web-ui/src/components/graph/graph-canvas.tsx

@@ -37,6 +37,14 @@ type FGLink = {
   treatment: string | null;
 };
 
+export type ColorBy =
+  | "type"
+  | "practice_area"
+  | "community"
+  | "precedent_level"
+  | "recency";
+export type SizeBy = "indegree" | "pagerank" | "betweenness";
+
 const NODE_COLORS: Record<string, string> = {
   precedent: "#1e3a5f", // navy
   halacha: "#b45309", // amber
@@ -51,8 +59,66 @@ const TREATMENT_COLORS: Record<string, string> = {
   distinguished: "#d97706",
 };
 
-function nodeRadius(n: GraphNode): number {
+// Distinct, cyclic palette for community (cluster) colouring.
+export const COMMUNITY_PALETTE = [
+  "#1e3a5f", "#a97d3a", "#3b7a57", "#8c3b4a", "#5b4b8a", "#b06a2c",
+  "#2f6f7a", "#9a4f6a", "#46688a", "#7d6b3a", "#6b7280", "#4d7c5a",
+];
+
+// Authority hierarchy: עליון darkest → ועדת ערר lightest.
+export const LEVEL_COLORS: Record<string, string> = {
+  עליון: "#13294b",
+  מנהלי: "#3b6ea5",
+  ועדת_ערר_מחוזית: "#8fb0cf",
+};
+
+export const PA_COLORS: Record<string, string> = {
+  rishuy_uvniya: "#1e3a5f",
+  betterment_levy: "#a97d3a",
+  compensation_197: "#3b7a57",
+};
+
+const FALLBACK_COLOR = "#94a3b8";
+
+/** Old (slate) → recent (gold) gradient over 1994–2026. */
+export function recencyColor(dateStr: string | null): string {
+  if (!dateStr) return FALLBACK_COLOR;
+  const y = Number(dateStr.slice(0, 4));
+  if (!y) return FALLBACK_COLOR;
+  const t = Math.max(0, Math.min(1, (y - 1994) / (2026 - 1994)));
+  const oldC = [100, 116, 139];
+  const newC = [169, 125, 58];
+  const c = oldC.map((o, i) => Math.round(o + (newC[i] - o) * t));
+  return `rgb(${c[0]},${c[1]},${c[2]})`;
+}
+
+export function colorForNode(n: GraphNode, colorBy: ColorBy): string {
+  // Hubs always keep their type colour — only precedents recolour.
+  if (n.type !== "precedent") return NODE_COLORS[n.type] ?? FALLBACK_COLOR;
+  switch (colorBy) {
+    case "practice_area":
+      return PA_COLORS[n.practice_area ?? ""] ?? FALLBACK_COLOR;
+    case "community":
+      return n.community != null
+        ? COMMUNITY_PALETTE[n.community % COMMUNITY_PALETTE.length]
+        : FALLBACK_COLOR;
+    case "precedent_level":
+      return LEVEL_COLORS[n.precedent_level ?? ""] ?? FALLBACK_COLOR;
+    case "recency":
+      return recencyColor(n.date);
+    default:
+      return NODE_COLORS.precedent;
+  }
+}
+
+export function radiusForNode(n: GraphNode, sizeBy: SizeBy): number {
   if (n.type === "topic" || n.type === "practice_area") return 5;
+  if (sizeBy === "pagerank" && n.pagerank != null) {
+    return 3 + Math.sqrt(n.pagerank) * 18;
+  }
+  if (sizeBy === "betweenness" && n.betweenness != null) {
+    return 3 + Math.sqrt(n.betweenness) * 18;
+  }
   return Math.min(22, 3 + Math.sqrt(Math.max(0, n.size)) * 1.7);
 }
 
@@ -76,10 +142,14 @@ export function GraphCanvas({
   data,
   selectedId,
   onNodeClick,
+  colorBy = "type",
+  sizeBy = "indegree",
 }: {
   data: CorpusGraph | undefined;
   selectedId: string | null;
   onNodeClick: (node: GraphNode) => void;
+  colorBy?: ColorBy;
+  sizeBy?: SizeBy;
 }) {
   const { ref, size } = useElementSize<HTMLDivElement>();
   // eslint-disable-next-line @typescript-eslint/no-explicit-any
@@ -143,9 +213,9 @@ export function GraphCanvas({
 
   const drawNode = useCallback(
     (node: FGNode, ctx: CanvasRenderingContext2D, globalScale: number) => {
-      const r = nodeRadius(node);
+      const r = radiusForNode(node, sizeBy);
       const dimmed = isDimmed(node.id);
-      const color = NODE_COLORS[node.type] ?? "#64748b";
+      const color = colorForNode(node, colorBy);
       ctx.globalAlpha = dimmed ? 0.18 : 1;
 
       ctx.beginPath();
@@ -194,17 +264,17 @@ export function GraphCanvas({
       }
       ctx.globalAlpha = 1;
     },
-    [activeId, isDimmed],
+    [activeId, isDimmed, colorBy, sizeBy],
   );
 
   const drawPointerArea = useCallback(
     (node: FGNode, color: string, ctx: CanvasRenderingContext2D) => {
       ctx.fillStyle = color;
       ctx.beginPath();
-      ctx.arc(node.x ?? 0, node.y ?? 0, nodeRadius(node) + 2, 0, 2 * Math.PI);
+      ctx.arc(node.x ?? 0, node.y ?? 0, radiusForNode(node, sizeBy) + 2, 0, 2 * Math.PI);
       ctx.fill();
     },
-    [],
+    [sizeBy],
   );
 
   const linkColor = useCallback(

web-ui/src/components/graph/graph-filter-panel.tsx

@@ -25,6 +25,7 @@ import {
   SelectValue,
 } from "@/components/ui/select";
 import type { GraphFacets } from "@/lib/api/graph";
+import type { ColorBy, SizeBy } from "@/components/graph/graph-canvas";
 
 export type GraphControls = {
   practiceArea: string;
@@ -37,6 +38,8 @@ export type GraphControls = {
   district: string;
   yearFrom: number;
   yearTo: number;
+  colorBy: ColorBy;
+  sizeBy: SizeBy;
   showTopics: boolean;
   showPracticeAreas: boolean;
   showHalachot: boolean;
@@ -45,6 +48,20 @@ export type GraphControls = {
 const ALL = "__all__";
 const YEARS = Array.from({ length: 2026 - 1994 + 1 }, (_, i) => 2026 - i);
 
+const COLOR_BY: { value: ColorBy; label: string }[] = [
+  { value: "type", label: "סוג נקודה" },
+  { value: "practice_area", label: "תחום" },
+  { value: "precedent_level", label: "דרגת סמכות" },
+  { value: "community", label: "אשכול (זיהוי אוטומטי)" },
+  { value: "recency", label: "עדכניות" },
+];
+
+const SIZE_BY: { value: SizeBy; label: string }[] = [
+  { value: "indegree", label: "ציטוטים נכנסים" },
+  { value: "pagerank", label: "השפעה (PageRank)" },
+  { value: "betweenness", label: "גשריות (Betweenness)" },
+];
+
 const PRACTICE_AREAS: { value: string; label: string }[] = [
   { value: "rishuy_uvniya", label: "רישוי ובנייה" },
   { value: "betterment_levy", label: "היטל השבחה" },
@@ -142,6 +159,46 @@ export function GraphFilterPanel({
           </Select>
         </div>
 
+        <Separator />
+
+        <div className="space-y-1.5">
+          <Label className="text-xs text-ink-muted">צביעה לפי</Label>
+          <Select
+            value={controls.colorBy}
+            onValueChange={(v) => onChange({ colorBy: v as GraphControls["colorBy"] })}
+          >
+            <SelectTrigger>
+              <SelectValue />
+            </SelectTrigger>
+            <SelectContent>
+              {COLOR_BY.map((c) => (
+                <SelectItem key={c.value} value={c.value}>
+                  {c.label}
+                </SelectItem>
+              ))}
+            </SelectContent>
+          </Select>
+        </div>
+
+        <div className="space-y-1.5">
+          <Label className="text-xs text-ink-muted">גודל נקודה לפי</Label>
+          <Select
+            value={controls.sizeBy}
+            onValueChange={(v) => onChange({ sizeBy: v as GraphControls["sizeBy"] })}
+          >
+            <SelectTrigger>
+              <SelectValue />
+            </SelectTrigger>
+            <SelectContent>
+              {SIZE_BY.map((s) => (
+                <SelectItem key={s.value} value={s.value}>
+                  {s.label}
+                </SelectItem>
+              ))}
+            </SelectContent>
+          </Select>
+        </div>
+
         <Accordion type="single" collapsible className="border-0">
           <AccordionItem value="advanced" className="border-0">
             <AccordionTrigger className="py-1 text-xs text-ink-muted hover:no-underline">

web-ui/src/components/graph/graph-view.tsx

@@ -3,12 +3,20 @@
 /**
  * Corpus graph orchestrator. Owns filter + selection state, decides whether to
  * render the full graph or a focused node neighborhood (the Obsidian "local
- * graph"), and wires the filter sidebar, canvas, and node detail panel.
+ * graph"), and wires the filter sidebar, canvas, ranking panel, and node panel.
+ *
+ * Analytics (PR B): when color-by=community or size-by=pagerank/betweenness,
+ * `metrics=true` is requested on the FULL graph (global importance). The
+ * neighborhood endpoint does not compute metrics — instead we overlay the
+ * cached full-graph metrics onto neighborhood nodes by id, so a node's
+ * PageRank doesn't change just because you zoomed into it.
  */
 
 import { useEffect, useMemo, useState } from "react";
 
 import { Button } from "@/components/ui/button";
+import { Card, CardContent } from "@/components/ui/card";
+import { Tabs, TabsContent, TabsList, TabsTrigger } from "@/components/ui/tabs";
 import {
   type CorpusGraph,
   type GraphNode,
@@ -20,7 +28,13 @@ import {
   type GraphControls,
   GraphFilterPanel,
 } from "@/components/graph/graph-filter-panel";
-import { GraphCanvas } from "@/components/graph/graph-canvas";
+import {
+  type ColorBy,
+  COMMUNITY_PALETTE,
+  GraphCanvas,
+  LEVEL_COLORS,
+  PA_COLORS,
+} from "@/components/graph/graph-canvas";
 import { GraphNodePanel } from "@/components/graph/graph-node-panel";
 
 const NODE_LIMIT = 400;
@@ -46,6 +60,8 @@ export function GraphView() {
     district: "",
     yearFrom: 0,
     yearTo: 0,
+    colorBy: "type",
+    sizeBy: "indegree",
     showTopics: true,
     showPracticeAreas: true,
     showHalachot: false,
@@ -65,6 +81,10 @@ export function GraphView() {
     return t.join(",");
   }, [controls.showTopics, controls.showPracticeAreas, controls.showHalachot]);
 
+  // Metrics are needed when colouring by cluster or sizing by a centrality.
+  const metricsOn =
+    controls.sizeBy !== "indegree" || controls.colorBy === "community";
+
   const debouncedQ = useDebouncedValue(controls.q, 350);
 
   const filters = useMemo(
@@ -81,6 +101,7 @@ export function GraphView() {
       district: controls.district,
       year_from: controls.yearFrom,
       year_to: controls.yearTo,
+      metrics: metricsOn,
     }),
     [
       controls.practiceArea,
@@ -92,19 +113,51 @@ export function GraphView() {
       controls.district,
       controls.yearFrom,
       controls.yearTo,
+      metricsOn,
       nodeTypes,
       debouncedQ,
     ],
   );
 
   const isFocused = !!focusNodeId;
-  const full = useCorpusGraph(filters, !isFocused);
+  // Keep the full query alive when metrics are on so the overlay map stays warm.
+  const full = useCorpusGraph(filters, !isFocused || metricsOn);
   const neighborhood = useNodeNeighborhood(focusNodeId, 1, nodeTypes);
 
   const active = isFocused ? neighborhood : full;
-  const data: CorpusGraph | undefined = active.data;
   const error = active.error as Error | undefined;
 
+  // Cache of global metrics by node id, overlaid onto neighborhood nodes.
+  const metricsMap = useMemo(() => {
+    const m = new Map<
+      string,
+      { pagerank: number | null; betweenness: number | null; community: number | null }
+    >();
+    for (const n of full.data?.nodes ?? []) {
+      if (n.pagerank != null || n.community != null) {
+        m.set(n.id, {
+          pagerank: n.pagerank,
+          betweenness: n.betweenness,
+          community: n.community,
+        });
+      }
+    }
+    return m;
+  }, [full.data]);
+
+  const data: CorpusGraph | undefined = useMemo(() => {
+    if (!isFocused) return full.data;
+    if (!neighborhood.data) return undefined;
+    if (metricsMap.size === 0) return neighborhood.data;
+    return {
+      ...neighborhood.data,
+      nodes: neighborhood.data.nodes.map((n) => {
+        const mv = metricsMap.get(n.id);
+        return mv ? { ...n, ...mv } : n;
+      }),
+    };
+  }, [isFocused, full.data, neighborhood.data, metricsMap]);
+
   const handleNodeClick = (node: GraphNode) => {
     setSelectedNode(node);
     setFocusNodeId(node.id);
@@ -115,13 +168,13 @@ export function GraphView() {
     setSelectedNode(null);
   };
 
+  const showRanking = metricsOn && !selectedNode && (full.data?.nodes.length ?? 0) > 0;
+
   return (
     <div className="space-y-3">
       <div className="flex items-center justify-between gap-3 text-xs text-ink-muted">
         <span>
-          {data
-            ? `${data.nodes.length} נקודות · ${data.edges.length} קשרים`
-            : "—"}
+          {data ? `${data.nodes.length} נקודות · ${data.edges.length} קשרים` : "—"}
         </span>
         {!isFocused && full.data?.truncated && (
           <span className="text-gold-deep">
@@ -158,6 +211,8 @@ export function GraphView() {
               data={data}
               selectedId={selectedNode?.id ?? null}
               onNodeClick={handleNodeClick}
+              colorBy={controls.colorBy}
+              sizeBy={controls.sizeBy}
             />
           )}
 
@@ -172,23 +227,123 @@ export function GraphView() {
             </Button>
           )}
 
-          <Legend />
+          <Legend colorBy={controls.colorBy} />
         </div>
 
-        {selectedNode && (
+        {selectedNode ? (
           <GraphNodePanel node={selectedNode} onClose={() => setSelectedNode(null)} />
-        )}
+        ) : showRanking ? (
+          <RankingPanel nodes={full.data!.nodes} onPick={handleNodeClick} />
+        ) : null}
       </div>
     </div>
   );
 }
 
-function Legend() {
-  const items = [
+function RankingPanel({
+  nodes,
+  onPick,
+}: {
+  nodes: GraphNode[];
+  onPick: (n: GraphNode) => void;
+}) {
+  const precedents = nodes.filter((n) => n.type === "precedent");
+  const byPagerank = [...precedents]
+    .filter((n) => n.pagerank != null)
+    .sort((a, b) => (b.pagerank ?? 0) - (a.pagerank ?? 0))
+    .slice(0, 12);
+  const byBetweenness = [...precedents]
+    .filter((n) => n.betweenness != null)
+    .sort((a, b) => (b.betweenness ?? 0) - (a.betweenness ?? 0))
+    .slice(0, 12);
+
+  return (
+    <Card className="bg-surface border-rule shadow-sm w-72 shrink-0 overflow-y-auto">
+      <CardContent className="p-4">
+        <Tabs defaultValue="pagerank">
+          <TabsList className="w-full">
+            <TabsTrigger value="pagerank" className="flex-1">
+              המשפיעות
+            </TabsTrigger>
+            <TabsTrigger value="betweenness" className="flex-1">
+              גשרים
+            </TabsTrigger>
+          </TabsList>
+          <TabsContent value="pagerank">
+            <RankList items={byPagerank} metric="pagerank" onPick={onPick} />
+          </TabsContent>
+          <TabsContent value="betweenness">
+            <RankList items={byBetweenness} metric="betweenness" onPick={onPick} />
+          </TabsContent>
+        </Tabs>
+      </CardContent>
+    </Card>
+  );
+}
+
+function RankList({
+  items,
+  metric,
+  onPick,
+}: {
+  items: GraphNode[];
+  metric: "pagerank" | "betweenness";
+  onPick: (n: GraphNode) => void;
+}) {
+  if (items.length === 0) {
+    return <p className="text-ink-muted text-xs mt-3">אין נתונים.</p>;
+  }
+  return (
+    <ol className="mt-2 space-y-1">
+      {items.map((n, i) => (
+        <li key={n.id}>
+          <button
+            type="button"
+            onClick={() => onPick(n)}
+            className="flex w-full items-baseline justify-between gap-2 rounded px-2 py-1 text-start text-sm hover:bg-gold-wash"
+          >
+            <span className="truncate">
+              <span className="text-ink-muted text-xs">{i + 1}.</span> {n.label}
+            </span>
+            <span className="text-ink-muted text-xs tabular-nums shrink-0">
+              {((n[metric] ?? 0) * 100).toFixed(0)}
+            </span>
+          </button>
+        </li>
+      ))}
+    </ol>
+  );
+}
+
+const LEGENDS: Record<ColorBy, { color: string; label: string }[]> = {
+  type: [
     { color: "#1e3a5f", label: "פסיקה" },
     { color: "#a97d3a", label: "נושא" },
     { color: "#475569", label: "תחום" },
-  ];
+  ],
+  practice_area: [
+    { color: PA_COLORS.rishuy_uvniya, label: "רישוי ובנייה" },
+    { color: PA_COLORS.betterment_levy, label: "היטל השבחה" },
+    { color: PA_COLORS.compensation_197, label: "פיצויים" },
+  ],
+  precedent_level: [
+    { color: LEVEL_COLORS["עליון"], label: "עליון" },
+    { color: LEVEL_COLORS["מנהלי"], label: "מנהלי" },
+    { color: LEVEL_COLORS["ועדת_ערר_מחוזית"], label: "ועדת ערר" },
+  ],
+  community: [
+    { color: COMMUNITY_PALETTE[0], label: "אשכול עיקרי" },
+    { color: COMMUNITY_PALETTE[1], label: "אשכול נוסף" },
+    { color: COMMUNITY_PALETTE[2], label: "אשכול נוסף" },
+  ],
+  recency: [
+    { color: "rgb(100,116,139)", label: "ישן (1994)" },
+    { color: "rgb(169,125,58)", label: "עדכני (2026)" },
+  ],
+};
+
+function Legend({ colorBy }: { colorBy: ColorBy }) {
+  const items = LEGENDS[colorBy] ?? LEGENDS.type;
   return (
     <div className="absolute bottom-3 end-3 flex flex-col gap-1 rounded-md bg-surface/85 backdrop-blur px-3 py-2 text-xs text-ink-muted">
       {items.map((i) => (

web-ui/src/lib/api/graph.ts

@@ -35,6 +35,9 @@ export type GraphNode = {
   court: string | null;
   date: string | null; // ISO date
   case_law_id: string | null;
+  pagerank: number | null; // normalized 0–1, only when metrics requested
+  betweenness: number | null; // normalized 0–1
+  community: number | null; // dense cluster id, 0 = largest
 };
 
 export type GraphFacets = {
@@ -72,6 +75,7 @@ export type GraphFilters = {
   district?: string;
   year_from?: number;
   year_to?: number;
+  metrics?: boolean;
 };
 
 export const graphKeys = {
@@ -95,6 +99,7 @@ function buildParams(f: GraphFilters): string {
   if (f.district) p.set("district", f.district);
   if (f.year_from) p.set("year_from", String(f.year_from));
   if (f.year_to) p.set("year_to", String(f.year_to));
+  if (f.metrics) p.set("metrics", "true");
   return p.toString();
 }
 

web/app.py

@@ -5780,6 +5780,7 @@ async def graph_corpus(
     district: str = "",
     year_from: int = 0,
     year_to: int = 0,
+    metrics: bool = False,
 ):
     """Full corpus graph under the given filters (most-cited nodes survive the cap)."""
     if practice_area and practice_area not in _PRACTICE_AREAS:
@@ -5799,6 +5800,7 @@ async def graph_corpus(
         district=district,
         year_from=year_from,
         year_to=year_to,
+        metrics=metrics,
     )
 
 

web/graph_api.py

@@ -35,6 +35,8 @@ from uuid import UUID
 import asyncpg
 from pydantic import BaseModel
 
+from web import graph_metrics
+
 # ── Node-type vocabulary ─────────────────────────────────────────────
 VALID_NODE_TYPES = {"precedent", "halacha", "topic", "practice_area"}
 DEFAULT_NODE_TYPES = ("precedent", "topic", "practice_area")
@@ -63,6 +65,10 @@ class GraphNode(BaseModel):
     court: str | None = None  # precedents only — for color-by / filter
     date: str | None = None  # precedents only — ISO date, for recency color/filter
     case_law_id: str | None = None  # canonical id for deep-link (precedents)
+    # Graph metrics — populated only when ``metrics=true`` (precedents only).
+    pagerank: float | None = None  # normalized 0–1 (global influence)
+    betweenness: float | None = None  # normalized 0–1 (bridge-ness)
+    community: int | None = None  # dense cluster id, 0 = largest
 
 
 class GraphFacets(BaseModel):
@@ -243,6 +249,7 @@ async def build_corpus_graph(
     district: str = "",
     year_from: int = 0,
     year_to: int = 0,
+    metrics: bool = False,
 ) -> CorpusGraph:
     """Assemble the full corpus graph under the given filters.
 
@@ -250,6 +257,10 @@ async def build_corpus_graph(
     so clipping never hides the structurally important nodes. ``truncated`` +
     ``total_available`` let the UI prompt the user to narrow filters. All
     filters are applied server-side in the WHERE clause (G5).
+
+    When ``metrics`` is true, PageRank / betweenness / community are computed
+    in-memory over the precedent citation subgraph (``graph_metrics``) and
+    stamped onto precedent nodes — no extra DB work (G2).
     """
     types = normalize_node_types(node_types)
     cap = max(1, min(int(limit), NODE_CAP_MAX))
@@ -298,6 +309,9 @@ async def build_corpus_graph(
         hub_nodes, edges = await _edges_and_hubs(conn, prec_rows, types)
         nodes.extend(hub_nodes)
 
+    if metrics:
+        _stamp_metrics(nodes, edges)
+
     return CorpusGraph(
         nodes=nodes,
         edges=edges,
@@ -306,6 +320,23 @@ async def build_corpus_graph(
     )
 
 
+def _stamp_metrics(nodes: list[GraphNode], edges: list[GraphEdge]) -> None:
+    """Compute PageRank/betweenness/community over the precedent subgraph and
+    stamp them onto precedent nodes in place (hubs stay ``None``)."""
+    prec_ids = [n.id for n in nodes if n.type == "precedent"]
+    if not prec_ids:
+        return
+    directed = [(e.source, e.target) for e in edges if e.type == "cites"]
+    undirected = [(e.source, e.target) for e in edges if e.type == "same_chain"]
+    m = graph_metrics.compute(prec_ids, directed, undirected)
+    for n in nodes:
+        mv = m.get(n.id)
+        if mv:
+            n.pagerank = mv["pagerank"]
+            n.betweenness = mv["betweenness"]
+            n.community = mv["community"]
+
+
 async def build_node_neighborhood(
     pool: asyncpg.Pool,
     node_id: str,

web/graph_metrics.py

@@ -0,0 +1,158 @@
+"""Graph metrics for the corpus graph — dependency-free (no networkx).
+
+Computed in-memory over the precedent citation subgraph that ``graph_api``
+already fetched (**G2**: no DB access here — pure functions over data the caller
+holds). The corpus graph is tiny (≤ ``NODE_CAP_MAX`` = 1500 nodes, sparse), so
+power-iteration PageRank, Brandes betweenness, and label-propagation communities
+all run synchronously well under a second.
+
+Edge model: ``cites`` is directional (authority flows citing → cited);
+``same_chain`` is non-directional. PageRank uses cites-direction + same_chain
+both ways; betweenness and communities treat the whole graph as undirected.
+Determinism (stable colors across requests): nodes are processed in sorted
+order and ties break by lowest label — no randomness.
+"""
+from __future__ import annotations
+
+from collections import Counter, defaultdict, deque
+
+
+def compute(
+    node_ids: list[str],
+    directed_edges: list[tuple[str, str]],
+    undirected_edges: list[tuple[str, str]] | None = None,
+) -> dict[str, dict]:
+    """Return ``{node_id: {pagerank, betweenness, community}}``.
+
+    ``pagerank`` / ``betweenness`` are normalized to max = 1.0 (easy client
+    scaling); ``community`` is a dense int 0..k-1 ordered by descending cluster
+    size (so the largest cluster is always colour 0).
+    """
+    nodes = list(node_ids)
+    node_set = set(nodes)
+    if not nodes:
+        return {}
+    undirected_edges = undirected_edges or []
+
+    de = [(s, t) for s, t in directed_edges if s in node_set and t in node_set and s != t]
+    ue = [(s, t) for s, t in undirected_edges if s in node_set and t in node_set and s != t]
+
+    pr = _normalize(_pagerank(nodes, de, ue))
+    bt = _normalize(_betweenness(nodes, de, ue))
+    comm = _communities(nodes, de, ue)
+
+    return {
+        n: {
+            "pagerank": round(pr[n], 4),
+            "betweenness": round(bt[n], 4),
+            "community": comm[n],
+        }
+        for n in nodes
+    }
+
+
+def _normalize(d: dict[str, float]) -> dict[str, float]:
+    m = max(d.values()) if d else 0.0
+    if m <= 0:
+        return {k: 0.0 for k in d}
+    return {k: v / m for k, v in d.items()}
+
+
+def _undirected_adj(nodes: list[str], de, ue) -> dict[str, set[str]]:
+    adj: dict[str, set[str]] = {n: set() for n in nodes}
+    for s, t in de:
+        adj[s].add(t)
+        adj[t].add(s)
+    for s, t in ue:
+        adj[s].add(t)
+        adj[t].add(s)
+    return adj
+
+
+def _pagerank(nodes, de, ue, d: float = 0.85, iters: int = 100, tol: float = 1e-9):
+    """Power-iteration PageRank. cites direction + same_chain both ways."""
+    out: dict[str, list[str]] = defaultdict(list)
+    for s, t in de:
+        out[s].append(t)
+    for s, t in ue:
+        out[s].append(t)
+        out[t].append(s)
+    n = len(nodes)
+    pr = {x: 1.0 / n for x in nodes}
+    for _ in range(iters):
+        dangling = sum(pr[x] for x in nodes if not out[x])
+        base = (1.0 - d) / n + d * dangling / n
+        new = {x: base for x in nodes}
+        for x in nodes:
+            deg = len(out[x])
+            if deg:
+                share = d * pr[x] / deg
+                for m in out[x]:
+                    new[m] += share
+        if sum(abs(new[x] - pr[x]) for x in nodes) < tol:
+            return new
+        pr = new
+    return pr
+
+
+def _betweenness(nodes, de, ue):
+    """Brandes betweenness on the undirected graph. O(V·(V+E))."""
+    adj = _undirected_adj(nodes, de, ue)
+    bc = {x: 0.0 for x in nodes}
+    for s in nodes:
+        stack: list[str] = []
+        preds: dict[str, list[str]] = {w: [] for w in nodes}
+        sigma = {w: 0.0 for w in nodes}
+        sigma[s] = 1.0
+        dist = {w: -1 for w in nodes}
+        dist[s] = 0
+        queue = deque([s])
+        while queue:
+            v = queue.popleft()
+            stack.append(v)
+            for w in adj[v]:
+                if dist[w] < 0:
+                    dist[w] = dist[v] + 1
+                    queue.append(w)
+                if dist[w] == dist[v] + 1:
+                    sigma[w] += sigma[v]
+                    preds[w].append(v)
+        delta = {w: 0.0 for w in nodes}
+        while stack:
+            w = stack.pop()
+            for v in preds[w]:
+                if sigma[w]:
+                    delta[v] += (sigma[v] / sigma[w]) * (1.0 + delta[w])
+            if w != s:
+                bc[w] += delta[w]
+    # Undirected: each shortest path counted from both endpoints.
+    return {x: v / 2.0 for x, v in bc.items()}
+
+
+def _communities(nodes, de, ue) -> dict[str, int]:
+    """Deterministic synchronous label propagation (+ dense renumbering).
+
+    Each node starts in its own community and repeatedly adopts the most common
+    label among its neighbours (ties → lowest label). Isolated nodes keep their
+    own singleton community. Labels are renumbered 0..k-1 by descending size.
+    """
+    adj = _undirected_adj(nodes, de, ue)
+    order = sorted(nodes)
+    label = {n: n for n in nodes}
+    for _ in range(30):
+        changed = False
+        for n in order:
+            neigh = adj[n]
+            if not neigh:
+                continue
+            counts = Counter(label[m] for m in neigh)
+            best = min(counts.items(), key=lambda kv: (-kv[1], kv[0]))[0]
+            if label[n] != best:
+                label[n] = best
+                changed = True
+        if not changed:
+            break
+    sizes = Counter(label.values())
+    ranked = [lab for lab, _ in sorted(sizes.items(), key=lambda kv: (-kv[1], kv[0]))]
+    remap = {lab: i for i, lab in enumerate(ranked)}
+    return {n: remap[label[n]] for n in nodes}