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Session 6 - Module A: Advanced Graph Algorithms

⚠️ ADVANCED OPTIONAL MODULE This is supplementary content for deeper specialization.
Prerequisites: Complete Session 6 core content first. Time Investment: 40 minutes Target Audience: Implementer path students and graph algorithm specialists

Module Learning Outcomes

After completing this module, you will master: - Advanced graph traversal algorithms for complex knowledge networks - Optimization techniques for large-scale graph operations - Custom algorithm development for specialized graph patterns - Performance analysis and scaling strategies for graph-based RAG

Code Files

Quick Start

# Test advanced graph algorithms
cd src/session6
python demo_session6.py

# Test graph traversal engine
python -c "from graph_traversal_engine import GraphTraversalEngine; print('Graph algorithms ready!')"

# Test hybrid graph-vector system
python -c "from hybrid_graph_vector_rag import HybridGraphVectorRAG; HybridGraphVectorRAG().test_system()"

Advanced Algorithms Content

PageRank Variants for Knowledge Graphs

Advanced PageRank implementations for different GraphRAG scenarios:

class AdvancedPageRankAlgorithms:
    """Advanced PageRank variants for specialized GraphRAG applications."""

    def __init__(self, graph: nx.Graph):
        self.graph = graph
        self.algorithm_variants = {
            'temporal_pagerank': self._temporal_pagerank,
            'topic_sensitive_pagerank': self._topic_sensitive_pagerank,
            'biased_pagerank': self._biased_pagerank,
            'multi_layer_pagerank': self._multi_layer_pagerank
        }

PageRank variants address different GraphRAG scenarios beyond basic link analysis. Temporal PageRank considers information freshness, topic-sensitive PageRank focuses on domain-specific entities, biased PageRank emphasizes query-relevant nodes, and multi-layer PageRank handles heterogeneous graph structures with different relationship types.

    async def temporal_pagerank(self, time_decay: float = 0.9) -> Dict[str, float]:
        """PageRank that considers temporal relationships and information freshness."""

        # Create time-weighted adjacency matrix
        temporal_weights = {}
        current_time = time.time()

        for edge in self.graph.edges(data=True):
            source, target, data = edge
            edge_time = data.get('timestamp', current_time)
            time_diff = current_time - edge_time

            # Apply exponential decay based on age
            temporal_weight = math.exp(-time_diff / (86400 * 30)) * time_decay  # 30-day decay
            temporal_weights[(source, target)] = temporal_weight

Temporal PageRank applies exponential decay to edge weights based on their age, ensuring newer information receives higher importance. The 30-day decay period means information older than 30 days has significantly reduced influence, making this ideal for news, research, or rapidly evolving domains where freshness matters.

        # Compute PageRank with temporal weights
        pagerank_scores = nx.pagerank(
            self.graph,
            weight=lambda u, v: temporal_weights.get((u, v), 0.1),
            alpha=0.85
        )

        return pagerank_scores

The temporal-weighted PageRank computation uses custom edge weights that reflect both structural importance and temporal relevance. The alpha parameter (0.85) controls the random walk's probability of following edges versus jumping to random nodes, balancing local and global importance.

    async def topic_sensitive_pagerank(self, topic_entities: List[str],
                                     topic_weight: float = 0.3) -> Dict[str, float]:
        """PageRank biased toward specific topic entities for domain-focused retrieval."""

        # Create personalization vector emphasizing topic entities
        personalization = {}
        base_weight = (1.0 - topic_weight) / len(self.graph.nodes())
        topic_boost = topic_weight / len(topic_entities) if topic_entities else 0

        for node in self.graph.nodes():
            if node in topic_entities:
                personalization[node] = base_weight + topic_boost
            else:
                personalization[node] = base_weight

Topic-sensitive PageRank creates a personalization vector that biases random walks toward topic-relevant entities. The topic_weight parameter (0.3) allocates 30% of the random jump probability to topic entities, making them more likely to be reached and thus receive higher scores for domain-focused queries.

        pagerank_scores = nx.pagerank(
            self.graph,
            personalization=personalization,
            alpha=0.85
        )

        return pagerank_scores

The personalized PageRank computation ensures that random walks frequently return to topic-relevant entities, effectively boosting their importance scores. This approach is particularly effective for domain-specific GraphRAG systems where certain entity types should receive preferential treatment in retrieval ranking.

Graph Embedding Algorithms

Advanced graph embedding for semantic graph search:

class GraphEmbeddingEngine:
    """Advanced graph embedding algorithms for semantic graph traversal."""

    def __init__(self, embedding_dim: int = 128):
        self.embedding_dim = embedding_dim
        self.node_embeddings = {}
        self.edge_embeddings = {}

Graph embeddings transform nodes into dense vector representations that capture structural and semantic relationships. These embeddings enable similarity search, clustering, and machine learning on graph structures, making them essential for semantic graph traversal and retrieval.

    async def compute_node2vec_embeddings(self, graph: nx.Graph,
                                        walk_params: Dict = None) -> Dict[str, np.ndarray]:
        """Compute Node2Vec embeddings for graph nodes."""

        params = walk_params or {
            'dimensions': self.embedding_dim,
            'walk_length': 30,
            'num_walks': 200,
            'workers': 4,
            'p': 1,  # Return parameter
            'q': 1   # In-out parameter
        }

Node2Vec parameters control the random walk behavior: walk_length determines how far each walk extends, num_walks affects embedding quality through more training data, while p and q parameters control exploration strategy. Higher p encourages staying in local neighborhoods, while higher q promotes exploring distant nodes.

        # Generate random walks
        walks = self._generate_biased_walks(graph, params)

        # Train Word2Vec on walks to get embeddings
        from gensim.models import Word2Vec

        model = Word2Vec(
            walks,
            vector_size=params['dimensions'],
            window=10,
            min_count=0,
            sg=1,  # Skip-gram
            workers=params['workers'],
            epochs=10
        )

Node2Vec treats random walks as sentences and applies Word2Vec to learn node embeddings. The skip-gram model (sg=1) predicts context nodes from target nodes, creating embeddings where nodes with similar neighborhoods have similar vectors. This captures both local and global graph structure.

        # Extract node embeddings
        node_embeddings = {}
        for node in graph.nodes():
            if str(node) in model.wv:
                node_embeddings[node] = model.wv[str(node)]
            else:
                # Random embedding for missing nodes
                node_embeddings[node] = np.random.normal(0, 0.1, params['dimensions'])

        return node_embeddings

Embedding extraction handles cases where nodes might not appear in random walks due to isolation or disconnection. Random embeddings with small variance (0.1) ensure all nodes have representations while maintaining the learned embedding space's properties.

    def _generate_biased_walks(self, graph: nx.Graph, params: Dict) -> List[List[str]]:
        """Generate biased random walks for Node2Vec."""

        walks = []
        nodes = list(graph.nodes())

        for _ in range(params['num_walks']):
            random.shuffle(nodes)

            for node in nodes:
                walk = self._biased_walk(graph, node, params)
                walks.append([str(n) for n in walk])

        return walks

Biased walk generation ensures comprehensive graph coverage by starting walks from every node multiple times. Shuffling node order prevents systematic bias in walk generation, while multiple walks per node capture different aspects of each node's neighborhood structure.

    def _biased_walk(self, graph: nx.Graph, start_node: str, params: Dict) -> List[str]:
        """Perform a single biased random walk."""

        walk = [start_node]

        for _ in range(params['walk_length'] - 1):
            current = walk[-1]
            neighbors = list(graph.neighbors(current))

            if not neighbors:
                break

            if len(walk) == 1:
                # First step - uniform random
                next_node = random.choice(neighbors)

The biased walk algorithm starts with a single node and iteratively extends the walk. The first step uses uniform random selection from neighbors to establish an initial direction without bias, providing a fair starting point for the subsequent biased exploration.

            else:
                # Biased step based on p and q parameters
                prev_node = walk[-2]
                next_node = self._choose_next_node(
                    current, prev_node, neighbors, params['p'], params['q']
                )

            walk.append(next_node)

        return walk

Subsequent steps apply the p and q bias parameters to control exploration strategy. The algorithm considers the relationship between the current node, previous node, and potential next nodes to make biased selections that capture different types of graph structure, enabling flexible control over local versus global structure capture.

Community Detection for Graph Clustering

Advanced community detection for graph-based knowledge organization:

class GraphCommunityDetector:
    """Advanced community detection algorithms for knowledge graph clustering."""

    def __init__(self):
        self.detection_algorithms = {
            'louvain': self._louvain_detection,
            'leiden': self._leiden_detection,
            'spectral': self._spectral_clustering,
            'label_propagation': self._label_propagation
        }

Community detection identifies clusters of densely connected nodes within knowledge graphs, enabling better organization and more targeted retrieval. Different algorithms excel in different scenarios: Louvain optimizes modularity, Leiden improves upon Louvain's limitations, spectral methods use eigenvalues, and label propagation spreads community labels.

    async def detect_knowledge_communities(self, graph: nx.Graph,
                                         algorithm: str = 'louvain') -> Dict[str, Any]:
        """Detect knowledge communities in the graph for better organization."""

        if algorithm not in self.detection_algorithms:
            raise ValueError(f"Unsupported algorithm: {algorithm}")

        # Apply community detection
        communities = self.detection_algorithms[algorithm](graph)

        # Analyze community characteristics
        community_analysis = self._analyze_communities(graph, communities)

        # Generate community summaries
        community_summaries = await self._generate_community_summaries(
            graph, communities
        )

The community detection pipeline applies the selected algorithm, analyzes the resulting community structure for quality metrics like modularity and size distribution, and generates human-readable summaries. This comprehensive approach supports both algorithmic processing and human understanding of the knowledge organization.

        return {
            'communities': communities,
            'analysis': community_analysis,
            'summaries': community_summaries,
            'algorithm_used': algorithm
        }

The structured output provides both the raw community assignments and interpretable analysis. Community summaries help users understand what topics or domains each cluster represents, while analysis metrics help evaluate the quality of the detected community structure.

    def _louvain_detection(self, graph: nx.Graph) -> Dict[str, int]:
        """Louvain algorithm for community detection."""
        import networkx.algorithms.community as nx_comm

        # Apply Louvain algorithm
        communities = nx_comm.louvain_communities(graph, seed=42)

        # Convert to node -> community mapping
        community_mapping = {}
        for i, community in enumerate(communities):
            for node in community:
                community_mapping[node] = i

        return community_mapping

The Louvain algorithm iteratively optimizes modularity by moving nodes between communities and merging communities. Using a fixed seed (42) ensures reproducible results across runs, which is important for consistent graph organization and user experience in production systems.

    async def _generate_community_summaries(self, graph: nx.Graph,
                                          communities: Dict[str, int]) -> Dict[int, str]:
        """Generate natural language summaries for each community."""

        community_summaries = {}

        # Group nodes by community
        community_nodes = {}
        for node, comm_id in communities.items():
            if comm_id not in community_nodes:
                community_nodes[comm_id] = []
            community_nodes[comm_id].append(node)

Community summary generation creates human-readable descriptions of what each community contains. This involves grouping nodes by their community assignment and extracting representative information like node types and relationships to create meaningful descriptions.

        # Generate summary for each community
        for comm_id, nodes in community_nodes.items():
            # Extract node types and relationships
            node_info = []
            for node in nodes[:10]:  # Limit for summary
                node_data = graph.nodes.get(node, {})
                node_type = node_data.get('type', 'unknown')
                node_info.append(f"{node} ({node_type})")

            # Create community summary
            summary = f"Community {comm_id}: Contains {len(nodes)} entities including " + \
                     ", ".join(node_info[:5])
            if len(nodes) > 5:
                summary += f" and {len(nodes) - 5} others"

            community_summaries[comm_id] = summary

        return community_summaries

The summary creation process extracts node metadata like entity types and creates descriptive text that shows what each community represents. Limiting to the first 5 entities prevents overly long summaries while still providing representative information about the community's contents.

Graph Neural Networks for RAG

Integration of Graph Neural Networks for advanced graph reasoning:

class GraphNeuralNetworkRAG:
    """Graph Neural Network integration for advanced GraphRAG reasoning."""

    def __init__(self, hidden_dim: int = 128, num_layers: int = 3):
        self.hidden_dim = hidden_dim
        self.num_layers = num_layers

Graph Neural Networks (GNNs) enable learning sophisticated node representations that capture both local neighborhood information and global graph structure. GNNs are particularly powerful for GraphRAG because they can learn to weigh different relationship types and propagate information along multiple hops.

    async def train_graph_attention_network(self, graph: nx.Graph,
                                          node_features: Dict[str, np.ndarray],
                                          training_config: Dict = None) -> Dict[str, Any]:
        """Train Graph Attention Network for enhanced node representations."""

        config = training_config or {
            'epochs': 100,
            'learning_rate': 0.001,
            'dropout': 0.1,
            'attention_heads': 8
        }

Graph Attention Networks (GATs) learn to focus on the most relevant neighbors for each node through attention mechanisms. Multiple attention heads (8) capture different types of relationships, while dropout (0.1) prevents overfitting. The learning rate (0.001) balances training speed with stability.

First, we prepare the graph data and import the necessary PyTorch Geometric components:

        # Convert NetworkX graph to PyTorch Geometric format
        edge_index, node_feature_matrix = self._prepare_graph_data(graph, node_features)

        # Create GAT model
        from torch_geometric.nn import GATConv
        import torch
        import torch.nn.functional as F

Next, we define the Graph Attention Network architecture with two attention layers:

        class GraphAttentionNetwork(torch.nn.Module):
            def __init__(self, input_dim, hidden_dim, output_dim, num_heads):
                super().__init__()
                self.conv1 = GATConv(input_dim, hidden_dim, heads=num_heads, dropout=0.1)
                self.conv2 = GATConv(hidden_dim * num_heads, output_dim, heads=1, dropout=0.1)

The GAT architecture uses two graph attention layers with different head configurations. The first layer uses multiple heads to capture diverse relationship patterns, while the second layer aggregates these patterns into final representations. PyTorch Geometric provides optimized implementations for large graphs.

The forward pass implements the attention mechanism with dropout regularization:

            def forward(self, x, edge_index):
                x = F.dropout(x, training=self.training)
                x = self.conv1(x, edge_index)
                x = F.elu(x)
                x = F.dropout(x, training=self.training)
                x = self.conv2(x, edge_index)
                return x

The forward pass applies dropout for regularization, computes attention-weighted neighbor aggregation in each layer, and uses ELU activation for smooth gradients. This architecture learns node representations that adaptively focus on the most relevant neighbors for each specific node.

Finally, we initialize the model with the appropriate dimensions and return the training results:

        # Initialize and train model
        model = GraphAttentionNetwork(
            input_dim=node_feature_matrix.shape[1],
            hidden_dim=self.hidden_dim,
            output_dim=self.hidden_dim,
            num_heads=config['attention_heads']
        )

        # Training loop would go here
        # (simplified for brevity)

        return {
            'model': model,
            'node_embeddings': {},  # Enhanced embeddings from trained model
            'training_loss': [],
            'attention_weights': {}  # Attention patterns for interpretability
        }

The trained GAT model produces enhanced node embeddings that incorporate learned attention patterns. The attention weights provide interpretability by showing which neighbors influenced each node's representation, helping users understand the reasoning behind GraphRAG retrieval decisions.

📝 Multiple Choice Test - Module A

Test your understanding of advanced graph algorithms:

Question 1: What is the main advantage of temporal PageRank over standard PageRank in GraphRAG?
A) Faster computation
B) Considers information freshness and time-based relationships
C) Requires less memory
D) Simpler implementation

Question 2: In Node2Vec embeddings, what do the parameters p and q control?
A) Embedding dimensions and learning rate
B) Walk length and number of walks
C) Return probability and in-out bias for walk exploration
D) Training epochs and batch size

Question 3: Why is community detection important for large-scale GraphRAG systems?
A) It reduces storage requirements
B) It organizes knowledge into coherent clusters for better retrieval
C) It speeds up all queries equally
D) It simplifies graph construction

Question 4: What advantage do Graph Attention Networks provide for GraphRAG?
A) Faster training than traditional neural networks
B) Learn adaptive importance weights for different relationships and neighbors
C) Require less data for training
D) Work only with directed graphs

Question 5: In GraphRAG applications, when would you use biased random walks over uniform walks?
A) When you want faster computation
B) When you need to explore specific regions or relationship types more thoroughly
C) When the graph is very small
D) When memory is limited

🗂️ View Test Solutions →

Related Modules: - Core Session: Session 6 - Graph-Based RAG - Related Module: Module B - Production Systems

🗂️ Code Files: All examples use files in src/session6/ - graph_traversal_engine.py - Advanced traversal algorithms - knowledge_graph_extractor.py - Graph construction and analysis - hybrid_graph_vector_rag.py - Combined graph and vector search

🚀 Quick Start: Run cd src/session6 && python demo_session6.py to see advanced graph algorithms in action