PageRank Algorithm

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Originally developed by Google to rank web pages, PageRank is now widely used for influence analysis, recommendation systems, and network centrality calculations. Geode provides native support for implementing PageRank through GQL queries, enabling efficient importance scoring across large graph datasets. <h3 id="understanding-pagerank" class="position-relative d-flex align-items-center group"> Understanding PageRank <button type="button" class="h-share btn btn-link p-0 text-decoration-none link-secondary opacity-50 hover-opacity-100 transition-all ms-1" data-share-target="understanding-pagerank" aria-haspopup="dialog" aria-label="Share link: Understanding PageRank"> Share link </button> </h3>PageRank assigns each node a numerical score representing its importance. The algorithm operates on the principle that important nodes receive links from other important nodes, creating a recursive definition of importance. <h4 id="core-concepts" class="position-relative d-flex align-items-center group"> Core Concepts <button type="button" class="h-share btn btn-link p-0 text-decoration-none link-secondary opacity-50 hover-opacity-100 transition-all ms-1" data-share-target="core-concepts" aria-haspopup="dialog" aria-label="Share link: Core Concepts"> Share link </button> </h4>Importance Propagation: Nodes distribute their importance scores to their outgoing neighbors, weighted by the number of outgoing edges. Damping Factor: A probability (typically 0.85) that represents the likelihood of continuing to follow links versus randomly jumping to any node. Iterative Convergence: PageRank calculations iterate until scores stabilize within a tolerance threshold, typically after 10-50 iterations. Initial Distribution: All nodes start with equal importance scores, usually 1/N where N is the total number of nodes. <h3 id="basic-pagerank-implementation" class="position-relative d-flex align-items-center group"> Basic PageRank Implementation <button type="button" class="h-share btn btn-link p-0 text-decoration-none link-secondary opacity-50 hover-opacity-100 transition-all ms-1" data-share-target="basic-pagerank-implementation" aria-haspopup="dialog" aria-label="Share link: Basic PageRank Implementation"> Share link </button> </h3> <h4 id="computing-pagerank-scores" class="position-relative d-flex align-items-center group"> Computing PageRank Scores <button type="button" class="h-share btn btn-link p-0 text-decoration-none link-secondary opacity-50 hover-opacity-100 transition-all ms-1" data-share-target="computing-pagerank-scores" aria-haspopup="dialog" aria-label="Share link: Computing PageRank Scores"> Share link </button> </h4><div class="highlight"><pre tabindex="0" class="chroma"><code class="language-gql" data-lang="gql">-- Create a temporary property to store PageRank scores MATCH (n:Page) SET n.pagerank = 1.0 / COUNT(*) RETURN n.id AS page, n.pagerank AS initial_score; -- Iterative PageRank update (run multiple times) MATCH (source:Page)-[r:LINKS_TO]->(target:Page) WITH target, SUM(source.pagerank / COUNT { (source)-[:LINKS_TO]->() }) AS incoming_rank SET target.pagerank_new = 0.15 + 0.85 * incoming_rank RETURN target.id AS page, target.pagerank_new AS updated_score; -- Copy new scores to current scores MATCH (n:Page) SET n.pagerank = n.pagerank_new REMOVE n.pagerank_new; </code></pre></div> <h4 id="finding-top-ranked-nodes" class="position-relative d-flex align-items-center group"> Finding Top-Ranked Nodes <button type="button" class="h-share btn btn-link p-0 text-decoration-none link-secondary opacity-50 hover-opacity-100 transition-all ms-1" data-share-target="finding-top-ranked-nodes" aria-haspopup="dialog" aria-label="Share link: Finding Top-Ranked Nodes"> Share link </button> </h4><div class="highlight"><pre tabindex="0" class="chroma"><code class="language-gql" data-lang="gql">-- Get nodes with highest PageRank scores MATCH (n:Page) WHERE n.pagerank IS NOT NULL RETURN n.id AS page, n.title AS title, n.pagerank AS importance_score ORDER BY n.pagerank DESC LIMIT 20; </code></pre></div> <h3 id="advanced-pagerank-patterns" class="position-relative d-flex align-items-center group"> Advanced PageRank Patterns <button type="button" class="h-share btn btn-link p-0 text-decoration-none link-secondary opacity-50 hover-opacity-100 transition-all ms-1" data-share-target="advanced-pagerank-patterns" aria-haspopup="dialog" aria-label="Share link: Advanced PageRank Patterns"> Share link </button> </h3> <h4 id="personalized-pagerank" class="position-relative d-flex align-items-center group"> Personalized PageRank <button type="button" class="h-share btn btn-link p-0 text-decoration-none link-secondary opacity-50 hover-opacity-100 transition-all ms-1" data-share-target="personalized-pagerank" aria-haspopup="dialog" aria-label="Share link: Personalized PageRank"> Share link </button> </h4>Personalized PageRank biases the random jump probability toward specific nodes, useful for recommendations. <div class="highlight"><pre tabindex="0" class="chroma"><code class="language-gql" data-lang="gql">-- Personalized PageRank from a seed node MATCH (seed:User {id: 'user123'}) MATCH (n:Item) SET n.ppr = CASE WHEN n = seed THEN 1.0 ELSE 0.0 END; -- Iterative update with personalization MATCH (source:Item)-[r:RELATES_TO]->(target:Item) WITH target, SUM(source.ppr / COUNT { (source)-[:RELATES_TO]->() }) AS incoming_ppr, 0.15 * CASE WHEN target IN [(seed:User {id: 'user123'})-[:LIKES]->() | seed] THEN 1.0 ELSE 0.0 END AS personalization SET target.ppr_new = personalization + 0.85 * incoming_ppr; </code></pre></div> <h4 id="topic-sensitive-pagerank" class="position-relative d-flex align-items-center group"> Topic-Sensitive PageRank <button type="button" class="h-share btn btn-link p-0 text-decoration-none link-secondary opacity-50 hover-opacity-100 transition-all ms-1" data-share-target="topic-sensitive-pagerank" aria-haspopup="dialog" aria-label="Share link: Topic-Sensitive PageRank"> Share link </button> </h4>Compute PageRank scores specific to different topics or categories. <div class="highlight"><pre tabindex="0" class="chroma"><code class="language-gql" data-lang="gql">-- PageRank within a topic MATCH (n:Article) WHERE n.category = 'technology' SET n.topic_rank = 1.0 / COUNT { MATCH (m:Article) WHERE m.category = 'technology' }; -- Iterate within topic boundaries MATCH (source:Article)-[:CITES]->(target:Article) WHERE source.category = 'technology' AND target.category = 'technology' WITH target, SUM(source.topic_rank / COUNT { (source)-[:CITES]->(:Article {category: 'technology'}) }) AS topic_incoming SET target.topic_rank_new = 0.15 + 0.85 * topic_incoming; </code></pre></div> <h4 id="weighted-pagerank" class="position-relative d-flex align-items-center group"> Weighted PageRank <button type="button" class="h-share btn btn-link p-0 text-decoration-none link-secondary opacity-50 hover-opacity-100 transition-all ms-1" data-share-target="weighted-pagerank" aria-haspopup="dialog" aria-label="Share link: Weighted PageRank"> Share link </button> </h4>Incorporate edge weights to reflect varying importance of connections. <div class="highlight"><pre tabindex="0" class="chroma"><code class="language-gql" data-lang="gql">-- Weighted PageRank calculation MATCH (source:Page)-[r:LINKS_TO]->(target:Page) WITH target, SUM(r.weight * source.pagerank / SUM { (source)-[rel:LINKS_TO]->() | rel.weight } ) AS weighted_incoming SET target.pagerank_new = 0.15 + 0.85 * weighted_incoming; </code></pre></div> <h3 id="practical-applications" class="position-relative d-flex align-items-center group"> Practical Applications <button type="button" class="h-share btn btn-link p-0 text-decoration-none link-secondary opacity-50 hover-opacity-100 transition-all ms-1" data-share-target="practical-applications" aria-haspopup="dialog" aria-label="Share link: Practical Applications"> Share link </button> </h3> <h4 id="web-page-ranking" class="position-relative d-flex align-items-center group"> Web Page Ranking <button type="button" class="h-share btn btn-link p-0 text-decoration-none link-secondary opacity-50 hover-opacity-100 transition-all ms-1" data-share-target="web-page-ranking" aria-haspopup="dialog" aria-label="Share link: Web Page Ranking"> Share link </button> </h4><div class="highlight"><pre tabindex="0" class="chroma"><code class="language-gql" data-lang="gql">-- Rank web pages by link importance MATCH (page:WebPage) SET page.pagerank = 1.0 / COUNT { MATCH (p:WebPage) }; -- Execute 20 iterations MATCH (source:WebPage)-[:LINKS_TO]->(target:WebPage) WITH target, SUM(source.pagerank / COUNT { (source)-[:LINKS_TO]->() }) AS rank_sum SET target.pagerank_new = 0.15 + 0.85 * rank_sum; MATCH (n:WebPage) SET n.pagerank = n.pagerank_new REMOVE n.pagerank_new; -- Get top pages MATCH (page:WebPage) RETURN page.url, page.title, page.pagerank ORDER BY page.pagerank DESC LIMIT 50; </code></pre></div> <h4 id="social-network-influence" class="position-relative d-flex align-items-center group"> Social Network Influence <button type="button" class="h-share btn btn-link p-0 text-decoration-none link-secondary opacity-50 hover-opacity-100 transition-all ms-1" data-share-target="social-network-influence" aria-haspopup="dialog" aria-label="Share link: Social Network Influence"> Share link </button> </h4><div class="highlight"><pre tabindex="0" class="chroma"><code class="language-gql" data-lang="gql">-- Identify influential users MATCH (user:User) SET user.influence = 1.0 / COUNT { MATCH (u:User) }; -- Propagate influence through follower relationships MATCH (follower:User)-[:FOLLOWS]->(influencer:User) WITH influencer, SUM(follower.influence / COUNT { (follower)-[:FOLLOWS]->() }) AS follower_influence SET influencer.influence_new = 0.15 + 0.85 * follower_influence; -- Top influencers MATCH (user:User) RETURN user.username, user.followers_count, user.influence AS influence_score ORDER BY user.influence DESC LIMIT 25; </code></pre></div> <h4 id="citation-network-analysis" class="position-relative d-flex align-items-center group"> Citation Network Analysis <button type="button" class="h-share btn btn-link p-0 text-decoration-none link-secondary opacity-50 hover-opacity-100 transition-all ms-1" data-share-target="citation-network-analysis" aria-haspopup="dialog" aria-label="Share link: Citation Network Analysis"> Share link </button> </h4><div class="highlight"><pre tabindex="0" class="chroma"><code class="language-gql" data-lang="gql">-- Rank academic papers by citation importance MATCH (paper:Paper) SET paper.citation_rank = 1.0 / COUNT { MATCH (p:Paper) }; -- Iterate based on citations MATCH (citing:Paper)-[:CITES]->(cited:Paper) WITH cited, SUM(citing.citation_rank / COUNT { (citing)-[:CITES]->() }) AS citation_score SET cited.citation_rank_new = 0.15 + 0.85 * citation_score; -- Most influential papers MATCH (paper:Paper) RETURN paper.title, paper.year, COUNT { (paper)<-[:CITES]-() } AS citation_count, paper.citation_rank AS impact_score ORDER BY paper.citation_rank DESC; </code></pre></div> <h3 id="performance-optimization" class="position-relative d-flex align-items-center group"> Performance Optimization <button type="button" class="h-share btn btn-link p-0 text-decoration-none link-secondary opacity-50 hover-opacity-100 transition-all ms-1" data-share-target="performance-optimization" aria-haspopup="dialog" aria-label="Share link: Performance Optimization"> Share link </button> </h3> <h4 id="efficient-iteration-strategy" class="position-relative d-flex align-items-center group"> Efficient Iteration Strategy <button type="button" class="h-share btn btn-link p-0 text-decoration-none link-secondary opacity-50 hover-opacity-100 transition-all ms-1" data-share-target="efficient-iteration-strategy" aria-haspopup="dialog" aria-label="Share link: Efficient Iteration Strategy"> Share link </button> </h4><div class="highlight"><pre tabindex="0" class="chroma"><code class="language-gql" data-lang="gql">-- Batch processing for large graphs MATCH (n:Node) WITH COUNT(n) AS total_nodes MATCH (n:Node) CALL { WITH n, total_nodes SET n.pagerank = 1.0 / total_nodes } IN TRANSACTIONS OF 10000 ROWS; -- Parallel iteration updates MATCH (source)-[r:CONNECTED_TO]->(target) WITH target, SUM(source.pagerank / source.out_degree) AS incoming_rank CALL { WITH target, incoming_rank SET target.pagerank_new = 0.15 + 0.85 * incoming_rank } IN TRANSACTIONS OF 10000 ROWS; </code></pre></div> <h4 id="convergence-detection" class="position-relative d-flex align-items-center group"> Convergence Detection <button type="button" class="h-share btn btn-link p-0 text-decoration-none link-secondary opacity-50 hover-opacity-100 transition-all ms-1" data-share-target="convergence-detection" aria-haspopup="dialog" aria-label="Share link: Convergence Detection"> Share link </button> </h4><div class="highlight"><pre tabindex="0" class="chroma"><code class="language-gql" data-lang="gql">-- Check convergence between iterations MATCH (n:Node) WITH SUM(ABS(n.pagerank - n.pagerank_new)) AS total_change, COUNT(n) AS node_count RETURN total_change / node_count AS average_change, CASE WHEN total_change / node_count < 0.0001 THEN 'CONVERGED' ELSE 'CONTINUE' END AS status; </code></pre></div> <h4 id="pre-computing-out-degrees" class="position-relative d-flex align-items-center group"> Pre-Computing Out-Degrees <button type="button" class="h-share btn btn-link p-0 text-decoration-none link-secondary opacity-50 hover-opacity-100 transition-all ms-1" data-share-target="pre-computing-out-degrees" aria-haspopup="dialog" aria-label="Share link: Pre-Computing Out-Degrees"> Share link </button> </h4><div class="highlight"><pre tabindex="0" class="chroma"><code class="language-gql" data-lang="gql">-- Cache out-degree counts for performance MATCH (n:Node) SET n.out_degree = COUNT { (n)-->() }; -- Use cached degrees in PageRank MATCH (source)-[r]->(target) WITH target, SUM(source.pagerank / source.out_degree) AS rank_contribution SET target.pagerank_new = 0.15 + 0.85 * rank_contribution; </code></pre></div> <h3 id="best-practices" class="position-relative d-flex align-items-center group"> Best Practices <button type="button" class="h-share btn btn-link p-0 text-decoration-none link-secondary opacity-50 hover-opacity-100 transition-all ms-1" data-share-target="best-practices" aria-haspopup="dialog" aria-label="Share link: Best Practices"> Share link </button> </h3> <h4 id="algorithm-configuration" class="position-relative d-flex align-items-center group"> Algorithm Configuration <button type="button" class="h-share btn btn-link p-0 text-decoration-none link-secondary opacity-50 hover-opacity-100 transition-all ms-1" data-share-target="algorithm-configuration" aria-haspopup="dialog" aria-label="Share link: Algorithm Configuration"> Share link </button> </h4>Iteration Count: Run 20-50 iterations for most graphs; monitor convergence to determine optimal count. Damping Factor: Use 0.85 as default; adjust to 0.80-0.90 based on graph characteristics and convergence behavior. Convergence Threshold: Set tolerance to 0.0001 for production systems; tighten to 0.00001 for research applications. Initial Values: Start with uniform distribution (1/N) to ensure consistent results across runs. <h4 id="memory-management" class="position-relative d-flex align-items-center group"> Memory Management <button type="button" class="h-share btn btn-link p-0 text-decoration-none link-secondary opacity-50 hover-opacity-100 transition-all ms-1" data-share-target="memory-management" aria-haspopup="dialog" aria-label="Share link: Memory Management"> Share link </button> </h4>Incremental Updates: Process PageRank in batches for graphs with millions of nodes. Property Cleanup: Remove temporary properties (pagerank_new) after iteration completion to reduce memory usage. Transaction Batching: Use IN TRANSACTIONS for large-scale updates to maintain database responsiveness. <h4 id="validation-techniques" class="position-relative d-flex align-items-center group"> Validation Techniques <button type="button" class="h-share btn btn-link p-0 text-decoration-none link-secondary opacity-50 hover-opacity-100 transition-all ms-1" data-share-target="validation-techniques" aria-haspopup="dialog" aria-label="Share link: Validation Techniques"> Share link </button> </h4><div class="highlight"><pre tabindex="0" class="chroma"><code class="language-gql" data-lang="gql">-- Verify PageRank sum equals 1.0 MATCH (n:Node) RETURN SUM(n.pagerank) AS total_rank, ABS(1.0 - SUM(n.pagerank)) AS deviation; -- Check for dangling nodes (no outgoing edges) MATCH (n:Node) WHERE NOT EXISTS { (n)-->() } RETURN COUNT(n) AS dangling_nodes; </code></pre></div> <h3 id="common-patterns" class="position-relative d-flex align-items-center group"> Common Patterns <button type="button" class="h-share btn btn-link p-0 text-decoration-none link-secondary opacity-50 hover-opacity-100 transition-all ms-1" data-share-target="common-patterns" aria-haspopup="dialog" aria-label="Share link: Common Patterns"> Share link </button> </h3> <h4 id="recommendation-systems" class="position-relative d-flex align-items-center group"> Recommendation Systems <button type="button" class="h-share btn btn-link p-0 text-decoration-none link-secondary opacity-50 hover-opacity-100 transition-all ms-1" data-share-target="recommendation-systems" aria-haspopup="dialog" aria-label="Share link: Recommendation Systems"> Share link </button> </h4><div class="highlight"><pre tabindex="0" class="chroma"><code class="language-gql" data-lang="gql">-- Recommend items based on personalized PageRank MATCH (user:User {id: 'user123'})-[:PURCHASED]->(seed:Product) MATCH (item:Product) SET item.recommendation_score = CASE WHEN item IN [seed] THEN 1.0 ELSE 0.0 END; -- Propagate scores through product relationships MATCH (source:Product)-[:SIMILAR_TO]->(target:Product) WITH target, SUM(source.recommendation_score / COUNT { (source)-[:SIMILAR_TO]->() }) AS score SET target.recommendation_score_new = 0.15 + 0.85 * score; -- Top recommendations MATCH (rec:Product) WHERE NOT EXISTS { (:User {id: 'user123'})-[:PURCHASED]->(rec) } RETURN rec.name, rec.recommendation_score ORDER BY rec.recommendation_score DESC LIMIT 10; </code></pre></div> <h4 id="knowledge-graph-importance" class="position-relative d-flex align-items-center group"> Knowledge Graph Importance <button type="button" class="h-share btn btn-link p-0 text-decoration-none link-secondary opacity-50 hover-opacity-100 transition-all ms-1" data-share-target="knowledge-graph-importance" aria-haspopup="dialog" aria-label="Share link: Knowledge Graph Importance"> Share link </button> </h4><div class="highlight"><pre tabindex="0" class="chroma"><code class="language-gql" data-lang="gql">-- Rank entities in knowledge graph MATCH (entity:Entity) SET entity.importance = 1.0 / COUNT { MATCH (e:Entity) }; MATCH (source:Entity)-[:RELATED_TO]->(target:Entity) WITH target, SUM(source.importance / COUNT { (source)-[:RELATED_TO]->() }) AS importance_sum SET target.importance_new = 0.15 + 0.85 * importance_sum; </code></pre></div> <h3 id="troubleshooting" class="position-relative d-flex align-items-center group"> Troubleshooting <button type="button" class="h-share btn btn-link p-0 text-decoration-none link-secondary opacity-50 hover-opacity-100 transition-all ms-1" data-share-target="troubleshooting" aria-haspopup="dialog" aria-label="Share link: Troubleshooting"> Share link </button> </h3> <h4 id="slow-convergence" class="position-relative d-flex align-items-center group"> Slow Convergence <button type="button" class="h-share btn btn-link p-0 text-decoration-none link-secondary opacity-50 hover-opacity-100 transition-all ms-1" data-share-target="slow-convergence" aria-haspopup="dialog" aria-label="Share link: Slow Convergence"> Share link </button> </h4>Issue: PageRank scores don’t stabilize after many iterations. Solution: Check for graph structure issues (disconnected components, large cycles). Increase damping factor to 0.90 or implement teleportation to all nodes. <div class="highlight"><pre tabindex="0" class="chroma"><code class="language-gql" data-lang="gql">-- Detect disconnected components MATCH (n:Node) WHERE NOT EXISTS { (n)--() } RETURN COUNT(n) AS isolated_nodes; </code></pre></div> <h4 id="rank-sinks" class="position-relative d-flex align-items-center group"> Rank Sinks <button type="button" class="h-share btn btn-link p-0 text-decoration-none link-secondary opacity-50 hover-opacity-100 transition-all ms-1" data-share-target="rank-sinks" aria-haspopup="dialog" aria-label="Share link: Rank Sinks"> Share link </button> </h4>Issue: PageRank accumulates in nodes with no outgoing edges. Solution: Redistribute dangling node rank equally to all nodes. <div class="highlight"><pre tabindex="0" class="chroma"><code class="language-gql" data-lang="gql">-- Handle dangling nodes MATCH (dangling:Node) WHERE NOT EXISTS { (dangling)-->() } WITH SUM(dangling.pagerank) / COUNT { MATCH (n:Node) } AS dangling_contribution MATCH (n:Node) SET n.pagerank_adjustment = dangling_contribution; </code></pre></div> <h4 id="memory-pressure" class="position-relative d-flex align-items-center group"> Memory Pressure <button type="button" class="h-share btn btn-link p-0 text-decoration-none link-secondary opacity-50 hover-opacity-100 transition-all ms-1" data-share-target="memory-pressure" aria-haspopup="dialog" aria-label="Share link: Memory Pressure"> Share link </button> </h4>Issue: PageRank computation exceeds available memory. Solution: Process in batches and use disk-backed temporary storage. <div class="highlight"><pre tabindex="0" class="chroma"><code class="language-gql" data-lang="gql">-- Batch processing pattern MATCH (n:Node) WHERE ID(n) % 10 = 0 -- Process 10% at a time WITH n CALL { ... PageRank update logic ... }; </code></pre></div> <h3 id="integration-examples" class="position-relative d-flex align-items-center group"> Integration Examples <button type="button" class="h-share btn btn-link p-0 text-decoration-none link-secondary opacity-50 hover-opacity-100 transition-all ms-1" data-share-target="integration-examples" aria-haspopup="dialog" aria-label="Share link: Integration Examples"> Share link </button> </h3> <h4 id="python-client" class="position-relative d-flex align-items-center group"> Python Client <button type="button" class="h-share btn btn-link p-0 text-decoration-none link-secondary opacity-50 hover-opacity-100 transition-all ms-1" data-share-target="python-client" aria-haspopup="dialog" aria-label="Share link: Python Client"> Share link </button> </h4><div class="highlight"><pre tabindex="0" class="chroma"><code class="language-python" data-lang="python">from geode_client import Client async def compute_pagerank(client, iterations=20, damping=0.85): # Initialize PageRank await client.execute(""" MATCH (n:Page) SET n.pagerank = 1.0 / COUNT { MATCH (p:Page) } """) # Iterate for i in range(iterations): result, _ = await client.query(f""" MATCH (s)-[:LINKS_TO]->(t) WITH t, SUM(s.pagerank / COUNT {{ (s)-[:LINKS_TO]->() }}) AS rank SET t.pr_new = {1-damping} + {damping} * rank WITH SUM(ABS(t.pagerank - t.pr_new)) AS change RETURN change """) await client.execute("MATCH (n) SET n.pagerank = n.pr_new REMOVE n.pr_new") if result.rows[0][0] < 0.0001: print(f"Converged after {i+1} iterations") break # Get top results top_pages, _ = await client.query(""" MATCH (n:Page) RETURN n.id, n.pagerank ORDER BY n.pagerank DESC LIMIT 20 """) return top_pages.rows </code></pre></div> <h4 id="rust-client" class="position-relative d-flex align-items-center group"> Rust Client <button type="button" class="h-share btn btn-link p-0 text-decoration-none link-secondary opacity-50 hover-opacity-100 transition-all ms-1" data-share-target="rust-client" aria-haspopup="dialog" aria-label="Share link: Rust Client"> Share link </button> </h4><div class="highlight"><pre tabindex="0" class="chroma"><code class="language-rust" data-lang="rust">use geode_client::Client; async fn pagerank_analysis(client: &Client) -> Result<Vec<(String, f64)>> { // Initialize client.execute("MATCH (n:Page) SET n.pr = 1.0 / COUNT { MATCH (p:Page) }").await?; // Iterate until convergence for iteration in 0..50 { let result = client.execute( "MATCH (s)-[:LINKS_TO]->(t) \ WITH t, SUM(s.pr / COUNT { (s)-[:LINKS_TO]->() }) AS r \ SET t.pr_new = 0.15 + 0.85 * r \ RETURN SUM(ABS(t.pr - t.pr_new)) AS change" ).await?; client.execute("MATCH (n) SET n.pr = n.pr_new REMOVE n.pr_new").await?; if let Some(change) = result.get_float(0, 0)? { if change < 0.0001 { println!("Converged at iteration {}", iteration); break; } } } // Extract results let results = client.execute( "MATCH (n:Page) RETURN n.id, n.pr ORDER BY n.pr DESC LIMIT 20" ).await?; Ok(results.into_iter() .map(|row| (row.get_string(0).unwrap(), row.get_float(1).unwrap())) .collect()) } </code></pre></div> <h3 id="related-topics" class="position-relative d-flex align-items-center group"> Related Topics <button type="button" class="h-share btn btn-link p-0 text-decoration-none link-secondary opacity-50 hover-opacity-100 transition-all ms-1" data-share-target="related-topics" aria-haspopup="dialog" aria-label="Share link: Related Topics"> Share link </button> </h3><ul> <li>Graph Algorithms: See graph-algorithms tag for comprehensive algorithm coverage</li> <li>Community Detection: Use community-detection for clustering and group identification</li> <li>Centrality Measures: Explore patterns tag for betweenness and closeness centrality</li> <li>Performance Tuning: Check optimization and performance tags for query optimization strategies</li> </ul> <h3 id="advanced-pagerank-variants" class="position-relative d-flex align-items-center group"> Advanced PageRank Variants <button type="button" class="h-share btn btn-link p-0 text-decoration-none link-secondary opacity-50 hover-opacity-100 transition-all ms-1" data-share-target="advanced-pagerank-variants" aria-haspopup="dialog" aria-label="Share link: Advanced PageRank Variants"> Share link </button> </h3> <h4 id="weighted-pagerank-with-damping" class="position-relative d-flex align-items-center group"> Weighted PageRank with Damping <button type="button" class="h-share btn btn-link p-0 text-decoration-none link-secondary opacity-50 hover-opacity-100 transition-all ms-1" data-share-target="weighted-pagerank-with-damping" aria-haspopup="dialog" aria-label="Share link: Weighted PageRank with Damping"> Share link </button> </h4><div class="highlight"><pre tabindex="0" class="chroma"><code class="language-gql" data-lang="gql">-- Edge-weighted PageRank MATCH (n:Page) SET n.pagerank = 1.0 / COUNT { MATCH (p:Page) }; MATCH (source:Page)-[link:LINKS_TO]->(target:Page) WITH target, SUM(link.weight * source.pagerank / SUM { (source)-[l:LINKS_TO]->() | l.weight }) AS weighted_contribution SET target.pagerank_new = 0.15 + 0.85 * weighted_contribution; </code></pre></div> <h4 id="topic-sensitive-pagerank-personalization" class="position-relative d-flex align-items-center group"> Topic-Sensitive PageRank (Personalization) <button type="button" class="h-share btn btn-link p-0 text-decoration-none link-secondary opacity-50 hover-opacity-100 transition-all ms-1" data-share-target="topic-sensitive-pagerank-personalization" aria-haspopup="dialog" aria-label="Share link: Topic-Sensitive PageRank (Personalization)"> Share link </button> </h4><div class="highlight"><pre tabindex="0" class="chroma"><code class="language-gql" data-lang="gql">-- Bias random jumps toward topic nodes MATCH (topic:Topic {name: $topic_name})<-[:BELONGS_TO]-(seed:Page) WITH COLLECT(seed) AS topic_pages MATCH (n:Page) SET n.personalized_pr = CASE WHEN n IN topic_pages THEN 1.0 / SIZE(topic_pages) ELSE 0.0 END; -- Iterate with personalization vector MATCH (source:Page)-[:LINKS_TO]->(target:Page) WITH target, SUM(source.personalized_pr / COUNT { (source)-[:LINKS_TO]->() }) AS propagated, target.personalized_pr AS teleport SET target.pr_new = 0.15 * teleport + 0.85 * propagated; </code></pre></div> <h3 id="large-scale-pagerank-optimization" class="position-relative d-flex align-items-center group"> Large-Scale PageRank Optimization <button type="button" class="h-share btn btn-link p-0 text-decoration-none link-secondary opacity-50 hover-opacity-100 transition-all ms-1" data-share-target="large-scale-pagerank-optimization" aria-haspopup="dialog" aria-label="Share link: Large-Scale PageRank Optimization"> Share link </button> </h3> <h4 id="block-striped-pagerank" class="position-relative d-flex align-items-center group"> Block Striped PageRank <button type="button" class="h-share btn btn-link p-0 text-decoration-none link-secondary opacity-50 hover-opacity-100 transition-all ms-1" data-share-target="block-striped-pagerank" aria-haspopup="dialog" aria-label="Share link: Block Striped PageRank"> Share link </button> </h4>Partition for parallel computation: <div class="highlight"><pre tabindex="0" class="chroma"><code class="language-gql" data-lang="gql">-- Partition nodes into blocks MATCH (n:Page) SET n.block_id = ID(n) % $num_blocks; -- Process blocks in parallel MATCH (source:Page {block_id: $block})-[:LINKS_TO]->(target:Page) WITH target, SUM(source.pagerank / source.out_degree) AS contribution SET target.pagerank_delta = contribution CALL IN TRANSACTIONS OF 100000 ROWS; -- Aggregate deltas MATCH (n:Page) SET n.pagerank = 0.15 + 0.85 * n.pagerank_delta, n.pagerank_delta = null; </code></pre></div> <h4 id="incremental-pagerank" class="position-relative d-flex align-items-center group"> Incremental PageRank <button type="button" class="h-share btn btn-link p-0 text-decoration-none link-secondary opacity-50 hover-opacity-100 transition-all ms-1" data-share-target="incremental-pagerank" aria-haspopup="dialog" aria-label="Share link: Incremental PageRank"> Share link </button> </h4>Update only affected subgraph: <div class="highlight"><pre tabindex="0" class="chroma"><code class="language-gql" data-lang="gql">-- After adding edge (u, v), identify affected nodes MATCH (u:Page {id: $new_source}), (v:Page {id: $new_target}) CREATE (u)-[:LINKS_TO]->(v); -- Mark affected region (forward reachable from v) MATCH (v:Page {id: $new_target})-[:LINKS_TO*0..3]->(affected:Page) SET affected.needs_update = true; -- Incremental update (only marked nodes) MATCH (source:Page)-[:LINKS_TO]->(target:Page) WHERE target.needs_update = true WITH target, SUM(source.pagerank / source.out_degree) AS new_contrib SET target.pagerank = 0.15 + 0.85 * new_contrib, target.needs_update = false; </code></pre></div> <h3 id="practical-applications-1" class="position-relative d-flex align-items-center group"> Practical Applications <button type="button" class="h-share btn btn-link p-0 text-decoration-none link-secondary opacity-50 hover-opacity-100 transition-all ms-1" data-share-target="practical-applications-1" aria-haspopup="dialog" aria-label="Share link: Practical Applications"> Share link </button> </h3> <h4 id="search-engine-ranking" class="position-relative d-flex align-items-center group"> Search Engine Ranking <button type="button" class="h-share btn btn-link p-0 text-decoration-none link-secondary opacity-50 hover-opacity-100 transition-all ms-1" data-share-target="search-engine-ranking" aria-haspopup="dialog" aria-label="Share link: Search Engine Ranking"> Share link </button> </h4><div class="highlight"><pre tabindex="0" class="chroma"><code class="language-gql" data-lang="gql">-- Combine PageRank with content relevance MATCH (page:Page) WHERE page.content CONTAINS $query WITH page, page.pagerank AS authority_score, // TF-IDF score from full-text index text_score(page, $query) AS relevance_score RETURN page.url, page.title, authority_score, relevance_score, 0.7 * relevance_score + 0.3 * authority_score AS combined_score ORDER BY combined_score DESC LIMIT 20; </code></pre></div> <h4 id="influence-propagation-in-social-networks" class="position-relative d-flex align-items-center group"> Influence Propagation in Social Networks <button type="button" class="h-share btn btn-link p-0 text-decoration-none link-secondary opacity-50 hover-opacity-100 transition-all ms-1" data-share-target="influence-propagation-in-social-networks" aria-haspopup="dialog" aria-label="Share link: Influence Propagation in Social Networks"> Share link </button> </h4><div class="highlight"><pre tabindex="0" class="chroma"><code class="language-gql" data-lang="gql">-- Find influential nodes for viral marketing MATCH (user:User) SET user.influence = 1.0 / COUNT { MATCH (u:User) }; -- Propagate through follower graph MATCH (follower:User)-[:FOLLOWS]->(leader:User) WITH leader, SUM(follower.influence / follower.following_count) AS follower_influence SET leader.influence_new = 0.15 + 0.85 * follower_influence; -- Select top influencers for campaign MATCH (influencer:User) WHERE influencer.influence > $threshold RETURN influencer.username, influencer.influence, COUNT { (influencer)<-[:FOLLOWS]-() } AS followers, influencer.influence * COUNT { (influencer)<-[:FOLLOWS]-() } AS reach_score ORDER BY reach_score DESC LIMIT 100; </code></pre></div> <h3 id="further-reading" class="position-relative d-flex align-items-center group"> Further Reading <button type="button" class="h-share btn btn-link p-0 text-decoration-none link-secondary opacity-50 hover-opacity-100 transition-all ms-1" data-share-target="further-reading" aria-haspopup="dialog" aria-label="Share link: Further Reading"> Share link </button> </h3><ul> <li>PageRank: The Original Algorithm and Modern Variants</li> <li>Personalized PageRank: Topic-Sensitive and Query-Dependent Ranking</li> <li>Scalable PageRank: Distributed and Incremental Computation</li> <li>Link Analysis: HITS, TrustRank, and Other Algorithms</li> </ul> Browse tagged content for complete PageRank documentation.

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