Weighted Graph Layouts with grip

Why weighted layouts deserve their own vignette

Classical GRIP is fundamentally combinatorial: it sees graph topology through hop-count neighborhoods. The unified grip() interface can instead make positive edge lengths a first-class geometric signal throughout the multiscale hierarchy:

grip(metric = "hop") selects the topology-first engine,
grip(metric = "edge_length") selects the edge-length-metric engine,
trace.grip() accepts the same metric choice, and
build.weighted.misf() exposes weighted hierarchy construction directly.

The package decision rule is:

use grip(metric = "hop") for ordinary unweighted or topology-first graphs,
use grip(metric = "edge_length") when edge lengths carry geometry you want to preserve,
add GKK/LGKK only after you already have weighted candidate layouts and need advanced experimental geodesic-aware scoring or polish.

library(grip)

plot.layout.triptych <- function(coords.list,
                                 edges,
                                 titles,
                                 projection = NULL,
                                 vertex.cols = rep("black", length(coords.list)),
                                 edge.col = "gray82") {
  op <- par(
    mfrow = c(1, length(coords.list)),
    mar = c(1.2, 1.2, 3, 1.2),
    bg = "white"
  )
  on.exit(par(op), add = TRUE)

  for (i in seq_along(coords.list)) {
    plot.layout(
      coords.list[[i]], edges,
      projection = projection,
      main = titles[[i]],
      vertex.col = vertex.cols[[i]],
      edge.col = edge.col
    )
  }
}

A first weighted surface example

The helper below creates a plain mesh topology whose edge lengths are induced by a curved 3D surface. The topology stays simple, but the intended metric is no longer the flat grid metric.

surface.mesh <- mesh.surface.graph(
  5, 5,
  surface = "saddle",
  amplitude = 0.9
)

coords.unweighted <- grip(
  surface.mesh$edges,
  n = surface.mesh$n,
  dim = 3,
  preset = "mesh",
  seed = 1
)

coords.weighted <- grip(metric = "edge_length",
  surface.mesh$edges,
  n = surface.mesh$n,
  edge_weights = surface.mesh$edge_weights,
  dim = 3,
  preset = "mesh",
  seed = 1
)

gkk.prepared <- prepare.geodesic.kk(
  surface.mesh$edges,
  n = surface.mesh$n,
  edge_weights = surface.mesh$edge_weights
)

surface.summary <- do.call(
  rbind,
  list(
    cbind(
      method = "Combinatorial GRIP",
      score.geodesic.kk(
        coords.unweighted,
        prepared = gkk.prepared
      )[, c(
        "gkk.weighted.rmse",
        "gkk.mean.abs.path.error",
        "gkk.mean.rel.path.error"
      )]
    ),
    cbind(
      method = "Weighted GRIP",
      score.geodesic.kk(
        coords.weighted,
        prepared = gkk.prepared
      )[, c(
        "gkk.weighted.rmse",
        "gkk.mean.abs.path.error",
        "gkk.mean.rel.path.error"
      )]
    )
  )
)

knitr::kable(surface.summary, digits = 3)

method	gkk.weighted.rmse	gkk.mean.abs.path.error	gkk.mean.rel.path.error
Combinatorial GRIP	7.943	7.392	0.109
Weighted GRIP	4.057	3.911	0.058

plot.layout.triptych(
  list(
    surface.mesh$coords_surface,
    coords.unweighted,
    coords.weighted
  ),
  edges = surface.mesh$edges,
  titles = c("Target geometry", "Combinatorial GRIP", "Weighted GRIP"),
  projection = "ortho",
  vertex.cols = c("#666666", "black", "#1F3B73")
)

The important pattern is not that one method always wins on every graph. It is that weighted GRIP is solving a different problem: it tries to respect the graph’s edge-length geometry, not only its combinatorial adjacency structure.

2D versus 3D on the same weighted graph

For many weighted geometric families, 3D is the more informative target space. The graph metric can be difficult or impossible to represent faithfully in 2D without substantial distortion.

coords.weighted.2d <- grip(metric = "edge_length",
  surface.mesh$edges,
  n = surface.mesh$n,
  edge_weights = surface.mesh$edge_weights,
  dim = 2,
  preset = "mesh",
  seed = 2
)

coords.weighted.3d <- grip(metric = "edge_length",
  surface.mesh$edges,
  n = surface.mesh$n,
  edge_weights = surface.mesh$edge_weights,
  dim = 3,
  preset = "mesh",
  seed = 2
)

dim.summary <- do.call(
  rbind,
  list(
    cbind(
      dim = "2D",
      score.geodesic.kk(
        coords.weighted.2d,
        prepared = gkk.prepared
      )[, c(
        "gkk.weighted.rmse",
        "gkk.mean.abs.path.error",
        "gkk.mean.rel.path.error"
      )]
    ),
    cbind(
      dim = "3D",
      score.geodesic.kk(
        coords.weighted.3d,
        prepared = gkk.prepared
      )[, c(
        "gkk.weighted.rmse",
        "gkk.mean.abs.path.error",
        "gkk.mean.rel.path.error"
      )]
    )
  )
)

knitr::kable(dim.summary, digits = 3)

dim	gkk.weighted.rmse	gkk.mean.abs.path.error	gkk.mean.rel.path.error
2D	4.058	3.911	0.058
3D	4.055	3.910	0.058

op <- par(mfrow = c(1, 2), mar = c(1.2, 1.2, 3, 1.2), bg = "white")
on.exit(par(op), add = TRUE)

plot.layout(
  coords.weighted.2d,
  surface.mesh$edges,
  main = "Weighted GRIP in 2D",
  vertex.col = "black",
  edge.col = "gray82"
)


plot.layout(
  coords.weighted.3d,
  surface.mesh$edges,
  projection = "ortho",
  main = "Weighted GRIP in 3D",
  vertex.col = "#1F3B73",
  edge.col = "gray82"
)

This is why the weighted benchmark work in grip treats 3D as the primary track and 2D as an informative limitation track.

Weighted presets

The weighted API keeps explicit presets tuned for the major weighted-family classes currently shipped with the package.

Family class	Good weighted preset	Typical use
Lifted mesh surfaces	`preset = "mesh"`	Rectangular weighted surfaces
Cylindrical grids	`preset = "cylinder"`	Open wrapped surfaces
Toroidal grids	`preset = "torus"`	Closed wrapped surfaces
Near-spherical surfaces	`preset = "sphere"`	Closed surface families
Irregular manifolds and porous families	`preset = "irregular"`	Non-lattice weighted manifolds
Intrinsic weighted trees	`preset = "tree"`	Edge-length-driven tree geometry
Recursive carpet-like lattices	`preset = "carpet"`	Recursive hole-rich weighted grids

These presets are starting points, not declarations that the graph belongs to a single correct family.

Intrinsic weighted trees

Weighted families do not need to come from ambient surfaces. They can also be intrinsically weighted. The example below keeps the topology of a binary tree but assigns edge lengths by depth and branch position.

tree.graph <- kary.tree.weighted.graph(
  k = 2,
  depth = 4,
  depth_rule = "geometric",
  depth_decay = 0.82,
  branch_rule = "linear",
  branch_spread = 0.25
)

tree.coords <- grip(metric = "edge_length",
  tree.graph$edges,
  n = tree.graph$n,
  edge_weights = tree.graph$edge_weights,
  dim = 2,
  preset = "tree",
  seed = 3
)

knitr::kable(
  head(tree.graph$edge_table[, c(
    "parent",
    "child",
    "child_depth",
    "branch_index",
    "edge_weight"
  )]),
  digits = 3
)

parent	child	child_depth	branch_index	edge_weight
1	2	1	1	1.411
1	3	1	2	1.814
2	4	2	1	1.157
2	5	2	2	1.487
3	6	2	1	1.157
3	7	2	2	1.487

plot.layout(
  tree.coords,
  tree.graph$edges,
  main = "Intrinsic weighted tree",
  vertex.col = "#1F3B73",
  edge.col = "gray80",
  pch = 16,
  cex = 0.55
)

This kind of example is useful because the geometry lives in the edge lengths themselves rather than in a chosen 3D embedding.

Trace and advanced geodesic hooks

The weighted API also supports:

trace.grip(metric = "edge_length") for traced weighted solves,
prepare.geodesic.kk() / score.geodesic.kk() for full GKK evaluation,
prepare.landmark.geodesic.kk() / score.landmark.geodesic.kk() for sparse LGKK evaluation,
lgkk_polish_rounds for post-layout landmark geodesic KK refinement,
lgkk_multiscale_rounds and the stage-specific LGKK controls for in-core multiscale refinement.

These GKK/LGKK tools are public, but they are not the default entry path. For most weighted problems, start with grip(metric = "edge_length") and use the geodesic tools only when you need a stronger metric comparison or an experimental polish step.

Here is the smallest traced weighted example pattern:

surface.trace <- trace.grip(metric = "edge_length",
  surface.mesh$edges,
  n = surface.mesh$n,
  edge_weights = surface.mesh$edge_weights,
  dim = 3,
  preset = "mesh",
  trace = "level",
  diagnostics = "light",
  seed = 1
)

head(surface.trace$meta)
head(surface.trace$diagnostics)

For small or medium weighted graphs where geodesic fidelity matters strongly, it is often worth comparing:

weighted GRIP alone,
weighted GRIP + experimental core LGKK,
weighted GRIP + experimental polish LGKK,
and the KK->GKK or KK->LGKK baselines.

Where to go next

Getting Started with grip shows how the weighted API fits into the broader package.
Choosing Layouts for Real Data focuses on candidate search and scoring.
Tracing and Diagnosing Layouts covers the trace APIs in more detail.
Synthetic Graph Families and Geometries explains the benchmark-family library that supports these weighted workflows.