centrality_lnc: Local neighbor contribution centrality

In cograph: Analysis and Visualization of Complex Networks

Local neighbor contribution centrality

Description

The local neighbor contribution (LNC) of Dai, Wang, Sheng, Sun, Khawaja, Ullah, Dejene and Duan multiplies what a node contributes on its own by what its neighborhood contributes to it: LNC(i)=d_i^{3}\,(1-1/d_i)^{d_i-1}\, \bigl(\sum_{j\in N(i)}d_j\bigr)/(n-1), with 0^0=1. The first two factors are the source's own contribution ownCon(i)=d_i(1-1/d_i)^{d_i-1}, the chance that a node picking one neighbor uniformly at random reaches a given one and misses the rest, scaled by its degree; the rest is the neighbor contribution neiCon(i)=d_i^{2}\sum_{j\in N(i)}d_j/(n-1), the source's cluster degree weighted by its neighbors' degree centralities.

Usage

centrality_lnc(x, ...)

Arguments

x	Network input accepted by centrality.
...	Additional arguments to centrality.

Details

The measure takes no parameters. The source calls this out as a feature, "Parameter-Free: LNC does not rely on prior knowledge and parameter adjustments", so none is offered.

Raw scores are not comparable across graphs of different order. The 1/(n-1) comes from the degree centrality of equation (1), where n is the vertex count of the whole network, not of the node's component. Adding a disconnected component therefore multiplies every score by (n-1)/(n'-1), leaving the ranking alone and the raw values not.

The source's printed equations do not literally give its printed numbers, and cograph follows the numbers. Equations (4) and (5) both sum a term over j=1,\dots,k, and k is described three incompatible ways: the prose calls it the number of nearest and next nearest neighbors, Algorithm 1 line 12 sets it to the degree, and equation (5) taken literally carries one factor of d_i too many. The printed intermediates D(v_5)=12, ownCon(v_5)=1.6875 and neiCon(v_5)=19.2, together with all eleven Table 1 influences, are reproduced by exactly one pair of factors, the one above: k acts as d_i in (5) and as d_i^2 in (4). The equally literal split that moves one d_i from the neighbor factor to the own factor gives the same product, so the measure itself is unambiguous.

This is not the Centrality Zoo's formula. Zoo section 2.238 writes the own contribution as d_i|N^{(\le 2)}(i)|\sum_{j\in N^{(\le 2)}(i)}(1/d_j) (1-1/d_j)^{|N^{(\le 2)}(i)|-1}, replacing the focal node's own contribution probability P(v_i) by each neighbor's P(v_j) and the binomial count d_i by the size of the two-hop neighborhood; its neighbor factor is right in form but uses that same two-hop size where the printed numbers need d_i^2. On the source's own Figure 1 the Zoo reading reproduces none of the eleven printed values and inverts the paper's headline ranking, scoring v_8 32.23 above v_5 28.90 where the paper prints 32.4 for v_5 and 29.7 for v_8, and lifting the degree-two nodes v_6, v_7 above the degree-three v_9. cograph implements the paper. No Zoo variant is offered.

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. Isolates score zero, and so does the single node of a singleton graph: the source has no value there, since P(v_i)=1/0 and the n-1 denominator vanishes, and zero is a cograph extension chosen because d_i^3 and the empty neighbor-degree sum are both zero. Empty graphs return no scores. The source states no normalization; normalized = TRUE max-scales the finished vector as elsewhere in centrality. Nothing overflows: the cubed degree is bounded by n^3, the neighbor-degree sum by twice the edge count, and the binomial factor lies in [1/4, 1]. Cost is one sparse matrix-vector product, O(n + m).

Numerical verification establishes agreement with the source's printed Table 1 and printed intermediates, 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

Dai, J., Wang, B., Sheng, J., Sun, Z., Khawaja, F. R., Ullah, A., Dejene, D. A. and Duan, G. (2019). Identifying influential nodes in complex networks based on local neighbor contribution. IEEE Access, 7, 131719-131731. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1109/ACCESS.2019.2939804")}.

See Also

centrality_semilocal and centrality_neighbor_distance for other neighborhood sums, and list_centralities for the catalogue.

Examples

# Every node of a ring has degree two and a neighbor-degree sum of four
centrality_lnc(igraph::make_ring(6))

# A star: the center carries the whole neighborhood
centrality_lnc(igraph::make_star(5, mode = "undirected"))

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