diff --git a/RELEASES.md b/RELEASES.md
index 45b7f7b1..983da445 100644
--- a/RELEASES.md
+++ b/RELEASES.md
@@ -50,6 +50,23 @@ no separate migration doc. **`8.0.0-rc.2` is live** — install with
 >   carry `visibility`; whole-graph/`getNouns` streaming no longer leaks `system`/`internal` entities;
 >   and small-page cursor walks over nouns no longer loop. No API change — just correct behavior.
 
+> **rc.3 additions (2026-06-23):**
+> - **Graph analytics on the `brain.graph` namespace.** Three intent-level reads that answer
+>   whole-graph questions in one call:
+>   - **`brain.graph.rank(opts?)`** → `{ id, score }[]` descending — "which entities matter most"
+>     (importance / centrality). `topK` to cap.
+>   - **`brain.graph.communities(opts?)`** → `{ groups: string[][], count }` — "which things group
+>     together". Weakly-connected components by default; `{ directed: true }` returns
+>     strongly-connected components.
+>   - **`brain.graph.path(from, to, opts?)`** → `{ nodes, relationships, cost } | null` — the best
+>     route between two entities. Fewest hops by default; `{ by: 'weight' }` minimizes summed edge
+>     weight (the 0–1 connection strength); `direction`, `type`, and `maxDepth` filters apply.
+>   These are **intent contracts, not algorithm contracts** — the question is the promise, the
+>   algorithm is the engine's choice. They transparently use the native `@soulcraft/cor` 3.0 graph
+>   engine when present, and fall back to pure-TS kernels (PageRank, connected components / Tarjan
+>   SCC, BFS / Dijkstra) otherwise. Both paths return identical shapes and respect the default
+>   visibility filter (opt in with `includeInternal` / `includeSystem`).
+
 ### Headline: Database as a Value
 
 8.0 replaces Brainy's two overlapping version-control subsystems (copy-on-write
diff --git a/src/brainy.ts b/src/brainy.ts
index 65a66239..1d1bf0e9 100644
--- a/src/brainy.ts
+++ b/src/brainy.ts
@@ -31,13 +31,26 @@ import { VirtualFileSystem } from './vfs/VirtualFileSystem.js'
 import { MetadataIndexManager } from './utils/metadataIndex.js'
 import { detectContentType, extractForHighlighting } from './utils/contentExtractor.js'
 import { GraphAdjacencyIndex } from './graph/graphAdjacencyIndex.js'
+import {
+  connectedComponents,
+  stronglyConnectedComponents,
+  pageRank,
+  MinHeap
+} from './graph/analyticsFallback.js'
 import { createPipeline } from './streaming/pipeline.js'
 import { configureLogger, LogLevel, prodLog } from './utils/logger.js'
 import { setGlobalCache } from './utils/unifiedCache.js'
 import type { UnifiedCache } from './utils/unifiedCache.js'
 import { rankIndicesByScore, reorderByIndices } from './utils/resultRanking.js'
 import { PluginRegistry, isVersionedIndexProvider, isGraphAccelerationProvider } from './plugin.js'
-import type { GraphAccelerationProvider, TraverseOptions, Subgraph } from './plugin.js'
+import type {
+  GraphAccelerationProvider,
+  TraverseOptions,
+  Subgraph,
+  RankOptions,
+  CommunitiesOptions,
+  PathOptions
+} from './plugin.js'
 import type {
   BrainyPlugin,
   BrainyPluginContext,
@@ -94,6 +107,12 @@ import {
   GraphNode,
   SubgraphOptions,
   GraphExportOptions,
+  GraphRankOptions,
+  GraphRankEntry,
+  GraphCommunitiesOptions,
+  GraphCommunitiesResult,
+  GraphPathOptions,
+  GraphPathResult,
   GetOptions,
   AddManyParams,
   RemoveManyParams,
@@ -3259,7 +3278,11 @@ export class Brainy<T = any> implements BrainyInterface<T> {
     return {
       subgraph: (seeds: string | string[], options?: SubgraphOptions) =>
         this.graphSubgraph(seeds, options),
-      export: (options?: GraphExportOptions) => this.graphExport(options)
+      export: (options?: GraphExportOptions) => this.graphExport(options),
+      rank: (options?: GraphRankOptions) => this.graphRank(options),
+      communities: (options?: GraphCommunitiesOptions) => this.graphCommunities(options),
+      path: (from: string, to: string, options?: GraphPathOptions) =>
+        this.graphPath(from, to, options)
     }
   }
 
@@ -3580,6 +3603,309 @@ export class Brainy<T = any> implements BrainyInterface<T> {
     }
   }
 
+  // ============= GRAPH ANALYTICS (rank / communities / path) =============
+
+  /**
+   * @description Load the whole visible graph into a dense integer adjacency for
+   * the rank / communities fallbacks: a node-id list, an id→index map, and
+   * `outAdj[i]` = the dense indices of node `i`'s out-edge targets (multiplicity
+   * preserved). Streamed via {@link graphExport} (cursor pagination — O(N+E), no
+   * re-scan), so hidden tiers are excluded the same way every other read excludes
+   * them. Isolated nodes (no edges) are retained so they form singleton groups.
+   */
+  private async loadAnalyticsGraph(opts: {
+    includeInternal?: boolean
+    includeSystem?: boolean
+  }): Promise<{ ids: string[]; outAdj: number[][] }> {
+    const ids: string[] = []
+    const index = new Map<string, number>()
+    const outAdj: number[][] = []
+    const idxOf = (id: string): number => {
+      let i = index.get(id)
+      if (i === undefined) {
+        i = ids.length
+        ids.push(id)
+        index.set(id, i)
+        outAdj.push([])
+      }
+      return i
+    }
+
+    for await (const chunk of this.graphExport({
+      includeInternal: opts.includeInternal,
+      includeSystem: opts.includeSystem
+    })) {
+      for (const node of chunk.nodes) idxOf(node.id)
+      for (const edge of chunk.edges) {
+        const s = idxOf(edge.from)
+        const t = idxOf(edge.to)
+        outAdj[s].push(t)
+      }
+    }
+
+    return { ids, outAdj }
+  }
+
+  /**
+   * @description {@link GraphApi.rank} dispatch: native ranking when a provider is
+   * present, else the TS PageRank fallback.
+   */
+  private async graphRank(options?: GraphRankOptions): Promise<GraphRankEntry[]> {
+    await this.ensureInitialized()
+    const accel = this.graphAccelerationProvider()
+    return accel ? this.graphRankNative(accel, options) : this.graphRankFallback(options)
+  }
+
+  /** TS PageRank over the dense visible adjacency; sorted descending, `topK`-capped. */
+  private async graphRankFallback(options?: GraphRankOptions): Promise<GraphRankEntry[]> {
+    const { ids, outAdj } = await this.loadAnalyticsGraph(options ?? {})
+    if (ids.length === 0) return []
+    const scores = pageRank(outAdj)
+    const entries: GraphRankEntry[] = ids.map((id, i) => ({ id, score: scores[i] }))
+    entries.sort((a, b) => b.score - a.score)
+    return options?.topK !== undefined ? entries.slice(0, options.topK) : entries
+  }
+
+  /** Native ranking → hydrate node-ints to ids (descending order preserved). */
+  private async graphRankNative(
+    accel: GraphAccelerationProvider,
+    options?: GraphRankOptions
+  ): Promise<GraphRankEntry[]> {
+    const excluded = this.excludedVisibilityTiers(options ?? {})
+    const opts: RankOptions = {
+      ...(options?.topK !== undefined && { topK: options.topK }),
+      ...(excluded && { excludeVisibility: excluded as unknown as string[] })
+    }
+    const result = await accel.rank(opts)
+    const idMapper = this.metadataIndex.getIdMapper()
+    const entries: GraphRankEntry[] = []
+    for (let i = 0; i < result.nodeInts.length; i++) {
+      const uuid = idMapper.getUuid(Number(result.nodeInts[i]))
+      if (uuid !== undefined) entries.push({ id: uuid, score: result.scores[i] })
+    }
+    return options?.topK !== undefined ? entries.slice(0, options.topK) : entries
+  }
+
+  /**
+   * @description {@link GraphApi.communities} dispatch: native grouping when a
+   * provider is present, else the TS connected-components fallback.
+   */
+  private async graphCommunities(
+    options?: GraphCommunitiesOptions
+  ): Promise<GraphCommunitiesResult> {
+    await this.ensureInitialized()
+    const accel = this.graphAccelerationProvider()
+    return accel
+      ? this.graphCommunitiesNative(accel, options)
+      : this.graphCommunitiesFallback(options)
+  }
+
+  /**
+   * @description TS grouping: weakly-connected components by default (union-find
+   * over the undirected projection), or strongly-connected components when
+   * `directed: true` (Tarjan). Largest group first.
+   */
+  private async graphCommunitiesFallback(
+    options?: GraphCommunitiesOptions
+  ): Promise<GraphCommunitiesResult> {
+    const { ids, outAdj } = await this.loadAnalyticsGraph(options ?? {})
+    if (ids.length === 0) return { groups: [], count: 0 }
+    const labels = options?.directed
+      ? stronglyConnectedComponents(outAdj)
+      : connectedComponents(outAdj)
+    return this.groupByLabel(ids, labels)
+  }
+
+  /** Collapse parallel `(id, label)` arrays into id-groups, largest first. */
+  private groupByLabel(ids: string[], labels: number[]): GraphCommunitiesResult {
+    const byLabel = new Map<number, string[]>()
+    for (let i = 0; i < ids.length; i++) {
+      const label = labels[i]
+      const existing = byLabel.get(label)
+      if (existing) existing.push(ids[i])
+      else byLabel.set(label, [ids[i]])
+    }
+    const groups = [...byLabel.values()].sort((a, b) => b.length - a.length)
+    return { groups, count: groups.length }
+  }
+
+  /** Native grouping → bucket node-ints by community label, hydrate to ids. */
+  private async graphCommunitiesNative(
+    accel: GraphAccelerationProvider,
+    options?: GraphCommunitiesOptions
+  ): Promise<GraphCommunitiesResult> {
+    const excluded = this.excludedVisibilityTiers(options ?? {})
+    const opts: CommunitiesOptions = {
+      ...(options?.directed !== undefined && { directed: options.directed }),
+      ...(excluded && { excludeVisibility: excluded as unknown as string[] })
+    }
+    const result = await accel.communities(opts)
+    const idMapper = this.metadataIndex.getIdMapper()
+    const byCommunity = new Map<number, string[]>()
+    for (let i = 0; i < result.nodeInts.length; i++) {
+      const uuid = idMapper.getUuid(Number(result.nodeInts[i]))
+      if (uuid === undefined) continue
+      const community = result.communityIds[i]
+      const existing = byCommunity.get(community)
+      if (existing) existing.push(uuid)
+      else byCommunity.set(community, [uuid])
+    }
+    const groups = [...byCommunity.values()].sort((a, b) => b.length - a.length)
+    return { groups, count: groups.length }
+  }
+
+  /**
+   * @description {@link GraphApi.path} dispatch: native pathfinding when a provider
+   * is present, else the TS BFS (hops) / Dijkstra (weight) fallback. Endpoints are
+   * id-normalized so callers may pass natural keys.
+   */
+  private async graphPath(
+    from: string,
+    to: string,
+    options?: GraphPathOptions
+  ): Promise<GraphPathResult | null> {
+    await this.ensureInitialized()
+    const fromId = resolveEntityId(from)
+    const toId = resolveEntityId(to)
+    const accel = this.graphAccelerationProvider()
+    return accel
+      ? this.graphPathNative(accel, fromId, toId, options)
+      : this.graphPathFallback(fromId, toId, options)
+  }
+
+  /**
+   * @description On-demand TS pathfinding — expands frontiers through the O(degree)
+   * `related()` adjacency (visibility / type filters applied there) rather than
+   * loading the whole graph, so a short path terminates early. BFS for fewest hops
+   * (default); Dijkstra (min-heap) for least summed edge weight (`by: 'weight'` —
+   * each edge costs its stored `weight`, the 0–1 connection strength, default 1.0).
+   */
+  private async graphPathFallback(
+    fromId: string,
+    toId: string,
+    options?: GraphPathOptions
+  ): Promise<GraphPathResult | null> {
+    if (fromId === toId) return { nodes: [fromId], relationships: [], cost: 0 }
+
+    const direction = options?.direction ?? 'both'
+    const maxDepth = options?.maxDepth ?? Number.POSITIVE_INFINITY
+    const subOptions: SubgraphOptions = {
+      ...(options?.type !== undefined && { type: options.type }),
+      ...(options?.includeInternal !== undefined && { includeInternal: options.includeInternal }),
+      ...(options?.includeSystem !== undefined && { includeSystem: options.includeSystem })
+    }
+    const pred = new Map<string, { prev: string; edgeId: string }>()
+
+    if (options?.by === 'weight') {
+      const dist = new Map<string, number>([[fromId, 0]])
+      const hops = new Map<string, number>([[fromId, 0]])
+      const settled = new Set<string>()
+      const heap = new MinHeap<string>()
+      heap.push(fromId, 0)
+      while (heap.size > 0) {
+        const node = heap.pop() as string
+        if (settled.has(node)) continue
+        settled.add(node)
+        if (node === toId) break
+        const depth = hops.get(node) as number
+        if (depth >= maxDepth) continue
+        const baseCost = dist.get(node) as number
+        const edges = await this.incidentEdges(node, direction, subOptions)
+        for (const edge of edges) {
+          const neighbor = edge.from === node ? edge.to : edge.from
+          if (settled.has(neighbor)) continue
+          // Cost = the edge's stored weight (0–1 connection strength, default 1.0).
+          // Non-negative by construction, so Dijkstra stays valid.
+          const weight = edge.weight ?? 1
+          const candidate = baseCost + weight
+          if (!dist.has(neighbor) || candidate < (dist.get(neighbor) as number)) {
+            dist.set(neighbor, candidate)
+            hops.set(neighbor, depth + 1)
+            pred.set(neighbor, { prev: node, edgeId: edge.id })
+            heap.push(neighbor, candidate)
+          }
+        }
+      }
+      if (!settled.has(toId)) return null
+      return this.reconstructPath(fromId, toId, pred, dist.get(toId) as number)
+    }
+
+    // Fewest hops — breadth-first, each edge cost 1.
+    const visited = new Set<string>([fromId])
+    const queue: Array<{ id: string; depth: number }> = [{ id: fromId, depth: 0 }]
+    let head = 0
+    while (head < queue.length) {
+      const { id, depth } = queue[head++]
+      if (depth >= maxDepth) continue
+      const edges = await this.incidentEdges(id, direction, subOptions)
+      for (const edge of edges) {
+        const neighbor = edge.from === id ? edge.to : edge.from
+        if (visited.has(neighbor)) continue
+        visited.add(neighbor)
+        pred.set(neighbor, { prev: id, edgeId: edge.id })
+        if (neighbor === toId) return this.reconstructPath(fromId, toId, pred, depth + 1)
+        queue.push({ id: neighbor, depth: depth + 1 })
+      }
+    }
+    return null
+  }
+
+  /** Walk predecessor links from `toId` back to `fromId` into a forward route. */
+  private reconstructPath(
+    fromId: string,
+    toId: string,
+    pred: Map<string, { prev: string; edgeId: string }>,
+    cost: number
+  ): GraphPathResult {
+    const nodes: string[] = []
+    const relationships: string[] = []
+    let cursor = toId
+    while (cursor !== fromId) {
+      nodes.push(cursor)
+      const step = pred.get(cursor) as { prev: string; edgeId: string }
+      relationships.push(step.edgeId)
+      cursor = step.prev
+    }
+    nodes.push(fromId)
+    nodes.reverse()
+    relationships.reverse()
+    return { nodes, relationships, cost }
+  }
+
+  /** Native pathfinding → resolve endpoints to ints, hydrate the returned route. */
+  private async graphPathNative(
+    accel: GraphAccelerationProvider,
+    fromId: string,
+    toId: string,
+    options?: GraphPathOptions
+  ): Promise<GraphPathResult | null> {
+    const fromInt = this.graphEntityInt(fromId)
+    const toInt = this.graphEntityInt(toId)
+    if (fromInt === undefined || toInt === undefined) return null
+
+    const verbTypes = options?.type
+      ? (Array.isArray(options.type) ? options.type : [options.type]).map((t) =>
+          TypeUtils.getVerbIndex(t)
+        )
+      : undefined
+    const excluded = this.excludedVisibilityTiers(options ?? {})
+    const pathOptions: PathOptions = {
+      ...(options?.direction !== undefined && { direction: options.direction }),
+      ...(options?.by !== undefined && { by: options.by }),
+      ...(verbTypes && { verbTypes }),
+      ...(excluded && { excludeVisibility: excluded as unknown as string[] }),
+      ...(options?.maxDepth !== undefined && { maxDepth: options.maxDepth })
+    }
+
+    const result = await accel.path(fromInt, toInt, pathOptions)
+    if (!result) return null
+    const nodes = this.entityIntsToUuids(Array.from(result.nodeInts))
+    const relationships = (
+      await this.graphIndex.verbIntsToIds(Array.from(result.edgeVerbInts))
+    ).filter((id): id is string => id !== null)
+    return { nodes, relationships, cost: result.cost }
+  }
+
   // ============= SEARCH & DISCOVERY =============
 
   /**
diff --git a/src/graph/analyticsFallback.ts b/src/graph/analyticsFallback.ts
new file mode 100644
index 00000000..cf863d74
--- /dev/null
+++ b/src/graph/analyticsFallback.ts
@@ -0,0 +1,244 @@
+/**
+ * @module graph/analyticsFallback
+ * @description Pure-TS graph-analytics kernels — the fallback implementations
+ * behind `brain.graph.rank` / `brain.graph.communities` when no native
+ * {@link import('../plugin.js').GraphAccelerationProvider} is registered. Each
+ * operates on a dense integer adjacency (`outAdj[i]` = the target indices of node
+ * `i`'s out-edges, multiplicity preserved) so callers can build it once from the
+ * UUID graph and run any kernel.
+ *
+ * These are INTENT-level fallbacks: the public API promises the QUESTION
+ * ("which nodes matter most", "which things group together") — these answer it
+ * with the standard textbook algorithm (PageRank, connected components,
+ * Tarjan SCC). A native provider may answer the same intent with a different
+ * algorithm (personalized PageRank, Louvain, etc.); correctness of the INTENT,
+ * not the algorithm, is the contract.
+ *
+ * Correct at small/medium scale (the native provider is the at-scale path). All
+ * graph walks are iterative (explicit stacks/queues) so a deep or wide graph
+ * never blows the call stack.
+ */
+
+/** Adjacency where `outAdj[i]` lists the target node indices of `i`'s out-edges. */
+export type DenseAdjacency = ReadonlyArray<ReadonlyArray<number>>
+
+/**
+ * @description Label each node with the id of its WEAKLY-connected component
+ *   (edges treated as undirected) via union-find with path compression + a single
+ *   pass over the adjacency. Two nodes share a label iff one can reach the other
+ *   ignoring edge direction.
+ * @param outAdj - Dense out-adjacency.
+ * @returns `labels[i]` = the component representative of node `i` (labels are
+ *   stable per component but NOT necessarily contiguous — group by equality).
+ */
+export function connectedComponents(outAdj: DenseAdjacency): number[] {
+  const n = outAdj.length
+  const parent = new Array<number>(n)
+  for (let i = 0; i < n; i++) parent[i] = i
+
+  const find = (x: number): number => {
+    let root = x
+    while (parent[root] !== root) root = parent[root]
+    // Path compression: point every node on the chain straight at the root.
+    while (parent[x] !== root) {
+      const next = parent[x]
+      parent[x] = root
+      x = next
+    }
+    return root
+  }
+
+  for (let i = 0; i < n; i++) {
+    const neighbors = outAdj[i]
+    for (let k = 0; k < neighbors.length; k++) {
+      const ra = find(i)
+      const rb = find(neighbors[k])
+      if (ra !== rb) parent[ra] = rb
+    }
+  }
+
+  const labels = new Array<number>(n)
+  for (let i = 0; i < n; i++) labels[i] = find(i)
+  return labels
+}
+
+/**
+ * @description Label each node with its STRONGLY-connected component (directed —
+ *   two nodes share a label iff each is reachable from the other following edge
+ *   direction) via an iterative Tarjan's algorithm.
+ * @param outAdj - Dense out-adjacency.
+ * @returns `labels[i]` = the SCC index of node `i` (contiguous `0..count-1`).
+ */
+export function stronglyConnectedComponents(outAdj: DenseAdjacency): number[] {
+  const n = outAdj.length
+  const index = new Array<number>(n).fill(-1)
+  const low = new Array<number>(n).fill(0)
+  const onStack = new Array<boolean>(n).fill(false)
+  const comp = new Array<number>(n).fill(-1)
+  const sccStack: number[] = []
+  let nextIndex = 0
+  let compCount = 0
+
+  for (let start = 0; start < n; start++) {
+    if (index[start] !== -1) continue
+    // Explicit DFS stack of frames; `pi` is the next out-edge to visit for `node`.
+    const callStack: Array<{ node: number; pi: number }> = [{ node: start, pi: 0 }]
+    while (callStack.length > 0) {
+      const frame = callStack[callStack.length - 1]
+      const v = frame.node
+      if (frame.pi === 0) {
+        index[v] = low[v] = nextIndex++
+        sccStack.push(v)
+        onStack[v] = true
+      }
+      if (frame.pi < outAdj[v].length) {
+        const w = outAdj[v][frame.pi]
+        frame.pi++
+        if (index[w] === -1) {
+          callStack.push({ node: w, pi: 0 })
+        } else if (onStack[w]) {
+          if (index[w] < low[v]) low[v] = index[w]
+        }
+      } else {
+        // All edges of v explored — if v roots an SCC, pop it off the SCC stack.
+        if (low[v] === index[v]) {
+          for (;;) {
+            const w = sccStack.pop() as number
+            onStack[w] = false
+            comp[w] = compCount
+            if (w === v) break
+          }
+          compCount++
+        }
+        callStack.pop()
+        if (callStack.length > 0) {
+          const parent = callStack[callStack.length - 1].node
+          if (low[v] < low[parent]) low[parent] = low[v]
+        }
+      }
+    }
+  }
+
+  return comp
+}
+
+/** Tuning knobs for {@link pageRank}. */
+export interface PageRankOptions {
+  /** Teleport/damping factor (default `0.85`). */
+  damping?: number
+  /** Max power-iterations before stopping (default `100`). */
+  maxIterations?: number
+  /** L1 convergence threshold on the score vector (default `1e-6`). */
+  tolerance?: number
+}
+
+/**
+ * @description PageRank importance score per node via power-iteration. Dangling
+ *   nodes (no out-edges) redistribute their mass uniformly so the score vector
+ *   stays a probability distribution (sums to 1). Edge multiplicity is honored
+ *   (a node with two edges to the same target sends it twice the share).
+ * @param outAdj - Dense out-adjacency.
+ * @param options - Damping / iteration / tolerance knobs.
+ * @returns `scores[i]` = node `i`'s PageRank (higher = more central).
+ */
+export function pageRank(outAdj: DenseAdjacency, options?: PageRankOptions): Float64Array {
+  const n = outAdj.length
+  if (n === 0) return new Float64Array(0)
+
+  const damping = options?.damping ?? 0.85
+  const maxIterations = options?.maxIterations ?? 100
+  const tolerance = options?.tolerance ?? 1e-6
+
+  const outDegree = new Array<number>(n)
+  for (let i = 0; i < n; i++) outDegree[i] = outAdj[i].length
+
+  let rank = new Float64Array(n)
+  rank.fill(1 / n)
+  const teleport = (1 - damping) / n
+
+  for (let iter = 0; iter < maxIterations; iter++) {
+    // Mass stranded on dangling nodes this round, spread uniformly to everyone.
+    let danglingMass = 0
+    for (let i = 0; i < n; i++) if (outDegree[i] === 0) danglingMass += rank[i]
+    const danglingShare = (damping * danglingMass) / n
+
+    const next = new Float64Array(n)
+    next.fill(teleport + danglingShare)
+
+    for (let i = 0; i < n; i++) {
+      const degree = outDegree[i]
+      if (degree === 0) continue
+      const share = (damping * rank[i]) / degree
+      const neighbors = outAdj[i]
+      for (let k = 0; k < neighbors.length; k++) next[neighbors[k]] += share
+    }
+
+    let delta = 0
+    for (let i = 0; i < n; i++) delta += Math.abs(next[i] - rank[i])
+    rank = next
+    if (delta < tolerance) break
+  }
+
+  return rank
+}
+
+/**
+ * @description A minimal binary min-heap (priority queue) — used by the weighted
+ *   shortest-path (Dijkstra) fallback to pop the lowest-cost frontier node in
+ *   O(log n). Stable enough for pathfinding; ties pop in unspecified order.
+ */
+export class MinHeap<T> {
+  private readonly items: Array<{ value: T; priority: number }> = []
+
+  /** Number of queued items. */
+  get size(): number {
+    return this.items.length
+  }
+
+  /**
+   * @description Insert `value` with the given `priority` (lower pops first).
+   * @param value - The payload.
+   * @param priority - Ordering key (ascending).
+   */
+  push(value: T, priority: number): void {
+    const items = this.items
+    items.push({ value, priority })
+    let i = items.length - 1
+    while (i > 0) {
+      const parent = (i - 1) >> 1
+      if (items[parent].priority <= items[i].priority) break
+      const tmp = items[parent]
+      items[parent] = items[i]
+      items[i] = tmp
+      i = parent
+    }
+  }
+
+  /**
+   * @description Remove and return the lowest-priority value.
+   * @returns The min value, or `undefined` when empty.
+   */
+  pop(): T | undefined {
+    const items = this.items
+    if (items.length === 0) return undefined
+    const top = items[0].value
+    const last = items.pop() as { value: T; priority: number }
+    if (items.length > 0) {
+      items[0] = last
+      let i = 0
+      for (;;) {
+        const left = 2 * i + 1
+        const right = 2 * i + 2
+        let smallest = i
+        if (left < items.length && items[left].priority < items[smallest].priority) smallest = left
+        if (right < items.length && items[right].priority < items[smallest].priority) smallest = right
+        if (smallest === i) break
+        const tmp = items[smallest]
+        items[smallest] = items[i]
+        items[i] = tmp
+        i = smallest
+      }
+    }
+    return top
+  }
+}
diff --git a/src/types/brainy.types.ts b/src/types/brainy.types.ts
index 2a6f58ac..4de62833 100644
--- a/src/types/brainy.types.ts
+++ b/src/types/brainy.types.ts
@@ -837,14 +837,87 @@ export interface GraphExportOptions {
   includeEdges?: boolean
 }
 
+/**
+ * Options for {@link GraphApi.communities}.
+ */
+export interface GraphCommunitiesOptions {
+  /** Treat edges as directed when grouping (default `false` — direction-agnostic). */
+  directed?: boolean
+  /** Also group through `internal`-visibility nodes/edges (hidden by default). */
+  includeInternal?: boolean
+  /** Also group through `system`-visibility nodes/edges (hidden by default). */
+  includeSystem?: boolean
+}
+
+/**
+ * Result of {@link GraphApi.communities} — the graph partitioned into connected
+ * groups of entity ids.
+ */
+export interface GraphCommunitiesResult {
+  /** Each group is the member entity ids; largest group first. */
+  groups: string[][]
+  /** Number of distinct groups (`= groups.length`). */
+  count: number
+}
+
+/**
+ * Options for {@link GraphApi.rank} (importance ranking). INTENT-level only — the
+ * ranking ALGORITHM is the provider's choice (the TS fallback uses PageRank), so
+ * no algorithm-tuning knobs are exposed.
+ */
+export interface GraphRankOptions {
+  /** Return only the top-K nodes by score (default: all, descending). */
+  topK?: number
+  /** Also rank through `internal`-visibility nodes/edges (hidden by default). */
+  includeInternal?: boolean
+  /** Also rank through `system`-visibility nodes/edges (hidden by default). */
+  includeSystem?: boolean
+}
+
+/** One ranked entity from {@link GraphApi.rank}, descending by `score`. */
+export interface GraphRankEntry {
+  /** The entity id. */
+  id: string
+  /** The importance score (relative; higher = more central). */
+  score: number
+}
+
+/**
+ * Options for {@link GraphApi.path} (best route between two entities).
+ */
+export interface GraphPathOptions {
+  /** Edge direction to follow (default `'both'`). */
+  direction?: 'in' | 'out' | 'both'
+  /** Optimize for fewest `'hops'` (default) or least summed edge `'weight'`. */
+  by?: 'hops' | 'weight'
+  /** Restrict the route to these relationship type(s). */
+  type?: VerbType | VerbType[]
+  /** Also route through `internal`-visibility nodes/edges (hidden by default). */
+  includeInternal?: boolean
+  /** Also route through `system`-visibility nodes/edges (hidden by default). */
+  includeSystem?: boolean
+  /** Abandon the search past this many hops. */
+  maxDepth?: number
+}
+
+/**
+ * Result of {@link GraphApi.path} — the route from `from` to `to`, or `null` when
+ * unreachable.
+ */
+export interface GraphPathResult {
+  /** The entity ids along the route, `from` … `to` inclusive. */
+  nodes: string[]
+  /** The relationship (verb) ids traversed, one fewer than `nodes`. */
+  relationships: string[]
+  /** Route cost: number of hops, or summed edge weight when `by: 'weight'`. */
+  cost: number
+}
+
 /**
  * The `brain.graph` namespace — graph-shaped reads over the knowledge graph.
  * Routes to a native {@link import('../plugin.js').GraphAccelerationProvider}
  * when one is registered, otherwise serves the same results from Brainy's
  * pure-TS adjacency (correct at small/medium scale).
- *
- * (Grows with the engine: `subgraph` + `export` ship first; `rank` / `path` /
- * `communities` follow.)
  */
 export interface GraphApi<T = any> {
   /**
@@ -874,6 +947,42 @@ export interface GraphApi<T = any> {
    * }
    */
   export(options?: GraphExportOptions): AsyncIterable<GraphView<T>>
+
+  /**
+   * @description Rank entities by graph importance (influence / centrality) — the
+   *   one-call answer to "which nodes matter most". The TS fallback uses PageRank;
+   *   a native provider may use personalized PageRank / eigenvector centrality.
+   * @param options - `topK`, visibility opt-ins.
+   * @returns Entities with scores, DESCENDING (most important first).
+   * @example
+   * const top = await brain.graph.rank({ topK: 10 })
+   * top[0] // { id, score } — the most central entity
+   */
+  rank(options?: GraphRankOptions): Promise<GraphRankEntry[]>
+
+  /**
+   * @description Partition the graph into connected communities (clusters of
+   *   related entities) — "which things group together".
+   * @param options - `directed`, visibility opt-ins.
+   * @returns The groups (member ids) + their count.
+   * @example
+   * const { groups, count } = await brain.graph.communities()
+   */
+  communities(options?: GraphCommunitiesOptions): Promise<GraphCommunitiesResult>
+
+  /**
+   * @description Find the best route between two entities — fewest hops (default)
+   *   or least summed edge weight (`by: 'weight'`). The pathfinding ALGORITHM is
+   *   the provider's choice (TS fallback: BFS for hops, Dijkstra for weight).
+   * @param from - Start entity id (natural keys are resolved).
+   * @param to - End entity id.
+   * @param options - Direction, `by`, type filter, `maxDepth`, visibility opt-ins.
+   * @returns The route (`nodes` + `relationships` + `cost`), or `null` if unreachable.
+   * @example
+   * const route = await brain.graph.path(aliceId, bobId)
+   * route?.nodes // [aliceId, …, bobId]
+   */
+  path(from: string, to: string, options?: GraphPathOptions): Promise<GraphPathResult | null>
 }
 
 // ============= Batch Operations =============
diff --git a/tests/unit/brainy/graph-analytics.test.ts b/tests/unit/brainy/graph-analytics.test.ts
new file mode 100644
index 00000000..bfec3347
--- /dev/null
+++ b/tests/unit/brainy/graph-analytics.test.ts
@@ -0,0 +1,211 @@
+/**
+ * @module tests/unit/brainy/graph-analytics
+ * @description Graph analytics surface — `brain.graph.rank()` (PageRank importance),
+ * `brain.graph.communities()` (connected-component grouping), and
+ * `brain.graph.path()` (best route). These exercise the pure-TS fallbacks (no native
+ * GraphAccelerationProvider is registered in CI); a native provider answers the same
+ * INTENT (which nodes matter most / which things group / best route) and returns the
+ * same public shapes, cross-layer-tested against the provider.
+ */
+
+import { describe, it, expect, beforeEach, afterEach } from 'vitest'
+import { Brainy } from '../../../src/index.js'
+import { NounType, VerbType } from '../../../src/types/graphTypes.js'
+import { createTestConfig } from '../../helpers/test-factory.js'
+
+describe('brain.graph.communities()', () => {
+  let brain: Brainy
+  let a: string, b: string, c: string, d: string, e: string, f: string
+
+  beforeEach(async () => {
+    brain = new Brainy(createTestConfig())
+    await brain.init()
+    // Two disjoint clusters + one isolated node:
+    //   cluster 1:  a → b → c
+    //   cluster 2:  d → e
+    //   isolated:   f (no edges)
+    a = await brain.add({ type: NounType.Person, data: 'A' })
+    b = await brain.add({ type: NounType.Person, data: 'B' })
+    c = await brain.add({ type: NounType.Person, data: 'C' })
+    d = await brain.add({ type: NounType.Person, data: 'D' })
+    e = await brain.add({ type: NounType.Person, data: 'E' })
+    f = await brain.add({ type: NounType.Person, data: 'F' })
+    await brain.relate({ from: a, to: b, type: VerbType.RelatedTo })
+    await brain.relate({ from: b, to: c, type: VerbType.RelatedTo })
+    await brain.relate({ from: d, to: e, type: VerbType.RelatedTo })
+  })
+
+  afterEach(async () => {
+    await brain.close()
+  })
+
+  it('partitions the graph into connected groups (isolated nodes are singletons)', async () => {
+    const { groups, count } = await brain.graph.communities()
+    expect(count).toBe(3)
+    const asSets = groups.map((g) => new Set(g))
+    expect(asSets).toContainEqual(new Set([a, b, c]))
+    expect(asSets).toContainEqual(new Set([d, e]))
+    expect(asSets).toContainEqual(new Set([f]))
+  })
+
+  it('orders groups largest-first', async () => {
+    const { groups } = await brain.graph.communities()
+    for (let i = 1; i < groups.length; i++) {
+      expect(groups[i - 1].length).toBeGreaterThanOrEqual(groups[i].length)
+    }
+    expect(groups[0]).toHaveLength(3) // the a-b-c cluster
+  })
+
+  it('directed mode groups by strong connectivity (a→b→c is NOT mutually reachable)', async () => {
+    // Undirected: {a,b,c} is one group. Directed: no cycle, so each is its own SCC.
+    const undirected = await brain.graph.communities()
+    expect(undirected.groups).toContainEqual(expect.arrayContaining([a, b, c]))
+
+    const directed = await brain.graph.communities({ directed: true })
+    // a,b,c split into singletons; d,e split too; f stays singleton → 6 groups.
+    expect(directed.count).toBe(6)
+    expect(directed.groups.every((g) => g.length === 1)).toBe(true)
+  })
+
+  it('directed mode keeps a cycle together as one strongly-connected community', async () => {
+    // x → y → z → x is a cycle: mutually reachable → one SCC even when directed.
+    const x = await brain.add({ type: NounType.Concept, data: 'X' })
+    const y = await brain.add({ type: NounType.Concept, data: 'Y' })
+    const z = await brain.add({ type: NounType.Concept, data: 'Z' })
+    await brain.relate({ from: x, to: y, type: VerbType.RelatedTo })
+    await brain.relate({ from: y, to: z, type: VerbType.RelatedTo })
+    await brain.relate({ from: z, to: x, type: VerbType.RelatedTo })
+
+    const directed = await brain.graph.communities({ directed: true })
+    const asSets = directed.groups.map((g) => new Set(g))
+    expect(asSets).toContainEqual(new Set([x, y, z]))
+  })
+
+  it('excludes internal-visibility edges by default; includeInternal re-links', async () => {
+    // A hidden edge bridging the two clusters is invisible by default.
+    await brain.relate({ from: c, to: d, type: VerbType.RelatedTo, visibility: 'internal' })
+
+    const hidden = await brain.graph.communities()
+    expect(hidden.count).toBe(3) // bridge hidden → still 3 groups
+
+    const surfaced = await brain.graph.communities({ includeInternal: true })
+    const asSets = surfaced.groups.map((g) => new Set(g))
+    expect(asSets).toContainEqual(new Set([a, b, c, d, e])) // bridge merges the clusters
+  })
+})
+
+describe('brain.graph.rank()', () => {
+  let brain: Brainy
+  let hub: string, a: string, b: string, c: string
+
+  beforeEach(async () => {
+    brain = new Brainy(createTestConfig())
+    await brain.init()
+    // a, b, c all point at hub → hub is the most "important" node.
+    hub = await brain.add({ type: NounType.Person, data: 'HUB' })
+    a = await brain.add({ type: NounType.Person, data: 'A' })
+    b = await brain.add({ type: NounType.Person, data: 'B' })
+    c = await brain.add({ type: NounType.Person, data: 'C' })
+    await brain.relate({ from: a, to: hub, type: VerbType.RelatedTo })
+    await brain.relate({ from: b, to: hub, type: VerbType.RelatedTo })
+    await brain.relate({ from: c, to: hub, type: VerbType.RelatedTo })
+  })
+
+  afterEach(async () => {
+    await brain.close()
+  })
+
+  it('ranks the most-pointed-to node highest, descending', async () => {
+    const ranked = await brain.graph.rank()
+    expect(ranked).toHaveLength(4)
+    expect(ranked[0].id).toBe(hub)
+    for (let i = 1; i < ranked.length; i++) {
+      expect(ranked[i - 1].score).toBeGreaterThanOrEqual(ranked[i].score)
+    }
+  })
+
+  it('scores form a probability distribution (PageRank sums to ~1)', async () => {
+    const ranked = await brain.graph.rank()
+    const total = ranked.reduce((sum, r) => sum + r.score, 0)
+    expect(total).toBeCloseTo(1, 5)
+  })
+
+  it('topK returns only the K highest', async () => {
+    const top1 = await brain.graph.rank({ topK: 1 })
+    expect(top1).toHaveLength(1)
+    expect(top1[0].id).toBe(hub)
+  })
+
+  it('returns [] on an empty graph', async () => {
+    const empty = new Brainy(createTestConfig())
+    await empty.init()
+    expect(await empty.graph.rank()).toEqual([])
+    await empty.close()
+  })
+})
+
+describe('brain.graph.path()', () => {
+  let brain: Brainy
+  let a: string, b: string, c: string, d: string, isolated: string
+
+  beforeEach(async () => {
+    brain = new Brainy(createTestConfig())
+    await brain.init()
+    // Chain a → b → c → d (light edges), plus a heavy direct shortcut a → d.
+    // weight is a 0–1 connection strength; `by:'weight'` minimizes summed weight.
+    a = await brain.add({ type: NounType.Person, data: 'A' })
+    b = await brain.add({ type: NounType.Person, data: 'B' })
+    c = await brain.add({ type: NounType.Person, data: 'C' })
+    d = await brain.add({ type: NounType.Person, data: 'D' })
+    isolated = await brain.add({ type: NounType.Person, data: 'ISO' })
+    await brain.relate({ from: a, to: b, type: VerbType.RelatedTo, weight: 0.1 })
+    await brain.relate({ from: b, to: c, type: VerbType.RelatedTo, weight: 0.1 })
+    await brain.relate({ from: c, to: d, type: VerbType.RelatedTo, weight: 0.1 })
+    await brain.relate({ from: a, to: d, type: VerbType.RelatedTo, weight: 0.9 }) // direct but heavy
+  })
+
+  afterEach(async () => {
+    await brain.close()
+  })
+
+  it('finds the fewest-hops route by default', async () => {
+    const route = await brain.graph.path(a, d)
+    expect(route).not.toBeNull()
+    // a → d direct edge is 1 hop, the shortest.
+    expect(route?.nodes).toEqual([a, d])
+    expect(route?.relationships).toHaveLength(1)
+    expect(route?.cost).toBe(1)
+  })
+
+  it('by:"weight" prefers the lighter multi-hop route over the heavy shortcut', async () => {
+    const route = await brain.graph.path(a, d, { by: 'weight' })
+    expect(route?.nodes).toEqual([a, b, c, d]) // 0.1+0.1+0.1 = 0.3 < 0.9
+    expect(route?.relationships).toHaveLength(3)
+    expect(route?.cost).toBeCloseTo(0.3, 6)
+  })
+
+  it('returns a zero-length route from a node to itself', async () => {
+    const route = await brain.graph.path(a, a)
+    expect(route).toEqual({ nodes: [a], relationships: [], cost: 0 })
+  })
+
+  it('returns null when the target is unreachable', async () => {
+    expect(await brain.graph.path(a, isolated)).toBeNull()
+  })
+
+  it('direction:"out" cannot walk backwards up the chain', async () => {
+    // d has no outgoing edges, so d → a is unreachable following only out-edges.
+    expect(await brain.graph.path(d, a, { direction: 'out' })).toBeNull()
+    // …but 'both' finds it.
+    const both = await brain.graph.path(d, a, { direction: 'both' })
+    expect(both?.nodes[0]).toBe(d)
+    expect(both?.nodes[both.nodes.length - 1]).toBe(a)
+  })
+
+  it('maxDepth abandons routes that are too long', async () => {
+    // The light route is 3 hops; cap at 1 hop → only the heavy direct shortcut.
+    const capped = await brain.graph.path(a, d, { by: 'weight', maxDepth: 1 })
+    expect(capped?.nodes).toEqual([a, d])
+    expect(capped?.cost).toBeCloseTo(0.9, 6)
+  })
+})
diff --git a/tests/unit/brainy/graph-native-routing.test.ts b/tests/unit/brainy/graph-native-routing.test.ts
index dc90a7e1..767e87fe 100644
--- a/tests/unit/brainy/graph-native-routing.test.ts
+++ b/tests/unit/brainy/graph-native-routing.test.ts
@@ -19,13 +19,28 @@ import { createTestConfig } from '../../helpers/test-factory.js'
 
 /** A stateful mock GraphAccelerationProvider; the test sets the columnar payloads. */
 function makeMockAccel() {
-  const calls = { traverse: 0, cursorOpen: 0, cursorNext: 0, cursorClose: 0 }
-  const state: { calls: typeof calls; traverseResult: any; cursorChunks: any[] } = {
+  const calls = { traverse: 0, cursorOpen: 0, cursorNext: 0, cursorClose: 0, rank: 0, communities: 0, path: 0 }
+  const state: {
+    calls: typeof calls
+    traverseResult: any
+    cursorChunks: any[]
+    rankResult: any
+    communitiesResult: any
+    pathResult: any
+  } = {
     calls,
     traverseResult: null,
-    cursorChunks: []
+    cursorChunks: [],
+    rankResult: null,
+    communitiesResult: null,
+    pathResult: null
   }
   const empty = { nodeInts: new BigInt64Array(0), scores: new Float64Array(0) }
+  const emptyCommunities = {
+    nodeInts: new BigInt64Array(0),
+    communityIds: new Uint32Array(0),
+    communityCount: 0
+  }
   const provider = {
     isInitialized: true,
     traverse: async () => {
@@ -45,9 +60,18 @@ function makeMockAccel() {
     graphCursorClose: async () => {
       calls.cursorClose++
     },
-    rank: async () => empty,
-    communities: async () => ({ nodeInts: new BigInt64Array(0), communityIds: new Uint32Array(0), communityCount: 0 }),
-    path: async () => null,
+    rank: async () => {
+      calls.rank++
+      return state.rankResult ?? empty
+    },
+    communities: async () => {
+      calls.communities++
+      return state.communitiesResult ?? emptyCommunities
+    },
+    path: async () => {
+      calls.path++
+      return state.pathResult
+    },
     sample: async () => state.traverseResult,
     mostConnected: async () => empty
   }
@@ -178,4 +202,63 @@ describe('brain.graph.* native routing + columnar hydration (native seam)', () =
     expect(new Set(view.nodes.map((n) => n.id))).toEqual(new Set([x, y]))
     await fb.close()
   })
+
+  it('rank() routes to the native provider and hydrates node ints -> ids, order preserved', async () => {
+    const gei = (id: string): bigint => (brain as any).graphEntityInt(id)
+    // Provider returns DESCENDING scores: c, then a, then b.
+    mock.state.rankResult = {
+      nodeInts: BigInt64Array.from([gei(c), gei(a), gei(b)]),
+      scores: Float64Array.from([0.5, 0.3, 0.2])
+    }
+    const ranked = await brain.graph.rank()
+
+    expect(mock.state.calls.rank).toBe(1) // native path, not the TS PageRank fallback
+    expect(ranked.map((r) => r.id)).toEqual([c, a, b]) // int -> id, provider order kept
+    expect(ranked[0].score).toBe(0.5)
+
+    const top2 = await brain.graph.rank({ topK: 2 })
+    expect(top2.map((r) => r.id)).toEqual([c, a])
+  })
+
+  it('communities() routes to the native provider and buckets ids by community label', async () => {
+    const gei = (id: string): bigint => (brain as any).graphEntityInt(id)
+    // a,b in community 0; c alone in community 1.
+    mock.state.communitiesResult = {
+      nodeInts: BigInt64Array.from([gei(a), gei(b), gei(c)]),
+      communityIds: Uint32Array.from([0, 0, 1]),
+      communityCount: 2
+    }
+    const { groups, count } = await brain.graph.communities()
+
+    expect(mock.state.calls.communities).toBe(1) // native path, not the TS fallback
+    expect(count).toBe(2)
+    const asSets = groups.map((g) => new Set(g))
+    expect(asSets).toContainEqual(new Set([a, b]))
+    expect(asSets).toContainEqual(new Set([c]))
+    expect(groups[0]).toHaveLength(2) // largest group first
+  })
+
+  it('path() routes to the native provider and hydrates node + verb ints', async () => {
+    const gei = (id: string): bigint => (brain as any).graphEntityInt(id)
+    const gi = (brain as any).graphIndex
+    const vAB = (await gi.getVerbIdsBySource(gei(a)))[0] as bigint
+    const vBC = (await gi.getVerbIdsBySource(gei(b)))[0] as bigint
+    mock.state.pathResult = {
+      nodeInts: BigInt64Array.from([gei(a), gei(b), gei(c)]),
+      edgeVerbInts: BigInt64Array.from([vAB, vBC]),
+      cost: 2
+    }
+    const route = await brain.graph.path(a, c)
+
+    expect(mock.state.calls.path).toBe(1) // native path, not the TS BFS/Dijkstra fallback
+    expect(route?.nodes).toEqual([a, b, c]) // node ints -> ids
+    expect(route?.relationships).toHaveLength(2) // verb ints -> verb-id strings
+    expect(route?.cost).toBe(2)
+  })
+
+  it('path() returns null when the native provider reports unreachable', async () => {
+    mock.state.pathResult = null
+    expect(await brain.graph.path(a, c)).toBeNull()
+    expect(mock.state.calls.path).toBe(1)
+  })
 })
diff --git a/tests/unit/graph/analyticsFallback.test.ts b/tests/unit/graph/analyticsFallback.test.ts
new file mode 100644
index 00000000..9686172e
--- /dev/null
+++ b/tests/unit/graph/analyticsFallback.test.ts
@@ -0,0 +1,163 @@
+/**
+ * @module tests/unit/graph/analyticsFallback
+ * @description Direct unit tests for the pure-TS graph-analytics kernels
+ * (`connectedComponents`, `stronglyConnectedComponents`, `pageRank`, `MinHeap`).
+ * These cover the algorithmic edge cases — empty graphs, dangling PageRank mass,
+ * nested SCCs, heap ordering — without the cost of a full Brainy instance. The
+ * `brain.graph.*` surface is tested end-to-end in graph-analytics.test.ts.
+ */
+
+import { describe, it, expect } from 'vitest'
+import {
+  connectedComponents,
+  stronglyConnectedComponents,
+  pageRank,
+  MinHeap
+} from '../../../src/graph/analyticsFallback.js'
+
+/** Group node indices that share a label into sorted sets for stable comparison. */
+function groupsOf(labels: number[]): number[][] {
+  const byLabel = new Map<number, number[]>()
+  labels.forEach((label, i) => {
+    const arr = byLabel.get(label)
+    if (arr) arr.push(i)
+    else byLabel.set(label, [i])
+  })
+  return [...byLabel.values()].map((g) => g.sort((a, b) => a - b)).sort((a, b) => a[0] - b[0])
+}
+
+describe('connectedComponents (weakly-connected, undirected projection)', () => {
+  it('labels an empty graph with no components', () => {
+    expect(connectedComponents([])).toEqual([])
+  })
+
+  it('treats every isolated node as its own component', () => {
+    expect(groupsOf(connectedComponents([[], [], []]))).toEqual([[0], [1], [2]])
+  })
+
+  it('merges across edge direction (0→1, 2→1 ⇒ one group)', () => {
+    // 0 → 1, 2 → 1, 3 isolated
+    const labels = connectedComponents([[1], [], [1], []])
+    expect(groupsOf(labels)).toEqual([[0, 1, 2], [3]])
+  })
+
+  it('keeps disjoint clusters separate', () => {
+    // 0↔1 , 2↔3
+    const labels = connectedComponents([[1], [0], [3], [2]])
+    expect(groupsOf(labels)).toEqual([
+      [0, 1],
+      [2, 3]
+    ])
+  })
+})
+
+describe('stronglyConnectedComponents (directed, Tarjan)', () => {
+  it('labels an empty graph with no components', () => {
+    expect(stronglyConnectedComponents([])).toEqual([])
+  })
+
+  it('splits a directed chain into singletons (no mutual reachability)', () => {
+    // 0 → 1 → 2
+    expect(groupsOf(stronglyConnectedComponents([[1], [2], []]))).toEqual([[0], [1], [2]])
+  })
+
+  it('collapses a directed cycle into one SCC', () => {
+    // 0 → 1 → 2 → 0
+    expect(groupsOf(stronglyConnectedComponents([[1], [2], [0]]))).toEqual([[0, 1, 2]])
+  })
+
+  it('separates a cycle from the acyclic tail that feeds it', () => {
+    // 3 → 0 → 1 → 2 → 0  : {0,1,2} is an SCC, 3 is its own SCC
+    const labels = stronglyConnectedComponents([[1], [2], [0], [0]])
+    expect(groupsOf(labels)).toEqual([[0, 1, 2], [3]])
+  })
+
+  it('handles two independent cycles', () => {
+    // 0↔1 cycle, 2↔3 cycle
+    const labels = stronglyConnectedComponents([
+      [1],
+      [0],
+      [3],
+      [2]
+    ])
+    expect(groupsOf(labels)).toEqual([
+      [0, 1],
+      [2, 3]
+    ])
+  })
+
+  it('produces contiguous community indices', () => {
+    const labels = stronglyConnectedComponents([[1], [2], [0], []])
+    const distinct = new Set(labels)
+    expect(Math.max(...labels)).toBe(distinct.size - 1)
+  })
+})
+
+describe('pageRank', () => {
+  it('returns an empty vector for an empty graph', () => {
+    expect(pageRank([]).length).toBe(0)
+  })
+
+  it('is uniform on a symmetric cycle', () => {
+    // 0 → 1 → 2 → 0 — every node identical by symmetry.
+    const scores = pageRank([[1], [2], [0]])
+    expect(scores[0]).toBeCloseTo(1 / 3, 6)
+    expect(scores[1]).toBeCloseTo(1 / 3, 6)
+    expect(scores[2]).toBeCloseTo(1 / 3, 6)
+  })
+
+  it('ranks a hub (high in-degree) above its sources', () => {
+    // 0 → 3, 1 → 3, 2 → 3 : node 3 is the hub.
+    const scores = pageRank([[3], [3], [3], []])
+    expect(scores[3]).toBeGreaterThan(scores[0])
+    expect(scores[3]).toBeGreaterThan(scores[1])
+    expect(scores[3]).toBeGreaterThan(scores[2])
+  })
+
+  it('redistributes dangling mass so scores sum to 1', () => {
+    // Node 3 is dangling (no out-edges) — its mass must not leak away.
+    const scores = pageRank([[3], [3], [3], []])
+    const total = scores.reduce((sum, s) => sum + s, 0)
+    expect(total).toBeCloseTo(1, 6)
+  })
+
+  it('sums to 1 even when every node is dangling (no edges)', () => {
+    const scores = pageRank([[], [], []])
+    const total = scores.reduce((sum, s) => sum + s, 0)
+    expect(total).toBeCloseTo(1, 6)
+    expect(scores[0]).toBeCloseTo(1 / 3, 6)
+  })
+})
+
+describe('MinHeap', () => {
+  it('pops in ascending priority order', () => {
+    const heap = new MinHeap<string>()
+    heap.push('c', 3)
+    heap.push('a', 1)
+    heap.push('b', 2)
+    heap.push('d', 4)
+    expect([heap.pop(), heap.pop(), heap.pop(), heap.pop()]).toEqual(['a', 'b', 'c', 'd'])
+  })
+
+  it('returns undefined when empty and tracks size', () => {
+    const heap = new MinHeap<number>()
+    expect(heap.size).toBe(0)
+    expect(heap.pop()).toBeUndefined()
+    heap.push(42, 0.5)
+    expect(heap.size).toBe(1)
+    expect(heap.pop()).toBe(42)
+    expect(heap.size).toBe(0)
+  })
+
+  it('handles interleaved push/pop correctly', () => {
+    const heap = new MinHeap<number>()
+    heap.push(5, 5)
+    heap.push(1, 1)
+    expect(heap.pop()).toBe(1)
+    heap.push(3, 3)
+    heap.push(2, 2)
+    expect(heap.pop()).toBe(2)
+    expect(heap.pop()).toBe(3)
+    expect(heap.pop()).toBe(5)
+  })
+})