centrality_series: Time-varying vertex centrality
In Dynet: Tidy Temporal Network Analysis
centrality_series R Documentation
Time-varying vertex centrality
Description
Centrality for every vertex at every time point. Ask for several measures
in one call and they arrive stacked in a single tidy frame, one row per
vertex, time point and measure.
Each value measures the network as it stands in one time bin, so the
result is a trajectory of ordinary centrality. Order within a bin is not
used: every tie active in the bin counts as present. Centrality computed
from time-respecting paths across the whole period is path_centrality().
Usage
centrality_series(
dn,
measure = "degree",
sessions = c("bounded", "collapse", "separate"),
sample = NULL,
damping = 0.85,
mode = c("all", "out", "in"),
start = NULL,
end = NULL,
step = NULL,
window = NULL,
exponent = 1,
prestige = "indegree",
rescale = FALSE,
lambda = 1,
plot = FALSE
)
Arguments
dn
A temporal network from dynet().
measure
One or more of "degree", "strength", "prestige", "closeness",
"betweenness", "eigenvector", "pagerank", "hub", "authority",
"coreness", "constraint", "power", "harary", "information",
"load", "flow_betweenness", or "diffusion".
The deprecated names "indegree" and
"outdegree", which warn with class dynet_deprecated and are replaced
by measure = "degree" with mode = "in" or mode = "out". Defaults to
"degree". Any other
name raises an error of class dynet_unknown_measure; the directed-only
measures "prestige", "hub", "authority" and the two deprecated
names "indegree" and "outdegree" raise dynet_needs_directed on an
undirected network.
sessions
How to treat sessions: "bounded" (the default) keeps
paths inside a
session, "collapse" ignores sessions, "separate" reports each session
on its own rows. "separate" on a network built without a session column
raises an error of class dynet_no_sessions.
sample
Deprecated. "instant" is equivalent to window = 0;
"window" uses the current positive/default window.
damping
Damping factor for PageRank; a single number strictly
between zero and one, 0.85 by default.
mode
Which edges count on a directed network: "all" both
directions, "out" outgoing only, "in" incoming only. Defaults to
"all"; any other string raises an error of class dynet_bad_input.
Name several at
once – mode = c("all", "in", "out") – to get degree, in-degree and
out-degree from a single call; the extra directions are then labelled
degree_in and degree_out in the measure column, while a call naming
one direction keeps the plain measure name. Applies to
"degree", "strength", "closeness", "coreness", "harary",
"eigenvector" and "diffusion"; the remaining measures have a single
directional definition and ignore it. Ignored
entirely on an undirected network. In-degree is therefore
mode = "in". The old "indegree" and "outdegree" measure names
remain as deprecated aliases.
start, end
First and last time at which to measure. Default to the
observed range. A network built from dates may be addressed with dates.
step
How often to measure. Defaults to the interval the network was
built with.
window
How much time each measurement covers. Defaults to step,
which tiles the period into disjoint bins. A larger value slides an
overlapping window; 0 samples the network at each point in time.
"all" measures the whole observed period as one window, closed on the
right so an event at the final instant is inside it; it cannot be combined
with step, and under sessions = "separate" or discontinuous
observation it gives one window per session or observed component.
exponent
Attenuation factor for Bonacich "power", a single finite
number, 1 by default. Positive rewards
being connected to well-connected others; negative rewards the opposite,
which is the bargaining reading.
prestige
Prestige definition, "indegree" by default.
"indegree" counts distinct active
incoming dyads. "indegree.rownorm" first gives every active sender one
unit split equally across its distinct outgoing dyads, then sums the
received mass. "indegree.rowcolnorm" balances a total-support binary
adjacency to doubly stochastic form; feasible scores are necessarily
uniform. "domain" counts the distinct other vertices with a directed
path into each vertex in the active snapshot. "domain.proximity"
discounts that incoming domain fraction by its mean directed hop distance.
"eigenvector" uses the unique nonnegative Perron ray of the transposed
binary adjacency. "eigenvector.rownorm" first divides every nonzero
binary sender row by its outgoing-dyad count and then solves the same
certified incoming Perron equation. "eigenvector.colnorm" instead
divides every nonzero binary receiver column by its incoming-dyad count
before solving. "eigenvector.rowcolnorm" first certifies total support
and balances binary adjacency to doubly stochastic form. Prestige is
directed and snapshot-only.
rescale
Whether to divide prestige by its total independently
inside every reported time/session block; FALSE by default. Zero-total
count/proximity
definitions return NaN; structurally undefined spectral definitions
return NA. This argument requires measure = "prestige", and raises
dynet_bad_input otherwise.
lambda
Nonnegative multiplier for "diffusion", 1 by default.
Diffusion degree is
the sum of the selected degree of a vertex and all of its one-step
neighbours, multiplied by lambda.
plot
Whether to draw the result as well as return it. Drawing is a
side effect in the manner of graphics::hist(): the verb still returns
its tidy table, invisibly when it has drawn, so plot = TRUE saves the
wrapping plot() call without changing what comes back. Use plot() on
the result when the figure needs arguments of its own.
Details
step and window are separate on purpose. step is how often you look;
window is how much of the timeline each look takes in. Setting them equal
partitions the period; setting window larger than step is a rolling
window, which keeps the resolution of the smaller step while smoothing over
the noise of a sparse bin. The arguments match tsna::tSnaStats(), where
they are called time.interval and aggregate.dur.
Snapshot "degree" counts distinct active binary dyads, so duplicate,
split and overlapping spells do not multiply it; mode = "all" on a
directed snapshot is in-degree plus out-degree. "strength" is the same
margin taken over spell weights rather than over binary dyads. In a
positive window each spell contributes its weight in proportion to the
share of its duration that falls inside the window, weight * overlap / duration, so a spell straddling two tiled windows splits its weight
between them and the pieces add back to the whole. A point contact has no
duration to split and contributes its full weight to the window holding
it. With window = 0 every active spell contributes its full weight at
that instant. The share uses the spell's recorded duration, so the part of
a spell outside the observation period is not reassigned to observed
windows. snapshots() and the network plots keep full weights per bin, so
their weight column is not the input to this strength.
Snapshot "closeness" is not Freeman's 1 / \sum_z d_{sz}, which
is
undefined once a snapshot is disconnected – and a time bin almost always
is. It is the reciprocal of the mean geodesic distance to the vertices a
vertex can actually reach: with R_s the reachable nonself set,
C(s) = |R_s| / \sum_{z \in R_s} d_{sz},
which is zero for an isolate and equals Freeman's normalised closeness
(n - 1) / \sum_z d_{sz} on a connected snapshot. "harary" is the
reciprocal of
eccentricity, zero for a vertex that cannot reach everything.
"eigenvector", "hub" and "authority" are certified the way eigenvector
prestige is: a snapshot whose spectral radius is zero (no cycle) or whose
Perron root is repeated (components of equal weight) has no single answer,
and every vertex of that block is NA under a warning of class
dynet_eigen_undefined.
"eigenvector" is uniquely determined when the Perron eigenvalue has a
one-dimensional eigenspace; strong connectivity is a sufficient condition.
Disconnected snapshots with equally dominant components can have more than
one correct eigenvector, so read the result as a within-snapshot ranking
rather than an automatically comparable number across the whole series.
Indegree prestige is the column sum of the directed binary active-dyad
adjacency matrix. It is exactly snapshot degree with mode = "in":
duplicate, split, and overlapping spells and edge weights do not multiply
the result, while an explicitly retained directed loop contributes once.
With rescale = TRUE, the column sums are divided by their block total. A
zero total is mathematically undefined and is returned as literal NaN.
Row-normalised indegree prestige first converts every nonzero binary
adjacency row to sum one; zero rows remain all zero. Its column sums are the
received sender-nomination mass, so their total is the number of active
senders. rescale = TRUE divides again by that block total. This closed-form
transform is the sna::prestige(cmode = "indegree.rownorm") definition on
binary matrices. Dynet deliberately ignores edge weights, whereas sna
uses their magnitudes on valued matrices.
Row-column-normalised prestige uses deterministic Sinkhorn–Knopp scaling
only when the full binary vertex matrix has total support: every active dyad
must belong to a perfect matching. It preserves all binary dyads and does
not remove isolates or unsupported edges. Infeasible blocks return NA for
every vertex with a classed warning. A feasible transform has every incoming
column sum equal to one, so raw prestige is uniformly one and rescaled
prestige uniformly 1 / n; this definition is a transform diagnostic, not
a vertex ranking. Dynet uses fixed-order sweeps, maximum absolute row/column
residual 1e-12, and at most 10,000 sweeps. It never returns a partial
iterate. This deliberately differs from the randomised loose-tolerance
annealer in sna 2.8.
Domain prestige is incoming indegree in the directed reachability graph
after excluding its reflexive diagonal. If H[i,j] records whether i = j
or a directed path from i to j exists, then
p[j] = sum(H[,j]) - 1. Every distinct reaching vertex counts once,
regardless of path length or multiplicity. Loops cannot add self credit,
isolates score zero, and a zero-total rescaling returns literal NaN.
Closure is computed on the binary active snapshot for each reporting block,
not on chronologically ordered temporal journeys through the raw spells.
Domain-proximity prestige additionally uses the shortest incoming hop
distances. For the nonself domain D[j], let r[j] be its size and s[j]
the sum of its finite distances into j. The score is zero when r[j] = 0
and otherwise r[j]^2 / ((n - 1) * s[j]): the incoming domain fraction
divided by mean hop distance. Unreachable vertices are omitted before the
distance sum. This deliberately fixes an arithmetic artefact in sna 2.8,
whose FALSE * Inf operation incorrectly zeros partial nonempty domains.
Eigenvector prestige solves t(B) %*% p = rho * p for the nonnegative
Perron ray of binary adjacency B. It requires positive spectral radius and
a one-dimensional Perron eigenspace. Raw scores have Euclidean norm one;
rescale = TRUE makes their sum one. Zero-radius or nonunique blocks return
all NA with a classed warning and diagnostics. Periodic cycles remain
valid even when negative or complex roots share the spectral radius. Dynet
uses direct eigenvalues plus an SVD nullity/residual check at tolerance
1e-10, orients the ray as nonnegative, and never applies elementwise
absolute value.
Row-normalised eigenvector prestige first forms binary adjacency B and
divides each nonzero sender row by its number of distinct outgoing dyads;
zero rows remain exactly zero. It then solves the certified incoming Perron
equation for the transpose of that row-stochastic matrix. Thus each active
sender distributes one unit of recursive nomination mass, with no
teleportation or dangling-row imputation. Binary session union and retained
loop policy occur before row normalisation. The positive-radius, geometric-
uniqueness, nonnegative-sign, L2/sum-scale, warning, and diagnostic rules
are otherwise exactly those of ordinary eigenvector prestige.
Column-normalised eigenvector prestige divides each nonzero binary receiver
column by its number of distinct incoming dyads; zero columns remain zero.
It solves the incoming Perron equation only after that transform. If every
vertex has positive indegree, the transformed transpose is row-stochastic
and every certified score is necessarily uniform. Nonuniform defined scores
therefore require a zero-indegree vertex. Binary union and retained-loop
policy precede normalisation; certification and scaling remain those above.
Row-column-normalised eigenvector prestige composes the total-support and
deterministic Sinkhorn–Knopp contract with the certified Perron contract.
Infeasible support and nonconvergent balancing terminate before the spectral
solve. A completed doubly stochastic transform always has the all-ones
Perron ray, but reducible transforms have several such rays and remain
undefined. Every fully certified score is therefore exactly uniform:
1 / sqrt(n) raw or 1 / n rescaled. This selector diagnoses support,
balance, and irreducibility; it is not a vertex ranking.
Declared vertex activity induces the eligible vertex population before any
kernel is evaluated. Positive windows independently use any-time vertex and
edge unions before induction, while window = 0 evaluates the exact state.
Results remain rectangular over the fixed vertex universe: inactive
vertices receive typed NA, while eligible isolates keep the centrality
kernel's ordinary static result.
Value
A dynet_metric: a tidy data frame with one row per vertex, time
point and measure. Columns are session (only under
sessions = "separate", which is the only mode that keeps session labels
apart), time, node, measure and value.
Print it, summary() it, plot() it, or take the plain frame with
as.data.frame(). Prestige stores its mathematical choices as direct
attributes for a prestige-only result and as named records under
measure_metadata otherwise. When a prestige
variant is structurally undefined or fails to converge, the affected
values are NA, a warning says how many reporting blocks were affected,
and a record naming the stage and reason for each comes out through
as.data.frame(x, what = "diagnostics").
Conditions
Errors: dynet_unknown_measure (a measure not listed above),
dynet_needs_directed ("prestige", "hub", "authority",
"indegree" or "outdegree" on an undirected network),
dynet_no_sessions (sessions = "separate" without a session column),
dynet_outside_observation (the requested range misses observed support;
it also carries dynet_bad_input),
and dynet_bad_input for every other broken contract – dn not a
dynet, an unknown mode, an out-of-range damping, exponent,
lambda, prestige, rescale, start, end, step or window, and
rescale = TRUE without measure = "prestige".
Warnings: dynet_deprecated (measure = "indegree"/"outdegree", or the
retired sample argument), dynet_eigen_undefined and
dynet_kernel_singular (both also carrying dynet_measure_undefined) when
a snapshot's eigenvector, hub, authority, Bonacich power or information
kernel has no unique answer, and dynet_prestige_infeasible,
dynet_prestige_nonconvergence and dynet_prestige_eigen_undefined when a
prestige variant is structurally undefined or fails to converge.
References
Holme, P., & Saramaki, J. (2012). Temporal networks. Physics Reports,
519(3), 97-125.
Wasserman, S., & Faust, K. (1994). Social Network Analysis: Methods and
Applications. Cambridge University Press, Chapter 5.
Butts, C. T. (2024). sna: Tools for Social Network Analysis, version 2.8.
doi:10.32614/CRAN.package.sna.
Lin, N. (1976). Foundations of Social Research. McGraw-Hill.
Freeman, L. C. (1979). Centrality in social networks: conceptual
clarification. Social Networks, 1(3), 215-239.
\Sexpr[results=rd]{tools:::Rd_expr_doi("10.1016/0378-8733(78)90021-7")}
Brandes, U. (2001). A faster algorithm for betweenness centrality.
Journal of Mathematical Sociology, 25(2), 163-177.
Bonacich, P. (1987). Power and centrality: a family of measures.
American Journal of Sociology, 92(5), 1170-1182.
Hage, P., & Harary, F. (1995). Eccentricity and centrality in networks.
Social Networks, 17(1), 57-63.
Stephenson, K., & Zelen, M. (1989). Rethinking centrality. Social
Networks, 11(1), 1-37.
Goh, K.-I., Kahng, B., & Kim, D. (2001). Universal behavior of load
distribution in scale-free networks. Physical Review Letters, 87(27),
278701.
Freeman, L. C., Borgatti, S. P., & White, D. R. (1991). Centrality in
valued graphs. Social Networks, 13(2), 141-154.
Page, L., Brin, S., Motwani, R., & Winograd, T. (1999). The PageRank
citation ranking: bringing order to the web. Technical Report 1999-66,
Stanford InfoLab.
Kleinberg, J. M. (1999). Authoritative sources in a hyperlinked
environment. Journal of the ACM, 46(5), 604-632.
doi:10.1145/324133.324140.
Seidman, S. B. (1983). Network structure and minimum degree. Social
Networks, 5(3), 269-287. doi:10.1016/0378-8733(83)90028-X.
Burt, R. S. (1992). Structural Holes: The Social Structure of
Competition. Harvard University Press.
Kundu, S., Murthy, C. A., & Pal, S. K. (2011). A new centrality measure for
influence maximization in social networks. In Pattern Recognition and
Machine Intelligence, Lecture Notes in Computer Science 6744 (pp. 242-247).
Springer. doi:10.1007/978-3-642-21786-9_40.
Bonacich, P. (1972). Factoring and weighting approaches to status scores
and clique identification. Journal of Mathematical Sociology, 2, 113-120.
doi:10.1080/0022250X.1972.9989806.
Berman, A., & Plemmons, R. J. (1994). Nonnegative Matrices in the
Mathematical Sciences. SIAM. doi:10.1137/1.9781611971262.
Sinkhorn, R. (1964). A relationship between arbitrary positive matrices
and doubly stochastic matrices. Annals of Mathematical Statistics, 35,
876-879. doi:10.1214/aoms/1177703591.
Sinkhorn, R., & Knopp, P. (1967). Concerning nonnegative matrices and
doubly stochastic matrices. Pacific Journal of Mathematics, 21, 343-348.
doi:10.2140/pjm.1967.21.343.
Knight, P. A. (2008). The Sinkhorn-Knopp algorithm: convergence and
applications. SIAM Journal on Matrix Analysis and Applications, 30,
261-275. doi:10.1137/060659624.
See Also
path_centrality() for closeness and betweenness on
time-respecting paths; reachability() for temporal reach.
Examples
dn <- dynet(school_contacts)
centrality_series(dn, measure = "degree")
centrality_series(dn, measure = c("degree", "betweenness"))
centrality_series(dn, measure = "prestige", rescale = TRUE)
centrality_series(dn, measure = "prestige",
prestige = "indegree.rownorm")
centrality_series(dn, measure = "prestige", prestige = "domain")
centrality_series(dn, measure = "prestige",
prestige = "domain.proximity")
centrality_series(dn, measure = "prestige", prestige = "eigenvector")
centrality_series(dn, measure = "prestige",
prestige = "eigenvector.rownorm")
centrality_series(dn, measure = "prestige",
prestige = "eigenvector.colnorm")
centrality_series(dn, measure = "prestige",
prestige = "eigenvector.rowcolnorm")
# A seven-day window, stepped one day at a time.
centrality_series(dn, measure = "degree", step = 1, window = 7)
degree <- centrality_series(dn, measure = "degree")
summary(degree)
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| centrality_series | R Documentation |
Time-varying vertex centrality
Description
Centrality for every vertex at every time point. Ask for several measures in one call and they arrive stacked in a single tidy frame, one row per vertex, time point and measure.
Each value measures the network as it stands in one time bin, so the
result is a trajectory of ordinary centrality. Order within a bin is not
used: every tie active in the bin counts as present. Centrality computed
from time-respecting paths across the whole period is path_centrality().
Usage
centrality_series(
dn,
measure = "degree",
sessions = c("bounded", "collapse", "separate"),
sample = NULL,
damping = 0.85,
mode = c("all", "out", "in"),
start = NULL,
end = NULL,
step = NULL,
window = NULL,
exponent = 1,
prestige = "indegree",
rescale = FALSE,
lambda = 1,
plot = FALSE
)
Arguments
dn |
A temporal network from |
measure |
One or more of |
sessions |
How to treat sessions: |
sample |
Deprecated. |
damping |
Damping factor for PageRank; a single number strictly
between zero and one, |
mode |
Which edges count on a directed network: |
start, end |
First and last time at which to measure. Default to the observed range. A network built from dates may be addressed with dates. |
step |
How often to measure. Defaults to the interval the network was built with. |
window |
How much time each measurement covers. Defaults to |
exponent |
Attenuation factor for Bonacich |
prestige |
Prestige definition, |
rescale |
Whether to divide prestige by its total independently
inside every reported time/session block; |
lambda |
Nonnegative multiplier for |
plot |
Whether to draw the result as well as return it. Drawing is a
side effect in the manner of |
Details
step and window are separate on purpose. step is how often you look;
window is how much of the timeline each look takes in. Setting them equal
partitions the period; setting window larger than step is a rolling
window, which keeps the resolution of the smaller step while smoothing over
the noise of a sparse bin. The arguments match tsna::tSnaStats(), where
they are called time.interval and aggregate.dur.
Snapshot "degree" counts distinct active binary dyads, so duplicate,
split and overlapping spells do not multiply it; mode = "all" on a
directed snapshot is in-degree plus out-degree. "strength" is the same
margin taken over spell weights rather than over binary dyads. In a
positive window each spell contributes its weight in proportion to the
share of its duration that falls inside the window, weight * overlap / duration, so a spell straddling two tiled windows splits its weight
between them and the pieces add back to the whole. A point contact has no
duration to split and contributes its full weight to the window holding
it. With window = 0 every active spell contributes its full weight at
that instant. The share uses the spell's recorded duration, so the part of
a spell outside the observation period is not reassigned to observed
windows. snapshots() and the network plots keep full weights per bin, so
their weight column is not the input to this strength.
Snapshot "closeness" is not Freeman's 1 / \sum_z d_{sz}, which
is
undefined once a snapshot is disconnected – and a time bin almost always
is. It is the reciprocal of the mean geodesic distance to the vertices a
vertex can actually reach: with R_s the reachable nonself set,
C(s) = |R_s| / \sum_{z \in R_s} d_{sz},
which is zero for an isolate and equals Freeman's normalised closeness
(n - 1) / \sum_z d_{sz} on a connected snapshot. "harary" is the
reciprocal of
eccentricity, zero for a vertex that cannot reach everything.
"eigenvector", "hub" and "authority" are certified the way eigenvector
prestige is: a snapshot whose spectral radius is zero (no cycle) or whose
Perron root is repeated (components of equal weight) has no single answer,
and every vertex of that block is NA under a warning of class
dynet_eigen_undefined.
"eigenvector" is uniquely determined when the Perron eigenvalue has a
one-dimensional eigenspace; strong connectivity is a sufficient condition.
Disconnected snapshots with equally dominant components can have more than
one correct eigenvector, so read the result as a within-snapshot ranking
rather than an automatically comparable number across the whole series.
Indegree prestige is the column sum of the directed binary active-dyad
adjacency matrix. It is exactly snapshot degree with mode = "in":
duplicate, split, and overlapping spells and edge weights do not multiply
the result, while an explicitly retained directed loop contributes once.
With rescale = TRUE, the column sums are divided by their block total. A
zero total is mathematically undefined and is returned as literal NaN.
Row-normalised indegree prestige first converts every nonzero binary
adjacency row to sum one; zero rows remain all zero. Its column sums are the
received sender-nomination mass, so their total is the number of active
senders. rescale = TRUE divides again by that block total. This closed-form
transform is the sna::prestige(cmode = "indegree.rownorm") definition on
binary matrices. Dynet deliberately ignores edge weights, whereas sna
uses their magnitudes on valued matrices.
Row-column-normalised prestige uses deterministic Sinkhorn–Knopp scaling
only when the full binary vertex matrix has total support: every active dyad
must belong to a perfect matching. It preserves all binary dyads and does
not remove isolates or unsupported edges. Infeasible blocks return NA for
every vertex with a classed warning. A feasible transform has every incoming
column sum equal to one, so raw prestige is uniformly one and rescaled
prestige uniformly 1 / n; this definition is a transform diagnostic, not
a vertex ranking. Dynet uses fixed-order sweeps, maximum absolute row/column
residual 1e-12, and at most 10,000 sweeps. It never returns a partial
iterate. This deliberately differs from the randomised loose-tolerance
annealer in sna 2.8.
Domain prestige is incoming indegree in the directed reachability graph
after excluding its reflexive diagonal. If H[i,j] records whether i = j
or a directed path from i to j exists, then
p[j] = sum(H[,j]) - 1. Every distinct reaching vertex counts once,
regardless of path length or multiplicity. Loops cannot add self credit,
isolates score zero, and a zero-total rescaling returns literal NaN.
Closure is computed on the binary active snapshot for each reporting block,
not on chronologically ordered temporal journeys through the raw spells.
Domain-proximity prestige additionally uses the shortest incoming hop
distances. For the nonself domain D[j], let r[j] be its size and s[j]
the sum of its finite distances into j. The score is zero when r[j] = 0
and otherwise r[j]^2 / ((n - 1) * s[j]): the incoming domain fraction
divided by mean hop distance. Unreachable vertices are omitted before the
distance sum. This deliberately fixes an arithmetic artefact in sna 2.8,
whose FALSE * Inf operation incorrectly zeros partial nonempty domains.
Eigenvector prestige solves t(B) %*% p = rho * p for the nonnegative
Perron ray of binary adjacency B. It requires positive spectral radius and
a one-dimensional Perron eigenspace. Raw scores have Euclidean norm one;
rescale = TRUE makes their sum one. Zero-radius or nonunique blocks return
all NA with a classed warning and diagnostics. Periodic cycles remain
valid even when negative or complex roots share the spectral radius. Dynet
uses direct eigenvalues plus an SVD nullity/residual check at tolerance
1e-10, orients the ray as nonnegative, and never applies elementwise
absolute value.
Row-normalised eigenvector prestige first forms binary adjacency B and
divides each nonzero sender row by its number of distinct outgoing dyads;
zero rows remain exactly zero. It then solves the certified incoming Perron
equation for the transpose of that row-stochastic matrix. Thus each active
sender distributes one unit of recursive nomination mass, with no
teleportation or dangling-row imputation. Binary session union and retained
loop policy occur before row normalisation. The positive-radius, geometric-
uniqueness, nonnegative-sign, L2/sum-scale, warning, and diagnostic rules
are otherwise exactly those of ordinary eigenvector prestige.
Column-normalised eigenvector prestige divides each nonzero binary receiver column by its number of distinct incoming dyads; zero columns remain zero. It solves the incoming Perron equation only after that transform. If every vertex has positive indegree, the transformed transpose is row-stochastic and every certified score is necessarily uniform. Nonuniform defined scores therefore require a zero-indegree vertex. Binary union and retained-loop policy precede normalisation; certification and scaling remain those above.
Row-column-normalised eigenvector prestige composes the total-support and
deterministic Sinkhorn–Knopp contract with the certified Perron contract.
Infeasible support and nonconvergent balancing terminate before the spectral
solve. A completed doubly stochastic transform always has the all-ones
Perron ray, but reducible transforms have several such rays and remain
undefined. Every fully certified score is therefore exactly uniform:
1 / sqrt(n) raw or 1 / n rescaled. This selector diagnoses support,
balance, and irreducibility; it is not a vertex ranking.
Declared vertex activity induces the eligible vertex population before any
kernel is evaluated. Positive windows independently use any-time vertex and
edge unions before induction, while window = 0 evaluates the exact state.
Results remain rectangular over the fixed vertex universe: inactive
vertices receive typed NA, while eligible isolates keep the centrality
kernel's ordinary static result.
Value
A dynet_metric: a tidy data frame with one row per vertex, time
point and measure. Columns are session (only under
sessions = "separate", which is the only mode that keeps session labels
apart), time, node, measure and value.
Print it, summary() it, plot() it, or take the plain frame with
as.data.frame(). Prestige stores its mathematical choices as direct
attributes for a prestige-only result and as named records under
measure_metadata otherwise. When a prestige
variant is structurally undefined or fails to converge, the affected
values are NA, a warning says how many reporting blocks were affected,
and a record naming the stage and reason for each comes out through
as.data.frame(x, what = "diagnostics").
Conditions
Errors: dynet_unknown_measure (a measure not listed above),
dynet_needs_directed ("prestige", "hub", "authority",
"indegree" or "outdegree" on an undirected network),
dynet_no_sessions (sessions = "separate" without a session column),
dynet_outside_observation (the requested range misses observed support;
it also carries dynet_bad_input),
and dynet_bad_input for every other broken contract – dn not a
dynet, an unknown mode, an out-of-range damping, exponent,
lambda, prestige, rescale, start, end, step or window, and
rescale = TRUE without measure = "prestige".
Warnings: dynet_deprecated (measure = "indegree"/"outdegree", or the
retired sample argument), dynet_eigen_undefined and
dynet_kernel_singular (both also carrying dynet_measure_undefined) when
a snapshot's eigenvector, hub, authority, Bonacich power or information
kernel has no unique answer, and dynet_prestige_infeasible,
dynet_prestige_nonconvergence and dynet_prestige_eigen_undefined when a
prestige variant is structurally undefined or fails to converge.
References
Holme, P., & Saramaki, J. (2012). Temporal networks. Physics Reports, 519(3), 97-125.
Wasserman, S., & Faust, K. (1994). Social Network Analysis: Methods and Applications. Cambridge University Press, Chapter 5.
Butts, C. T. (2024). sna: Tools for Social Network Analysis, version 2.8. doi:10.32614/CRAN.package.sna.
Lin, N. (1976). Foundations of Social Research. McGraw-Hill.
Freeman, L. C. (1979). Centrality in social networks: conceptual clarification. Social Networks, 1(3), 215-239. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1016/0378-8733(78)90021-7")}
Brandes, U. (2001). A faster algorithm for betweenness centrality. Journal of Mathematical Sociology, 25(2), 163-177.
Bonacich, P. (1987). Power and centrality: a family of measures. American Journal of Sociology, 92(5), 1170-1182.
Hage, P., & Harary, F. (1995). Eccentricity and centrality in networks. Social Networks, 17(1), 57-63.
Stephenson, K., & Zelen, M. (1989). Rethinking centrality. Social Networks, 11(1), 1-37.
Goh, K.-I., Kahng, B., & Kim, D. (2001). Universal behavior of load distribution in scale-free networks. Physical Review Letters, 87(27), 278701.
Freeman, L. C., Borgatti, S. P., & White, D. R. (1991). Centrality in valued graphs. Social Networks, 13(2), 141-154.
Page, L., Brin, S., Motwani, R., & Winograd, T. (1999). The PageRank citation ranking: bringing order to the web. Technical Report 1999-66, Stanford InfoLab.
Kleinberg, J. M. (1999). Authoritative sources in a hyperlinked environment. Journal of the ACM, 46(5), 604-632. doi:10.1145/324133.324140.
Seidman, S. B. (1983). Network structure and minimum degree. Social Networks, 5(3), 269-287. doi:10.1016/0378-8733(83)90028-X.
Burt, R. S. (1992). Structural Holes: The Social Structure of Competition. Harvard University Press.
Kundu, S., Murthy, C. A., & Pal, S. K. (2011). A new centrality measure for influence maximization in social networks. In Pattern Recognition and Machine Intelligence, Lecture Notes in Computer Science 6744 (pp. 242-247). Springer. doi:10.1007/978-3-642-21786-9_40.
Bonacich, P. (1972). Factoring and weighting approaches to status scores and clique identification. Journal of Mathematical Sociology, 2, 113-120. doi:10.1080/0022250X.1972.9989806.
Berman, A., & Plemmons, R. J. (1994). Nonnegative Matrices in the Mathematical Sciences. SIAM. doi:10.1137/1.9781611971262.
Sinkhorn, R. (1964). A relationship between arbitrary positive matrices and doubly stochastic matrices. Annals of Mathematical Statistics, 35, 876-879. doi:10.1214/aoms/1177703591.
Sinkhorn, R., & Knopp, P. (1967). Concerning nonnegative matrices and doubly stochastic matrices. Pacific Journal of Mathematics, 21, 343-348. doi:10.2140/pjm.1967.21.343.
Knight, P. A. (2008). The Sinkhorn-Knopp algorithm: convergence and applications. SIAM Journal on Matrix Analysis and Applications, 30, 261-275. doi:10.1137/060659624.
See Also
path_centrality() for closeness and betweenness on
time-respecting paths; reachability() for temporal reach.
Examples
dn <- dynet(school_contacts)
centrality_series(dn, measure = "degree")
centrality_series(dn, measure = c("degree", "betweenness"))
centrality_series(dn, measure = "prestige", rescale = TRUE)
centrality_series(dn, measure = "prestige",
prestige = "indegree.rownorm")
centrality_series(dn, measure = "prestige", prestige = "domain")
centrality_series(dn, measure = "prestige",
prestige = "domain.proximity")
centrality_series(dn, measure = "prestige", prestige = "eigenvector")
centrality_series(dn, measure = "prestige",
prestige = "eigenvector.rownorm")
centrality_series(dn, measure = "prestige",
prestige = "eigenvector.colnorm")
centrality_series(dn, measure = "prestige",
prestige = "eigenvector.rowcolnorm")
# A seven-day window, stepped one day at a time.
centrality_series(dn, measure = "degree", step = 1, window = 7)
degree <- centrality_series(dn, measure = "degree")
summary(degree)
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