hypernets: Higher-Order Network Analysis

A higher-order network is one in which a relation reaches beyond a single pair of nodes at a single moment: either it binds more than two nodes at once, or it depends on more than the current node. The literature (Battiston et al. 2020; Bianconi 2021) organizes that idea into three structure families, and hypernets implements all three behind one taxonomy.

Ownership

The memory-network and simplicial-complex estimators are imported from a sibling estimation package; hypernets wraps them under its own names (hon(), honem(), mogen(), hypa(), markov_order(), memory(), hg_markov_stability(), simplicial(), hg_homology(), hg_landscape(), hg_bottleneck(), hg_betti(), hg_euler(), hg_qanalysis(), hg_degree()) and returns their result objects unchanged, so every number is the estimator's own. The wrappers add one sequence-input contract (long, wide, list or model input; see sequence-input), the hg_get() reader for every result class, and the verbs hypernets computes itself on top of them (hg_bootstrap(), hg_compare(), hg_centrality(), hg_communities(), hg_wasserstein()). The hypergraph and text families are hypernets' own.

The three families

Memory networks: A node is a state plus the memory of how it was reached, so a relation depends on history rather than only on the present state. Built from categorical sequences. Constructors hon(), honem(), hypa(), mogen(); diagnostics markov_order(), memory(); inference hg_bootstrap(), hg_compare(); measures hg_centrality().
Simplicial complexes: A relation is a set of nodes that are all mutually related, together with every one of its subsets, which gives the object a geometry and hence a topology. Constructor simplicial(); measures hg_betti(), hg_euler(), hg_degree(), hg_qanalysis(); topology hg_homology(), hg_landscape(), hg_bottleneck(), hg_wasserstein().
Hypergraphs: A relation is an arbitrary set of nodes bound as a unit, with no requirement that its subsets also be relations. Constructors network_hypergraph(), window_hypergraph(), group_hypergraph(), temporal_hypergraph(); random models hg_sample_gnp(), hg_sample_sbm(), hg_sample_uniform(); measures hg_measures(), hg_centrality(); spectral methods hg_laplacian(), hg_cluster(), hg_classify(); PageRank hg_pagerank(); embeddings hg_embed(); projections hg_clique_expansion(), hg_project(), hg_line_graph(), dual_hypergraph(); temporal views hg_snapshot(), hg_snapshots(); hyperedge tables hg_edges() and s-centrality hg_edge_centrality(); communities hg_communities() and hg_community_quality(); motifs hg_motifs(); null models hg_null_test(); neural networks hg_neural(), text_hypergat(), heterogeneous_hgat(), hg_hypergcn(), hg_hnhn(), hg_allset(); embedding constructor knn_hypergraph().
Text hypergraphs: A corpus is a bipartite document-word structure, which is a hypergraph in either orientation: documents as nodes bound by shared words, or words as nodes bound by shared documents. Constructor text_hypergraph() (bag of words, token windows, or embedding nearest neighbours); tidy verbs hg_measures(), hg_centrality(), hg_cluster(), hg_keywords(), hg_classify(), hg_stability(), hg_agreement(), hg_seeds(). Every verb also accepts any net_hg.

Verb grammar

The same naming rules hold across all families:

Constructors are nouns: hon(), honem(), mogen(), simplicial(), markov_order(), memory(), text_hypergraph(), window_hypergraph(), group_hypergraph(), ... Where a family admits several construction routes, they are selected with type =.
Everything else is hg_*(): Measures return a tidy data.frame, one row per node or per structure (hg_centrality(), hg_measures(), hg_degree()); inference returns a result object carrying estimates, intervals and p-values (hg_bootstrap(), hg_compare(), hg_null_test()). One verb names one idea and dispatches on its input: hg_centrality(), hg_communities() and hypa() take a memory network or a hypergraph.
hg_get() is the one reader: Every result object hands over its tables through hg_get(x, what = ) – never through $. what = selects a secondary table; filters, sort_by and top are named arguments. hypernets' result classes also have print(), summary() and plot() methods.

Crossing between families

The families are entry points into one another, not islands. The text family is the corpus front end of the hypergraph family: a text_hypergraph is a net_hg, so every hypergraph verb takes it, and every hg_*() verb takes any net_hg in return. simplicial() with type = "pathway" turns a memory network into a simplicial complex; window_hypergraph() turns the same sequences into a hypergraph; hg_clique_expansion() projects a hypergraph back to a pairwise network that any of the first-order tools accept. hg_get(x, what = "pathways") on a memory network hands its sequence-derived path strings to the other two.

References

Battiston, F., Cencetti, G., Iacopini, I., Latora, V., Lucas, M., Patania, A., Young, J.-G., & Petri, G. (2020). Networks beyond pairwise interactions: Structure and dynamics. Physics Reports, 874, 1-92.

Bianconi, G. (2021). Higher-Order Networks. Cambridge University Press.

Author

Maintainer: Mohammed Saqr saqr@saqr.me (ORCID) [copyright holder]