Centrality measures in simplicial complexes: applications of Topological Data Analysis to Network Science

• Multi-interactions in social, biological, brain, and data systems motivate representing networks with simplicial complexes rather than graphs.Simplices encode higher-dimensional relations, with binary and ternary interactions represented by edges and triangles, respectively.Source: §1 · p. 2
• Centrality measures for simplices require new adjacency definitions because simplices can share lower-dimensional faces or belong to higher-dimensional simplices.Existing lower, upper, and general adjacency definitions do not compare simplices of different dimensions sharing arbitrary faces or nested in larger simplices.Source: §1 · p. 3
• The generalized framework in [20] defines p-lower, p-upper, and p-adjacency across dimensions, with associated degrees that generalize usual notions.A higher-order multi-combinatorial Laplacian, built from a multi-parameter boundary operator, computes these degrees.Source: §1 · p. 3
• The paper applies results given in [20] to introduce new centrality measures and proposes a generalized simplicial eigenvector centrality while examining higher-order connectivity in several real-world datasets.Source: §1 · p. 4
• It also defines simplicial walks, distances, closeness, betweenness, and a new clustering coefficient to study connectivity, information transmission, simplicial density, and well-connected hubs.The clustering coefficient differs from, generalizes graph vertex clustering, and measures clustering around simplicial communities rather than only nodes.Source: §1 · p. 4
• Earlier sections introduce lower, upper, and general adjacency, higher-order degrees, and computational methods using multiparameter boundary and combinatorial Laplacian operators.Source: §1 · p. 4
• Section 3 introduces simplicial degree centralities and p-order simplicial eigenvector centrality based on higher-order simplicial degrees.These measures are among the paper’s centrality contributions for simplicial complexes.Source: §1 · p. 5
• The paper presents walks and distances in simplicial complexes to study connectivity and generalize closeness and betweenness centralities.Source: §1 · p. 5
• It defines a simplicial clustering coefficient using higher-order degree and generalized distance in a simplicial complex.Source: §1 · p. 5

2.1. Higher order adjacency and simplicial degrees.

• The framework requires lower and upper adjacency notions to compare simplices of arbitrary dimensions and define corresponding generalized degrees.Lower adjacency is based on shared p-faces, whereas upper adjacency is based on containment in a common p-simplex; strict and (h, p)-degree variants are also defined.Source: §2 · p. 5, §2.1 · pp. 6–8

2. General adjacency, general simplicial degree and multi combinatorial Laplacian

• The multi combinatorial Laplacian of [20] generalizes graph and q-combinatorial Laplacians and provides a computational method for generalized simplicial degrees, requiring oriented simplices and chain/cochain constructions.Orientation distinguishes the two equivalence classes of vertex orderings, while the f-vector records the numbers of simplices by dimension.Source: §2 · pp. 5–6

2.1. Higher order adjacency and simplicial degrees.

• The generalized degrees recover standard simplicial and graph degrees in special cases, including (1, q −1)-lower degree, (1, q + 1)-upper degree, and the usual vertex degree.For equal-dimensional simplices, the relevant lower and upper adjacencies recover the ordinary notions described in.Source: §2.1 · pp. 7–9
• Unlike ordinary adjacency, generalized upper and lower adjacencies need not have unique common simplices, and p-adjacency degrees may overcount simplices.The section explicitly notes that ordinary implications between upper and lower adjacency can fail and illustrates overcounting when multiple shared faces induce adjacency.Source: §2.1 · p. 9
• General p-adjacency combines strict lower adjacency with exclusion of higher upper adjacency, yielding p-adjacency and maximal p-adjacency degrees for simplices.For vertices, adjacency is defined through 1-upper adjacency, agreeing with graph theory.Source: §2.1 · p. 9
• Higher-order adjacency and simplicial degrees count collaborations across smaller subcommunities and larger simplices containing a given simplex, including two-parameter and maximal degree variants.Maximal degrees retain only maximal p-adjacent communities and maximal containing communities, avoiding redundant counts from nested faces.Source: §2.1 · p. 10

2.2. Generalized boundary operator and multi combinatorial Laplacian.

• The generalized (q, h)-boundary operator removes h vertices with orientation signs, extending the ordinary boundary operator recovered when h = 1.Upper and lower similarity orientations, signs, and oriented degrees define the corresponding higher-order adjacency structure.Source: §2.2 · pp. 11, 13
• The (q, h, h′)-Laplacian combines generalized boundary and coboundary operators, with h = h′ = 1 recovering the ordinary q-Laplacian.Its matrix representation is defined on oriented simplex bases, and a theorem specifies entries of the associated multi-Laplacian matrices.Source: §2.2 · pp. 13–15

2.3. How to explicitly compute higher order degrees.

• The multi combinatorial Laplacian does not compute every higher-order degree, so explicit sign-function and generalized boundary/coboundary matrix formulas are provided instead.In particular, it computes a lower degree but not the general p-lower degree; Theorem 2 gives an explicit computation framework for higher-order degrees.Source: §2.3 · p. 15

3. Centrality measures in simplicial complexes

• The section proposes simplicial centrality measures based on higher-order degrees, including maximal simplicial degree centrality and p-adjacency eigenvector centrality.These measures target both the relevance of simplicial communities and the quantity or quality of their higher-order connections.Source: §3 · p. 16
• The section further defines simplicial walks and distances to generalize closeness and betweenness centralities, and introduces a clustering coefficient for clustering around simplices.These measures address connectivity, information transmission, and clustering beyond individual nodes; the new coefficient differs from that of.Source: §3 · p. 16

3.1. Degree centralties.

• Higher-degree centralities count normalized higher-order communities containing a simplex, while p-adjacency centralities capture communities collaborating through its faces.The definitions distinguish upper, strict upper, and p-adjacency relations; maximal simplicial centrality combines face collaborations with communities containing the simplex.Source: §3.1 · pp. 17–18
• The higher-degree vertex centrality reduces to ordinary degree centrality when h = 1 and measures normalized popularity in communities of at least h agents.For q-simplices, the analogous upper-degree centrality counts adjacent (q+h)-simplices relative to all possible such extensions.Source: §3.1 · p. 17
• Maximal simplicial degree centrality measures a simplex’s normalized relevance using non-overcounted relevance from its subcommunities rather than summing individual relevance.It combines maximal p-adjacency degree for face collaborations with strict upper degree for communities containing the simplex.Source: §3.1 · pp. 18–19
• In [20], maximal simplicial degree distributions across real-world datasets revealed rich, varied higher-order connectivity structures and similar patterns among datasets of the same type.The analyzed datasets included coauthor, bill-cosponsorship, school-contact, drug-abuse-warning, email, publication, and online-forum networks.Source: §3.1 · p. 19
• The degree-centrality subsection defines simplicial degrees and normalized average-degree measures using strict upper, maximal adjacency, and p-adjacency relations.For q = 0 and h = 1, the maximal normalized average degree coincides with the graph network’s normalized average degree.Source: §3.1 · pp. 21–22

3.2. Eigenvector Centralities.

• The eigenvector-centrality subsection assigns each q-simplex a principal-eigenvector score and extends the measure to higher-order p-adjacency via A(q, p).The normalized p-adjacency simplicial eigenvector centrality is the corresponding component of the eigenvector for the largest eigenvalue, with components summing to 1; p = q − 1 recovers the definition in.Source: §3.2 · pp. 24–25

3.3. Centrality measures associated with walks and distances.

• The walks-and-distances subsection begins by defining two simplices as p-near when they share a p-face.Source: §3.3 · p. 25
• Maximal p-nearness and p-walks generalize earlier connectivity notions by linking consecutive simplices through maximal p-adjacency, with p equal to the minimum adjacency level along the walk.When all simplex dimensions are q and p = q − 1, the definition recovers the sq-walk of [13, Def. 9].Source: §3.3 · pp. 25–26
• The resulting maximal p-connected components refine ordinary p-connectedness, because maximal p-connectedness is not itself reflexive and therefore requires a separately defined equivalence relation.The number of maximal p-connected components yields the Q*-vector generalizing the topological Q-vector.Source: §3.3 · pp. 26–27
• The proposed p-distance is the minimum length of a p-walk between two simplices, with unreachable pairs assigned distance ∞, and is a generalized distance on K.This framework also defines average simplicial shortest p-walk length and p-diameter.Source: §3.3 · pp. 27–28
• The section defines p-closeness and p-betweenness centralities for simplices, using shortest p-walks between simplices and restricting comparisons to those in the same maximal p-connected component.Source: §3.3 · p. 28

3.4. Simplicial clustering: a centrality measure associated with both the higher order degree and the p-distance.

• The paper rejects an existing clustering definition as potentially nonzero when neighboring tetrahedra share only a common vertex, despite lacking a separate interconnection.This motivates a different simplicial clustering coefficient whose value can reflect whether neighboring simplex communities are linked beyond the shared face.Source: §3.3 · p. 29
• The new simplicial clustering coefficient generalizes the standard vertex clustering coefficient by combining maximal p-adjacency degree with strictly upper degree.The graph coefficient is defined as the ratio of links among a vertex’s neighbors to all possible links among them.Source: §3.3 · p. 29

3.3. Centrality measures associated with walks and distances.

• In the vertex example, triangles upper-adjacent to v have no qualifying links, so the coefficient’s numerator is 0; a second example has five maximal neighbours and two links.The five neighbours comprise one edge and four triangles, while the two links connect t1–t2 and t2–t3.Source: §3.3 · pp. 30–31
• The simplicial clustering coefficient counts links among a simplex’s maximal neighbours, with neighbours defined through lower adjacency or strict upper adjacency and links formed by shared faces or qualifying edges.For vertices, maximal neighbours are strictly upper-adjacent simplices; for q-simplices with q > 0, both lower and upper adjacency are included.Source: §3.3 · p. 31
• The construction generalises graph clustering by representing the target simplex and its maximal neighbours as vertices, with neighbour links becoming graph edges; the resulting coefficient equals standard graph clustering.This correspondence is explicitly established for the associated vertex representation of any q-simplex.Source: §3.3 · p. 32
• For a 1-simplex example, the maximal-neighbour links number 4, and its simplicial clustering coefficient agrees with the standard graph clustering of the simplex viewed as a vertex.The links arise from a 0-walk and shared faces among the maximal neighbours.Source: §3.3 · pp. 30, 32, 33
• The definition can be adapted for different applications by changing the link notion or allowed s-distances, and weights may be needed because shared faces of different dimensions are treated uniformly.The authors note that any shared face is interpreted simply as a connection under the basic definition.Source: §3.3 · p. 33

Centrality measures

• The higher-order degree framework supports centrality measures for identifying important simplicial communities and studying topological and dynamical properties of complex systems.The authors connect these measures to real-world simplicial connectivity studies and higher-order degree distributions.Source: §4 · p. 33

Future work

• Future work includes configuration models and relating the simplicial clustering coefficient to the first two Betti numbers of a simplicial network.The paper also points toward adapting centrality indices to study spreading capability in higher-order contagion processes.Source: §4 · pp. 33–34

Centrality measures

• The proposed eigenvector centrality weights relationships to more highly interconnected q-simplices more strongly, targeting relevant same-sized communities at a fixed adjacency order.This measure is intended for applications that compare simplicial communities of the same size.Source: §4 · p. 33

Walks and distances

• Walks and distances in simplicial complexes generalize closeness and betweenness centralities and open a way to study information flow through simplicial communities.The paper frames these notions as tools for analyzing connectivity and transmission across higher-order structures.Source: §4 · p. 33

Simplicial clustering

• The simplicial clustering coefficient generalizes graph vertex clustering to measure clustering around nodes and entire simplicial communities.The authors propose using it to understand simplicial-network topology in terms of community density.Source: §4 · pp. 33–34

Future applications

• The proposed centrality measures may help analyze how cliques and simplicial communities evolve and support analogues of clique-percolation techniques.These extensions are presented as future applications of simplicial degree-based centrality.Source: §4 · p. 34