centrality_iec: Immediate Effects Centrality
In cograph: Analysis and Visualization of Complex Networks
View source: R/centrality-batch49.R
centrality_iec R Documentation
Immediate Effects Centrality
Description
Friedkin's immediate effects centrality scores a node by how quickly the
rest of the network's influence reaches it. Actors whose effects travel
over long sequences of interpersonal influence are more dependent on
intervening actors than those whose effects travel over short ones, so
the measure is the reciprocal of the mean length of the influence
sequences that end at a node. Writing W for the row-stochastic
influence matrix, c for its left eigenvector at eigenvalue one,
Z=(I-W+\mathbf{1}c')^{-1} for the fundamental matrix, Z_{dg}
for Z with its off-diagonal entries set to zero and E for the
all-ones matrix, the mean lengths are
M=(I-Z+EZ_{dg})\,\mathrm{diag}(1/c) and the score is
c_{IEC}(j)=(n-1)/\sum_{i\neq j}m_{ij}. M is the mean first
passage time matrix of the chain, so the sum runs down column
j and a high score marks a node the network reaches fast.
Usage
centrality_iec(x, ...)
Arguments
x
Network input accepted by centrality.
...
Additional arguments to centrality.
Details
The influence matrix carries a unit self-loop, and the self-loop
is load-bearing. The source builds W by setting the diagonal of
the adjacency matrix to one and dividing each row by its sum,
w_{ij}=a_{ij}/\sum_j a_{ij} with a_{ii}=1, a construction it
attributes to French (1956) and states twice on page 1494, once in the
body and once in the note to Table 1. Its footnote 10 says why the
diagonal is there: a strong network with w_{ii}>0 must be regular,
meaning aperiodic, and its footnote 9 gives the two-cycle
counterexample that a zero diagonal admits. An implementation that drops
the self-loop is not computing this measure on a different scale, it is
computing a different measure.
This is not cograph's centrality_markov, and the
difference is not a rescaling. The two are both built from mean first
passage times and are easy to confuse – cograph's own candidate ledger
confused them for several rounds – but they differ twice over.
markov normalizes A without adding the diagonal, and it
divides the column sum by n, counting the excluded diagonal entry,
where equation (20) divides by n-1. The second difference is a
constant factor n/(n-1) and cannot reorder anything; the first can
and does. On the five-node star markov gives
1.25, 0.161, 0.161, 0.161, 0.161 where iec gives
0.5, 0.08, 0.08, 0.08, 0.08, and the two rank the nodes
differently on 2 of the 21 connected five-node graphs. Both are kept:
markov is the older behavior that existing results depend on,
iec is Friedkin's published measure.
Reducible input is refused, not extended. Equation (11) needs an
irreducible chain. Without one the eigenvector of equation (9) has a
dimension per closed class, so c is not determined, and
\mathrm{diag}(1/c) is undefined wherever c vanishes. The
danger is that the closed form does not announce the failure: for
i and j in different blocks z_{ij}=0, and equation (11)
then returns the entirely finite m_{ij}=z_{jj}/c_j in place of an
infinite mean first passage time. Rather than publish a finite wrong
number, cograph tests the chain first and returns NA at every node
with a cograph_undefined_measure warning. In practice the test is
connectedness of an undirected graph and strong connectedness of a
directed one, since the mandated self-loops settle aperiodicity for free.
Friedkin restricts his own analysis to regular networks and never defines
the measure outside them. centrality_rsp_betweenness
answers on disconnected input because its source states a rule for
an unreachable pair; this one states none, and a component-wise reading
would additionally have to invent whether the n-1 of equation (20)
counts the component or the network.
A singleton is NA and an empty graph returns no scores.
Equation (20) divides by n-1, which is zero when n=1; the
same NA and the same warning follow. An isolate never appears on
its own, because a graph containing one is reducible and is already
NA everywhere.
Direction is kept; weights, loops and parallel edges are not.
W is a matrix of directed influence, row i being what actor
i attends to, so a directed input is used as it stands and the
measure needs a strongly connected one. There is no in/out/all variant to
choose between, so mode, cutoff and invert_weights
are ignored. Weights are dropped, deliberately: a_{ii}=1 is
calibrated against a_{ij}=1, so multiplying every weight by a
constant would silently re-weight each actor's self-reliance against the
network, and the source demonstrates only the binary case. Loops in the
input are absorbed by the mandated unit diagonal and parallel edges
collapse, since a_{ij}=1 "wherever a line exists between two
points". The source states no normalization, so normalized = TRUE
max-scales the finished vector as elsewhere in centrality.
The source prints a complete numerical fixture. Table 1, pages
1492-1494, gives this measure to three decimals for every node of all 21
connected non-isomorphic five-node graphs. All 105 printed values are
reproduced by this implementation; see the batch 49 published audit in
the package's verification directory.
Value
Named numeric vector in input node order, NA at every node
when the influence chain is reducible or the graph has one node.
References
Friedkin, N. E. (1991). Theoretical foundations for centrality measures.
American Journal of Sociology, 96(6), 1478-1504. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1086/229694")}.
See Also
centrality_markov for the older, and different,
mean-first-passage measure, centrality_random_walk for
another chain-based score, and list_centralities for the
catalogue.
Examples
# On a complete graph W = J/n, so Z = I, every mean first passage time is
# n, and the score is (n - 1) / (n (n - 1)) = 1/n. Friedkin's Table 1
# prints .200 for the five-node case.
centrality_iec(igraph::make_full_graph(5))
# The five-node star is row 1 of that table: .500 at the center and .080
# at each leaf.
centrality_iec(igraph::make_star(5, mode = "undirected"))
# A disconnected graph has no answer: the influence chain is reducible,
# so every node is NA and a warning says why.
two <- matrix(0, 4, 4)
two[1, 2] <- two[2, 1] <- two[3, 4] <- two[4, 3] <- 1
tryCatch(centrality_iec(two), warning = conditionMessage)
cograph documentation built on Sept. 30, 2026, 5:08 p.m.
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| centrality_iec | R Documentation |
Immediate Effects Centrality
Description
Friedkin's immediate effects centrality scores a node by how quickly the
rest of the network's influence reaches it. Actors whose effects travel
over long sequences of interpersonal influence are more dependent on
intervening actors than those whose effects travel over short ones, so
the measure is the reciprocal of the mean length of the influence
sequences that end at a node. Writing W for the row-stochastic
influence matrix, c for its left eigenvector at eigenvalue one,
Z=(I-W+\mathbf{1}c')^{-1} for the fundamental matrix, Z_{dg}
for Z with its off-diagonal entries set to zero and E for the
all-ones matrix, the mean lengths are
M=(I-Z+EZ_{dg})\,\mathrm{diag}(1/c) and the score is
c_{IEC}(j)=(n-1)/\sum_{i\neq j}m_{ij}. M is the mean first
passage time matrix of the chain, so the sum runs down column
j and a high score marks a node the network reaches fast.
Usage
centrality_iec(x, ...)
Arguments
x |
Network input accepted by |
... |
Additional arguments to |
Details
The influence matrix carries a unit self-loop, and the self-loop
is load-bearing. The source builds W by setting the diagonal of
the adjacency matrix to one and dividing each row by its sum,
w_{ij}=a_{ij}/\sum_j a_{ij} with a_{ii}=1, a construction it
attributes to French (1956) and states twice on page 1494, once in the
body and once in the note to Table 1. Its footnote 10 says why the
diagonal is there: a strong network with w_{ii}>0 must be regular,
meaning aperiodic, and its footnote 9 gives the two-cycle
counterexample that a zero diagonal admits. An implementation that drops
the self-loop is not computing this measure on a different scale, it is
computing a different measure.
This is not cograph's centrality_markov, and the
difference is not a rescaling. The two are both built from mean first
passage times and are easy to confuse – cograph's own candidate ledger
confused them for several rounds – but they differ twice over.
markov normalizes A without adding the diagonal, and it
divides the column sum by n, counting the excluded diagonal entry,
where equation (20) divides by n-1. The second difference is a
constant factor n/(n-1) and cannot reorder anything; the first can
and does. On the five-node star markov gives
1.25, 0.161, 0.161, 0.161, 0.161 where iec gives
0.5, 0.08, 0.08, 0.08, 0.08, and the two rank the nodes
differently on 2 of the 21 connected five-node graphs. Both are kept:
markov is the older behavior that existing results depend on,
iec is Friedkin's published measure.
Reducible input is refused, not extended. Equation (11) needs an
irreducible chain. Without one the eigenvector of equation (9) has a
dimension per closed class, so c is not determined, and
\mathrm{diag}(1/c) is undefined wherever c vanishes. The
danger is that the closed form does not announce the failure: for
i and j in different blocks z_{ij}=0, and equation (11)
then returns the entirely finite m_{ij}=z_{jj}/c_j in place of an
infinite mean first passage time. Rather than publish a finite wrong
number, cograph tests the chain first and returns NA at every node
with a cograph_undefined_measure warning. In practice the test is
connectedness of an undirected graph and strong connectedness of a
directed one, since the mandated self-loops settle aperiodicity for free.
Friedkin restricts his own analysis to regular networks and never defines
the measure outside them. centrality_rsp_betweenness
answers on disconnected input because its source states a rule for
an unreachable pair; this one states none, and a component-wise reading
would additionally have to invent whether the n-1 of equation (20)
counts the component or the network.
A singleton is NA and an empty graph returns no scores.
Equation (20) divides by n-1, which is zero when n=1; the
same NA and the same warning follow. An isolate never appears on
its own, because a graph containing one is reducible and is already
NA everywhere.
Direction is kept; weights, loops and parallel edges are not.
W is a matrix of directed influence, row i being what actor
i attends to, so a directed input is used as it stands and the
measure needs a strongly connected one. There is no in/out/all variant to
choose between, so mode, cutoff and invert_weights
are ignored. Weights are dropped, deliberately: a_{ii}=1 is
calibrated against a_{ij}=1, so multiplying every weight by a
constant would silently re-weight each actor's self-reliance against the
network, and the source demonstrates only the binary case. Loops in the
input are absorbed by the mandated unit diagonal and parallel edges
collapse, since a_{ij}=1 "wherever a line exists between two
points". The source states no normalization, so normalized = TRUE
max-scales the finished vector as elsewhere in centrality.
The source prints a complete numerical fixture. Table 1, pages 1492-1494, gives this measure to three decimals for every node of all 21 connected non-isomorphic five-node graphs. All 105 printed values are reproduced by this implementation; see the batch 49 published audit in the package's verification directory.
Value
Named numeric vector in input node order, NA at every node
when the influence chain is reducible or the graph has one node.
References
Friedkin, N. E. (1991). Theoretical foundations for centrality measures. American Journal of Sociology, 96(6), 1478-1504. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1086/229694")}.
See Also
centrality_markov for the older, and different,
mean-first-passage measure, centrality_random_walk for
another chain-based score, and list_centralities for the
catalogue.
Examples
# On a complete graph W = J/n, so Z = I, every mean first passage time is
# n, and the score is (n - 1) / (n (n - 1)) = 1/n. Friedkin's Table 1
# prints .200 for the five-node case.
centrality_iec(igraph::make_full_graph(5))
# The five-node star is row 1 of that table: .500 at the center and .080
# at each leaf.
centrality_iec(igraph::make_star(5, mode = "undirected"))
# A disconnected graph has no answer: the influence chain is reducible,
# so every node is NA and a warning says why.
two <- matrix(0, 4, 4)
two[1, 2] <- two[2, 1] <- two[3, 4] <- two[4, 3] <- 1
tryCatch(centrality_iec(two), warning = conditionMessage)
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