centrality_ira: Iterative resource allocation (IRA)

In cograph: Analysis and Visualization of Complex Networks

Iterative resource allocation (IRA)

Description

Every node starts with one unit of resource and hands it to its neighbors in proportion to the receiver's centrality, repeatedly, until the amounts stop moving. The share node j sends to a neighbor i is a_{ij}=\theta_i^{\alpha}/\sum_{u\in\Gamma(j)}\theta_u^{\alpha}, the recursion is I(t+1)=AI(t) from I(0)=(1,\dots,1), and the steady state I ranks the spreaders. Because every non-isolate column of A sums to one, the total resource is conserved: \sum_i I_i(t)=n at every step on a graph with no isolates, and each connected component keeps its own vertex count.

Usage

centrality_ira(
  x,
  ira_mass = "coreness",
  ira_alpha = 1,
  ira_tol = 1e-06,
  ira_max_iter = 1000,
  ...
)

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". The source also mentions closeness and betweenness.
ira_alpha	Exponent \alpha on the mass, a single finite number; default one, the only value the source uses.
ira_tol	Stopping tolerance \varepsilon on the largest absolute change between successive iterates; default 1e-6, the source's own value.
ira_max_iter	Iteration bound, a single whole number of at least one; default 1000. Reaching it raises cograph_no_converge.
...	Additional arguments to centrality.

Details

The equilibrium has a closed form. Writing s_i=\sum_{u\in\Gamma(i)} \theta_u^{\alpha}, the limit is I_i\propto\theta_i^{\alpha}s_i within each component, scaled so the component's scores sum to its size. On the source's own figure 1(a) that reproduces the printed [15/8, 5/4, 5/4, 5/16, 5/16] exactly. cograph nevertheless iterates, because the iteration is what the source defines and what its table reports, and because the closed form is a limit that need not exist; see the next paragraph.

The iteration does not always converge, and cograph says so. A is the transition matrix of a reversible walk, so on a bipartite component it has an eigenvalue of exactly -1. The coefficient of that eigenvector in I(0)=(1,\dots,1) is the difference in size between the component's two vertex classes, so the iteration settles into a period-two cycle, never meets ira_tol, and returns a value that depends on the parity of the last step. The three-star alternates for ever between (3,1/3,1/3,1/3) and (1,1,1,1), while the four-path, whose classes are equal, converges to (2/3,4/3,4/3,2/3). Neither the source nor the Centrality Zoo mentions this. cograph runs the source's own rule, stops at ira_max_iter, raises a cograph_no_converge warning naming the largest remaining change, and returns I at ira_max_iter. It does not silently report that iterate as an equilibrium, and it does not substitute the average of the two alternating iterates, which would converge but is not the source's rule. Every graph in the source's own figure 1 carries a triangle and converges.

The Centrality Zoo (section 2.204) states the transpose, p_{ij}=a_{ij}c_j^{\alpha}/\sum_k a_{ik}c_k^{\alpha}, and asks for the principal left eigenvector of P. That is the same object up to scale on a graph where the limit exists, but it is not the source's finite iteration: it sidesteps the parity problem instead of reporting it, and it carries no \sum_i I_i=n scale.

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 is in nobody's neighborhood, so it receives nothing and its own unit is not passed on: it scores zero from the first step, which is the value of the source's empty sum and not an accidental zero, and it is the reason \sum_i I_i=n is stated only for graphs with no isolates. Empty graphs return no scores. Cost is one dense n^2 matrix plus one matrix-vector product per iteration.

Numerical verification establishes agreement with the source equations and with every value printed in the source's table 1, not parity with author software, which does not exist, and not any claim about spreading performance.

Value

Named numeric vector in input node order.

References

Ren, Z.-M., Zeng, A., Chen, D.-B., Liao, H. and Liu, J.-G. (2014). Iterative resource allocation for ranking spreaders in complex networks. EPL (Europhysics Letters), 106(4), 48005. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1209/0295-5075/106/48005")}.

See Also

centrality_iira for the improved variant, and list_centralities for the catalogue.

Examples

# The source's figure 1(a): a triangle with two pendants on one corner.
# The printed steady state is 15/8, 5/4, 5/4, 5/16, 5/16.
fig1a <- igraph::make_graph(c(1, 2, 1, 3, 2, 3, 1, 4, 1, 5),
                            directed = FALSE)
centrality_ira(fig1a)

# The source's other mass, and a nonlinear exponent
centrality_ira(fig1a, ira_mass = "degree", ira_alpha = 2)

cograph documentation built on Sept. 30, 2026, 5:08 p.m.