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.
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.
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. Definitions 1-5, equations (1)-(6) and Algorithm 1, journal pages 131721-131723, with the Figure 1 graph and Table 1 on page 131720. doi:10.1109/ACCESS.2019.2939804 .
See also
centrality_semilocal and
centrality_neighbor_distance for other neighborhood
sums, and list_centralities for the catalogue.