pagerank(): Fix problem with sink nodes

Nodes with out-degree zero are now assumed to connect to every other
node.

src/graph/centrality/graph_pagerank.hh

@@ -41,12 +41,14 @@ struct get_pagerank
         RankMap deg(vertex_index, num_vertices(g));
 
         // init degs
-        parallel_vertex_loop
-            (g,
-             [&](auto v)
-             {
-                 put(deg, v, out_degreeS()(v, g, weight));
-             });
+        std::vector<size_t> sinks;
+        for (auto v : vertices_range(g))
+        {
+            auto k = out_degreeS()(v, g, weight);
+            put(deg, v, k);
+            if (k == 0)
+                sinks.push_back(v);
+        }
 
         rank_type delta = epsilon + 1;
         rank_type d = damping;
@@ -54,13 +56,21 @@ struct get_pagerank
         while (delta >= epsilon)
         {
             delta = 0;
-            #pragma omp parallel if (num_vertices(g) > OPENMP_MIN_THRESH) \
+
+            double p_sink = 0;
+            #pragma omp parallel if (sinks.size() > OPENMP_MIN_THRESH)      \
+                reduction(+:p_sink)
+            parallel_loop_no_spawn
+                (sinks,
+                 [&](auto, auto v){ p_sink += get(rank, v); });
+
+            #pragma omp parallel if (num_vertices(g) > OPENMP_MIN_THRESH)   \
                 reduction(+:delta)
             parallel_vertex_loop_no_spawn
                 (g,
                  [&](auto v)
                  {
-                     rank_type r = 0;
+                     rank_type r = p_sink * get(pers, v);
                      for (const auto& e : in_or_out_edges_range(v, g))
                      {
                          auto s = source(e, g);

src/graph_tool/centrality/__init__.py

@@ -61,8 +61,7 @@ __all__ = ["pagerank", "betweenness", "central_point_dominance", "closeness",
 
 def pagerank(g, damping=0.85, pers=None, weight=None, prop=None, epsilon=1e-6,
              max_iter=None, ret_iter=False):
-    r"""
-    Calculate the PageRank of each vertex.
+    r"""Calculate the PageRank of each vertex.
 
     Parameters
     ----------
@@ -109,8 +108,8 @@ def pagerank(g, damping=0.85, pers=None, weight=None, prop=None, epsilon=1e-6,
         PR(v) = \frac{1-d}{N} + d \sum_{u \in \Gamma^{-}(v)}
                 \frac{PR (u)}{d^{+}(u)}
 
-    where :math:`\Gamma^{-}(v)` are the in-neighbors of v, :math:`d^{+}(w)` is
-    the out-degree of w, and d is a damping factor.
+    where :math:`\Gamma^{-}(v)` are the in-neighbors of v, :math:`d^{+}(u)` is
+    the out-degree of u, and d is a damping factor.
 
     If a personalization property :math:`p(v)` is given, the definition becomes:
 
@@ -129,6 +128,10 @@ def pagerank(g, damping=0.85, pers=None, weight=None, prop=None, epsilon=1e-6,
     where :math:`d^{+}(u)=\sum_{y}A_{u,y}w_{u\to y}` is redefined to be the sum
     of the weights of the out-going edges from u.
 
+    If a node has out-degree zero, it is assumed to connect to every other node
+    with a weight proportional to :math:`p(v)` or a constant if no
+    personalization is given.
+
     The implemented algorithm progressively iterates the above equations, until
     it no longer changes, according to the parameter epsilon. It has a
     topology-dependent running time.