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* @fileoverview Graph algorithms for MEP relationship-network OSINT.
*
* Pure, deterministic helpers consumed by `network_analysis`:
* - {@link buildAdjacency} — adjacency map from a weighted edge list.
* - {@link bfsLimited} — depth-bounded BFS ego-network extraction.
* - {@link weightedDegree} — weighted-degree centrality per node.
* - {@link betweennessCentrality} — Brandes' algorithm (weighted, Dijkstra-based, O(V²·log V) with the current array-backed priority queue).
* - {@link labelPropagation} — deterministic asynchronous label-propagation community detection.
* - {@link modularity} — Newman's modularity Q for a partition.
*
* Algorithms are deterministic — every iteration orders nodes/edges by string
* comparison so identical input always produces identical output, satisfying
* the OSINT reproducibility requirement.
*
* ISMS Policy: SC-002 (Input Validation), AC-003 (Least Privilege).
*
* @module utils/graphAlgorithms
* @since 1.4.0
*/
/** Simple weighted undirected edge representation. */
export interface WeightedEdge {
/** Stable identifier of the first endpoint. */
sourceId: string;
/** Stable identifier of the second endpoint. */
targetId: string;
/** Edge weight in `[0, 1]`. */
weight: number;
}
/** Adjacency representation: `adj.get(node)` → neighbour → weight. */
export type AdjacencyMap = Map<string, Map<string, number>>;
/**
* Build an undirected adjacency map from a weighted edge list.
*
* Self-loops are dropped. Parallel edges between the same pair are merged
* by taking the **maximum** weight (so combined networks remain
* well-defined when both committee and voting edges describe the same pair).
*
* @param nodeIds - Full node universe; nodes without edges still appear as empty maps.
* @param edges - Weighted edges (order-independent).
* @returns Adjacency map keyed by node id.
*/
export function buildAdjacency(nodeIds: readonly string[], edges: readonly WeightedEdge[]): AdjacencyMap {
const adj: AdjacencyMap = new Map();
for (const id of nodeIds) adj.set(id, new Map());
for (const e of edges) {
if (e.sourceId === e.targetId) continue;
const a = adj.get(e.sourceId);
const b = adj.get(e.targetId);
Iif (a === undefined || b === undefined) continue;
const existing = a.get(e.targetId) ?? 0;
if (e.weight > existing) {
a.set(e.targetId, e.weight);
b.set(e.sourceId, e.weight);
}
}
return adj;
}
/**
* Breadth-first traversal capped at `maxDepth` hops from one or more seeds.
*
* Returns the set of node ids reachable within `maxDepth` hops (inclusive).
* `maxDepth=1` returns only the seeds plus their direct neighbours;
* `maxDepth=0` returns only the seeds themselves.
*
* @param adj - Adjacency map produced by {@link buildAdjacency}.
* @param seeds - One or more starting node ids.
* @param maxDepth - Non-negative hop limit.
* @returns Set of reachable node ids.
*/
/** Lex-compare helper for deterministic ordering. */
function lexCompare(a: string, b: string): number {
if (a < b) return -1;
Eif (a > b) return 1;
return 0;
}
/** Expand one BFS frontier into the next within an adjacency map. */
function expandFrontier(
adj: AdjacencyMap,
frontier: readonly string[],
visited: Set<string>
): string[] {
const next: string[] = [];
for (const node of frontier) {
const neighbours = adj.get(node);
Iif (neighbours === undefined) continue;
const sortedNeighbours = [...neighbours.keys()].sort(lexCompare);
for (const n of sortedNeighbours) {
if (!visited.has(n)) {
visited.add(n);
next.push(n);
}
}
}
return next;
}
export function bfsLimited(adj: AdjacencyMap, seeds: readonly string[], maxDepth: number): Set<string> {
const visited = new Set<string>();
Iif (maxDepth < 0) return visited;
let frontier: string[] = [];
for (const s of seeds) {
if (adj.has(s) && !visited.has(s)) {
visited.add(s);
frontier.push(s);
}
}
for (let depth = 0; depth < maxDepth && frontier.length > 0; depth++) {
frontier.sort(lexCompare);
frontier = expandFrontier(adj, frontier, visited);
}
return visited;
}
/**
* Weighted-degree centrality: sum of incident edge weights per node.
*
* @param nodeIds - Node universe (so isolates return 0 rather than be omitted).
* @param edges - Weighted edges.
* @returns Map from node id → summed incident weight.
*/
export function weightedDegree(nodeIds: readonly string[], edges: readonly WeightedEdge[]): Map<string, number> {
const out = new Map<string, number>();
for (const id of nodeIds) out.set(id, 0);
for (const e of edges) {
Iif (e.sourceId === e.targetId) continue;
Eif (out.has(e.sourceId)) out.set(e.sourceId, (out.get(e.sourceId) ?? 0) + e.weight);
Eif (out.has(e.targetId)) out.set(e.targetId, (out.get(e.targetId) ?? 0) + e.weight);
}
return out;
}
/**
* Brandes' betweenness centrality on a weighted undirected graph.
*
* Implements Dijkstra-based shortest-path counting from each source as
* described in U. Brandes, *A Faster Algorithm for Betweenness Centrality*
* (2001). Edge weights are interpreted as **similarity** — internally
* converted to distance `1 / weight` so high-weight edges are "shorter"
* and therefore preferred routes for brokers.
*
* Complexity: `O(V² · log V)` per source with the current array-backed
* priority queue (each `popClosest` performs a full `Array.prototype.sort`
* and `relaxShorter` uses `Array.prototype.includes`). A binary-heap
* priority queue would lower this to the textbook `O(V · (V + E) · log V)`
* bound but is not required for the small graphs this module is designed
* for (V is capped at a few hundred nodes by the calling tool).
*
* The returned scores are normalised by `1 / ((V-1)(V-2))` for V≥3 (undirected
* graph — each unordered pair is traversed twice during Brandes) so they
* land in `[0, 1]`. For V<3, raw scores (all zero) are returned.
*
* @param nodeIds - Node universe (deterministic order is enforced internally).
* @param edges - Weighted edges (similarity in `(0, 1]`).
* @returns Map from node id → normalised betweenness in `[0, 1]`.
*/
/** Brandes per-source Dijkstra state. */
interface BrandesState {
stack: string[];
pred: Map<string, string[]>;
sigma: Map<string, number>;
dist: Map<string, number>;
}
function initBrandesState(nodes: readonly string[], source: string): BrandesState {
const pred = new Map<string, string[]>();
const sigma = new Map<string, number>();
const dist = new Map<string, number>();
for (const v of nodes) {
pred.set(v, []);
sigma.set(v, 0);
dist.set(v, Number.POSITIVE_INFINITY);
}
sigma.set(source, 1);
dist.set(source, 0);
return { stack: [], pred, sigma, dist };
}
function popClosest(queue: string[], dist: Map<string, number>): string | undefined {
queue.sort((a, b) => {
const da = dist.get(a) ?? Number.POSITIVE_INFINITY;
const db = dist.get(b) ?? Number.POSITIVE_INFINITY;
if (da !== db) return da - db;
return lexCompare(a, b);
});
return queue.shift();
}
function relaxShorter(state: BrandesState, queue: string[], v: string, w: string, alt: number): void {
state.dist.set(w, alt);
state.sigma.set(w, state.sigma.get(v) ?? 0);
state.pred.set(w, [v]);
Eif (!queue.includes(w)) queue.push(w);
}
function relaxEqual(state: BrandesState, v: string, w: string): void {
state.sigma.set(w, (state.sigma.get(w) ?? 0) + (state.sigma.get(v) ?? 0));
state.pred.get(w)?.push(v);
}
function relaxEdge(
state: BrandesState,
queue: string[],
v: string,
w: string,
weight: number
): void {
Iif (weight <= 0) return;
const dv = state.dist.get(v) ?? Number.POSITIVE_INFINITY;
const alt = dv + 1 / weight;
const dw = state.dist.get(w) ?? Number.POSITIVE_INFINITY;
if (alt < dw - 1e-12) relaxShorter(state, queue, v, w, alt);
else if (Math.abs(alt - dw) < 1e-12) relaxEqual(state, v, w);
}
function runDijkstra(adj: AdjacencyMap, state: BrandesState, source: string): void {
const queue: string[] = [source];
while (queue.length > 0) {
const v = popClosest(queue, state.dist);
Iif (v === undefined) break;
state.stack.push(v);
const neighbours = adj.get(v);
Iif (neighbours === undefined) continue;
// Iterate neighbours in lex-sorted order so tie-breaking is fully
// deterministic regardless of the underlying Map insertion order
// (which depends on the input edge-list ordering).
const sortedNeighbours = [...neighbours.keys()].sort(lexCompare);
for (const w of sortedNeighbours) {
const weight = neighbours.get(w);
Iif (weight === undefined) continue;
relaxEdge(state, queue, v, w, weight);
}
}
}
function propagateDelta(state: BrandesState, w: string, delta: Map<string, number>): void {
const sw = state.sigma.get(w) ?? 0;
const dw = delta.get(w) ?? 0;
Iif (sw <= 0) return;
for (const v of state.pred.get(w) ?? []) {
const sv = state.sigma.get(v) ?? 0;
delta.set(v, (delta.get(v) ?? 0) + (sv / sw) * (1 + dw));
}
}
function accumulateBetweenness(
state: BrandesState,
source: string,
nodes: readonly string[],
cb: Map<string, number>
): void {
const delta = new Map<string, number>();
for (const v of nodes) delta.set(v, 0);
while (state.stack.length > 0) {
const w = state.stack.pop();
Iif (w === undefined) break;
propagateDelta(state, w, delta);
if (w !== source) cb.set(w, (cb.get(w) ?? 0) + (delta.get(w) ?? 0));
}
}
export function betweennessCentrality(
nodeIds: readonly string[],
edges: readonly WeightedEdge[]
): Map<string, number> {
const sortedNodes = [...nodeIds].sort(lexCompare);
const adj = buildAdjacency(sortedNodes, edges);
const cb = new Map<string, number>();
for (const v of sortedNodes) cb.set(v, 0);
for (const s of sortedNodes) {
const state = initBrandesState(sortedNodes, s);
runDijkstra(adj, state, s);
accumulateBetweenness(state, s, sortedNodes, cb);
}
const n = sortedNodes.length;
if (n >= 3) {
// Undirected graph: each unordered pair is traversed twice by Brandes
// (once from each endpoint as source), so the normalisation factor is
// 1 / ((n-1)(n-2)) rather than the directed 2 / ((n-1)(n-2)).
const norm = 1 / ((n - 1) * (n - 2));
for (const [k, v] of cb) cb.set(k, Math.round(v * norm * 10000) / 10000);
}
return cb;
}
/**
* Deterministic asynchronous label propagation for community detection.
*
* Each node is initialised with its own label (its id). On every iteration
* each node — visited in sorted id order — adopts the label that maximises
* the **sum of edge weights** to neighbours sharing that label, with ties
* broken by lexicographic label order. Labels are updated **in place** so
* downstream nodes in the same sweep already see their predecessors'
* decisions. This asynchronous variant avoids the oscillation that pure
* synchronous label propagation exhibits on symmetric / bipartite graphs.
*
* As an additional safety bound the algorithm fingerprints the full label
* vector after each sweep and stops if either:
* - no label changed during the sweep (true convergence), or
* - the fingerprint matches the previous-previous sweep (2-cycle).
*
* Sorted-order traversal plus deterministic tie-breaking guarantees
* reproducible clusters across runs.
*
* @param nodeIds - Node universe.
* @param edges - Weighted edges.
* @param maxIterations - Safety bound (default 30, typically converges in <10).
* @returns Map from node id → community label.
*/
function runLabelIteration(
sorted: readonly string[],
adj: AdjacencyMap,
labels: Map<string, string>
): boolean {
let changed = false;
// Asynchronous: update labels in place so each node in `sorted` sees the
// most recent labels assigned earlier in the same sweep.
for (const node of sorted) {
const neighbours = adj.get(node);
if (neighbours === undefined || neighbours.size === 0) continue;
const score = aggregateNeighbourLabels(neighbours, labels);
const bestLabel = pickBestLabel(score);
if (bestLabel !== undefined && bestLabel !== labels.get(node)) {
labels.set(node, bestLabel);
changed = true;
}
}
return changed;
}
/** Build a deterministic fingerprint of the current label vector. */
function fingerprintLabels(sorted: readonly string[], labels: ReadonlyMap<string, string>): string {
const parts: string[] = [];
for (const id of sorted) parts.push(`${id}=${labels.get(id) ?? ''}`);
return parts.join('|');
}
export function labelPropagation(
nodeIds: readonly string[],
edges: readonly WeightedEdge[],
maxIterations = 30
): Map<string, string> {
const sorted = [...nodeIds].sort(lexCompare);
const adj = buildAdjacency(sorted, edges);
const labels = new Map<string, string>();
for (const id of sorted) labels.set(id, id);
// Track the previous two fingerprints to detect 2-cycles (label vector
// alternating between two states without converging).
let prevPrint = '';
let prevPrevPrint = '';
for (let iter = 0; iter < maxIterations; iter++) {
if (!runLabelIteration(sorted, adj, labels)) break;
const print = fingerprintLabels(sorted, labels);
Iif (print === prevPrevPrint) break;
prevPrevPrint = prevPrint;
prevPrint = print;
}
return labels;
}
/** Pick the label with the highest aggregated weight; ties broken lex-smallest. */
function pickBestLabel(score: Map<string, number>): string | undefined {
Iif (score.size === 0) return undefined;
const sortedScores = [...score.entries()].sort((a, b) => lexCompare(a[0], b[0]));
let bestLabel: string | undefined;
let bestScore = -Infinity;
for (const [lab, s] of sortedScores) {
if (s > bestScore + 1e-12) {
bestScore = s;
bestLabel = lab;
}
}
return bestLabel;
}
/** Aggregate weighted neighbour labels for a single node during label propagation. */
function aggregateNeighbourLabels(
neighbours: Map<string, number>,
labels: Map<string, string>
): Map<string, number> {
const score = new Map<string, number>();
for (const [n, w] of neighbours) {
const lab = labels.get(n);
Iif (lab === undefined) continue;
score.set(lab, (score.get(lab) ?? 0) + w);
}
return score;
}
/**
* Newman's modularity Q for a weighted partition.
*
* `Q = (1 / 2m) Σ_{ij} [A_ij - k_i k_j / 2m] δ(c_i, c_j)`
*
* where `A_ij` is edge weight, `k_i` weighted degree, and `δ` the Kronecker
* indicator for same-community. Q ∈ `[-0.5, 1]`; values >0.3 indicate
* meaningful community structure.
*
* @param nodeIds - Node universe.
* @param edges - Weighted edges.
* @param labels - Community label per node (from {@link labelPropagation}).
* @returns Modularity Q rounded to 4 decimal places (`0` when the graph is empty).
*/
export function modularity(
nodeIds: readonly string[],
edges: readonly WeightedEdge[],
labels: Map<string, string>
): number {
let m2 = 0;
for (const e of edges) m2 += 2 * e.weight;
if (m2 === 0) return 0;
const deg = weightedDegree(nodeIds, edges);
let q = 0;
// Edge contributions (A_ij).
for (const e of edges) {
if (labels.get(e.sourceId) === labels.get(e.targetId)) {
q += 2 * e.weight; // undirected pair counted both directions
}
}
// Expected-degree subtraction over all same-community pairs (including self).
// Iterate in sorted order for deterministic floating-point accumulation.
const sortedNodeIds = [...nodeIds].sort();
for (const i of sortedNodeIds) {
const li = labels.get(i);
const ki = deg.get(i) ?? 0;
for (const j of sortedNodeIds) {
if (labels.get(j) !== li) continue;
const kj = deg.get(j) ?? 0;
q -= (ki * kj) / m2;
}
}
return Math.round((q / m2) * 10000) / 10000;
}
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