Package 'cograph'

Description

Abbreviates labels to a maximum length, adding ellipsis if truncated.

Usage

abbrev_label(label, abbrev = NULL, n_labels = NULL)

label_abbrev(label, abbrev = NULL, n_labels = NULL)

Arguments

Value

Character vector of (possibly abbreviated) labels.

Examples

abbrev_label(colnames(regulation_net), abbrev = 4)

Description

Appends edges between existing nodes. An endpoint that is not a node of the network raises a cograph_bad_selection error.

Usage

add_edges(x, from, to, weight = 1, ..., keep_format = FALSE, directed = NULL)

Arguments

Value

A cograph_network with the new edges, or the input format when keep_format = TRUE. An edge that already exists has its weight replaced, and a cograph_edges_replaced warning says how many.

Note

When the igraph package is attached it masks this function with igraph::add_edges(), which takes an igraph object. Use cograph::add_edges() to be explicit.

See Also

remove_edges, add_nodes, bind_networks

Examples

add_edges(regulation_net, from = "Share", to = "Explore", weight = 0.5)

Description

Appends new nodes, identified by label, to a network. A label that already exists raises a cograph_bad_selection error.

Usage

add_nodes(x, labels, ..., keep_format = FALSE, directed = NULL)

Arguments

Value

A cograph_network with the new nodes appended (isolated until edges are added), or the input format when keep_format = TRUE.

See Also

remove_nodes, add_edges, mutate_nodes

Examples

add_nodes(regulation_net, labels = "Revise")

Description

Combines multiple network layers into a single network.

Usage

aggregate_layers(
  layers,
  method = c("sum", "mean", "max", "min", "union", "intersection"),
  weights = NULL
)

lagg(
  layers,
  method = c("sum", "mean", "max", "min", "union", "intersection"),
  weights = NULL
)

Arguments

Value

The aggregated adjacency matrix, with the dimnames of the first layer. A list with a single layer is aggregated the same way, so "union" and "intersection" binarize it and "sum" multiplies it by its layer weight.

Examples

layers <- list(forward = regulation_net, backward = t(regulation_net))
aggregate_layers(layers, method = "mean")

Description

Aggregates a vector of edge weights into a single value. The method names follow those of igraph's edge.attr.comb argument.

Usage

aggregate_weights(w, method = "sum", n_possible = NULL)

wagg(w, method = "sum", n_possible = NULL)

Arguments

Value

A single numeric value, or 0 when no non-zero, non-NA weight remains.

Examples

aggregate_weights(regulation_net, method = "mean")

Description

as_cograph() creates a cograph_network object from a matrix, an edge list, or a network object of another package. to_cograph() is an alias. The object is a named list that every cograph function accepts.

Usage

as_cograph(x, directed = NULL, simplify = FALSE, ...)

to_cograph(x, directed = NULL, ...)

## S3 method for class 'cograph_network'
as.data.frame(
  x,
  row.names = NULL,
  optional = FALSE,
  ...,
  what = c("edges", "nodes")
)

Arguments

Details

A cograph_network prints a short description of its nodes, edges and source, summary() reports counts and edge weight statistics, plot() plots it with sn_render, and as.data.frame() returns its edge or node table. The accessor and setter functions are documented in get_nodes.

Producer packages may attach plotting hints under meta$splot. The recognized fields are renderer (the cograph renderer to use), weight (the edge column or matrix plotted as weight), and defaults (a named list of renderer arguments). Entries in defaults are defaults only, and arguments passed to splot override them. renderer and weight define which view is plotted and are not overridden by plot arguments.

Value

as_cograph() and to_cograph() return a cograph_network object, a named list with components:

nodes

Data frame with id, label, and optional layout or metadata columns.

edges

Data frame with integer from and to columns (row numbers into nodes), weight, and extra columns such as session and time for tna input. Repeated rows of an edge list are kept as separate edges.

directed

Logical. Whether the network is directed.

weights

The n x n weight matrix for matrix and tna input, or NULL (for example for an edge list).

data

The original estimation data (sequence data, edge list), or NULL.

meta

Metadata list with source (input type), a tna entry for tna input (type, group name and group index) and, optionally, splot (rendering hints read by splot).

node_groups

Optional data frame of node groupings.

as.data.frame() returns a base data frame. For what = "edges" it has one row per edge with columns from and to (node labels), weight, and any extra edge columns the network carries, such as session or columns computed by mutate_edges. For what = "nodes" it has one row per node with the node table columns (id, label, layout coordinates and any custom columns). to_df returns only from, to and weight.

See Also

get_nodes, splot, to_df

Examples

net <- as_cograph(regulation_net)
as.data.frame(net, what = "edges")

Description

Converts an object to the mcml class, a representation of a multi-cluster network that does not depend on the tna package.

Usage

as_mcml(x, ...)

## S3 method for class 'cluster_summary'
as_mcml(x, ...)

## S3 method for class 'group_tna'
as_mcml(x, clusters = NULL, method = "sum", type = "tna", directed = TRUE, ...)

## S3 method for class 'mcml'
as_mcml(x, ...)

## Default S3 method:
as_mcml(x, ...)

Arguments

Value

An mcml object with components macro, clusters, cluster_members, and meta.

An mcml object.

An mcml object. When clusters is provided, macro$data contains the cluster assignments and macro$weights is NULL.

The input mcml object unchanged.

See Also

summarize_clusters, as_tna

Examples

clusters <- list(C1 = c("Explore", "Reflect", "Discuss"),
                 C2 = c("Plan", "Create", "Share"),
                 C3 = c("Monitor", "Adapt", "Synthesize", "Evaluate"))
as_mcml(csum(regulation_net, clusters = clusters, type = "tna"))

Description

Converts a cluster_summary or mcml object to tna models that can be used with the functions of the tna package. The result holds a macro (cluster-level) model and one model of the internal transitions of each cluster, as a flat group_tna object.

Usage

as_tna(x)

## S3 method for class 'cluster_summary'
as_tna(x)

## S3 method for class 'mcml'
as_tna(x)

## Default S3 method:
as_tna(x)

Arguments

Details

Requirements

The tna package must be installed. Without it, the function raises an error.

Excluded Clusters

A per-cluster tna cannot be created when:

• The cluster has only 1 node (no internal transitions possible)

• Some nodes in the cluster have no outgoing edges (row sums to 0)

These clusters are left out of the result with a warning of class cograph_cluster_dropped, which names each cluster and the nodes that have no transition within it. The macro (cluster-level) model still includes all clusters.

Value

A group_tna object, a flat named list of tna objects. The first element is named "macro" and holds the cluster-level transitions. The remaining elements are named by cluster and hold the internal transitions of each cluster.

macro

A tna object of cluster-level transitions, with weights (k x k transition matrix), inits (initial distribution) and labels (cluster names).

<cluster_name>

One tna object per cluster, with weights (n_i x n_i matrix), inits (initial distribution) and labels (node labels). A cluster that cannot become a tna model is left out with a warning (see Excluded Clusters).

A group_tna object (flat list of tna objects: macro + per-cluster).

A group_tna object (flat list of tna objects: macro + per-cluster).

A tna object constructed from the input.

See Also

csum to create the input object, plot_mcml for visualization without conversion, tna::tna for the underlying tna constructor

Examples

clusters <- list(C1 = c("Explore", "Reflect", "Discuss"),
                 C2 = c("Plan", "Create", "Share"),
                 C3 = c("Monitor", "Adapt", "Synthesize", "Evaluate"))
as_tna(csum(regulation_net, clusters = clusters, type = "tna"))

Description

Computes the degree assortativity coefficient, measuring the tendency of nodes to connect to other nodes with similar degree. Positive values indicate assortative mixing (high-degree nodes connect to high-degree nodes), negative values indicate disassortative mixing.

Usage

assortativity(x, directed = NULL, type = NULL, digits = NULL, ...)

Arguments

Details

The degree assortativity coefficient is defined as the Pearson correlation coefficient between the degrees of nodes at either end of each edge (Newman 2002):

$r = \frac{\sum_{jk} jk(e_{jk} - q_j q_k)}{\sigma_q^2}$

where $e_{jk}$ is the fraction of edges connecting degree-$j$ to degree-$k$ vertices, $q_k$ is the excess degree distribution, and $\sigma_q^2$ its variance.

Because the Pearson correlation is invariant to subtracting a constant, the implementation computes the correlation of the raw degrees at the two ends of each edge, counting every undirected edge in both orientations. This is numerically identical to the formula above. Degrees are unweighted counts of edges, so edge weights do not enter the coefficient.

For directed networks, the coefficient is the Pearson correlation between the source-end and target-end degrees over each edge in its stored orientation, with the degree mode at each end chosen by type (Foster et al. 2010).

The coefficient is NA when the network has no edges or when either degree vector has zero variance.

Value

An object of class "cograph_assortativity" with components:

coefficient

Numeric scalar: the assortativity coefficient in $[-1, 1]$.

type

Character: the degree type used.

directed

Logical: whether the network was treated as directed.

n_nodes

Numeric: number of nodes.

n_edges

Numeric: number of edges.

network

The original input network.

References

Newman, M.E.J. (2002). Assortative mixing in networks. Physical Review Letters, 89(20), 208701. doi:10.1103/PhysRevLett.89.208701

Foster, J.G., Foster, D.V., Grassberger, P., & Paczuski, M. (2010). Edge direction and the structure of networks. PNAS, 107(24), 10815-10820. doi:10.1073/pnas.0912671107

See Also

assortativity_attribute, centrality, network_summary

Examples

cograph::assortativity(regulation_net)

Description

Computes assortativity with respect to a node attribute, measuring the tendency of nodes to connect to others with similar attribute values. For categorical attributes, this computes the modularity-based nominal assortativity. For numeric attributes, this computes the Pearson correlation between attribute values at edge endpoints.

Usage

assortativity_attribute(x, values, directed = NULL, digits = NULL, ...)

homophily(x, values, directed = NULL, digits = NULL, ...)

Arguments

Details

For categorical (nominal) attributes, the coefficient is:

$r = \frac{\text{tr}(\mathbf{e}) - \|\mathbf{e}^2\|}{1 - \|\mathbf{e}^2\|}$

where $\mathbf{e}$ is the mixing matrix with $e_{ij}$ = fraction of edges connecting type $i$ to type $j$.

For numeric (scalar) attributes, the coefficient is the Pearson correlation between attribute values at edge endpoints (computed over both orientations of every edge when the network is undirected). Any non-numeric values vector (character or factor) is treated as nominal.

The coefficient is NA when the network has no edges, when a nominal attribute has a single category, or when either value vector has zero variance.

Value

An object of class "cograph_assortativity" with components:

coefficient

Numeric scalar: assortativity coefficient.

type

Character: "nominal" or "scalar".

directed

Logical: whether the network was treated as directed.

n_nodes

Numeric: number of nodes.

n_edges

Numeric: number of edges.

attribute_values

The attribute values used, named by node and in node order.

network

The original input network.

References

Newman, M.E.J. (2003). Mixing patterns in networks. Physical Review E, 67(2), 026126. doi:10.1103/PhysRevE.67.026126

See Also

assortativity, detect_communities

Examples

cograph::assortativity_attribute(regulation_net, values = rep(c("self", "social"), each = 5))

Description

Replaces every surviving weight with 1 and drops edges whose weight is at or below the threshold. The operation corresponds to sna::event2dichot() with an absolute threshold.

Usage

binarize(
  x,
  threshold = 0,
  absolute = TRUE,
  signed = FALSE,
  keep_isolates = TRUE,
  keep_format = FALSE,
  directed = NULL
)

Arguments

Value

A cograph_network whose weights are all 1 (or, when signed = TRUE, 1 for a positive edge and -1 for a negative one), or the input format when keep_format = TRUE. Nodes left without edges are kept and reported in a cograph_isolates_created warning, unless keep_isolates = FALSE.

References

Butts, C. T. (2008). Social network analysis with sna. Journal of Statistical Software, 24(6), 1–51.

See Also

threshold_edges, normalize_weights

Examples

binarize(regulation_net, threshold = 0.1)

Description

Aligns two networks on node labels and combines their edges.

Usage

bind_networks(
  x,
  y,
  method = c("union", "intersection", "difference"),
  weight = c("sum", "mean", "max", "min", "first"),
  keep_format = FALSE,
  directed = NULL
)

Arguments

Value

A cograph_network over the combined node set, or x's format when keep_format = TRUE. Nodes are ordered with x's first, then any node only y has.

See Also

add_edges, plot_difference

Examples

bind_networks(regulation_net, t(regulation_net))

Description

Computes centrality measures for nodes in a network and returns a tidy data frame. Accepts matrices, edge-list data frames, igraph objects, cograph_network, or tna objects.

Usage

centrality(
  x,
  type = c("basic", "extended", "all"),
  measures = NULL,
  include = NULL,
  mode = "all",
  normalized = FALSE,
  weighted = TRUE,
  directed = NULL,
  loops = TRUE,
  simplify = "sum",
  digits = NULL,
  sort_by = NULL,
  cutoff = -1,
  invert_weights = NULL,
  alpha = 1,
  damping = 0.85,
  personalized = NULL,
  transitivity_type = "local",
  isolates = "nan",
  lambda = 1,
  diffusion_method = NULL,
  k = 3,
  states = NULL,
  decay_parameter = 0.5,
  dmnc_epsilon = 1.7,
  membership = NULL,
  katz_alpha = 0.1,
  hubbell_weight = 0.5,
  shapley_k = 2,
  shapley_cutoff = 2,
  s_shell_a = 0.5,
  discount_p = 0.01,
  ncvote_theta = 0.5,
  comm_r = "max_intra",
  ld_radius = 2,
  enrenew_depth = 2,
  voterank_lambda = 0.1,
  contraction_rho = 5,
  wks_alpha = 1,
  wks_beta = 1,
  renewed_threshold = 2,
  kpath_k = 3,
  kpath_len = 3,
  epc_threshold = 0.5,
  epc_runs = 1000,
  epc_seed = NULL,
  betweenness_delta = 1,
  closeness_delta = 1,
  gravity_mass = "kshell",
  gravity_radius = 3,
  mdd_lambda = 0.7,
  volume_radius = 2,
  diffusion_q = 1,
  diffusion_steps = 3,
  ds_beta = 0.1,
  ds_mu = 1,
  ds_steps = 5,
  cda_alpha = 0.5,
  icc_alpha = 0.2,
  exogenous_base = "reverse_closeness",
  wlr_alpha = 1,
  alr_h_mode = "all",
  grc_gamma = 1,
  rwd_decay = 0.5,
  rwd_node_weights = NULL,
  linerank_aggregation = "probability",
  bridging_steps = 2,
  bridging_values = NULL,
  proximal_variant = "source",
  exf_alpha = 2,
  beta_direction = "positive",
  ninl_order = 3,
  ninl_radius = NULL,
  map_flow = "unrecorded",
  map_convention = "paper",
  sr_prior = 0,
  mcgm_radius = 2,
  mcgm_alpha = NULL,
  dkgm_radius = 2,
  nd_order = 2,
  nd_decay = 0.2,
  nd_mass = "degree",
  ira_mass = "coreness",
  ira_alpha = 1,
  ira_tol = 1e-06,
  ira_max_iter = 1000,
  iira_beta = 0.2,
  iira_steps = 50,
  hcc_delta = 0.5,
  lhc_radius = 2,
  tpr_alpha = 0.85,
  tpr_k = 0.85,
  tpr_decay = 1,
  tpr_tol = 1e-14,
  tpr_max_iter = 1000,
  rsp_beta = 0.01,
  rsp_cost = c("inverse", "weight"),
  re_indexes = c("degree", "closeness", "betweenness", "constraint"),
  re_negative = NULL,
  tna_network = NULL,
  psych_network = NULL,
  ...
)

Arguments

Details

The following centrality measures are available:

degree

Count of edges (supports mode: in/out/all)

strength

Weighted degree (supports mode: in/out/all)

betweenness

Shortest path centrality

closeness

Inverse distance centrality (supports mode: in/out/all)

eigenvector

Influence-based centrality

pagerank

Random walk centrality (supports damping and personalization)

authority

HITS authority score

hub

HITS hub score

eccentricity

Maximum distance to other nodes (supports mode). Distances use the raw edge weights whatever invert_weights says, and hop counts with weighted = FALSE.

coreness

K-core membership (supports mode: in/out/all)

constraint

Burt's constraint (structural holes)

transitivity

Local clustering coefficient (supports multiple types)

harmonic

Harmonic centrality, the sum of inverse distances. It stays finite on disconnected graphs (supports mode: in/out/all)

diffusion

Diffusion degree centrality. With diffusion_method = "kandhway_kuri" it is the scaled degree of the node plus the scaled degrees of its neighbors (supports mode: in/out/all, lambda scaling)

leverage

Leverage centrality. Influence over neighbors based on relative degree differences (supports mode: in/out/all)

kreach

K-reach centrality. Number of nodes reachable within k steps (supports mode: in/out/all)

alpha

Alpha centrality. Influence through paths, attenuated by length, with a unit exogenous contribution at every node (supports mode: in/out/all)

power

Bonacich power centrality. Influence based on connections to other influential nodes (supports mode: in/out/all)

subgraph

Subgraph centrality. Participation in closed walks, with shorter walks weighted more heavily

laplacian

Laplacian centrality with the local formula of Qi et al. (2012). Matches NetworkX and centiserve::laplacian()

load

Load centrality. Share of shortest-path load passing through the node, with load split evenly at each branching point

current_flow_closeness

Information centrality. Closeness based on electrical current flow (requires a connected graph)

current_flow_betweenness

Random-walk betweenness. Betweenness based on electrical current flow (requires a connected graph)

voterank

VoteRank. Influential spreaders selected by iterative voting. The value is the election order rescaled so that the first elected node scores 1 and the last scores $1/n$

percolation

Percolation centrality. Shortest-path betweenness weighted by the node states in states. With equal states it is betweenness divided by $(n-1)(n-2)$

radiality

Radiality centrality (centiserve). Sum of (diam + 1 - d) normalized by n-1.

lin

Lin's centrality. Reachable nodes squared divided by sum of distances.

decay

Decay centrality. Sum of $\delta^d$ over all nodes, the node itself included, with $\delta$ = decay_parameter.

residual_closeness

Residual closeness. Sum of $1/2^d$ over all nodes, the node itself included.

dangalchev

Dangalchev closeness. Same values as residual_closeness.

generalized_closeness

Generalized closeness. Same formula as decay, using decay_parameter.

harary

Harary centrality. Sum of 1/d^2 for all reachable pairs.

average_distance

Average distance (centiserve). Sum of distances / (n+1).

barycenter

Barycenter centrality. 1 / sum of distances.

wiener

Wiener index. Total sum of shortest path distances from node.

closeness_vitality

Closeness vitality. Drop in Wiener index when node removed.

communicability

Total communicability. Row sums of matrix exponential.

communicability_betweenness

Communicability betweenness. Fraction of communicability through each node.

random_walk

Random walk centrality. Inverse sum of random walk distances (requires connected graph).

stress

Stress centrality. Number of shortest paths through node.

flow_betweenness

Flow betweenness. Max-flow based betweenness.

lobby

Lobby index (h-index of neighborhood).

entropy

Graph entropy centrality. Entropy change on node removal.

semilocal

Semi-local centrality. Triple-nested neighborhood sum.

clusterrank

ClusterRank. Clustering coefficient times neighbor degree sum.

bottleneck

Bottleneck centrality. Count of shortest path trees where node is critical.

centroid

Centroid value. Minimum f(v,i) across all nodes.

mnc

Maximum Neighborhood Component size.

dmnc

Density of Maximum Neighborhood Component.

topological_coefficient

Topological coefficient. Shared neighbor ratio.

bridging

Bridging centrality. Betweenness times bridging coefficient.

local_bridging

Local bridging. (1/degree) times bridging coefficient.

effective_size

Burt's effective size. Degree minus redundancy.

diversity

Diversity centrality. Shannon entropy of edge weight distribution.

cross_clique

Cross-clique connectivity. Count of cliques containing node.

markov

Markov centrality. Inverse mean first passage time (requires connected graph).

integration

Integration centrality. Distance-based influence.

expected

Expected centrality. Sum of neighbor degrees.

gilschmidt

Gil-Schmidt power index. Sum of 1/d normalized by n-1.

salsa

SALSA authority scores (directed graphs only).

leaderrank

LeaderRank. PageRank with ground node (directed graphs only).

participation

Participation coefficient. Diversity of inter-community connections (requires membership).

within_module_z

Within-module degree z-score. Intra-community connectivity (requires membership).

gateway

Gateway coefficient. Inter-community brokerage weighted by centrality (requires membership).

distance_entropy

Normalized Shannon entropy of a node's hop-distance profile. It is 1 when the distances are spread evenly and 0 when all lie at one distance.

local_dimension

Growth exponent of the ball around a node (slope of $\ln B_i(r)$ on $\ln r$). Lower values mark more influential nodes.

local_information_dimension

Entropy-weighted local dimension over boxes up to half the node's eccentricity. Higher values mark more influential nodes.

neighborhood_connectivity

Mean degree of a node's neighbors (average neighbor degree); isolates score 0.

modularity_vitality

Drop in modularity when the node is removed under a fixed partition. Positive values mark community hubs and negative values mark bridges (requires membership).

shapley_game1, shapley_game2, shapley_game3

Shapley value of the node in the coverage games of Michalak et al. (2013): one-hop coverage, shapley_k-neighbor coverage, and coverage within shapley_cutoff hops. Values sum to the node count.

access_information

Mean bits needed to reach every other node along shortest paths without a map. Low values mark well-connected nodes.

hide_information

Mean bits others need to find the node. High values mark hidden nodes.

rumor

Log rumor centrality on the node's BFS tree: log of the number of spreading orders that could start there.

community_hub_bridge

Community size times intra-community degree plus number of other communities touched times inter-community degree (requires membership).

entropy_variation_degree, entropy_variation_betweenness

Drop in the Shannon entropy of the degree (by mode) or betweenness distribution when the node is deleted; signed, nats.

s_shell

Shell index of the strength-based peeling with asymmetric topological link weights, exponent s_shell_a.

degree_discount, single_discount

Greedy seed-selection order under degree discounting (discount_p) or unit discounting, scored 1 for the first selected down to 1/n.

ncvoterank

VoteRank with voters weighted by normalized neighborhood coreness (ncvote_theta); election order scored like voterank.

community_based, comm_centrality, community_mediator

Links weighted by the size of the community they reach; Gupta's scaled intra/inter-degree combination (comm_r); base-2 entropy of the link distribution over communities times degree share (all require membership).

local_dimension_fixed, fuzzy_local_dimension, local_volume_dimension

Silva-Costa estimator at ld_radius; slope of the fuzzy ball, where higher values mark more influential nodes; slope of the degree volume, where lower values mark more important nodes.

wvoterank, enrenew, voterank_plus

Election orders of the weighted, entropy-based (enrenew_depth) and degree-weighted (voterank_lambda) VoteRank variants, scored like voterank.

node_contraction, node_contraction_improved

One minus the agglomeration ratio after contracting the node with its neighbors; the improved form adds the same score of its edges on the line graph (contraction_rho).

two_way_rw

Number of node pairs whose most likely two-way random-walk route passes through the node.

heatmap

Farness minus mean neighbor farness. Lower values mark more central nodes.

flow_coefficient

Share of neighbor pairs linked through the node but not directly.

local_entropy

$-\sum_{j \in N(i)} k_j \ln k_j$. Lower values mark more central nodes.

weighted_h_index

h-index over topological link weights $k_i k_j$ repeated $k_j$ times.

redundancy

Mean degree of the neighbors inside the ego network; degree minus effective size.

weighted_kshell

k-shell on $(k^\alpha s^\beta)^{1/(\alpha + \beta)}$ after Garas' weight normalization (wks_alpha, wks_beta).

renewed_coreness

k-core of the graph after removing links whose diffusion importance is below renewed_threshold.

geodesic_kpath

Number of shortest paths of length at most kpath_k starting at the node.

local_efficiency

Global efficiency of the subgraph induced on the node's neighbors, the node itself removed. This differs from igraph::local_efficiency(), which measures the distances between those neighbors through the rest of the network.

s_core

Largest strength threshold whose s-core still contains the node; the k-core number when weights are absent.

fragmentation

Distance-weighted fragmentation of the network after deleting the node. Higher means a more disruptive removal.

kpath

Number of simple paths of length at most kpath_len that the node lies on, endpoints included.

epc

Edge percolated component: mean size of the node's component over epc_runs bond-percolation realizations, as a share of the network. A Monte Carlo estimate; epc_seed makes it reproducible.

length_scaled_betweenness

Betweenness with each separated pair weighted by $1 / d(s,t)$.

delta_betweenness

Betweenness with the pair weight $(h(s,t) - 1)^{-\delta}$ (betweenness_delta), where $h(s,t)$ is the number of edges on a shortest path.

ego_betweenness

Betweenness inside the node's own ego network.

delta_closeness

$\sum_j d_{ij}^{-\delta} / (n-1)$ (closeness_delta).

truss, mdd

Node truss number (k-2 triangles convention) and mixed-degree shell threshold (mdd_lambda). Both use the simple undirected skeleton; see centrality_truss.

bridging_coefficient, godfather, support

Reciprocal-degree ratio, count of unconnected neighbor pairs, and count of triangle-supported relationships on the simple undirected skeleton.

volume

Sum of degrees in the closed volume_radius-hop neighborhood on the simple undirected skeleton.

mcc

Maximal clique centrality: sum of $(|C|-1)!$ over incident maximal cliques of size at least two. Costly; see centrality_mcc for isolate and precision conventions.

diffusion_centrality

Finite-horizon weighted outgoing walks: $\sum_{t=1}^{T}(qA)^t\mathbf{1}$, with diffusion_q and diffusion_steps. Distinct from diffusion degree.

dynamical_importance

Relative spectral-radius loss on vertex deletion, evaluated by repeated eigendecomposition. Costly; see centrality_dynamical_importance for zero-radius graphs.

dynamics_sensitive

Finite-time spreading score including ds_beta, ds_mu and ds_steps; uses the simple undirected skeleton.

malatya

Sum of focal-to-neighbor degree ratios on the simple undirected skeleton; the reciprocal of the bridging coefficient on nonisolated vertices.

resistance_curvature

One minus half the incident conductance times effective-resistance sum. Weighted, componentwise and costly; see centrality_resistance_curvature.

extended_coreness

Sum of neighbors' neighborhood coreness; equivalently the squared simple adjacency times core numbers.

dkgm

Gravity with the degree k-shell index as the mass at both ends, default radius two; see centrality_dkgm.

neighbor_distance

Benchmark centrality plus its decayed sums over non-backtracking walks of up to nd_order steps. See centrality_neighbor_distance.

ira

Steady state of a unit resource repeatedly reallocated to neighbors in proportion to their ira_mass; conserved, so the scores of a component sum to its size. Warns cograph_no_converge where no steady state exists. See centrality_ira.

iira

The same recursion with each share scaled by $1-(1-\beta)^{k_i}$ for the iira_beta spreading rate, run iira_steps times. Decays geometrically, so only the order is meaningful. See centrality_iira.

lnc

Local neighbor contribution: the cubed degree times the binomial own-contribution factor $(1-1/d_i)^{d_i-1}$ times the neighbors' degree sum over $n-1$. Parameter-free; raw scores depend on the whole graph's order. See centrality_lnc.

ked

KED method: the degree times one plus the normalized entropy of the neighbors' degrees times $\exp(K_i/N)$ for the neighbor-degree sum $K_i$ and the whole graph's order $N$. Parameter-free. See centrality_ked.

hcc

Hybrid characteristic centrality: the extended degree $\delta k_i+(1-\delta)\sum_{j\in N(i)}k_j$ over its maximum, plus the E-shell peeling round in which the node leaves over the number of rounds. Raw scores lie in $[0,2]$ and are not component-local. See centrality_hcc.

ehcc

Extended hybrid characteristic centrality: the closed-neighborhood sum of hcc, the focal node counted once. See centrality_ehcc.

lhc

Lhc index: the degree-and-triangle-share influence $C(v)=\sum_{u\in\Phi(v)}k_u(1+TP(u))/d^2(uv)$ over the ball of radius lhc_radius, summed over the open neighborhood. The triangle share is normalized by $TNTS=\sum_u NTS(u)$, three times the number of distinct triangles, and is written as zero on a triangle-free graph. Raw scores are not component-local. See centrality_lhc.

iec

Immediate effects centrality: the reciprocal mean length of the influence sequences that end at a node, $(n-1)/\sum_{i\neq j}m_{ij}$ for the mean first passage times $M=(I-Z+EZ_{dg})\mathrm{diag}(1/c)$ of the influence chain $W=A/\mathrm{rowSums}(A)$ built with $a_{ii}=1$. Direction-sensitive and costly (one eigenproblem and two dense solves). NA at every node when the chain is reducible or the graph has one node. Not the same measure as markov. See centrality_iec.

dil

Degree and importance of lines: the degree plus the share of each incident line's importance $I_e=(k_m-p-1)(k_n-p-1)/ (p/2+1)$ that the node's own degree claims, $k_i+\sum_{j\in\Gamma_i}I_{e_{ij}}(k_i-1)/(k_i+k_j-2)$, with $p$ the number of triangles on the line. Two-hop local and component-local; never below the node's degree. See centrality_dil.

trust_pagerank

Trust-PageRank: a damped PageRank whose split of a node's score among its neighbors is the column-stochastic trust-value $T(i,j)=(1-k)s(i,j)/\sum_{l\in N_j}s(j,l)+ k\,d_i/\sum_{l\in N_j}d_l$, with $s$ the fixed point of SimRank restricted to the lines of the graph. Scores sum to one when no node is isolated. NA at every node of a component that has lines but no triangle, where the similarity vanishes and the ratio is undefined. Costly (two fixed-point recursions over dense matrices). See centrality_trust_pagerank.

rsp_betweenness

Simple randomized shortest paths betweenness: the expected number of visits a node receives over the Boltzmann distribution on absorbing walks, summed over every ordered source-target pair. rsp_beta interpolates between the random-walk and shortest-path readings. Direction-sensitive, component-local, and costly (one dense inverse). See centrality_rsp_betweenness.

relative_entropy

Normalized geometric mean of several index distributions, the minimum-relative-entropy integration of re_indexes; sums to one. See centrality_relative_entropy.

mixed_gravity

Gravity with focal core-number and partner-degree masses, default radius three.

extended_mixed_gravity

Sum of immediate neighbors' raw mixed gravitational centralities.

extended_gravity

Sum of neighbors' raw k-shell gravity scores, with gravity_radius applied around each neighbor.

cda

Weighted degree and strength, adjusted by Barrat clustering, plus weighted neighbor contributions; uses cda_alpha.

improved_closeness

Closeness using distances divided by the number of shortest paths raised to icc_alpha.

exogenous

Contribution to all other nodes' base centrality, measured by deletion. Selects a base using exogenous_base.

global_structure

Exponential focal coreness times distance-discounted partner coreness (GSM).

hybrid_global_structure

Exponential degree-coreness influences with an adaptive distance exponent (H-GSM).

improved_global_structure

Exponential focal degree with partner degrees discounted by a global mean-degree distance exponent (IGSM).

weighted_leaderrank

Stationary scores with ground-node outgoing weights determined by original in-degree and wlr_alpha.

linerank

PageRank on the line graph, aggregated at endpoints; uses damping and linerank_aggregation.

expected_force

Entropy of onward boundary degrees over all two-event transmission sequences.

mcgm

Multi-characteristics gravity with degree, coreness and eigenvector masses; default radius two.

spectralrank

Outgoing Perron eigenvector with a unit-linked ground node; sr_prior supplies optional diagonal information.

controlrank

Smallest eigenvalue of each grounded symmetric row-Laplacian; see centrality_controlrank.

map_equation

Codelength saving on silencing a node, conditional on the supplied partition, flow model and coding convention.

ninl

Finite neighbor propagation of closed-neighborhood degree volume; uses ninl_order and ninl_radius.

beta_measure

BG power shared by successors among predecessors; beta_direction selects positive or negative orientation.

localized_bridging, extended_local_bridging

Betweenness in one-hop or two-hop ego networks times the original bridging coefficient.

modified_expected_force

Expected Force multiplied by log degree with the scaling parameter exf_alpha.

proximal_betweenness

First/last shortest-path intermediaries; uses proximal_variant on the directed unweighted skeleton.

x_degree

Counts four-edge nonbacktracking walks with each node at the middle, using original neighbor excess degrees.

coleman_theil

Concentration of dyadic Burt constraints across contacts; isolates zero and single-contact nodes one.

bridging_capital

Information-walk loss under single-entry deletion; uses bridging_steps and bridging_values.

random_walk_decay

Weighted sum of discounted first arrivals from random walks; uses rwd_decay and rwd_node_weights.

graph_regularization

Reciprocal diagonal of the inverse regularized weighted Laplacian, using grc_gamma.

adaptive_leaderrank

Stationary scores with destination weights determined by original H-indices using alr_h_mode.

Value

A base data.frame with one row per node, in the input's node order unless sort_by is given, and the columns:

• node: character, the node labels (the index as a string when the input carried no names)

• One numeric column per requested measure, with a mode suffix for the mode-aware measures (e.g., degree_in, closeness_all); see list_centralities for which measures carry a suffix. A measure that a tier supplied but that has no value on this input is an all-NA column.

Measures without a value on a given input

The community-aware measures warn and return an all-NA column when membership is missing, and the brokerage roles do the same on an undirected graph. "relative_entropy" has no value when one of its constituent indexes is zero at every node. Named in measures or include, it then raises a cograph_undefined_index error. Supplied by a tier, it gives a cograph_undefined_measure warning and an all-NA column, and the other measures of the tier are still computed. "flow_betweenness" requires the igraph package. Without igraph, naming it raises a cograph_needs_igraph error, and a tier gives the same warning and all-NA column.

Examples

centrality(regulation_net)

Description

Access and hide information (Rosvall et al. 2005; Sneppen, Trusina and Rosvall 2005) measure the number of bits a walker needs to follow a shortest path without a map. The search information from $i$ to $j$ sums over all shortest paths $p(i, j)$, with $k_i$ the degree of the source and $k_l - 1$ the choices left at each intermediate node:

$S(i \to j) = -\log_2 \sum_{p(i, j)} \frac{1}{k_i} \prod_{l \in p,\, l \ne i, j} \frac{1}{k_l - 1}.$

Access information $A_i$ averages $S(i \to j)$ over targets $j$, and hide information $H_i$ averages $S(j \to i)$ over sources.

Usage

centrality_access_information(x, ...)

centrality_hide_information(x, ...)

Arguments

Details

Distances are hop counts, so edge weights are ignored. On a directed network every step uses the out-degree. The average runs over the nodes a walker can reach, or be reached from, so values stay finite on a disconnected network and equal the source's $1/N$ average on a connected one. Hubs score high on access information and low on hide information. On a star with five leaves the hub has access information 1.93 bits and hide information 0, and each leaf has access information 1.33 bits. The Centrality Zoo states the star case in reverse, and the values above follow the formulas and the source papers.

Value

A named numeric vector with one score per node, in input node order, in bits.

References

Rosvall, M., Trusina, A., Minnhagen, P., & Sneppen, K. (2005). Networks and cities: An information perspective. Physical Review Letters, 94, 028701.

Sneppen, K., Trusina, A., & Rosvall, M. (2005). Hide-and-seek on complex networks. Europhysics Letters, 69(5), 853-859.

See Also

centrality_closeness, centrality.

Examples

centrality_access_information(regulation_net)
centrality_hide_information(regulation_net)

Description

Adaptive LeaderRank (Xu and Wang 2017) adds a ground node with H-index 1 that is linked in both directions to every node, and weights each arc from $j$ to $i$ by the H-index $h_i$ of its target. The H-index of a node is the largest $h$ such that at least $h$ of its neighbors have degree at least $h$. The score is the stationary resource of the random walk on the row-normalized weights, started with one unit on every node and none on the ground.

Usage

centrality_adaptive_leaderrank(x, alr_h_mode = "all", ...)

Arguments

Details

The random walk keeps the direction of the arcs, and an undirected edge counts as two opposite arcs. Edge weights, loops and parallel arcs are ignored, and mode has no effect. H-indices are computed once on the original network. With alr_h_mode = "all" they use the simple undirected skeleton, "out" uses the out-degrees of out-neighbors and "in" the in-degrees of in-neighbors. The returned scores omit the ground, so they sum to less than $N$. A node with H-index 0 scores 0, and when every H-index is 0 all scores are NaN with a cograph_undefined_measure warning.

Value

A named numeric vector with one score per node, in input node order.

References

Xu, S., & Wang, P. (2017). Identifying important nodes by adaptive LeaderRank. Physica A, 469, 654-664. doi:10.1016/j.physa.2016.11.034.

See Also

centrality_weighted_leaderrank, centrality_lobby, centrality.

Examples

centrality_adaptive_leaderrank(regulation_net)

Description

Alpha centrality (Bonacich and Lloyd 2001) gives every node an exogenous score of one plus the scores of the nodes that point to it:

$x = \alpha A^{T} x + e, \qquad x = (I - \alpha A^{T})^{-1} e.$

The attenuation is fixed at $\alpha = 1$ and $e$ is a vector of ones, the defaults of igraph::alpha_centrality().

Usage

centrality_alpha(x, mode = "all", ...)

Arguments

Details

$A$ holds the edge weights, or ones with weighted = FALSE, with the diagonal set to zero. The alpha argument of centrality is the weight-inversion exponent, which this measure does not read. The scores are positive when the spectral radius of $A$ is below one and can be negative otherwise. A singular system or a negative edge weight raises an error of class cograph_singular_system.

On a directed network mode = "in" sums over incoming ties as above and equals igraph::alpha_centrality(), mode = "out" uses $A$ in place of $A^{T}$ and so sums over outgoing ties, and mode = "all" (default) uses the symmetrized weights $A + A^{T}$. On an undirected network the three modes agree.

Value

A named numeric vector with one score per node, in input node order.

References

Bonacich, P., & Lloyd, P. (2001). Eigenvector-like measures of centrality for asymmetric relations. Social Networks, 23(3), 191-201. doi:10.1016/S0378-8733(01)00038-7.

See Also

centrality_katz, centrality_eigenvector, centrality.

Examples

centrality_alpha(regulation_net)

Description

The HITS algorithm (Kleinberg 1999) assigns each node an authority score from the hubs that point to it and a hub score from the authorities it points to:

$a = \lambda^{-1} A^{T} h, \qquad h = \lambda^{-1} A a .$

Authorities are the dominant eigenvector of $A^{T} A$ and hubs the dominant eigenvector of $A A^{T}$.

Usage

centrality_authority(x, ...)

centrality_hub(x, ...)

Arguments

Details

$A$ holds the edge weights, or ones with weighted = FALSE. Both scores are scaled to a maximum of one, so they lie between 0 and 1. On an undirected network both equal eigenvector centrality. A network without edges gives every node a score of one. centrality_hub() returns the hub scores.

Value

A named numeric vector with one score per node, in input node order.

References

Kleinberg, J. M. (1999). Authoritative sources in a hyperlinked environment. Journal of the ACM, 46(5), 604-632. doi:10.1145/324133.324140.

See Also

centrality_eigenvector, centrality_pagerank, centrality.

Examples

centrality_authority(regulation_net)
centrality_hub(regulation_net)

Description

Average distance is the sum of the shortest-path distances from a node to every node, divided by $n + 1$ as in the centiserve package:

$AD(v) = \frac{1}{n + 1} \sum_{w} d(v, w).$

Lower values mark more central nodes.

Usage

centrality_average_distance(x, mode = "all", ...)

Arguments

Details

Edge weights are read as path lengths. invert_weights = TRUE uses $1/w^\alpha$ as the length, and weighted = FALSE counts hops. mode sets the direction of the paths. A node that cannot reach every other node scores Inf, so on a disconnected network every score is Inf.

Value

A named numeric vector with one score per node, in input node order.

See Also

centrality_barycenter, centrality_closeness, centrality.

Examples

centrality_average_distance(regulation_net)

Description

Barycenter centrality is the reciprocal of the total shortest-path distance from a node to the nodes it reaches:

$BC(v) = \frac{1}{\sum_{w \ne v} d(v, w)}.$

Unreachable nodes are left out of the sum.

Usage

centrality_barycenter(x, mode = "all", ...)

Arguments

Details

Edge weights are read as path lengths. invert_weights = TRUE uses $1/w^\alpha$ as the length, and weighted = FALSE counts hops. mode sets the direction of the paths. The formula is the one centrality_closeness computes, and with the default weighted = TRUE the two agree on every node that reaches another node. A node that reaches no other node scores 0, where closeness returns NaN.

Value

A named numeric vector with one score per node, in input node order.

See Also

centrality_closeness, centrality_average_distance, centrality.

Examples

centrality_barycenter(regulation_net)

Description

The beta measure, or BG-index (van den Brink and Gilles 2000), gives node $i$ the sum, over its successors $j$, of one divided by the in-degree of $j$. Each node with predecessors thus shares one unit of domination power equally among them:

$\beta_i = \sum_{j : i \to j} \frac{1}{d^{in}_j}.$

The negative variant applies the measure to the reversed network (Boldi and Vigna 2014).

Usage

centrality_beta_measure(x, beta_direction = "positive", ...)

Arguments

Details

The measure is computed on the simple unweighted network with direction kept, so weights, loops and parallel edges are ignored. An undirected edge is a pair of reciprocal arcs, so on an undirected network both variants equal the sum of the reciprocal degrees of the neighbors. A node without successors has positive score zero, and a node without predecessors has negative score zero. The positive scores sum to the number of nodes with nonzero in-degree, and the negative scores sum to the number of nodes with nonzero out-degree.

Value

A named numeric vector with one score per node, in input node order.

References

van den Brink, R. and Gilles, R. P. (2000). Measuring domination in directed networks. Social Networks, 22, 141-157. doi:10.1016/S0378-8733(00)00019-8.

Boldi, P. and Vigna, S. (2014). Axioms for centrality. Internet Mathematics, 10, 222-262. doi:10.1080/15427951.2013.865686.

See Also

centrality_indegree, centrality_prestige_domain, centrality.

Examples

centrality_beta_measure(regulation_net)

Description

Betweenness centrality (Freeman 1977) sums, over pairs of other nodes, the fraction of shortest paths between them that pass through the node:

$B(v) = \sum_{s \ne v \ne t} \frac{\sigma_{st}(v)}{\sigma_{st}},$

where $\sigma_{st}$ is the number of shortest paths from $s$ to $t$ and $\sigma_{st}(v)$ the number of those through $v$.

Usage

centrality_betweenness(x, ...)

Arguments

Details

On a directed network the sum runs over ordered pairs along the edge direction, and on an undirected network over unordered pairs. Edge weights are read as path lengths, and invert_weights = TRUE uses $1/w^\alpha$ instead. weighted = FALSE uses hop counts. cutoff drops paths longer than the given length. normalized = TRUE divides the scores by their maximum.

Value

A named numeric vector with one score per node, in input node order.

References

Freeman, L. C. (1977). A set of measures of centrality based on betweenness. Sociometry, 40(1), 35-41. doi:10.2307/3033543.

See Also

centrality_load, centrality_stress, centrality.

Examples

centrality_betweenness(regulation_net)

Description

Bottleneck centrality (Przulj, Wigle and Jurisica 2004) counts the shortest-path trees in which a node is a bottleneck. For each source $s$, every shortest path from $s$ to every reachable node is enumerated, and a node $v \ne s$ scores one for that source when it lies on more than $n/4$ of these paths.

Usage

centrality_bottleneck(x, mode = "all", ...)

Arguments

Details

Distances are hop counts, so edge weights are ignored. mode sets the direction in which the paths leave the source. Paths that end at $v$ count toward its total. A network with one node gives that node a score of one.

Value

A named integer vector with one score per node, in input node order.

References

Przulj, N., Wigle, D. A., & Jurisica, I. (2004). Functional topology in a network of protein interactions. Bioinformatics, 20(3), 340-348. doi:10.1093/bioinformatics/btg415.

See Also

centrality_stress, centrality_betweenness, centrality.

Examples

centrality_bottleneck(regulation_net)

Description

Bridging centrality (Hwang et al. 2008) is the product of betweenness $B(v)$ and the bridging coefficient, which compares the inverse degree of a node with the inverse degrees of its neighbors:

$BrC(v) = B(v) \frac{1/k_v}{\sum_{u \in N(v)} 1/k_u}.$

High values mark nodes that lie on many shortest paths and connect densely linked regions.

Usage

centrality_bridging(x, ...)

Arguments

Details

The betweenness factor reads edge weights as path lengths and follows the edge direction of a directed network. weighted = FALSE uses hop counts, and invert_weights has no effect. The degrees are total degrees, and on a directed network a reciprocated neighbor enters the sum twice. An isolated node scores 0.

Value

A named numeric vector with one score per node, in input node order.

References

Hwang, W., Kim, T., Ramanathan, M., & Zhang, A. (2008). Bridging centrality: Graph mining from element level to group level. In Proceedings of the 14th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (pp. 336-344). doi:10.1145/1401890.1401934.

See Also

centrality_local_bridging, centrality_betweenness, centrality.

Examples

centrality_bridging(regulation_net)

Description

Bridging capital (Jackson 2020, section 3.3) credits node $i$ with the expected number of walks of length at most $T$ that are lost when one entry $P_{ij}$ of the transmission matrix is deleted, summed over $j$ and weighted by source-destination values $v_{st}$:

$Brid_i = \sum_j \sum_{s,t} v_{st} \sum_{h=1}^{T} \left[P^h - (P - P_{ij} E_{ij})^h\right]_{st}.$

Usage

centrality_bridging_capital(x, bridging_steps = 2, bridging_values = NULL, ...)

Arguments

Details

Edge weights are the transmission probabilities in $P$ and must lie between zero and one, and an unweighted edge has probability one. A weight above one raises an error. Direction and loops are kept, and the two entries of an undirected edge are deleted separately. Walks may repeat nodes and edges, and a walk that uses the deleted entry several times is counted once. Isolated nodes score zero, and bridging_steps = 0 gives zero scores. The function computes the expected walk count EInf of the source paper.

Value

A named numeric vector with one score per node, in input node order.

References

Jackson, M. O. (2020). A typology of social capital and associated network measures. Social Choice and Welfare, 54, 311-336. doi:10.1007/s00355-019-01189-3.

See Also

centrality_bridging, centrality_diffusion_centrality, centrality.

Examples

centrality_bridging_capital(regulation_net)

Description

Coordinator brokerage (Gould and Fernandez 1989) counts the open two-paths $a \to v \to c$ through node $v$ in which $a$, $v$ and $c$ all belong to the same group. A two-path is open when the network has no edge from $a$ to $c$. This role is $w_I$ in the notation of the source.

Usage

centrality_brokerage_coordinator(x, membership = NULL, ...)

Arguments

Details

The measure is defined for directed networks. On an undirected network it returns NA with a cograph_undefined_measure warning, and the same happens when membership is missing, where the warning also has class cograph_bad_membership. A membership whose length differs from the number of nodes raises a cograph_bad_membership error. Group labels may be numbers or strings. Edge weights and self-loops are ignored. The other four roles are centrality_brokerage_itinerant, centrality_brokerage_representative, centrality_brokerage_gatekeeper and centrality_brokerage_liaison.

Value

A named integer vector with one count per node, in input node order. normalized = TRUE returns a numeric vector.

References

Gould, R. V., & Fernandez, R. M. (1989). Structures of mediation: A formal approach to brokerage in transaction networks. Sociological Methodology, 19, 89-126. doi:10.2307/270949.

See Also

centrality_brokerage_gatekeeper, centrality_gateway, centrality.

Examples

centrality_brokerage_coordinator(regulation_net, membership = rep(1:2, each = 5))

Description

Gatekeeper brokerage (Gould and Fernandez 1989) counts the open two-paths $a \to v \to c$ through node $v$ in which $v$ and $c$ belong to the same group and $a$ to another group. The broker admits contact from outside to a member of its own group. A two-path is open when the network has no edge from $a$ to $c$. This role is $b_{OI}$ in the notation of the source.

Usage

centrality_brokerage_gatekeeper(x, membership = NULL, ...)

Arguments

Details

The measure is defined for directed networks. On an undirected network it returns NA with a cograph_undefined_measure warning, and the same happens when membership is missing, where the warning also has class cograph_bad_membership. A membership whose length differs from the number of nodes raises a cograph_bad_membership error. Group labels may be numbers or strings. Edge weights and self-loops are ignored.

Value

A named integer vector with one count per node, in input node order. normalized = TRUE returns a numeric vector.

References

Gould, R. V., & Fernandez, R. M. (1989). Structures of mediation: A formal approach to brokerage in transaction networks. Sociological Methodology, 19, 89-126. doi:10.2307/270949.

See Also

centrality_brokerage_representative, centrality_brokerage_coordinator, centrality.

Examples

centrality_brokerage_gatekeeper(regulation_net, membership = rep(1:2, each = 5))

Description

Itinerant brokerage (Gould and Fernandez 1989), also called the consultant role, counts the open two-paths $a \to v \to c$ through node $v$ in which $a$ and $c$ belong to the same group and $v$ to another group. A two-path is open when the network has no edge from $a$ to $c$. This role is $w_O$ in the notation of the source.

Usage

centrality_brokerage_itinerant(x, membership = NULL, ...)

Arguments

Details

The measure is defined for directed networks. On an undirected network it returns NA with a cograph_undefined_measure warning, and the same happens when membership is missing, where the warning also has class cograph_bad_membership. A membership whose length differs from the number of nodes raises a cograph_bad_membership error. Group labels may be numbers or strings. Edge weights and self-loops are ignored.

Value

A named integer vector with one count per node, in input node order. normalized = TRUE returns a numeric vector.

References

Gould, R. V., & Fernandez, R. M. (1989). Structures of mediation: A formal approach to brokerage in transaction networks. Sociological Methodology, 19, 89-126. doi:10.2307/270949.

See Also

centrality_brokerage_coordinator, centrality_brokerage_liaison, centrality.

Examples

centrality_brokerage_itinerant(regulation_net, membership = rep(1:2, each = 5))

Description

Liaison brokerage (Gould and Fernandez 1989) counts the open two-paths $a \to v \to c$ through node $v$ in which $a$, $v$ and $c$ belong to three different groups. A two-path is open when the network has no edge from $a$ to $c$. This role is $b_O$ in the notation of the source.

Usage

centrality_brokerage_liaison(x, membership = NULL, ...)

Arguments

Details

The measure is defined for directed networks. On an undirected network it returns NA with a cograph_undefined_measure warning, and the same happens when membership is missing, where the warning also has class cograph_bad_membership. A membership whose length differs from the number of nodes raises a cograph_bad_membership error. Group labels may be numbers or strings. With fewer than three groups every count is 0. Edge weights and self-loops are ignored.

Value

A named integer vector with one count per node, in input node order. normalized = TRUE returns a numeric vector.

References

Gould, R. V., & Fernandez, R. M. (1989). Structures of mediation: A formal approach to brokerage in transaction networks. Sociological Methodology, 19, 89-126. doi:10.2307/270949.

See Also

centrality_brokerage_itinerant, centrality_brokerage_coordinator, centrality.

Examples

centrality_brokerage_liaison(regulation_net, membership = rep(1:3, length.out = 10))

Description

Representative brokerage (Gould and Fernandez 1989) counts the open two-paths $a \to v \to c$ through node $v$ in which $a$ and $v$ belong to the same group and $c$ to another group. The broker passes contact from its own group to the outside. A two-path is open when the network has no edge from $a$ to $c$. This role is $b_{IO}$ in the notation of the source.

Usage

centrality_brokerage_representative(x, membership = NULL, ...)

Arguments

Details

The measure is defined for directed networks. On an undirected network it returns NA with a cograph_undefined_measure warning, and the same happens when membership is missing, where the warning also has class cograph_bad_membership. A membership whose length differs from the number of nodes raises a cograph_bad_membership error. Group labels may be numbers or strings. Edge weights and self-loops are ignored.

Value

A named integer vector with one count per node, in input node order. normalized = TRUE returns a numeric vector.

References

Gould, R. V., & Fernandez, R. M. (1989). Structures of mediation: A formal approach to brokerage in transaction networks. Sociological Methodology, 19, 89-126. doi:10.2307/270949.

See Also

centrality_brokerage_gatekeeper, centrality_brokerage_coordinator, centrality.

Examples

centrality_brokerage_representative(regulation_net, membership = rep(1:2, each = 5))

Description

The clustering degree algorithm (CDA; Wang et al. 2018) scores a node by its clustering degree $CD_i = [\alpha d_i + (1-\alpha) s_i] / [1 + \exp(-C_i^w)]$ plus the clustering degrees of its neighbors, each scaled by the edge weight:

$PC_i = CD_i + \sum_{j \in N(i)} \frac{w_{ij}}{w_{\max}} CD_j.$

Here $d_i$ is the degree, $s_i$ the strength, $C_i^w$ the weighted clustering coefficient of Barrat et al., and $w_{\max}$ the largest edge weight in the network.

Usage

centrality_cda(x, cda_alpha = 0.5, ...)

Arguments

Details

The measure works on an undirected weighted network. On a directed network the weights of the two arcs between a pair are added, and weighted = FALSE uses the simple undirected skeleton. Self-loops are removed, and mode and invert_weights have no effect. A node with fewer than two neighbors has clustering 0, and an isolated node scores 0. Negative or non-finite weights raise an error. With cda_alpha = 1 the scores are unchanged when every weight is multiplied by a constant, and with cda_alpha = 0 they scale by that constant. On an unweighted network cda_alpha has no effect.

Value

A named numeric vector with one score per node, in input node order.

References

Wang, Q., Ren, J., Wang, Y., Zhang, B., Cheng, Y., & Zhao, X. (2018). CDA: A Clustering Degree Based Influential Spreader Identification Algorithm in Weighted Complex Network. IEEE Access, 6, 19550-19559. doi:10.1109/ACCESS.2018.2822844.

See Also

centrality_strength, centrality_transitivity, centrality.

Examples

centrality_cda(regulation_net)

Description

The centroid value (Koschuetzki et al. 2005) compares a node with every other node by the number of nodes each one is closer to. With $\gamma(v, u)$ the number of nodes strictly closer to $v$ than to $u$,

$CV(v) = \min_{u} \left[ \gamma(v, u) - \gamma(u, v) \right].$

The minimum includes $u = v$, so the score is at most 0, and values closer to 0 mark more central nodes.

Usage

centrality_centroid(x, mode = "all", ...)

Arguments

Details

Edge weights are read as path lengths. invert_weights = TRUE uses $1/w^\alpha$ as the length, and weighted = FALSE counts hops. mode sets the direction of the paths.

Value

A named numeric vector with one score per node, in input node order.

References

Koschuetzki, D., Lehmann, K. A., Peeters, L., Richter, S., Tenfelde-Podehl, D., & Zlotowski, O. (2005). Centrality indices. In U. Brandes & T. Erlebach (Eds.), Network Analysis: Methodological Foundations (pp. 16-61). Springer. doi:10.1007/978-3-540-31955-9_3.

See Also

centrality_closeness, centrality_closeness_vitality, centrality.

Examples

centrality_centroid(regulation_net)

Description

Closeness centrality (Sabidussi 1966) is the reciprocal of the total shortest-path distance from a node to the nodes it reaches:

$C(v) = \frac{1}{\sum_{w \ne v} d(v, w)}.$

Unreachable nodes are left out of the sum, as in igraph::closeness().

Usage

centrality_closeness(x, mode = "all", ...)

centrality_incloseness(x, ...)

centrality_outcloseness(x, ...)

Arguments

Details

Edge weights are read as path lengths, and invert_weights = TRUE uses $1/w^\alpha$ instead. weighted = FALSE uses hop counts. mode = "out" follows paths leaving the node and mode = "in" paths arriving at it. centrality_outcloseness() and centrality_incloseness() are these two forms. A node that reaches no other node returns NaN. normalized = TRUE multiplies each score by the number of other nodes the node reaches, as igraph does.

Value

A named numeric vector with one score per node, in input node order.

References

Sabidussi, G. (1966). The centrality index of a graph. Psychometrika, 31(4), 581-603. doi:10.1007/BF02289527.

See Also

centrality_harmonic, centrality_barycenter, centrality.

Examples

centrality_closeness(regulation_net)

Description

Closeness vitality (Koschuetzki et al. 2005) is the drop in the Wiener index when a node is removed:

$CV(v) = W(G) - W(G - v), \qquad W(G) = \sum_{s \ne t} d(s, t).$

The Wiener index sums the finite distances over ordered pairs, as in networkx::closeness_vitality().

Usage

centrality_closeness_vitality(x, mode = "all", ...)