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.
Arguments
- x
Network input accepted by
centrality.- ...
Additional arguments to
centrality.
Value
Named numeric vector in input node order, NA at every node
when the influence chain is reducible or the graph has one node.
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.
References
Friedkin, N. E. (1991). Theoretical foundations for centrality measures. American Journal of Sociology, 96(6), 1478-1504. Equation (9) on page 1485, equation (11) on page 1486, equation (20) on page 1489, the construction of \(W\) and Table 1 on pages 1492-1494. doi:10.1086/229694 . The fundamental matrix and the mean first passage form are Kemeny, J. G. and Snell, J. L. (1960). Finite Markov Chains, page 79.
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))
#> 1 2 3 4 5
#> 0.2 0.2 0.2 0.2 0.2
# 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"))
#> 1 2 3 4 5
#> 0.50 0.08 0.08 0.08 0.08
# 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)
#> [1] "`iec` has no value on this input, so its column is NA: with the unit self-loops the source mandates, the influence chain is still reducible, so the left eigenvector of equation (9) is not determined and the mean first passage times of equation (11) are infinite between classes. Restrict the input to a connected undirected graph or a strongly connected directed one."