{R}$ is considered. Then, $\mathcal{S}$ and $\mathcal{R}$ are transformed in two vectors, say $\mathbf{s}$ and $\mathbf{r}$, by counting the occurrences of simplices in the unique set within the two simplicial complexes. Finally, $WJK(\mathcal{S},\mathcal{R})=\frac{{\sum}_{i}\mathrm{min}({\mathbf{s}}_{i},{\mathbf{r}}_{i})}{{\sum}_{i}\mathrm{max}({\mathbf{s}}_{i},{\mathbf{r}}_{i})}$. The kernel matrix obtained by evaluating the pairwise weighted Jaccard similarity between any two pairs of simplicial complexes in the available dataset is finally fed to a $\nu $-SVM.**GRALG**. Originally proposed in Ref. [43] and later used in Refs. [44,79] for image classification, GRALG is a Granular Computing-based classification system for graphs. Despite the fact that it considers network motifs rather than simplices, it is still based on the same embedding procedure by means of symbolic histograms. In synthesis, GRALG extracts network motifs from the training data and runs a clustering procedure on such subgraphs by using a graph edit distance as the core (dis)similarity measure. The medoids (MinSODs [39,40,41,42]) of these clusters form the alphabet on top of which the embedding space is built. Two genetic algorithms take care of tuning the alphabet synthesis and the feature selection procedure, respectively. GRALG, however, suffers from an heavy computational burden which may become unfeasible for large datasets. In order to overcome this problem, the random walk-based variant proposed in Ref. [80] has been used.

#### 3.2. On Real-world Proteomic Data

#### 3.2.1. Experiment #1: Protein Function Classification

#### Data Retrieval and Preprocessing

- the entire Escherichia coli (str. K12) list of proteins has been retrieved from UniProt [84]
- the list has been cross-checked with Protein Data Bank [85] in order to download PDB files for resolved proteins
- proteins with multiple EC numbers have been discarded
- in PDB files containing multiple structure models, only the first model is retained; similarly, for atoms having alternate coordinate locations, only the first location is retained.

#### Computational Results

- at $\alpha =1$: ${\ell}_{1}$-SVMs outperform the kernelised counterpart in terms of SNS (all classes) and NPV (all classes), whereas $\nu $-SVMs outperform the former in terms of SPC (all classes) and PPV (all classes). The overall ACC sees ${\ell}_{1}$-SVMs outperforming $\nu $-SVMs only for class 7, the two classifiers perform equally for classes 2 and 4 and for the remaining classes $\nu $-SVMs perform better. Regardless of which performs the best in an absolute manner, the performance shifts are rather small as far as ACC, SPC and NPV are concerned ($\approx 3.3\%$ or less), whereas interesting shifts include SNS (${\ell}_{1}$-SVMs outperforming by $\approx 10\%$ on class 4) and PPV ($\nu $-SVMs outperforming by $\approx 10\%$ on class 3 and $\approx 22\%$ on class 5);
- at $\alpha =0.5$: ${\ell}_{1}$-SVMs outperform the kernelised counterpart in terms of SNS (all classes) and NPV (all classes), whereas $\nu $-SVMs outperform the former in terms of SPC (all classes), PPV (all classes) and ACC (all classes). While the performance shifts are rather small for ACC (≈1–2%) and SPC ($\approx 3-4\%$), there are remarkable shifts regarding PPV ($\nu $-SVMs outperform up to $36\%$ for class 5) and SNS (${\ell}_{1}$-SVMs outperform up to $13\%$ for class 4).

- at $\alpha =1$: ${\ell}_{1}$-SVMs select fewer symbols with respect to $\nu $-SVMs only for classes 1 and 7
- at $\alpha =0.5$: ${\ell}_{1}$-SVMs outperform $\nu $-SVMs for all classes.

#### 3.2.2. Experiment #2: Protein Solubility Classification

#### Data Retrieval and Preprocessing

- from the eSOL database (eSOL database http://tp-esol.genes.nig.ac.jp/)) developed in the Targeted Proteins Research Project., containing the solubility degree (in percentage) for the E. coli proteins using the chaperone-free PURE system [87], the entire dump has been collected
- proteins with no information about their solubility degree have been discarded
- in order to enlarge the number of samples (From the entire dump, only 432 proteins had their corresponding PDB ID.), we reversed the JW-to-PDB relation by downloading all structure files (if any) related to each JW entry from eSOL. Each structure will inherit the solubility degree from the JW entry
- inconsistent data (e.g., the same PDB with different solubility values) have been discarded; duplicates have been removed in case of redundant data (e.g., one solubility per PDB but multiple JWs)
- proteins that have a solubility degree greater than $100\%$ have been set as $100\%$. The (small) deviations from $100\%$ can be ascribed to minor experimental errors. After straightforward normalisation, the solubility degree can be considered a real-valued number in range $[0,1]$.

#### Computational Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

ACC | Accuracy |

EC | Enzyme Commission |

NPV | Negative Predictive Value |

PCN | Protein Contact Network |

PDB | Protein Data Bank |

PPV | Positive Predictive Value |

SNS | Sensitivity |

SPC | Specificity |

SVM | Support Vector Machine |

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**Figure 1.**Average accuracy on the test set amongst the dummy classifier, GRALG, WJK and the proposed embedding technique. Results are given in percentage. The colour scale has been normalised row-wise (i.e., for each dataset) from yellow (lower values) towards green (higher values, preferred).

**Figure 2.**Resolution distribution within the initial 6685 proteins set. Proteins with no resolution information are not considered.

Class | ||||||
---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 |

$12,036\pm 35$ | $12,025\pm 25$ | $12,025\pm 18$ | $12,028\pm 11$ | $12,038\pm 24$ | $12,013\pm 44$ | $12,012\pm 34$ |

**Table 2.**Average results (in percentage) on Test Set for ${\ell}_{1}$-SVM. In bold, the best between the two fitness function tradeoff values for $\alpha $.

Class | $\mathit{\alpha}$ | ACC | SPC | SNS | NPV | PPV | Sparsity |
---|---|---|---|---|---|---|---|

1 | 0.5 | 95.3 | 96.5 | 87.1 | 98.2 | 77.7 | 3.3 |

1 | 97 | 98.4 | 87.3 | 98.3 | 88 | 16 | |

2 | 0.5 | 92.7 | 94.2 | 86.7 | 96.5 | 79.6 | 4.5 |

1 | 94.5
| 97.3 | 83.8 | 95.9 | 88.9 | 22.7 | |

3 | 0.5 | 92.1 | 93.8 | 84.4 | 96.3 | 76.2 | 4 |

1 | 93.3 | 95.5 | 83.9 | 96.3 | 81.9 | 17.4 | |

4 | 0.5 | 96.6 | 97.7 | 82.5 | 98.7 | 72.9 | 2.8 |

1 | 97.3 | 98.6 | 79.8 | 98.5 | 81.6 | 7.8 | |

5 | 0.5 | 96.9 | 97.8 | 71.7 | 99 | 56.9 | 1.8 |

1 | 97.9 | 98.9 | 70.4 | 98.9 | 75.3 | 5.1 | |

6 | 0.5 | 97.5 | 97.9 | 88.8 | 99.4 | 71.5 | 2.2 |

1 | 98.7 | 99.4 | 86.2 | 99.3 | 87.8 | 9.6 | |

7 | 0.5 | 86.6 | 89.9 | 80.1 | 89.9 | 80.5 | 4.8 |

1 | 88.8 | 91.6 | 83.4 | 91.6 | 83.6 | 36.3 |

**Table 3.**Average results (in percentage) on Test Set for $\nu $-SVM. In bold, the best between the two fitness function tradeoff values for $\alpha $.

Class | $\mathit{\alpha}$ | ACC | SPC | SNS | NPV | PPV | Sparsity |
---|---|---|---|---|---|---|---|

1 | 0.5 | 96.8 | 99 | 80.4 | 97.4 | 92 | 9.9 |

1 | 97.2 | 99.2 | 81.9 | 97.6 | 93.6 | 11.5 | |

2 | 0.5 | 93.9 | 98 | 77.8 | 94.5 | 90.9 | 6.8 |

1 | 94.5 | 98.3 | 79.7 | 95 | 92.2 | 26.1 | |

3 | 0.5 | 94 | 98.5 | 74.3 | 94.3 | 92.1 | 6.8 |

1 | 94.7 | 98.5 | 78.2 | 95.1 | 92.3 | 18.6 | |

4 | 0.5 | 97.3 | 99.3 | 69.6 | 97.8 | 88.4 | 12.8 |

1 | 97.3 | 99.4 | 69.2 | 97.8 | 89.9 | 19.1 | |

5 | 0.5 | 98.5 | 99.8 | 61.3 | 98.6 | 93 | 13.6 |

1 | 98.7 | 99.9 | 63.8 | 98.7 | 97.1 | 31.7 | |

6 | 0.5 | 98.9 | 99.9 | 80.3 | 99 | 97.1 | 23.3 |

1 | 99.1 | 99.9 | 83.5 | 99.2 | 97.2 | 28.7 | |

7 | 0.5 | 87.4 | 93 | 76.5 | 88.6 | 84.7 | 6.5 |

1 | 87.4 | 93.4 | 75.7 | 88.3 | 85.3 | 6.9 |

Polarity (avg) | Hydrophilicity (avg) | Polarity (std) | Hydrophilicity (std) | |
---|---|---|---|---|

Polarity (avg) | 1 | 0.99818 | −0.01869 | −0.06879 |

Hydrophilicity (avg) | 0.99818 | 1 | −0.03705 | −0.08582 |

Polarity (std) | −0.01869 | −0.03705 | 1 | 0.99397 |

Hydrophilicity (std) | −0.06879 | −0.08582 | 0.99397 | 1 |

EC1 | EC2 | EC3 | EC4 | EC5 | EC6 | Not Enzymes | |
---|---|---|---|---|---|---|---|

${R}^{2}$ | 0.0250 | 0.0239 | 0.0212 | 0.0199 | 0.0239 | 0.0170 | 0.0250 |

p | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 |

Hydrophilicity (avg) | Hydrophilicity (std) | |||||
---|---|---|---|---|---|---|

Class | t-Value | p | Coefficient | t-Value | p | Coefficient |

1 | 11.55 | <0.0001 | −4.17734 | 0.92 | 0.3563 | 0.24438 |

2 | 10.52 | <0.0001 | −3.73211 | 0.0647 | 1.85 | 0.47999 |

3 | 10.61 | <0.0001 | −3.38981 | 0.0651 | 1.84 | 0.43182 |

4 | 11.08 | <0.0001 | −2.98596 | 2.11 | 0.0352 | 0.41574 |

5 | 12.13 | <0.0001 | −2.43624 | 2.49 | 0.0127 | 0.36671 |

6 | 10.73 | <0.0001 | −2.65512 | 2.57 | 0.01 | 0.46672 |

7 | 11.55 | <0.0001 | −4.17734 | 0.92 | 0.3563 | 0.24438 |

Polarity (avg) | Polarity (std) | |||||
---|---|---|---|---|---|---|

Class | t-Value | p | Coefficient | t-Value | p | Coefficient |

1 | 11.27 | <0.0001 | 1.51515 | 1.77 | 0.0762 | −0.17376 |

2 | 10.26 | <0.0001 | 1.35280 | 2.52 | 0.0118 | −0.24206 |

3 | 10.43 | <0.0001 | 1.23898 | 2.62 | 0.0089 | −0.22655 |

4 | 10.83 | <0.0001 | 1.08515 | 2.72 | 0.0066 | −0.19836 |

5 | 11.84 | <0.0001 | 0.88388 | 3.16 | 0.0016 | −0.17190 |

6 | 10.52 | <0.0001 | 0.96768 | 3.14 | 0.0017 | −0.21080 |

7 | 11.27 | <0.0001 | 1.51515 | 1.77 | 0.0762 | −0.17376 |

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**MDPI and ACS Style**

Martino, A.; Giuliani, A.; Rizzi, A.
(Hyper)Graph Embedding and Classification via Simplicial Complexes. *Algorithms* **2019**, *12*, 223.
https://doi.org/10.3390/a12110223

**AMA Style**

Martino A, Giuliani A, Rizzi A.
(Hyper)Graph Embedding and Classification via Simplicial Complexes. *Algorithms*. 2019; 12(11):223.
https://doi.org/10.3390/a12110223

**Chicago/Turabian Style**

Martino, Alessio, Alessandro Giuliani, and Antonello Rizzi.
2019. "(Hyper)Graph Embedding and Classification via Simplicial Complexes" *Algorithms* 12, no. 11: 223.
https://doi.org/10.3390/a12110223