IIRA is centrality_ira with the receiver's share scaled by
how much of a spreading process that receiver could actually carry:
\(a_{ij}=[1-(1-\beta)^{k_i}]\,\theta_i
(\sum_{u\in\Gamma(j)}\theta_u)^{-1}\), where \(k_i\) is the degree of
\(i\) and \(\beta\) the spreading rate. The recursion and the
initial condition \(I(0)=(1,\dots,1)\) are unchanged; there is no
\(\alpha\) exponent, and the denominator keeps the plain masses.
Arguments
- x
Network input accepted by
centrality.- ira_mass
Node centrality \(\theta\):
"coreness"(default, the k-shell index the source's worked example uses) or"degree". Shared withcentrality_ira.- iira_beta
Spreading rate \(\beta\), a single number in \((0,1]\); default 0.2, the source's worked-example value. The source sweeps \(\beta\) in its experiments and recommends no other default.
- iira_steps
Number of iterations \(t\), a single nonnegative whole number; default 50, the source's worked-example value. Zero returns \(I(0)\).
- ...
Additional arguments to
centrality.
Details
The scores are tiny and only their order means anything. The
factor \(\psi_i=1-(1-\beta)^{k_i}\) is strictly below one, so every
column of \(A\) sums to less than one, the spectral radius is below
one, and \(I(t)\to 0\) geometrically. The source runs exactly
\(t=50\) steps and prints an \(I(50)\) of order \(10^{-20}\);
cograph returns that raw vector, so the printed example is reproducible,
and normalized = TRUE max-scales it into \([0,1]\) for reading.
Never compare raw IIRA scores across connected components: each
component decays at its own rate, so after iira_steps steps they
sit on different exponential scales. A large iira_steps underflows
to zero.
The Centrality Zoo entry is not this formula. Section 2.185 prints \(p_{ij}=(1-(1-\beta)^{d_i})a_{ij}c_i/\sum_k a_{ik}c_k\), which pairs the numerator's index with the denominator's own neighborhood; the source pairs them with opposite sets. As printed, the Zoo's row sums are \(\psi_i c_i d_i/\sum_{k\in N(i)}c_k\), so its matrix is stochastic in neither direction although the entry calls it stochastic, and it does not reproduce the source's printed matrix or its printed \(I(50)\). cograph implements the source.
Uses the simple undirected unweighted skeleton, which is the source
domain: either arc creates one edge, parallel edges count once and loops
are removed. Edge weights, mode, cutoff and path-weight inversion are
ignored. An isolate has an empty neighbor sum and \(\psi=0\), so it
scores zero from the first step; that is the value of the source's empty
sum, not an accidental zero. iira_steps = 0 returns the initial
\(I(0)\), a vector of ones. Empty graphs return no scores. Cost is one
dense \(n^2\) matrix plus iira_steps matrix-vector products.
The version of record was not read: what was read is the author preprint arXiv:1505.03214v1, whose method section, worked example and figures carry the definition reproduced here. Numerical verification establishes agreement with those equations and with every value printed in the preprint's figure 2 example, not parity with author software, which does not exist, and not any claim about spreading performance.
References
Zhong, L.-F., Liu, J.-G. and Shang, M.-S. (2015). Iterative resource allocation based on propagation feature of node for identifying the influential nodes. Physics Letters A, 379(38), 2272-2276. Equations 1, 2 and 4 and figure 2 on page 2 of the author preprint arXiv:1505.03214v1, which is what was read. doi:10.1016/j.physleta.2015.05.021 .
See also
centrality_ira for the measure this improves, and
list_centralities for the catalogue.
Examples
# The source's figure 2, whose printed I(50) is
# 8.19e-20, 4.32e-20, 4.32e-20, 6.7e-21, 6.7e-21
fig2 <- igraph::make_graph(c(1, 2, 1, 3, 2, 3, 1, 4, 1, 5),
directed = FALSE)
centrality_iira(fig2)
#> 1 2 3 4 5
#> 8.193120e-20 4.317938e-20 4.317938e-20 6.698728e-21 6.698728e-21
# Only the order carries meaning, so max-scale for reading
centrality_iira(fig2, normalized = TRUE)
#> 1 2 3 4 5
#> 1.00000000 0.52702007 0.52702007 0.08176041 0.08176041