Fix the pagerank and triangle centrality examples (#162)

* Fix the pagerank and triangle centrality examples

- pagerank: drop the removed `GBArray` annotation and import `second`.
- triangle_centrality: pass the Descriptor as `desc=` and import
  `second`. Use `-` (eadd) instead of `.-`, which broadcasts as emul and
  dropped vertices that are in no triangle but have triangle neighbours.
- triangle_centrality1: multiply by the pattern of T (`(+, second)`)
  instead of its triangle counts, as in the definition.
- Add reference headers.

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>

* Test pagerank and triangle centrality and list them in the docs

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>

---------

Co-authored-by: Claude Opus 5.5 <noreply@anthropic.com>

Author: ViralBShah

docs/src/examples.md

@@ -22,13 +22,15 @@ Unless noted otherwise `A` is the adjacency matrix of an undirected graph with `
 | Triangle counting (Sandia) | `Sandia.jl` | `sandia(A)` | |
 | Triangle counting (CMU) | `CMU.jl` | `cmu(A, n)` | |
 | Triangles per vertex | `NodeWiseTriangleCount.jl` | `nodewise_tc(A)` | |
+| Triangle centrality | `triangle_centrality.jl` | `triangle_centrality(A)`, `triangle_centrality1(A)` | `A` has unit weights. |
 | k-truss | `KTruss.jl` | `ktruss(A, k)` | Returns the truss with edge supports as values. |
 | BFS parents | `BFSParents.jl` | `bfs_parents(A, s, n)` | The parent of the source is `0`. |
 | Multi-source BFS parents | `BFSParentsMultiSource.jl` | `ms_bfsp(A, S, n, s)` | Row `i` of the `s × n` matrix `S` marks source `i`. |
 | Multi-source BFS levels | `BFSLevelsMultiSource.jl` | `ms_bfsl(A, S, n, s)` | `A` and `S` are `Bool`. Sources are level `1`, unreachable vertices `0`. |
 | Bellman-Ford | `BellmanFord.jl` | `bellmanford(A, s, n)` | `A[i,j]` is the weight of edge `i → j`, with explicit zeros on the diagonal. |
 | Floyd-Warshall | `FloydWarshall.jl` | `floydwarshall(A, n)` | Same input as Bellman-Ford. |
 | Prim's minimum spanning tree | `BasicPrim.jl`, `AdvancedPrim.jl` | `b_prim(A, n, zeros(n))`, `a_prim(A, n, zeros(n))` | `A` is dense, with `Inf` where there is no edge and zeros on the diagonal. |
+| PageRank | `pagerank.jl` | `pagerank(A)` | `A[i,j]` is the edge `i → j`. Returns `(ranks, iterations)`. Ranks of dangling vertices are not redistributed. |
 | Community detection by label propagation | `CDLP.jl` | `cdlp(A, n, t)` | Runs `t` iterations. Every vertex must have a neighbour. |
 
 The graph below has edges `1-2`, `1-3`, `2-3`, `2-4`, `3-4` and `4-5`, and two triangles, `{1,2,3}` and `{2,3,4}`.

examples/pagerank.jl

@@ -1,6 +1,14 @@
+#=
+PageRank (GAP benchmark variant, dangling vertices are not redistributed):
+- Reference: https://doi.org/10.1109/HPEC49654.2021.9622789, https://en.wikipedia.org/wiki/PageRank
+=#
+
 using SuiteSparseGraphBLAS
+using SuiteSparseGraphBLAS: second
+
+#A is the adjacency matrix (A[i,j] is the edge i->j), d the out-degrees, Output: ranks and number of iterations
 function pagerank(
-    A::SuiteSparseGraphBLAS.GBArray,
+    A,
     d = reduce(+, A; dims=2),
     α = 0.85,
     maxiters = 100,

examples/triangle_centrality.jl

@@ -1,15 +1,22 @@
+#=
+Triangle centrality:
+- Reference: https://arxiv.org/abs/2105.00110, https://doi.org/10.1109/HPEC49654.2021.9622789
+=#
+
 using SuiteSparseGraphBLAS
-using SuiteSparseGraphBLAS: pair
+using SuiteSparseGraphBLAS: pair, second
+
+#A is the adjacency matrix of an undirected graph with unit weights, Output: triangle centrality of each vertex
 function triangle_centrality1(A)
     T = *(A, A', (+, *), mask=A)
     y = reduce(+, T, dims=2)
     k = reduce(+, y)
-    return (3 * *(A, y) - 2 * *(T, y) .+ y) ./ k
+    return (3 * *(A, y) - 2 * *(T, y, (+, second)) .+ y) ./ k
 end
 
 function triangle_centrality(A)
-    T = *(A, A', (+, pair), mask=A, Descriptor(structural_mask=true))
+    T = *(A, A', (+, pair), mask=A, desc=Descriptor(structural_mask=true))
     y = reduce(+, T, dims=2)
     k = reduce(+, y)
-    return (3.0 * *(A, y) .- (2.0 * *(T, y, (+, second))) .+ y) ./ k
+    return (3.0 * *(A, y) - (2.0 * *(T, y, (+, second))) .+ y) ./ k
 end

test/examples.jl

@@ -3,7 +3,7 @@ using SuiteSparseGraphBLAS, SparseArrays, LinearAlgebra, Test
 
 for f in ["AdvancedPrim", "BasicPrim", "BellmanFord", "BFSLevelsMultiSource", "BFSParents",
           "BFSParentsMultiSource", "CDLP", "CMU", "Cohen", "FloydWarshall", "KTruss",
-          "NodeWiseTriangleCount", "Sandia"]
+          "NodeWiseTriangleCount", "Sandia", "pagerank", "triangle_centrality"]
     include(joinpath(@__DIR__, "..", "examples", "$f.jl"))
 end
 
@@ -59,4 +59,16 @@ end
     @test diag(Matrix(cdlp(T, 6, 2))) == [1, 1, 1, 4, 4, 4]
     @test diag(Matrix(cdlp(A, 5, 1))) == [2, 1, 1, 2, 4]
 end
+
+@testset "centrality" begin
+    Af = GBMatrix(sym(ones(6)))
+    # Vertex 5 is in no triangle and only sees the triangles of its neighbour 4.
+    @test collect(triangle_centrality(Af)) ≈ [5/6, 1, 1, 5/6, 1/2]
+    @test collect(triangle_centrality1(Af)) ≈ [5/6, 1, 1, 5/6, 1/2]
+    # PageRank is uniform on a directed cycle, otherwise compare with dense power iteration.
+    @test collect(pagerank(GBMatrix([1, 2, 3, 4], [2, 3, 4, 1], 1.0))[1]) ≈ fill(0.25, 4)
+    M = Matrix(sym(ones(6))); x = fill(0.2, 5)
+    for _ in 1:100; x = 0.15 / 5 .+ 0.85 .* (M' * (x ./ vec(sum(M, dims=2)))); end
+    @test collect(pagerank(Af)[1]) ≈ x atol=1e-5
+end
 end