Simplicial degree in complex networks. Applications of Topological Data Analysis to Network Science

• Many real-world systems contain higher-order group interactions that pairwise graph models cannot represent, motivating simplicial-complex methods for structural connectivity.Simplicial complexes preserve many-body interactions and distinguish filled higher-dimensional relations from merely pairwise links.Source: §1 · pp. 2–3
• The paper generalizes lower, upper, and general simplicial adjacency and defines associated degrees that compare simplices across arbitrary dimensions, including strict faces.Closed formulas explicitly compute strict lower, upper, and general degrees, including the previously proposed simplex-to-facets degree.Source: §1 · p. 4
• A multi-parameter boundary operator and multi-combinatorial Laplacian extend graph and q-combinatorial Laplacians while encoding higher-order simplicial degrees.The associated matrix entries compute some defined degrees, and boundary and coboundary operators yield closed effective formulas for all higher-order degrees.Source: §1 · pp. 4–5
• The applied framework proposes maximal upper and maximal simplicial degrees to measure a community’s relevance through maximal communities containing a simplex and, additionally, its strict faces.The maximal simplicial degree counts distinct maximal communities containing the simplex or its strict faces, whereas maximal upper degree counts communities containing the simplex itself.Source: §1 · p. 5
• The study analyzes higher-order connectivity and associated degree distributions in 17 real-world datasets, represented as simplices with corresponding nodes, bounded to a maximum of 25 nodes, across diverse domains.Source: §1 · p. 5
• The study finds rich, varied higher-order connectivity structures, while datasets of the same type exhibit similar higher-order patterns and classical degree distributions.Collaboration sizes vary across categories, whereas facet sizes are similar across most datasets.Source: §1 · p. 5
• The maximal simplicial degree of Definition 14 counts containing facets for a simplex and its strict subcommunities, producing distributions generally different from classical node degrees.For every dataset, the maximal upper simplicial-degree distribution is closer to a power law with a more pronounced decay.Source: §1 · p. 5
• The paper introduces new notions of simplicial adjacencies and degrees that generalize usual notions and provides closed formulas for computing these higher-order degrees.Source: §1 · p. 6
• It defines a higher-order multi-combinatorial Laplacian using a novel multi-parameter boundary operator and expresses its matrix entries through simplicial degrees.Source: §1 · p. 6
• The theoretical results are applied through statistical analysis of simplicial-degree distributions across several varied real-world datasets.Source: §1 · p. 6

• Simplicial complexes represent multi-interaction relations in networks by including simplices together with all their faces.A q-simplex generalizes points, edges, triangles, and higher-dimensional structures; a complex is closed under taking faces.Source: §2 · pp. 6–7
• Higher-dimensional simplices require lower and upper adjacency because a simplex can relate both to shared faces and to higher-dimensional simplices containing it.Lower adjacency means sharing a common (q −1)-face, while upper adjacency means being faces of the same common (q +1)-simplex; upper adjacency implies lower adjacency.Source: §2 · p. 7
• The lower and upper degrees of a q-simplex count its adjacency through common lower simplices and its containment in (q +1)-simplices, respectively.These definitions generalize vertex degree to simplices of arbitrary dimension.Source: §2 · pp. 7–8
• The q-combinatorial Laplacian extends the graph Laplacian to q-simplices and encodes their degrees and adjacency relations.Its construction uses oriented simplices, the q-boundary operator, the adjoint coboundary operator, and associated boundary matrices.Source: §2 · p. 8
• Orientations distinguish the two equivalence classes of vertex orderings and determine the signs used in boundary, coboundary, and Laplacian constructions.The section also distinguishes similarly and dissimilarly oriented adjacent simplices and separates upper and lower Laplacian matrices.Source: §2 · pp. 8–10
• The q-Laplacian matrices are defined through entries that depend on whether simplices have similar or dissimilar orientations.Source: §2 · p. 10
• The off-diagonal entries of the q-adjacency matrix in [15] equal the absolute values of the corresponding off-diagonal entries of L_q.Source: §2 · p. 11
• L_0 = D − A is the traditional Laplacian matrix of a network.The construction gives L_0 = B_1B_1^t.Source: §2 · p. 11

3. Higher-order adjacency and simplicial degree

• Earlier work [31] generalized vertex-to-simplex and q-simplex-to-(q+h)-simplex degrees, but found no clear models for the latter and gave no explicit formula for q-simplex-to-facets degree.The alternative vertex-to-facets degree counts distinct maximal collaborative groups containing a vertex, and the generalized facet degree was described as computable by searching and counting a facet list.Source: §3 · pp. 11–12
• The paper introduces higher-order lower, upper, and general adjacency notions for simplices of arbitrary dimensions, together with properties, closed degree formulae, and an explicit route to computing q-simplex-to-facets degree.These definitions extend and redefine the q-simplex-to-(q+h)-simplex degree and address the absence of an explicit facet-degree formula.Source: §3 · p. 12

3.1. General adjacencies for simplices of different dimensions.

• General adjacency distinguishes p-lower adjacency through shared p-faces, p-upper adjacency through containment in a common p-simplex, and p-adjacency by strict lower adjacency without corresponding upper adjacency.Strict p-adjacency excludes relationships that also occur at the next dimension; unlike ordinary adjacency, common lower or upper simplices need not be unique, and upper adjacency does not generally imply lower adjacency.Source: §3.1 · p. 12

3.2. Generalized lower degree for simplices.

• The p-lower degree of a q-simplex counts q′-simplices that are p-lower adjacent, while the (h,p)-lower degree counts (q−h)-simplices; when h=1 and p=q−1, it recovers the lower degree.Source: §3.2 · pp. 14–15
• The lower-degree example aggregates incident edges from several higher-dimensional simplices and a triangle, yielding a total of 18.The contributions are 3, 4, 3, and 2 incident edges, plus 6 edges in a tetrahedron.Source: §3.2 · p. 16

3.3. Generalized upper degree for simplices.

• The generalized upper degree counts higher-dimensional simplices p-upper adjacent to a simplex, with strict variants excluding adjacency at the next level.The framework also defines (h,p)-upper degrees for (q+h)-simplices and recovers the usual vertex degree and vertex-to-simplex degrees as special cases.Source: §3.3 · pp. 17–18
• A single multi-parameter combinatorial Laplacian computes generalized upper degrees as diagonal entries, replacing products of entries from several boundary-operator matrices.The construction provides an alternative to computing these quantities from distinct q-boundary operators, as stated in.Source: §3.3 · pp. 19–20

3.4. Generalized adjacency degree for simplices and simplicial degree.

• The generalized adjacency degree counts p-adjacent simplices, while the maximal p-adjacency degree counts only maximal collaborative communities to avoid overcounting faces.The text notes that ordinary p-adjacency can count a neighboring triangle together with its two adjacent edges, motivating the maximal variant.Source: §3.4 · p. 20
• The maximal simplicial degree combines maximal communities adjacent to a simplex’s faces with distinct maximal communities containing the simplex itself.For an example edge e, the reported maximal simplicial degree is deg*(e) = 4.Source: §3.4 · p. 21

4. The multi-combinatorial Laplacian

• Generalized boundary and coboundary operators provide a framework for computing higher-order degrees in oriented simplicial complexes.The coboundary operator exists uniquely and is adjoint to the generalized boundary operator.Source: §4 · p. 21, §4.1 · p. 26

4.2. The multi-combinatorial Laplacian.

• The (q, h, h′)-Laplacian generalizes the combinatorial Laplacian and computes higher-order degrees through upper and lower operators.When h = h′ = 1, it recovers the q-combinatorial Laplacian.Source: §4.2 · pp. 26–28
• The multi-combinatorial Laplacian can contain zero entries from non-adjacency or cancellation between oppositely oriented common faces.Thus, zero matrix entries do not necessarily indicate that simplices are non-adjacent.Source: §4.2 · p. 28
• Theorem 1’s Laplacian does not compute all higher-order simplex degrees, including the general p-lower degree.The section therefore gives explicit degree computations using the generalized boundary and coboundary operators and Propositions 2 and 3.Source: §4.2 · p. 29
• The multi-combinatorial Laplacian framework defines p-lower and p-upper adjacency orders by counting shared p-faces or p-cofaces, respectively.These orders address cases where the common lower or upper simplex is not unique, preventing repeated counting in degree calculations.Source: §4.2 · p. 30
• Theorem-based formulas express p-lower, p-upper, and maximal p-adjacency degrees through matrices of boundary and coboundary operators.The explicit adjacency-degree formulas are stated for simplices across dimensions q, q′, and p.Source: §4.2 · pp. 30–31

4.4. On some potential theoretical applications.

• The multi-parameter boundary operator and multi-combinatorial Laplacian extend potential spectral applications to generalized random walks, harmonic functions, and eigenvalue analysis.Alternating q,h operators recover the usual random walk when h = 1; the Laplacian’s kernel enables harmonic-function and spectral studies.Source: §4.4 · p. 32
• A key limitation is that the multi-parameter boundary operator δq,h does not always satisfy δq,h◦δq+h,h = 0, leaving its topological and spectral invariants for future study.The paper states that further research is needed to determine what the multi-combinatorial Laplacian and its spectra measure in a simplicial network.Source: §4.4 · p. 32

5. Analysis of higher-order connectivity on real world networks

• Across 17 real-world datasets, the analysis finds a rich variety of higher-order connectivity structures and similar connectivity patterns among datasets of the same type.The datasets span coauthorship, Congress bills, school contacts, drug networks, email, and online forums.Source: §5 · p. 33
• For every analysed dataset, maximal upper simplicial degree follows a power-law distribution resembling classical degree distributions but with more pronounced decay and less small-degree saturation.This contrasts with maximal simplicial degree, whose distribution is generally surprisingly different and has higher small-degree saturation.Source: §5 · p. 33
• The analysis identifies “simplicial hubs,” namely simplices with many simplicial connections, as relevant structures for studying real-world simplicial-network topology.This extends the organizing role of high-connection nodes in classical Network Science to higher-order connectivity.Source: §5 · p. 33

5.1. Dataset description and summary statistics.

• The study analyzes 17 publicly available real-world datasets comprising simplices bounded to a maximum of 25 nodes, without accounting for simplex repetitions.The datasets were obtained from. Weighted simplicial degree is outside this work’s scope and is identified as a future research direction.Source: §5.1 · pp. 33, 35
• Despite substantial variation in the numbers of nodes and simplices across categories, facet-size distributions are similar in almost all categories.Table 2 reports maximum, average, median, and most probable facet sizes for the datasets.Source: §5.1 · pp. 35–36, Table 2 · p. 36
• Facet sizes are generally low in median and most-probable values, but analyses impose a 25-node maximum whose impact on tails remains unverified.The limitation is especially relevant for congress-bill data, whose facet-size tail does not appear to vanish at 25; incompleteness may also exaggerate tail decay.Source: §5.1 · p. 37

5.2. Simplicial degree distributions.

• The maximal upper simplicial-degree distribution is consistently closer to a power law, with stretched-exponential cutoffs and fewer, smaller simplicial hubs.This tendency persists even when classical node degrees are not power-law distributed, while power-law node-degree datasets show more pronounced decay.Source: §5.2 · p. 41
• The maximal simplicial degree tracks connections of a simplex and its strict faces, making it a proposed measure of community relevance and influential simplicial communities.Its distributions show smaller values, pronounced tail cutoffs, and fewer and smaller hubs than classical node-degree distributions.Source: §5.2 · pp. 41–42
• Across datasets, higher-order connectivity shows rich patterns, with maximal simplicial-degree distributions often differing markedly from classical node-degree distributions.They exhibit either greater small-degree saturation or bell-shaped curves, suggesting more homogeneous topology and a possible random component.Source: §5.2 · pp. 41–42
• Coauthor networks show low-degree saturation and high-degree structural cutoffs resembling sublinear preferential attachment, but with faster decay than classical node degrees.High maximal simplicial-degree communities are more probable than high-degree individuals, whereas maximal simplicial hubs are less probable than usual hubs.Source: §5.2 · pp. 42–43
• Contact-school networks display random-like classical and node-to-facets distributions, while maximal simplicial degree is more clearly random-like and maximal upper simplicial degree approaches a power law.The power-law-like maximal upper simplicial distribution remains unexplained.Source: §5.2 · p. 43
• Maximal simplicial degree distributions can diverge sharply from classical node-degree distributions, including random-like bell shapes that transition to power laws with exponential cutoffs.This pattern appears in DAWN, tags-ask-ubuntu, and tags-math-sx, whereas classical node degree reflects sublinear preferential attachment.Source: §5.2 · p. 44
• Thread-based forum networks suggest superlinear maximal upper simplicial degree, while maximal simplicial degree may produce winner-takes-all behavior through higher saturation and an increasing tail cutoff.These distributions still require proper interpretation.Source: §5.2 · p. 44
• Higher-order distribution models may depend on simplex dimension, becoming closer to random models as simplex dimension increases, but bounded and incomplete data limit reliable fitting.Generative simplicial models would be needed to analytically predict expected distributions.Source: §5.2 · pp. 42, 44

6. Conclusions and future research

• The paper introduces higher-order simplicial adjacencies and two simplicial degrees, with explicit computation methods for comparisons across simplicial dimensions.These notions extend adjacency and degree beyond pairwise network structure.Source: §6 · p. 45
• A multi-parameter combinatorial Laplacian generalizes graph and combinatorial Laplacians and supports proposed studies of generalized random walks and spectral topological invariants.The paper presents these as potential theoretical applications rather than completed empirical analyses.Source: §6 · p. 45
• Across many diverse real-world datasets, the simplicial-degree distributions reveal rich higher-order connectivity structures and similar collaboration patterns among datasets of the same type.The analyzed data span social, biological, biomedical, and computer-science settings.Source: §6 · p. 45
• Maximal simplicial degree generally produces distributions surprisingly different from classical and upper adjacency distributions, including greater small-degree saturation and an added random component.The maximal degree counts distinct facets containing a simplex and its strict faces.Source: §6 · p. 45
• The analysis identifies simplicial hubs, or influential communities, as a deep organizing principle in simplicial-network topology.The paper connects these hubs to influencer-like roles in social networks.Source: §6 · p. 45
• Future work requires statistical studies of simplicial centralities, generative models for expected simplicial-degree distributions, and methods addressing incomplete empirical data.The authors note that incompleteness can affect degree-distribution tail cutoffs and that rigorous analytic models are still needed.Source: §6 · pp. 45–46

7.1. Proof of Theorem 2.

• Theorem 2 is proved by representing boundary operators with matrices whose compositions encode shared p-faces and p-lower adjacency counts.The proof uses fixed bases and sign matrices to avoid multiple counting of adjacent simplices.Source: §7.1 · pp. 48–49

7.2. Proof of Theorem 3.

• Theorem 3 extends the analogous matrix argument to p-upper adjacency, counting p-simplices that contain both compared simplices.The proof identifies upper adjacency through common containing p-simplices.Source: §7.2 · p. 50

7.3. Proof of Theorem 4.

• Theorem 4 characterizes p-adjacency and maximal p-adjacency by excluding higher-dimensional simplices that contain already counted adjacent faces.The maximal degree counts q′-simplices that are p-adjacent but are not faces of further p-adjacent simplices.Source: §7.3 · p. 51
• Sign-based criteria determine when two simplices are p-lower adjacent, providing the orientation-sensitive condition used in the adjacency proofs.The criterion is expressed through the lower sign of the simplices relative to a p-simplex basis.Source: §7.3 · p. 51