Centrality catalogue

Notation

Throughout, \(G = (V, E)\) is a graph with \(n = |V|\) nodes and adjacency matrix \(A = (a_{ij})\); \(w_{ij}\) are edge weights, \(N(v)\) is the neighbour set of \(v\), \(k_v = |N(v)|\) its degree, and \(s(v) = \sum_u w_{vu}\) its strength. We write \(d(u, v)\) for the shortest-path distance, \(\Delta\) for the graph diameter, and \(\sigma_{st}\) (resp. \(\sigma_{st}(v)\)) for the number of shortest \(s\)–\(t\) paths (resp. those passing through \(v\)). \(L = D - A\) is the graph Laplacian and \(\mathbf{1}\) the all-ones vector.

Degree Centrality

Degree centrality counts the number of direct ties incident on a node. In directed graphs, it can be separated into incoming and outgoing ties.

\[ C_D(v) = \sum_{u \in V} a_{vu} = k_v \]

Meaning. A high degree value indicates many direct observed connections. It is a local connectivity measure and does not imply brokerage or global reach.

centrality_degree(student_interactions)

Equivalence reference. Equivalent to igraph::degree() under matching mode and loop settings.

Indegree Centrality

Indegree counts ties pointing into a node under a directed interpretation.

\[ k_v^{\mathrm{in}} = \sum_{u \in V} a_{uv} \]

Meaning. A high indegree value indicates many incoming observed ties. Its substantive meaning depends on what edge direction represents.

centrality_indegree(student_interactions)

Equivalence reference. Equivalent to igraph::degree(mode = "in").

Outdegree Centrality

Outdegree counts ties leaving a node under a directed interpretation.

\[ k_v^{\mathrm{out}} = \sum_{u \in V} a_{vu} \]

Meaning. A high outdegree value indicates many outgoing observed ties. In transition networks, this can mean many possible next states.

centrality_outdegree(student_interactions)

Equivalence reference. Equivalent to igraph::degree(mode = "out").

Strength Centrality

Strength centrality is weighted degree: it sums edge weights instead of counting edges.

\[ s(v) = \sum_{u \in V} w_{vu} \]

Meaning. A high strength value indicates high total incident edge weight. This should be interpreted according to how weights were defined.

centrality_strength(student_interactions)

Equivalence reference. Equivalent to igraph::strength() when weights are supplied.

Instrength Centrality

Instrength sums incoming edge weights.

\[ s^{\mathrm{in}}(v) = \sum_{u \in V} w_{uv} \]

Meaning. A high instrength value indicates high total incoming weight. It is a weighted incoming-connectivity measure.

centrality_instrength(student_interactions)

Equivalence reference. Equivalent to igraph::strength(mode = "in").

Outstrength Centrality

Outstrength sums outgoing edge weights.

\[ s^{\mathrm{out}}(v) = \sum_{u \in V} w_{vu} \]

Meaning. A high outstrength value indicates high total outgoing weight. It does not show whether that weight is concentrated or distributed.

centrality_outstrength(student_interactions)

Equivalence reference. Equivalent to igraph::strength(mode = "out").

Expected Centrality

Expected centrality sums the degrees of a node’s neighbours.

\[ C_{\mathrm{exp}}(v) = \sum_{u \in N(v)} k_u \]

Meaning. A high value indicates adjacency to well-connected nodes.

centrality_expected(student_interactions)

Equivalence reference. Implemented as a native neighbourhood centrality in cograph.

Leverage Centrality

Leverage centrality compares a node’s degree with the degrees of its neighbours.

\[ \ell(v) = \frac{1}{k_v} \sum_{u \in N(v)} \frac{k_v - k_u}{k_v + k_u} \]

Meaning. A positive high value indicates that the node is more connected than its neighbours under this degree comparison.

centrality_leverage(student_interactions)

Equivalence reference. Validated against centiserve::leverage().

Lobby Centrality

Lobby centrality is the h-index of neighbour degrees.

\[ h(v) = \max\bigl\{ h : |\{ u \in \{v\} \cup N(v) : k_u \ge h \}| \ge h \bigr\} \]

Meaning. A high value indicates many neighbours that themselves have reasonably high degree.

centrality_lobby(student_interactions)

Equivalence reference. Validated against centiserve::lobby().

H-Index Strength Centrality

H-index strength is a weighted h-index-style neighbourhood measure.

\[ h_s(v) = \max\bigl\{ h : |\{ u \in \{v\} \cup N(v) : s(u) \ge h \}| \ge h \bigr\} \]

Meaning. A high value indicates strong ties to neighbours meeting a weighted connectivity threshold.

centrality(student_interactions, measures = "hindex_strength")

Equivalence reference. Equivalent to lobby centrality on unweighted graphs in package tests.

Local H-Index Centrality

Local h-index centrality iteratively applies h-index logic to local neighbourhoods.

\[ h^{(t+1)}(v) = \mathcal{H}\bigl(\{ h^{(t)}(u) : u \in N(v) \}\bigr), \quad h^{(0)}(v) = k_v \]

Meaning. A high value indicates a locally robust neighbourhood position under the h-index updating rule.

centrality(student_interactions, measures = "local_hindex")

Equivalence reference. Implemented following local h-index centrality formulations and checked in package tests.

Semi-Local Centrality

Semi-local centrality extends degree-like counting beyond immediate neighbours.

\[ C_{SL}(v) = \sum_{u \in N(v)} \sum_{w \in N(u)} \bigl| \{\, x : d(w, x) \le 2 \,\} \bigr| \]

Meaning. A high value indicates proximity to a locally well-connected neighbourhood.

centrality_semilocal(student_interactions)

Equivalence reference. Validated against centiserve::semilocal().

ClusterRank Centrality

ClusterRank combines neighbour degree information with local clustering.

\[ C_{CR}(v) = c(v) \sum_{u \in N(v)} \bigl( k_u + 1 \bigr), \quad c(v) = \text{local clustering coefficient} \]

Meaning. A high value indicates local connectedness adjusted for clustering-related redundancy.

centrality_clusterrank(student_interactions)

Equivalence reference. Validated against centiserve::clusterrank().

Collective Influence Centrality

Collective influence combines a node’s excess degree with excess degree on a local boundary.

\[ \mathrm{CI}_\ell(v) = (k_v - 1) \sum_{u \,\in\, \partial B(v, \ell)} (k_u - 1), \quad \ell = 2 \]

Meaning. A high value indicates potential importance under the collective influence model.

centrality(student_interactions, measures = "collective_influence")

Equivalence reference. Based on Morone and Makse’s collective influence formulation.

MNC Centrality

Maximum neighbourhood component centrality is the size of the largest connected component in a node’s neighbourhood.

\[ \mathrm{MNC}(v) = \max_{C \,\in\, \mathcal{C}(G[N(v)])} |C| \]

Meaning. A high value indicates that many neighbours belong to one connected local component.

centrality_mnc(student_interactions)

Equivalence reference. Validated against centiserve::mnc().

DMNC Centrality

DMNC is a density-adjusted version of maximum neighbourhood component centrality.

\[ \mathrm{DMNC}(v) = \frac{E_c}{N_c^{\,\varepsilon}}, \quad \varepsilon = 1.7 \]

Meaning. A high value indicates a large and dense local neighbour component under the chosen density exponent. Here \(E_c\) and \(N_c\) are the edges and nodes of the largest component of \(G[N(v)]\).

centrality_dmnc(student_interactions)

Equivalence reference. Validated against centiserve::dmnc().

LAC Centrality

LAC, or local average connectivity, measures connectivity in a node’s neighbourhood.

\[ \mathrm{LAC}(v) = \frac{1}{k_v} \sum_{u \in N(v)} \deg_{G[N(v)]}(u) \]

Meaning. A high value indicates that the node’s neighbours form a relatively connected local structure.

centrality_lac(student_interactions)

Equivalence reference. Corresponds to CytoHubba-style local average connectivity.

Gravity Centrality

Gravity centrality treats node degree or mass as attracting over graph distance.

\[ G(v) = \sum_{u \ne v,\; d(u,v) < \infty} \frac{k_u \cdot ks(u)}{d(u, v)^2}, \quad ks = \text{k-shell} \]

Meaning. A high value indicates proximity to high-degree nodes under the gravity formula.

centrality(student_interactions, measures = "gravity")

Equivalence reference. Formula verified in package tests against the degree-mass-over-distance definition.

Closeness Centrality

Closeness centrality summarises how short a node’s paths are to other reachable nodes.

\[ C_C(v) = \frac{1}{\sum_{u \ne v} d(v, u)} \]

Meaning. A high value indicates short graph distances to other nodes. In disconnected graphs, harmonic centrality may be easier to interpret.

centrality_closeness(student_interactions)

Equivalence reference. Equivalent to igraph::closeness() under matching mode and weight settings.

Incloseness Centrality

Incloseness computes closeness over incoming directed paths.

\[ C_C^{\mathrm{in}}(v) = \frac{1}{\sum_{u \ne v} d(u, v)} \]

Meaning. A high value indicates that other nodes can reach the focal node through short directed paths.

centrality_incloseness(student_interactions)

Equivalence reference. Equivalent to igraph::closeness(mode = "in").

Outcloseness Centrality

Outcloseness computes closeness over outgoing directed paths.

\[ C_C^{\mathrm{out}}(v) = \frac{1}{\sum_{u \ne v} d(v, u)} \]

Meaning. A high value indicates that the focal node can reach other nodes through short directed paths.

centrality_outcloseness(student_interactions)

Equivalence reference. Equivalent to igraph::closeness(mode = "out").

Harmonic Centrality

Harmonic centrality sums reciprocal shortest-path distances. Unreachable nodes contribute zero.

\[ C_H(v) = \sum_{u \ne v} \frac{1}{d(v, u)} \]

Meaning. A high value indicates broad reach through short paths while handling disconnected components more gracefully than classical closeness.

centrality_harmonic(student_interactions)

Equivalence reference. Equivalent to igraph::harmonic_centrality() under matching settings.

Inharmonic Centrality

Inharmonic centrality computes harmonic centrality over incoming directed paths.

\[ C_H^{\mathrm{in}}(v) = \sum_{u \ne v} \frac{1}{d(u, v)} \]

Meaning. A high value indicates that the node is reachable from many others through short directed paths.

centrality_inharmonic(student_interactions)

Equivalence reference. Equivalent to igraph::harmonic_centrality(mode = "in").

Outharmonic Centrality

Outharmonic centrality computes harmonic centrality over outgoing directed paths.

\[ C_H^{\mathrm{out}}(v) = \sum_{u \ne v} \frac{1}{d(v, u)} \]

Meaning. A high value indicates that many nodes can be reached from the focal node through short directed paths.

centrality_outharmonic(student_interactions)

Equivalence reference. Equivalent to igraph::harmonic_centrality(mode = "out").

Residual Closeness Centrality

Residual closeness sums \(1 / 2^d\) over shortest-path distances (the focal node contributes \(1\)).

\[ C_R(v) = \sum_{u \in V} 2^{-d(v, u)} \]

Meaning. A high value indicates many close reachable nodes with rapid distance decay.

centrality_residual_closeness(student_interactions)

Equivalence reference. Validated against centiserve::closeness.residual().

Dangalchev Closeness Centrality

Dangalchev closeness is the residual-closeness distance-decay measure.

\[ C_D(v) = \sum_{u \in V} 2^{-d(v, u)} \]

Meaning. A high value indicates short-distance reach with distant nodes down-weighted.

centrality_dangalchev(student_interactions)

Equivalence reference. Implemented as the residual closeness equivalent used in cograph.

Generalized Closeness Centrality

Generalized closeness sums \(\alpha^d\), where \(d\) is the shortest-path distance.

\[ C_G(v) = \sum_{u \in V} \alpha^{\,d(v, u)}, \quad \alpha = 0.5 \]

Meaning. A high value indicates reach under the chosen distance-decay parameter.

centrality_generalized_closeness(student_interactions)

Equivalence reference. Corresponds to generalized distance-decay closeness formulations.

Harary Centrality

Harary centrality sums inverse squared shortest-path distances.

\[ C_{\mathrm{Har}}(v) = \sum_{u \ne v} \frac{1}{d(v, u)^2} \]

Meaning. A high value indicates many nearby reachable nodes, with distant nodes strongly down-weighted.

centrality_harary(student_interactions)

Equivalence reference. Based on the Harary distance family.

Average Distance Centrality

Average distance centrality summarises mean shortest-path distance from a node (the focal node’s \(d = 0\) term is included; the centiserve convention divides by \(n + 1\)).

\[ \bar{d}(v) = \frac{\sum_{u \in V} d(v, u)}{n + 1} \]

Meaning. Lower values indicate shorter average graph distance. The direction of interpretation should be checked because it is a distance quantity.

centrality_average_distance(student_interactions)

Equivalence reference. Validated against the corresponding centiserve average-distance measure.

Barycenter Centrality

Barycenter centrality is the inverse of the sum of shortest-path distances.

\[ C_{\mathrm{Bary}}(v) = \frac{1}{\sum_{u \ne v} d(v, u)} \]

Meaning. A high value indicates a small total distance to other reachable nodes.

centrality_barycenter(student_interactions)

Equivalence reference. Validated against centiserve::barycenter().

Wiener Centrality

Wiener centrality records total shortest-path distance from a node (unreachable pairs contribute zero).

\[ W(v) = \sum_{u \ne v} d(v, u) \]

Meaning. Lower values indicate shorter total distance. It is better read as distance burden than influence.

centrality_wiener(student_interactions)

Equivalence reference. Based on the Wiener index distance formulation.

Lin Centrality

Lin centrality combines reachable nodes and total distance to those nodes.

\[ C_{\mathrm{Lin}}(v) = \frac{|R(v)|^2}{\sum_{u \in R(v)} d(v, u)}, \quad R(v) = \{ u \ne v : d(v,u) < \infty \} \]

Meaning. A high value indicates many reachable nodes through relatively short paths.

centrality_lin(student_interactions)

Equivalence reference. Validated against centiserve::lincent().

Decay Centrality

Decay centrality sums reach discounted by distance (the focal node contributes \(1\)).

\[ C_{\delta}(v) = \sum_{u \in V} \delta^{\,d(v, u)}, \quad \delta = 0.5 \]

Meaning. A high value indicates access to many nodes, especially nearby nodes. The value depends on the decay parameter.

centrality_decay(student_interactions)

Equivalence reference. Validated against centiserve::decay().

Radiality Centrality

Radiality compares node distances to the graph diameter.

\[ \mathrm{Rad}(v) = \frac{\sum_{u \ne v} \bigl( \Delta + 1 - d(v, u) \bigr)}{n - 1} \]

Meaning. A high value indicates short distances to others relative to overall graph diameter.

centrality_radiality(student_interactions)

Equivalence reference. Validated against centiserve::radiality().

Gil-Schmidt Centrality

Gil-Schmidt centrality sums reciprocal distances and normalises by graph size.

\[ C_{GS}(v) = \frac{1}{n - 1} \sum_{u \ne v} \frac{1}{d(v, u)} \]

Meaning. A high value indicates broad reciprocal-distance access to the network.

centrality_gilschmidt(student_interactions)

Equivalence reference. Validated against sna Gil-Schmidt style centrality.

Integration Centrality

Integration centrality is a distance-based measure of how integrated a node is within the graph.

\[ \mathrm{Int}(v) = \sum_{u \ne v} \left( 1 - \frac{d(v, u) - 1}{d_{\max}} \right), \quad d(v,u) = d_{\max} + 1 \text{ if unreachable} \]

Meaning. A high value indicates broad distance-based access to the network.

centrality_integration(student_interactions)

Equivalence reference. Implemented as a native distance-based centrality in cograph.

Eccentricity Centrality

Eccentricity is the maximum shortest-path distance from a node to any reachable node.

\[ \varepsilon(v) = \max_{u \in V} d(v, u) \]

Meaning. Lower eccentricity generally indicates smaller worst-case distance. A high value means at least one reachable node is far away.

centrality_eccentricity(student_interactions)

Equivalence reference. Equivalent to igraph::eccentricity().

Ineccentricity Centrality

Ineccentricity computes eccentricity over incoming directed paths.

\[ \varepsilon^{\mathrm{in}}(v) = \max_{u \in V} d(u, v) \]

Meaning. It summarises the largest directed distance from other nodes into the focal node.

centrality_ineccentricity(student_interactions)

Equivalence reference. Equivalent to igraph::eccentricity(mode = "in").

Outeccentricity Centrality

Outeccentricity computes eccentricity over outgoing directed paths.

\[ \varepsilon^{\mathrm{out}}(v) = \max_{u \in V} d(v, u) \]

Meaning. It summarises the largest directed distance from the focal node to reachable others.

centrality_outeccentricity(student_interactions)

Equivalence reference. Equivalent to igraph::eccentricity(mode = "out").

Closeness Vitality

Closeness vitality measures how total network distance changes when a node is removed, using the Wiener index \(W(G) = \sum_{s, t} d(s, t)\).

\[ \mathrm{CV}(v) = W(G) - W(G \setminus v) \]

Meaning. A high value indicates that removing the node substantially changes shortest-path structure.

centrality_closeness_vitality(student_interactions)

Equivalence reference. Corresponds to NetworkX closeness vitality conventions.

Entropy Centrality

Entropy centrality measures the change in graph entropy associated with a node, from the distribution of finite shortest-path distances after the node is removed.

\[ H(v) = -\sum_{w} Y_w \log_2 Y_w, \quad Y_w = \frac{|\{ x : d(w, x) < \infty \}|}{\sum_{w'} |\{ x : d(w', x) < \infty \}|} \]

Meaning. A high value indicates strong contribution under the graph entropy definition being used.

centrality_entropy(student_interactions)

Equivalence reference. Validated against centiserve::entropy().

Centroid Centrality

Centroid centrality compares distance dominance between pairs of nodes.

\[ f(v, u) = \gamma(v, u) - \gamma(u, v), \quad \gamma(v, u) = |\{ w : d(v, w) < d(u, w) \}|, \quad C_{\mathrm{cen}}(v) = \min_{u \ne v} f(v, u) \]

Meaning. A high value indicates a favourable position in pairwise distance comparisons.

centrality_centroid(student_interactions)

Equivalence reference. Implemented natively because some reference implementations have known edge-case issues.

Betweenness Centrality

Betweenness centrality measures how often a node lies on shortest paths between other pairs of nodes.

\[ C_B(v) = \sum_{s \ne v \ne t} \frac{\sigma_{st}(v)}{\sigma_{st}} \]

Meaning. A high value is consistent with a bridge or brokerage position under the shortest-path model. It is not direct evidence of intentional mediation.

centrality_betweenness(student_interactions)

Equivalence reference. Equivalent to igraph::betweenness() under matching directedness, weights, and normalization conventions.

Stress Centrality

Stress centrality counts shortest paths passing through a node without fractional normalization across tied paths.

\[ C_S(v) = \sum_{s \ne v \ne t} \sigma_{st}(v) \]

Meaning. A high value indicates that many shortest paths include the node.

centrality_stress(student_interactions)

Equivalence reference. Validated against sna::stresscent().

Load Centrality

Load centrality distributes shortest-path load across alternative shortest routes: at each branch point a unit packet is split evenly among next hops on shortest paths.

\[ C_L(v) = \sum_{s \ne v \ne t} \mathrm{load}_{st}(v) \]

Meaning. A high value indicates that a node carries a large share of geodesic routing load.

centrality_load(student_interactions)

Equivalence reference. Validated against sna::loadcent().

Bottleneck Centrality

Bottleneck centrality counts cases where a node is critical in shortest-path tree structures.

\[ \mathrm{BN}(v) = \sum_{s \in V} p_s(v), \quad p_s(v) = 1 \text{ if } v \text{ carries} > \tfrac{n}{4} \text{ of the shortest-path tree } T_s \]

Meaning. A high value indicates frequent bottleneck position in local shortest-path trees.

centrality_bottleneck(student_interactions)

Equivalence reference. Validated against centiserve::bottleneck().

Bridging Centrality

Bridging centrality combines betweenness with a bridging coefficient.

\[ \mathrm{Br}(v) = C_B(v) \cdot \beta(v), \quad \beta(v) = \frac{1/k_v}{\sum_{u \in N(v)} 1/k_u} \]

Meaning. A high value indicates a possible bridge position between locally distinct areas.

centrality_bridging(student_interactions)

Equivalence reference. Based on bridging centrality formulations by Hwang and colleagues.

Local Bridging Centrality

Local bridging centrality measures bridge-like position using local neighbourhood information.

\[ \mathrm{LBr}(v) = \frac{1}{k_v} \cdot \beta(v), \quad \beta(v) = \frac{1/k_v}{\sum_{u \in N(v)} 1/k_u} \]

Meaning. A high value indicates that the node connects neighbours that are not strongly connected through local alternatives.

centrality_local_bridging(student_interactions)

Equivalence reference. Based on local bridging coefficient formulations.

Percolation Centrality

Percolation centrality weights shortest-path brokerage by node states in a percolation process.

\[ \mathrm{PC}(v) = \frac{1}{n - 2} \sum_{s \ne v \ne t} \frac{\sigma_{st}(v)}{\sigma_{st}} \cdot \frac{x_s}{\sum_{i \ne v} x_i} \]

Meaning. A high value indicates state-dependent path importance under the supplied state vector \(x\).

centrality_percolation(student_interactions)

Equivalence reference. Based on Piraveenan, Prokopenko, and Hossain’s percolation centrality; equivalent to betweenness-like behavior when states are equal.

Flow Betweenness Centrality

Flow betweenness measures brokerage using maximum flow rather than only shortest paths.

\[ C_{FB}(v) = \sum_{s \ne v \ne t} f_{st}(v), \quad f_{st}(v) = \text{flow through } v \text{ in a max } s\text{-}t \text{ flow} \]

Meaning. A high value indicates importance for potential flow capacity between other nodes under the graph model.

centrality_flow_betweenness(student_interactions)

Equivalence reference. Validated against sna::flowbet().

Current-Flow Betweenness Centrality

Current-flow betweenness measures how much electrical current between node pairs passes through a node.

\[ C_{CFB}(v) = \frac{1}{(n - 1)(n - 2)} \sum_{s \ne t} \tau_{st}(v), \quad \tau_{st}(v) = \text{current through } v \]

Meaning. A high value indicates an intermediary position under an all-path current-flow model.

centrality_current_flow_betweenness(student_interactions)

Equivalence reference. Corresponds to NetworkX current-flow betweenness centrality.

Current-Flow Closeness Centrality

Current-flow closeness uses electrical-network (effective-resistance) distances rather than only shortest paths.

\[ C_{CFC}(v) = \frac{n - 1}{\sum_{u \ne v} R_{vu}}, \quad R_{vu} = \text{effective resistance between } v \text{ and } u \]

Meaning. A high value indicates closeness under an all-path flow model. It generally requires connected graphs.

centrality_current_flow_closeness(student_interactions)

Equivalence reference. Corresponds to NetworkX current-flow closeness centrality.

Eigenvector Centrality

Eigenvector centrality gives high scores to nodes connected to other high-scored nodes.

\[ A\,\mathbf{x} = \lambda_{\max}\,\mathbf{x}, \quad C_E(v) = x_v \]

Meaning. A high value indicates embeddedness in a central neighbourhood. It should not be interpreted as influence unless the relation supports that interpretation.

centrality_eigenvector(student_interactions)

Equivalence reference. Equivalent to igraph::eigen_centrality().

PageRank Centrality

PageRank is a damped random-walk centrality. Nodes score highly when a random walker reaches them often.

\[ \mathrm{PR}(v) = \frac{1 - \alpha}{n} + \alpha \sum_{u \,:\, u \to v} \frac{\mathrm{PR}(u)}{k_u^{\mathrm{out}}}, \quad \alpha = 0.85 \]

Meaning. A high value indicates random-walk prominence under the chosen damping, direction, and weight conventions.

centrality_pagerank(student_interactions)

Equivalence reference. Equivalent to igraph::page_rank() under matching parameters; original method by Page, Brin, Motwani, and Winograd.

Authority Centrality

Authority centrality is part of HITS. Authorities are nodes pointed to by good hubs.

\[ \mathbf{a} = A^{\top} \mathbf{h}, \quad \mathbf{h} = A\,\mathbf{a} \;\Rightarrow\; \mathbf{a} = \text{principal eigenvector of } A^{\top} A \]

Meaning. A high value indicates incoming support from nodes that point to good authorities.

centrality_authority(student_interactions)

Equivalence reference. Equivalent to igraph::authority_score(); based on Kleinberg’s HITS algorithm.

Hub Centrality

Hub centrality is the HITS counterpart to authority. Hubs point to good authorities.

\[ \mathbf{h} = \text{principal eigenvector of } A\,A^{\top} \]

Meaning. A high value indicates outgoing ties to high-authority nodes.

centrality_hub(student_interactions)

Equivalence reference. Equivalent to igraph::hub_score(); based on Kleinberg’s HITS algorithm.

SALSA Centrality

SALSA is a stochastic link-analysis method related to HITS for directed graphs.

\[ \mathbf{a} = \text{principal eigenvector of } A_c^{\top} A_r, \quad A_r, A_c = \text{row- and column-normalised } A \]

Meaning. A high value indicates directed authority under the SALSA random-walk model.

centrality_salsa(student_interactions)

Equivalence reference. Validated against centiserve::salsa() where applicable.

LeaderRank Centrality

LeaderRank is a PageRank-like directed ranking method that adds a ground node \(g\) linked both ways to every node; the stationary random-walk mass on \(g\) is then redistributed equally.

\[ \mathbf{s}^{*} = \text{stationary distribution of the walk on } G \cup \{g\}, \quad C(v) = s^{*}_v + \tfrac{1}{n} s^{*}_g \]

Meaning. A high value indicates directed prestige under the LeaderRank model.

centrality_leaderrank(student_interactions)

Equivalence reference. Validated against centiserve::leaderrank() where applicable.

Alpha Centrality

Alpha centrality is an eigenvector-like measure that includes exogenous input.

\[ \mathbf{x} = (I - \alpha A^{\top})^{-1} \mathbf{e} \]

Meaning. A high value indicates recursive prominence under the chosen attenuation and exogenous assumptions.

centrality_alpha(student_interactions)

Equivalence reference. Equivalent to igraph::alpha_centrality() under matching parameters.

Bonacich Power Centrality

Bonacich power centrality scores nodes using the centrality of their neighbours and a parameter \(\beta\) controlling dependence.

\[ \mathbf{c}(\alpha, \beta) = \alpha (I - \beta A)^{-1} A\,\mathbf{1} \]

Meaning. A high value indicates a favourable recursive position under the selected Bonacich parameterization.

centrality_power(student_interactions)

Equivalence reference. Equivalent to igraph::power_centrality() under matching parameters.

Katz Centrality

Katz centrality counts walks from all nodes to a focal node, attenuating longer walks.

\[ \mathbf{x} = (I - \alpha A^{\top})^{-1} \mathbf{1}, \quad \alpha = 0.1 \]

Meaning. A high value indicates that many short and longer walks reach the node. The attenuation parameter must be valid for the graph.

centrality_katz(student_interactions)

Equivalence reference. Validated against igraph::alpha_centrality(exo = 1) and NetworkX Katz centrality conventions.

Hubbell Centrality

Hubbell centrality is an input-output centrality where status is recursively reinforced through ties.

\[ \mathbf{x} = (I - w W)^{-1} \mathbf{1}, \quad w = 0.5,\; W = \text{weighted adjacency} \]

Meaning. A high value indicates recursive prominence under the chosen weight factor. Some parameter settings are not solvable.

centrality_hubbell(student_interactions)

Equivalence reference. Based on Hubbell’s input-output centrality.

Subgraph Centrality

Subgraph centrality measures participation in closed walks, with shorter closed walks weighted more strongly.

\[ \mathrm{SC}(v) = \sum_{k=0}^{\infty} \frac{(A^k)_{vv}}{k!} = (e^{A})_{vv} \]

Meaning. A high value indicates embeddedness in many closed walk structures.

centrality_subgraph(student_interactions)

Equivalence reference. Equivalent to igraph::subgraph_centrality(); based on Estrada’s formulation.

Laplacian Centrality

Laplacian centrality measures a node’s contribution to the graph’s Laplacian energy.

\[ C_L(v) = \frac{E_L(G) - E_L(G \setminus v)}{E_L(G)}, \quad E_L(G) = \sum_i \mu_i^2 \;\; (\mu_i = \text{Laplacian eigenvalues}) \]

Meaning. A high value indicates a large local structural contribution under the Laplacian energy definition.

centrality_laplacian(student_interactions)

Equivalence reference. Matches NetworkX and centiserve::laplacian() conventions.

Communicability Centrality

Communicability centrality uses the matrix exponential to summarise walk-based communication potential.

\[ C_{\mathrm{Comm}}(v) = \sum_{u \in V} (e^{A})_{vu} \]

Meaning. A high value indicates many walk-based routes to other nodes, with shorter walks weighted more strongly.

centrality_communicability(student_interactions)

Equivalence reference. Based on Estrada communicability.

Communicability Betweenness Centrality

Communicability betweenness measures walk-based communicability between other pairs through a node.

\[ C_{CB}(v) = \frac{1}{(n-1)(n-2)} \sum_{s \ne t \ne v} \frac{G_{st} - G_{st}^{(v)}}{G_{st}}, \quad G = e^{A} \]

Meaning. A high value indicates walk-based intermediary position, not limited to shortest paths. \(G^{(v)}\) is computed with \(v\) removed.

centrality_communicability_betweenness(student_interactions)

Equivalence reference. Based on Estrada, Higham, and Hatano’s communicability betweenness.

Random Walk Centrality

Random walk centrality uses expected random-walk access times rather than shortest-path distances.

\[ C_{RW}(v) = \frac{1}{\sum_{u \ne v} \tilde{m}_{vu}}, \quad \tilde{m}_{vu} = \tfrac{1}{2}\bigl( m_{vu} + m_{uv} \bigr) \]

Meaning. A high value indicates that the node is reached efficiently under a random-walk process. \(m_{uv}\) is the mean first-passage time.

centrality_random_walk(student_interactions)

Equivalence reference. Corresponds to random-walk centrality based on mean first-passage or resistance-distance formulations.

Markov Centrality

Markov centrality is based on mean first-passage times in a Markov process on the graph.

\[ C_{\mathrm{Mk}}(v) = \frac{1}{\frac{1}{n} \sum_{u} m_{uv}}, \quad m_{uv} = \text{mean first-passage time } u \to v \]

Meaning. A high value indicates that the node is reached quickly on average under the Markov process.

centrality_markov(student_interactions)

Equivalence reference. Validated against centiserve::markovcent().

Second-Order Centrality

Second-order centrality summarises variability in random-walk return times.

\[ \mathrm{SO}(v) = \operatorname{sd}_{u}\bigl( m_{uv} \bigr) \]

Meaning. The statistic describes random-walk regularity rather than direct connectivity (lower is more central).

centrality(student_interactions, measures = "second_order")

Equivalence reference. Rank behavior is checked against NetworkX second-order centrality in package tests.

Information Centrality

Information centrality measures centrality through resistance-distance or information-flow ideas (Stephenson and Zelen).

\[ C_I(v) = \left( C_{vv} + \frac{T - 2 R_v}{n} \right)^{-1}, \quad C = (D - A + J)^{-1},\; T = \operatorname{tr} C,\; R_v = \textstyle\sum_j C_{vj} \]

Meaning. A high value indicates a central position under the information-flow model.

centrality_information(student_interactions)

Equivalence reference. Based on Stephenson and Zelen’s information centrality.

Nonbacktracking Centrality

Nonbacktracking centrality scores nodes using walks that do not immediately return along the edge just traversed (the Hashimoto matrix \(B\)).

\[ B_{(i \to j),\,(k \to l)} = \delta_{jk}\,(1 - \delta_{il}), \quad C(v) \propto \text{aggregated leading eigenvector of } B \]

Meaning. A high value indicates walk-based prominence after reducing immediate backtracking inflation.

centrality(student_interactions, measures = "nonbacktracking")

Equivalence reference. Validated against NetworkX-style nonbacktracking behavior in package tests.

Diffusion Centrality

Diffusion centrality estimates reach through a diffusion process over the graph.

\[ \mathbf{DC} = \sum_{t=1}^{T} (\lambda A)^{t}\,\mathbf{1}, \quad \lambda = 1 \]

Meaning. A high value indicates a structurally favourable position for spreading under the selected diffusion model.

centrality_diffusion(student_interactions)

Equivalence reference. Validated against centiserve::diffusion() where applicable and TNA-style diffusion conventions for transition networks.

Infection Centrality

Infection centrality estimates spreading potential through infection-style self-avoiding walks with attenuation.

\[ \mathrm{Inf}(v) = \sum_{d=1}^{L} \beta^{\,d+1} (1 - \mu)^{d}\, w_d(v), \quad \beta = 0.8,\; \mu = 0,\; L = 6 \]

Meaning. A high value indicates a favourable position under the assumed infection process, not observed contagion. \(w_d(v)\) counts length-\(d\) self-avoiding walks from \(v\).

centrality(student_interactions, measures = "infection")

Equivalence reference. Implemented according to infection centrality formulas used in the network centrality literature.

VoteRank Centrality

VoteRank is an iterative voting algorithm for ranking spreader candidates: each round the highest-voted node is selected, its voting ability is zeroed, and its neighbours’ voting abilities are reduced.

Meaning. A high value indicates early selection by the VoteRank rule. It is a heuristic model result, not observed spread.

centrality_voterank(student_interactions)

Equivalence reference. Corresponds to NetworkX VoteRank behavior.

Expected Influence 1-Step

Expected influence 1-step sums signed edge weights adjacent to a node.

\[ \mathrm{EI}_1(v) = \sum_{u \in V} W_{vu}, \quad W = \text{signed weight matrix} \]

Meaning. A high positive value indicates strong positive immediate signed connectivity. This is mainly meaningful for signed networks.

centrality_expected_influence_1(student_interactions)

Equivalence reference. Based on Robinaugh, Millner, and McNally’s expected influence measure.

Expected Influence 2-Step

Expected influence 2-step extends signed influence to one- and two-step paths.

\[ \mathrm{EI}_2(v) = \mathrm{EI}_1(v) + \sum_{u \in V} W_{vu}\,\mathrm{EI}_1(u) \]

Meaning. A high positive value indicates positive signed connectivity through direct and indirect paths. Interpretation requires a signed network.

centrality_expected_influence_2(student_interactions)

Equivalence reference. Based on Robinaugh, Millner, and McNally’s expected influence measure.

Spanning Tree Centrality

Spanning tree centrality summarises a node’s contribution across spanning-tree structures via the Laplacian pseudoinverse \(L^{+}\).

\[ \mathrm{ST}(v) = \frac{1}{L^{+}_{vv}}, \quad L^{+} = \text{Moore-Penrose pseudoinverse of } L \]

Meaning. A high value indicates structural participation across many tree-like ways of connecting the graph.

centrality(student_interactions, measures = "spanning_tree")

Equivalence reference. Based on matrix-tree theorem centrality formulations.

Transitivity Centrality

Transitivity measures local clustering: whether a node’s neighbours are connected to each other.

\[ c(v) = \frac{2\, t_v}{k_v (k_v - 1)}, \quad t_v = \text{number of triangles through } v \]

Meaning. A high value indicates locally closed neighbourhoods. This can represent cohesion or redundancy, depending on the relation.

centrality_transitivity(student_interactions)

Equivalence reference. Equivalent to igraph::transitivity() for standard types; cograph also supports weighted variants.

Constraint Centrality

Constraint measures how redundant a node’s contacts are, following Burt’s structural holes framework.

\[ C(v) = \sum_{u \in N(v)} \left( p_{vu} + \sum_{q \ne v, u} p_{vq}\, p_{qu} \right)^2, \quad p_{vu} = \text{proportional tie strength} \]

Meaning. A high value indicates a locally constrained ego network. Lower constraint may indicate structural holes, depending on theory and relation type.

centrality_constraint(student_interactions)

Equivalence reference. Equivalent to igraph::constraint().

Effective Size Centrality

Effective size estimates the number of nonredundant contacts in a node’s ego network (Burt).

\[ \mathrm{ES}(v) = k_v - \frac{1}{k_v} \sum_{u \in N(v)} |N(v) \cap N(u)| \]

Meaning. A high value indicates relatively nonoverlapping contacts.

centrality_effective_size(student_interactions)

Equivalence reference. Validated against influenceR::effective_size().

Topological Coefficient Centrality

Topological coefficient centrality measures shared-neighbour overlap.

\[ T(v) = \frac{\operatorname{avg}_{u}\, J(v, u)}{k_v}, \quad J(v, u) = \text{number of neighbours shared by } v \text{ and } u \]

Meaning. A high value indicates neighbourhood overlap or topological similarity.

centrality_topological_coefficient(student_interactions)

Equivalence reference. Validated against centiserve::topocoefficient().

Diversity Centrality

Diversity centrality measures entropy in the distribution of edge weights around a node.

\[ \mathrm{Div}(v) = \frac{-\sum_{u \in N(v)} p_{vu} \log_2 p_{vu}}{\log_2 k_v}, \quad p_{vu} = \frac{w_{vu}}{\sum_{u'} w_{vu'}} \]

Meaning. A high value indicates that weighted ties are relatively evenly distributed rather than concentrated.

centrality_diversity(student_interactions)

Equivalence reference. Validated against igraph::diversity().

Cross-Clique Centrality

Cross-clique centrality counts how many cliques contain a node.

\[ X(v) = |\{\, Q \in \mathcal{Q}(G) : v \in Q \,\}|, \quad \mathcal{Q}(G) = \text{set of cliques} \]

Meaning. A high value indicates participation in many fully connected local groups.

centrality_cross_clique(student_interactions)

Equivalence reference. Validated against centiserve::crossclique().

Coreness Centrality

Coreness assigns nodes to k-core shells.

\[ C_{\mathrm{core}}(v) = \max \{\, k : v \in (k\text{-core of } G) \,\} \]

Meaning. A high value indicates membership in a dense core. It does not imply brokerage or short paths to all nodes.

centrality_coreness(student_interactions)

Equivalence reference. Equivalent to igraph::coreness().

Onion Centrality

Onion centrality assigns nodes to layers from onion decomposition, a fine-grained extension of k-core decomposition.

\[ \text{layer}(v) = \text{iteration index at which } v \text{ is peeled by the onion decomposition} \]

Meaning. A high layer indicates that the node remains until later stages of the peeling process.

centrality(student_interactions, measures = "onion")

Equivalence reference. Validated against NetworkX onion layer behavior in package tests.

K-Reach Centrality

K-reach centrality counts nodes reachable within path length \(k\).

\[ C_{kR}(v) = |\{\, u : 0 < d(v, u) \le k \,\}|, \quad k = 3 \]

Meaning. A high value indicates broad reach within a fixed local radius. Results depend on the chosen \(k\).

centrality_kreach(student_interactions)

Equivalence reference. Validated against k-path reach conventions.

Prestige Domain Centrality

Prestige domain centrality counts how many nodes can reach a focal node in a directed graph.

\[ P(v) = |\{\, u \ne v : u \rightsquigarrow v \,\}| \]

Meaning. A high value indicates a large incoming reach domain.

centrality_prestige_domain(student_interactions)

Equivalence reference. Based on Wasserman-Faust prestige concepts and sna-style prestige measures.

Prestige Domain Proximity Centrality

Prestige domain proximity measures how close nodes in the prestige domain are to the focal node.

\[ C(v) = \frac{|R^{-}(v)|^2}{(n - 1) \sum_{u \in R^{-}(v)} d(u, v)}, \quad R^{-}(v) = \{ u \ne v : u \rightsquigarrow v \} \]

Meaning. A high value indicates that many predecessors can reach the node through short directed paths.

centrality_prestige_domain_proximity(student_interactions)

Equivalence reference. Based on Wasserman-Faust prestige proximity concepts.

Local Reaching Centrality

Local reaching centrality measures how much of the network is reachable from a node through directed paths.

\[ \mathrm{LRC}(v) = \frac{|\{\, u : v \rightsquigarrow u \,\}|}{n - 1} \]

Meaning. A high value indicates broad directed reach from the node.

centrality_reaching_local(student_interactions)

Equivalence reference. Matches networkx.local_reaching_centrality in package equivalence tests.

Pairwise Disconnectivity Centrality

Pairwise disconnectivity measures how directed reachability between pairs changes when a node is removed.

\[ \mathrm{Dis}(v) = \frac{P(G) - P(G \setminus v)}{P(G)}, \quad P(G) = |\{ (s, t) : s \rightsquigarrow t \}| \]

Meaning. A high value indicates structural importance for preserving directed reachability.

centrality_pairwisedis(student_interactions)

Equivalence reference. Validated against reference implementations in package tests.

Trophic Level Centrality

Trophic level estimates hierarchical position in a directed flow network.

\[ (I - W)\,\mathbf{s} = \mathbf{1}, \quad W_{ji} = \frac{a_{ij}}{k_j^{\mathrm{in}}}, \quad \text{basal nodes have level } 1 \]

Meaning. A higher value indicates a higher position in the inferred directed hierarchy. This is appropriate only for flow-like relations.

centrality(student_interactions, measures = "trophic_level")

Equivalence reference. Checked against NetworkX trophic-level behavior in package tests.

The measures below need a community membership vector. Build one with detect_communities() (walktrap works on the directed student_interactions graph), then pass it as membership =:

comm   <- detect_communities(student_interactions, method = "walktrap")
groups <- setNames(comm$community, comm$node)
centrality_participation(student_interactions, membership = groups)
#>        Ac        Ad        Fi        Ik        Vx        Rt        Km        Gj 
#> 0.6078972 0.5450000 0.4965278 0.5612245 0.6213018 0.5150000 0.4628099 0.5540166 
#>        Bd        Ce        Oq        Ya        Mo        Hj        Tv        Eg 
#> 0.4861111 0.4600000 0.4897959 0.4733728 0.5416667 0.6250000 0.5400000 0.5400000 
#>        Pr        Qs        Xz        Np        Dg        Hk        Wy        Jl 
#> 0.4297521 0.5244444 0.4897959 0.4687500 0.5562130 0.6015625 0.6446281 0.4177778 
#>        Fh        Zb        Eh        Be        Df        Cf        Su        Ln 
#> 0.5396825 0.6938776 0.6938776 0.6020408 0.3750000 0.6805556 0.4444444 0.4081633 
#>        Gi        Uw 
#> 0.6666667 0.6250000

Participation Coefficient

Participation coefficient measures how evenly a node’s ties are distributed across communities.

\[ P(v) = 1 - \sum_{m} \left( \frac{k_v^{(m)}}{k_v} \right)^2, \quad k_v^{(m)} = \text{ties from } v \text{ to module } m \]

Meaning. A high value indicates ties spread across multiple communities. It requires a meaningful membership vector.

centrality_participation(student_interactions, membership = groups)

Equivalence reference. Validated against brainGraph participation coefficient conventions.

Within-Module Z Centrality

Within-module z centrality standardises a node’s within-community degree against other nodes in the same community.

\[ z(v) = \frac{k_v^{(m_v)} - \mu_{m_v}}{\sigma_{m_v}}, \quad m_v = \text{community of } v \]

Meaning. A high value indicates unusually high within-module connectivity.

centrality_within_module_z(student_interactions, membership = groups)

Equivalence reference. Validated against brainGraph within-module z-score conventions.

Gateway Centrality

Gateway centrality measures inter-community brokerage weighted by centrality of communities or nodes involved.

\[ g(v) = 1 - \frac{1}{k_v^2} \sum_{s} k_{vs}^2\, \gamma_{vs}^2, \quad k_{vs} = \text{ties from } v \text{ to module } s \]

Meaning. A high value indicates a structurally prominent position between communities.

centrality_gateway(student_interactions, membership = groups)

Equivalence reference. Validated against brainGraph::gateway() conventions.

Brokerage Coordinator Centrality

Coordinator brokerage occurs when a node mediates between two nodes in its own group. It counts open directed two-paths \(a \to v \to c\) (with no direct \(a \to c\)).

\[ \mathrm{Coord}(v) = |\{\, a \to v \to c : g(a) = g(v) = g(c) \,\}| \]

Meaning. A high value indicates within-group mediation under the supplied membership vector.

centrality_brokerage_coordinator(student_interactions, membership = groups)

Equivalence reference. Based on Gould-Fernandez brokerage roles.

Brokerage Itinerant Centrality

Itinerant brokerage occurs when a node mediates between two nodes in another group.

\[ \mathrm{Itin}(v) = |\{\, a \to v \to c : g(a) = g(c) \ne g(v) \,\}| \]

Meaning. A high value indicates brokerage among members outside the node’s own group.

centrality_brokerage_itinerant(student_interactions, membership = groups)

Equivalence reference. Based on Gould-Fernandez brokerage roles.

Brokerage Representative Centrality

Representative brokerage occurs when a node mediates from its own group to another group.

\[ \mathrm{Rep}(v) = |\{\, a \to v \to c : g(a) = g(v) \ne g(c) \,\}| \]

Meaning. A high value indicates outward brokerage from the node’s group.

centrality_brokerage_representative(student_interactions, membership = groups)

Equivalence reference. Based on Gould-Fernandez brokerage roles.

Brokerage Gatekeeper Centrality

Gatekeeper brokerage occurs when a node mediates from another group into its own group.

\[ \mathrm{Gate}(v) = |\{\, a \to v \to c : g(a) \ne g(v) = g(c) \,\}| \]

Meaning. A high value indicates inward brokerage into the node’s group.

centrality_brokerage_gatekeeper(student_interactions, membership = groups)

Equivalence reference. Based on Gould-Fernandez brokerage roles.

Brokerage Liaison Centrality

Liaison brokerage occurs when a node mediates between two groups, neither of which is its own.

\[ \mathrm{Liai}(v) = |\{\, a \to v \to c : g(a),\, g(v),\, g(c) \text{ all distinct} \,\}| \]

Meaning. A high value indicates between-group brokerage outside the node’s own group.

centrality_brokerage_liaison(student_interactions, membership = groups)

Equivalence reference. Based on Gould-Fernandez brokerage roles.