centrality_lhc: Lhc Index

In cograph: Analysis and Visualization of Complex Networks

Lhc Index

Description

Wang, Yang, Liu and Ma's Lhc index is a semi-local hybrid: it reads a node's neighbor information from degree and its topological location from the share of the network's triangles that sit on it, then spreads both over a small ball and collects the result one step out. The influence of a node is C(v)=\sum_{u\in\Phi(v)}k_u(1+TP(u))/d^2(uv), a sum over the ball \Phi(v) of radius lhc_radius in which each member contributes its degree, inflated by its triangle share, discounted by the square of its distance; and the index itself is Lhc(v)=\sum_{w\in\tau(v)}C(w), the influence summed over the open neighborhood \tau(v)=N(v). The triangle share is TP(u)=NTS(u)/TNTS, with NTS(u) the number of triangles containing u and TNTS=\sum_u NTS(u).

Usage

centrality_lhc(x, ...)

Arguments

x	Network input accepted by centrality.
...	Additional arguments to centrality, including lhc_radius.

Details

The denominator is TNTS, not the number of triangles, and the paper settles it rather than the Zoo. Immediately after defining TNTS the source writes that "the total number of triangle structure exists in the network are \frac{1}{3}*TNTS", so TNTS=3\Delta for \Delta distinct triangles and TP sums to exactly one over the nodes – it really is a share. Entry 2.221 of the Centrality Zoo transcribes the structure of both equations correctly but names the denominator "\Delta, the total number of triangular structures in the network", which read literally is three times too small. The two readings are not related by a monotone transform in general, and they differ substantially: on the Krackhardt kite the paper's reading scores node 1 at 100.15 where the Zoo's literal wording gives 125.45. cograph follows the paper.

lhc_radius is the source's own parameter, exposed with the source's default. The paper writes it d, states on page 4 that "the distance ranged d is set to be 2, namely, only the nearest neighbors and the next-nearest neighbors are taken into consideration", and then sweeps it in section 3 over eleven real networks, reporting that "the optimal value of d is about 2-3" and that the correlation stabilizes beyond 3. It is therefore a genuine modeling knob rather than an implementation detail, and it is exposed with the paper's 2 as the default. At lhc_radius = 1 the ball collapses to the neighbors and C(v) becomes \sum_{u\in N(v)}k_u(1+TP(u)); a radius at or above the graph's diameter takes in everything reachable and the score stops moving. The domain is a whole number of at least one; anything else is refused with a cograph_bad_parameter error.

Both neighborhoods are open, and a node contributes to its own score. \Phi(v) is 1\le d(u,v)\le lhc_radius: the focal node is outside it, because d^2(vv)=0 would divide by zero, and unreachable nodes fall outside the radius so no infinity arises. \tau(v) is the open neighborhood. It follows – the paper does not remark on it, but its equations say so – that v does enter its own Lhc(v), since v lies in \Phi(w) at distance 1 for every neighbor w.

Triangle-free graphs are a cograph decision, taken explicitly. Every tree, star, path, even cycle and bipartite graph has TNTS=0, and TP(u) is then 0/0 everywhere. The source never mentions the case. Since TNTS is a sum of nonnegative counts, it vanishes exactly when every numerator NTS(u) vanishes too, so there is no share to distribute and no node with a claim on one: TP is written as zero, and the index reduces to the pure degree-over-squared-distance sum, which is the neighbor and location half of the hybrid with the triangle half contributing nothing. The test is made on TNTS before any division, so no 0/0 is evaluated; NA or an error would refuse every tree, which the source's own construction handles perfectly well.

Raw scores are not component-local. TNTS is a global sum, so attaching a disconnected component that carries a triangle rescales every TP and moves every score. Attaching a component with no triangle – an isolate included – changes nothing, since it changes no degree, no triangle and no finite distance inside the existing components. An isolate itself scores zero because \tau(v) is empty and equation (2) is an empty sum; a singleton graph and every node of an edgeless graph score zero for the same reason, and an empty graph returns no scores.

Direction, weights, loops and parallel edges are dropped to the simple undirected skeleton the source defines on: k_u is a count, d(uv) a hop count and NTS(u) a combinatorial quantity, and the paper's eleven networks are simple and undirected. There is no in/out/all variant to select, so the measure sits in the no-mode family, and cutoff and invert_weights are ignored as well. The source states no normalization, so normalized = TRUE max-scales the finished vector as elsewhere in centrality.

The source prints no numerical example. There is no toy graph with a table of scores anywhere in the paper – its Table 1 lists network statistics and its figures are aggregate SIR and Kendall plots – so there is no published per-node fixture to reproduce. Verification rests instead on independent reference implementations and on hand-derived closed forms for stars, complete graphs, rings and paths.

Value

Named numeric vector in input node order.

References

Wang, X., Yang, Q., Liu, M. and Ma, X. (2021). Comprehensive influence of topological location and neighbor information on identifying influential nodes in complex networks. PLoS ONE, 16(5), e0251208. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1371/journal.pone.0251208")}.

See Also

centrality_hcc and centrality_ked for other degree-and-position hybrids, centrality_neighbor_distance for another distance-discounted neighborhood sum, and list_centralities for the catalogue.

Examples

# The path 1-2-3 is triangle-free, so the triangle share drops out and
# the scores are the hand-derived 2, 4.5, 2.
centrality_lhc(igraph::make_graph(c(1, 2, 2, 3), directed = FALSE))

# On a complete graph every node scores (n-1)^3 (n+1) / n; for n = 5
# that is 76.8.
centrality_lhc(igraph::make_full_graph(5))

# Widening the ball can only raise the score, and it stops moving once
# the radius reaches the diameter.
ring <- igraph::make_ring(9)
centrality_lhc(ring, lhc_radius = 1)
centrality_lhc(ring)
centrality_lhc(ring, lhc_radius = 4)

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