Simplex2Vec embeddings for community detection in simplicial complexes

I. INTRODUCTION

• Simplicial complexes capture higher-order interactions that pairwise networks can miss, including group relationships requiring more than two participants.They have been applied to social, collaboration, neural, and dynamical systems.Source: §I · p. 1
• Community structure in simplicial complexes has received scarce attention despite the development of many network community-detection methods.Source: §I · p. 1
• The paper combines symbolic embeddings with biased and unbiased random walks to map, detect, and visualize simplicial-complex communities.Source: §I · p. 1

A. Simplices and simplicial complexes

• A k-simplex is a set of k + 1 vertices, so edges encode pairwise interactions while higher-dimensional simplices encode group interactions.Source: A. Simplices and simplicial complexes · p. 1
• A simplicial complex contains every subsimplex of each simplex, and any two simplices intersect either trivially or in a shared face.Source: A. Simplices and simplicial complexes · p. 2
• Clique complexes promote network cliques to simplices, but cannot represent an empty triangle whose three edges lack the filled triangle.Source: A. Simplices and simplicial complexes · p. 2

B. Hasse diagrams and persistent homology

• A Hasse diagram is a directed acyclic graph whose vertices are simplices and whose edges connect inclusions between consecutive dimensions.It provides the backbone of inclusions among simplices of different dimensions.Source: B. Hasse diagrams and persistent… · p. 2
• Persistent homology quantifies multiscale shape by tracking which topological features persist across a sequence of simplicial complexes.Persistence diagrams encode the resulting feature intervals for comparison.Source: B. Hasse diagrams and persistent… · p. 2
• The paper uses sliced Wasserstein distance dSWS to compare persistence diagrams.Source: B. Hasse diagrams and persistent… · p. 2

C. Random Walks and symbolic embedding

• Random walks on the Hasse diagram generate symbolic sequences from simplex neighborhoods, which can be processed by word2vec for embedding.The walk discards inclusion direction when choosing jumps.Source: C. Random Walks and symbolic embedding · pp. 2–4
• The unbiased walk moves uniformly among neighboring Hasse-diagram nodes, whereas biased walks make transition probability proportional to destination-node weight.Source: C. Random Walks and symbolic embedding · p. 2, Figure 1 · p. 2
• The higher-order weighting scheme assigns weight 1 to an n-simplex and recursively distributes inverse combinatorial factors to reachable lower-dimensional simplices.Source: C. Random Walks and symbolic embedding · p. 3
• Lower-order bias favors 0-simplices, while higher-order bias pushes exploration toward higher-dimensional simplices to capture higher-order structure.Source: C. Random Walks and symbolic embedding · p. 3
• Walk length L and walk count n_w influence the analysis and are generally chosen so that L n_w is approximately |K|, the number of simplices.Source: C. Random Walks and symbolic embedding · p. 3
• Word2vec places symbols with similar sampled contexts near one another in a vector space, building on the random-walk embedding strategy used by DeepWalk and node2vec.Source: C. Random Walks and symbolic embedding · pp. 3–4

D. Simplex2Vec

• Simplex2Vec uses Hasse-diagram random walks as symbolic words and embeds their contexts in a metric space with word2vec.The walks preserve high-order information from the simplicial complex.Source: D. Simplex2Vec · p. 4
• The method simultaneously represents vertices and all higher-dimensional simplices up to the chosen maximal simplicial dimension.Source: D. Simplex2Vec · p. 4

1. Costa-Farber Model

• The Costa-Farber construction retains vertices with p0, adds edges with p1, and fills cliques with higher simplices using probabilities p2 through pk.The resulting complex depends on (p0, p1, ..., pk), with each probability between 0 and 1.Source: §1 · p. 4

2. Sociopatterns Data

• Social simplicial complexes aggregate face-to-face interactions into 15-minute windows, treating maximal cliques as facets and retaining simplices up to order 3.Lower-order simplices are reconstructed from the aggregated facets, while sub-cliques up to size 4 are counted even when higher-dimensional cliques are excluded.Source: §2 · p. 4

3. Functional Connectivity

• Brain complexes use correlations among 200 ICA components, retain the upper 5% of absolute correlations, and construct facets from maximal cliques and their sub-cliques.Equal transition weights are assigned between facets connected in the Hasse graph.Source: §3 · p. 5

F. Clustering

• Hierarchical clustering is applied to node embeddings and evaluated against known group labels using normalized mutual information.This procedure is used to assess whether the embedding representations support meaningful community partitions.Source: F. Clustering · p. 5
• Higher-order effects are measured with sliced Wasserstein distances between one-dimensional persistence diagrams of embeddings from pure networks and simplicial complexes.Figure 3 varies p2, network size N, and walk length L, with L = 30 in panel (a).Source: Figure 3 · p. 5, F. Clustering · p. 5

A. Evolution of embedding structure

• The Costa-Farber experiment fixes N and edge probability p1 while increasing p2 from the graph case p2 = 0 to introduce higher-order simplices.At p2 = 0, the complex is one-dimensional and corresponds to a standard graph.Source: A. Evolution of embedding structure · pp. 5–6
• Large embedding-topology changes occur already at small p2, exceeding variability across Costa-Farber realizations with the same p2.The effect is observed through one-dimensional persistent homology and sliced Wasserstein distances between persistence diagrams.Source: A. Evolution of embedding structure · p. 6
• The higher-order interaction effect persists across different numbers of vertices N and embedding walk lengths L.The comparison uses a standard Rips-Vietoris filtration followed by sliced Wasserstein distance computation.Source: A. Evolution of embedding structure · p. 6

B. Effects of higher-order interactions on clustering stability

• Adding small amounts of higher-order interactions makes the resulting partitions highly dissimilar from those obtained using only edge structure.The same pattern appears when comparing embeddings across different values of p2.Source: B. Effects of higher-order… · p. 6
• Average partition similarity is approximately 0.4 across different triangle densities.Realizations with the same triangle density are much closer to one another than partitions produced at different triangle densities.Source: B. Effects of higher-order… · p. 6

C. Simplicial communities in social systems

• The datasets comprise temporal face-to-face interactions among students and teachers in an elementary school and among students in a high school.Interactions were aggregated into 15-minute windows, with fully connected subgraphs represented as simplices.Source: C. Simplicial communities in social… · p. 6
• Higher-order-biased embeddings achieved better NMI, approximately 0.5, for detecting student communities in both school datasets.The embeddings used simplices up to dimension k = 4 and were clustered with standard techniques.Source: C. Simplicial communities in social… · p. 6
• Increasing the maximal simplicial dimension produces little to no NMI change under lower-order bias but a clear NMI increase under higher-order bias.This complexity ladder compares embeddings built from edges, triangles, tetrahedra, and higher-dimensional simplices.Source: C. Simplicial communities in social… · p. 6

D. Communities of fMRI functional connectivity

• Small increases in triangle probability produce partitions very dissimilar from those obtained using only edges, and partitions also differ across triangle probabilities.Figure 4 compares clustering similarity for N = 200 synthetic complexes.Source: Figure 4 · p. 7
• In primary-school embeddings, higher-order bias yields more distinguishable communities and reveals teacher attribution more clearly than lower-order bias.Colors indicate class attribution and crosses indicate teachers in the primary-school data.Source: Figure 5 · p. 7
• In high-school embeddings, higher-order bias likewise produces more distinguishable communities, with class labels used as ground truth.The high-school dataset does not provide separate teacher labels.Source: Figure 6 · p. 8
• Higher-order-biased random walks improve NMI with class ground truth as maximal simplicial dimension increases, unlike lower-order-biased walks.Figure 7 reports this trend for two face-to-face interaction datasets.Source: Figure 7 · p. 8
• The cosine-based stability score s_i identifies brain regions whose community participation changes across maximal simplicial orders.Low s_i marks regions more affected by higher-order interactions; affected regions include reticular, visual, somatosensory-thalamic, and brain-stem areas, whereas many cerebellar regions remain stable.Source: D. Communities of fMRI functional… · p. 9

• Simplex2Vec provides a symbolic-embedding framework for community detection in systems whose interactions involve more than pairs.Source: §IV · p. 9
• Higher-order simplices can strongly alter community structure, so their contribution should be considered or tested in interacting systems.Source: §IV · p. 9
• Social-system evidence indicates that simplicial contributions can improve correspondence between detected clusters and ground truth.Source: §IV · p. 9
• A parsimonious network-neuroscience example assesses which nodes are more or less affected by higher-order interactions.Source: §IV · p. 9
• Future work includes extending Simplex2Vec to other dynamical systems and predicting missing interactions for arbitrary group sizes.Source: §IV · p. 9
• Further directions include higher-order simplex communities, broader random-walk and simplex-weighting schemes, and comparisons with combinatorial-Laplacian-constrained walks.Source: §IV · p. 9