Clustering & Community Operators

Operator Category: Clustering & Community (clustering coefficient, community detection, transitivity, cycle structures)

Number of Algorithms: 23

Applicable Stages: network structure insight, clique/group identification, community quality evaluation, closed-loop/cycle detection, relationship density quantification, graph partitioning optimization, bipartite graph community analysis

Product Positioning: Provide a unified operator capability base for answering questions such as: "Does the network have tight subgroups?", "How can we partition people/entities into communities?", "Is this partition good?", "Are there closed loops/cycles?", "How to perform bipartite or multi-community partitioning?", "How to identify triadic structure patterns?"

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I. Operator Set Overview

The Clustering & Community operator set covers four major categories:

1. Local Density & Network Clustering

   • Are a node’s neighbors connected to each other?
   • How closed are triangles at the network level?
   • Are there cliques, gangs, or locally dense structures?
   • Typical algorithms: clustering, average_clustering, triangles, transitivity, square_clustering

2. Community Detection & Graph Partitioning

   • How to automatically discover communities, factions, or circles without labels?
   • How to perform fast community detection on large-scale networks?
   • How to perform bipartition or partition with a specified number of communities?
   • Typical algorithms: greedy_modularity_communities, naive_greedy_modularity_communities, louvain_communities, leiden_communities, girvan_newman, label_propagation_communities, asyn_lpa_communities, asyn_fluidc, kernighan_lin_bisection

3. Overlapping Communities & Bipartite Graph Community Analysis

   • Can a node belong to multiple communities simultaneously?
   • How to perform modularity-based bipartition on bipartite graphs?
   • Typical algorithms: k_clique_communities, spectral_modularity_bipartition

4. Cycle Structures & Loop Detection

   • What are the simple cycles in the graph?
   • What is a cycle basis for an undirected graph?
   • Is there a minimum weight cycle basis?
   • Typical algorithms: simple_cycles, cycle_basis, minimum_cycle_basis

5. Community Evaluation & Validation

   • How good is a community partition?
   • Is it a strict partition?
   • Are within-group edges sufficient and between-group edges few?
   • Typical algorithms: modularity, partition_quality, is_partition

6. Triadic Structure Pattern Analysis

   • What are the typical 3-node structure patterns in directed graphs?
   • Are there closed triads, transitive triples, or cyclic triples?
   • Typical algorithm: triadic_census

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II. Operator Capability Classification

Capability Type	Operator	Description
Node clustering coefficient	clustering	Compute local clustering coefficient for specified nodes or all nodes
Average clustering coefficient	average_clustering	Compute average clustering coefficient for the whole graph or a node subset
Triangle count	triangles	Count number of triangles each node participates in
Global transitivity	transitivity	Compute the overall triangle closure ratio of the network
Square clustering	square_clustering	Measure node’s tendency to participate in 4-cycles (redundant local structure)
Community - greedy modularity	greedy_modularity_communities	Clauset-Newman-Moore greedy merge to maximize modularity
Community - naive greedy modularity	naive_greedy_modularity_communities	Naive greedy modularity optimization, suitable for teaching or small graphs
Community - divisive (GN)	girvan_newman	Iteratively remove critical edges to obtain hierarchical community structure
Community - synchronous LPA	label_propagation_communities	Communities formed by majority label diffusion among neighbors
Community - asynchronous LPA	asyn_lpa_communities	Asynchronous label propagation supporting weights and random seeds
Community - k-clique percolation	k_clique_communities	Overlapping communities based on k-clique percolation
Community - Louvain	louvain_communities	Multi-level modularity optimization, fast community detection on large graphs
Community - Leiden	leiden_communities	Improved version of Louvain, more stable and enforces community connectivity
Community - asynchronous fluid	asyn_fluidc	Fluid-based diffusion to partition into a specified number of communities
Graph bipartition	kernighan_lin_bisection	Kernighan-Lin heuristic to partition graph into two parts
Bipartite graph spectral modularity bipartition	spectral_modularity_bipartition	Spectral method for modularity bipartition of bipartite graphs
Cycle – simple cycles enumeration	simple_cycles	Enumerate all simple cycles in a graph
Cycle – cycle basis	cycle_basis	Return a cycle basis for an undirected graph
Cycle – minimum cycle basis	minimum_cycle_basis	Return a cycle basis with minimal total weight for an undirected graph
Triadic census	triadic_census	Count the different 3-node structure patterns in a directed graph
Quality – modularity	modularity	Compute modularity Q for a given community partition
Quality – coverage/performance	partition_quality	Compute coverage and performance for a community partition
Validity – is partition	is_partition	Check whether a list of communities forms a strict partition

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III. General Input/Output Conventions

• Input G: NetworkX Graph / DiGraph

  • Clustering measures are mostly for undirected graphs, but some support directed or weighted forms.
  • Simple cycle enumeration is typically for directed graphs but can be used for some undirected scenarios.
  • Community detection is mostly for undirected graphs.
  • Bipartite algorithms require the input to conform to bipartite structure.

• Common Outputs

  • Metrics: float
  • Counts: dict / int
  • Community results: list[set] / iterable[set] / iterator[tuple[set]]
  • Cycle structures: list[list[node]] or iterator
  • Bisection result: tuple[set, set]
  • Validation: bool

Notes:

• Community detection algorithm outputs are usually lists of node sets.
• k_clique_communities allows overlapping communities, so its result may not be a strict partition.
• Modularity, coverage, performance generally require a partition covering all nodes without overlap.
• Cycle enumeration and clique-based algorithms may produce exponential results; set limits or filter subgraphs for large graphs.

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IV. Detailed Operator Descriptions

4.1 Local Density & Clustering

1. clustering — Node Clustering Coefficient

Description
Compute the local clustering coefficient for a node, measuring the proportion of connections among its neighbors – i.e., whether neighbors know each other.

Product Value

• Identify tightly-knit friend circles or gang cores
• Discover high-density relationship regions and key nodes
• Serve as structural features for fraud detection, social analysis, and recommendation systems

Typical Scenarios

• Social networks: find users with the tightest friend circles
• Transaction networks: identify accounts in closed transaction relationships
• Collaboration networks: discover members with high collaboration density
• Risk control networks: identify local gang structures

Key Parameters

• nodes: single node, list of nodes, or all nodes
• weight: edge weight representing relationship strength

Applicability & Characteristics

• Graph type: primarily undirected
• Output: {node: score} or single node score
• Complexity: typically related to square of node degree

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2. average_clustering — Average Clustering Coefficient

Description
Return the average clustering coefficient for the whole graph or a specified set of nodes. Values usually range between 0 and 1, summarizing the overall cliquishness of the network.

Product Value

• Provide a one-number summary of network clustering
• Suitable for comparisons across networks, time windows, or regions
• Monitor trends in network cliquishness or circle formation

Typical Scenarios

• Social network: overall clique density assessment
• Risk control: monitor gang formation trends
• City road networks: compare local redundancy levels
• Organizational collaboration: compare density of collaboration

Key Parameters

• nodes: only compute for a subgroup
• weight: incorporate relationship strength
• count_zeros: whether to include nodes with zero clustering coefficient in the average

Applicability & Characteristics

• Graph type: primarily undirected
• Output: float
• Complexity: typically related to square of node degree

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3. transitivity — Global Transitivity

Description
Compute the global transitivity of the network, i.e., the ratio of closed triads to all triads, measuring the overall triangle closure tendency.

Product Value

• Determine whether the network tends to be random, small-world, or cliquish
• Suitable for global structural characterization
• Support network modeling and comparison

Typical Scenarios

• Social networks: overall triangle closure degree
• Collaboration networks: tendency for three-way collaborations
• Risk control: closed structures in funds or accounts
• Knowledge graphs: transitive relationships among concepts

Applicability & Characteristics

• Graph type: primarily undirected
• Output: float
• Complexity: related to triangle counting

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4. triangles — Triangle Count

Description
Count the number of triangles each node participates in, or return the triangle count for specified nodes.

Product Value

• More triangles indicate a node lies in a tight circle
• Useful as a feature for gang identification, local density, and community core detection
• When combined with clustering coefficient, distinguishes “many connections” from “tight connections”

Typical Scenarios

• Social networks: users involved in many closed friend circles
• Transaction networks: accounts in closed transaction relationships
• Collaboration networks: group members with frequent three-way collaboration
• Recommendation networks: analysis of closed common-interest loops

Key Parameters

• nodes: single node, list, or all nodes

Applicability & Characteristics

• Graph type: undirected
• Output: dict or int
• Complexity: related to node degree and edge count

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5. square_clustering — Square Clustering Coefficient

Description
Measure the tendency of a node to participate in 4-cycles. A 4-cycle often represents local redundant relationships where two different paths connect the same targets.

Product Value

• Suitable for analyzing closed relationships in bipartite or near-bipartite structures
• Discover local alternative paths and structural redundancy
• More interpretable than triangles in user-item, author-paper, account-device networks

Typical Scenarios

• User-item network: multiple users purchasing multiple items
• Author-paper network: multiple authors co-authoring multiple papers
• Account-device network: multiple accounts sharing multiple devices
• Road network: local alternative route redundancy

Key Parameters

• nodes: compute only for a subset of nodes

Applicability & Characteristics

• Graph type: primarily undirected
• Output: {node: score}
• Complexity: related to local neighborhood size

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4.2 Community Detection & Graph Partitioning

6. greedy_modularity_communities — Greedy Modularity Communities

Description
Maximize modularity using a greedy merging strategy, outputting a list of communities. The algorithm starts from each node as its own community and repeatedly merges pairs that give the largest increase in modularity.

Product Value

• Classic, stable, and interpretable
• Default choice for mid-to-large scale undirected graphs
• Supports resolution parameter to control community size

Typical Scenarios

• Social networks: natural circle detection
• Transaction networks: initial gang identification
• Corporate relationship networks: group detection
• Knowledge networks: topic cluster identification

Key Parameters

• weight: edge weight
• resolution: community scale (larger yields smaller communities)
• cutoff: minimum number of communities
• best_n: maximum number of communities

Applicability & Characteristics

• Graph type: undirected
• Output: list[set]
• Suitable for: mid-to-large scale community detection

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7. girvan_newman — Girvan-Newman Hierarchical Communities

Description
Iteratively remove the most critical edges (by default, edges with highest betweenness centrality) to progressively split the graph, producing a hierarchical community structure from coarse to fine.

Product Value

• Highly interpretable, suitable for illustrating how communities are split
• Produces multi-level community partitions
• Good for small graphs or scenarios requiring an interpretable splitting path

Typical Scenarios

• Small-scale organizational relationship splitting
• Teaching complex network concepts
• Critical-edge-driven community fission analysis
• Risk control investigations needing an interpretable split path

Key Parameters

• most_valuable_edge: custom strategy to select the most valuable edge

Applicability & Characteristics

• Graph type: undirected
• Output: iterator over partitions
• Note: High complexity, not suitable for very large graphs

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8. label_propagation_communities — Synchronous Label Propagation Communities

Description
Form communities by diffusing labels to neighbors. Nodes iteratively adopt the majority label among neighbors; nodes with the same label form a community.

Product Value

• No need to specify number of communities
• Fast, suitable for coarse clustering on large graphs
• Can serve as a preprocessing step for more refined community detection

Typical Scenarios

• Large-scale social network rapid circle detection
• Recommendation system initial user interest grouping
• Communication network coarse partitioning
• Risk control candidate gang discovery

Applicability & Characteristics

• Graph type: undirected
• Output: generator[set(node)]
• Complexity: typically O(V + E) per iteration

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9. asyn_lpa_communities — Asynchronous Label Propagation Communities

Description
Asynchronous label propagation where node update order affects the result. Random seeds can improve reproducibility, and edge weights can influence label importance.

Product Value

• More flexible than synchronous LPA
• Supports weights and randomness control
• Fast community detection on large graphs

Typical Scenarios

• Weighted social network circle detection
• Large-scale transaction network preliminary gang identification
• Organizational communication network rapid grouping
• Recommendation network interest community partitioning

Key Parameters

• weight: edge weight influences label propagation
• seed: random seed for reproducibility

Applicability & Characteristics

• Graph type: undirected
• Output: generator[set(node)]
• Complexity: typically O(V + E) per iteration

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10. k_clique_communities — k-Clique Percolation Communities

Description
Use k-cliques as basic units; if two k-cliques share k-1 nodes, they belong to the same community. Nodes can belong to multiple communities.

Product Value

• Ideal for detecting very dense core circles
• Supports overlapping communities, more realistic for social and collaboration networks
• Highly interpretable for gangs, collusion, and small group identification

Typical Scenarios

• Social networks: allowing one person to belong to multiple friend circles
• Collaboration networks: authors belonging to multiple research teams
• Risk control: accounts involved in multiple gang structures
• Biological networks: proteins participating in multiple functional modules

Key Parameters

• k: minimum clique size (larger k yields stricter communities)
• cliques: optional precomputed clique list to avoid recomputation

Applicability & Characteristics

• Graph type: undirected
• Output: generator[frozenset(node)]
• Note: Results can overlap, not necessarily a strict partition

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11. louvain_communities — Louvain Communities

Description
The classic multi-level modularity optimization method. It first performs local node moves to improve modularity, then compresses communities into super-nodes for further optimization.

Product Value

• Industry standard for large-scale community detection
• Fast and generally good performance
• Supports multi-level community structures and resolution control

Typical Scenarios

• Large-scale social network community detection
• Transaction network gang identification
• Communication network structural partitioning
• Knowledge graph topic community discovery

Key Parameters

• weight: edge weight
• resolution: community scale
• threshold: modularity improvement threshold
• max_level: maximum number of levels
• seed: random seed

Applicability & Characteristics

• Graph type: undirected
• Output: list[set]
• Characteristics: results may be affected by randomness; set seed for reproducibility

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12. leiden_communities — Leiden Communities

Description
Leiden is an improved version of Louvain, generally more stable and emphasizes community internal connectivity, avoiding partitions where a community is not well-connected internally.

Product Value

• Usually more robust community quality than Louvain
• Suitable for scenarios requiring strong internal connectivity
• Better for rigorous community analysis and production outputs

Typical Scenarios

• Large-scale social network precise grouping
• Financial risk control gang identification
• Biological network functional module identification
• Organizational network stable circle detection

Key Parameters

• weight: edge weight
• resolution: community scale
• max_level: maximum number of levels
• seed: random seed

Applicability & Characteristics

• Graph type: undirected
• Output: list[set]
• Characteristics: typically more stable than Louvain

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13. naive_greedy_modularity_communities — Naive Greedy Modularity Communities

Description
Optimize modularity using a naive greedy approach. Less efficient than the optimized version, but more intuitive. Suitable for teaching, small graph validation, or comparison.

Product Value

• Easy to explain algorithm logic
• Good for small-scale community detection validation
• Can serve as a reference or baseline against optimized greedy modularity

Typical Scenarios

• Small network community detection
• Teaching demonstrations of modularity maximization
• Comparing with optimized greedy modularity results
• Debugging community detection logic

Key Parameters

• weight: edge weight
• resolution: community scale

Applicability & Characteristics

• Graph type: undirected
• Output: list of community sets
• Note: Not recommended for very large graphs

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14. spectral_modularity_bipartition — Spectral Modularity Bipartition

Description
Use spectral methods to perform modularity-based bipartition on bipartite graphs, dividing nodes into two modules.

Product Value

• Suitable for community detection on bipartite graphs
• Preserves the original two-sided structure of user-item, author-paper, etc.
• Avoids information loss caused by projection onto one side

Typical Scenarios

• User-item network community detection
• Author-paper network research direction grouping
• Institution-project network module identification
• Account-device network risk cluster splitting

Applicability & Characteristics

• Graph type: bipartite
• Output: bipartition result (two sets)
• Suitable for: modularity analysis on bipartite graphs

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15. asyn_fluidc — Asynchronous Fluid Communities

Description
Community detection based on fluid diffusion. The user specifies the number of communities k, and the algorithm asynchronously updates to form the specified number of communities.

Product Value

• Explicit control over the number of communities
• Fast, suitable for scenarios requiring a fixed number of groups
• Like label propagation, good for rapid community partitioning

Typical Scenarios

• Networks needing fixed K partitions
• Recommendation systems with known number of user segments
• Operations network fixed zoning
• Risk control candidate gang generation with specified count

Key Parameters

• k: number of communities to partition into
• max_iter: maximum number of iterations
• seed: random seed

Applicability & Characteristics

• Graph type: typically requires a connected undirected graph
• Output: iterable[set(node)]
• Note: Requires advance specification of community count

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16. kernighan_lin_bisection — Kernighan-Lin Bisection

Description
Use the Kernighan-Lin heuristic to partition the graph into two parts, aiming to minimize the cut edge weight between them.

Product Value

• Suitable for graph bipartition, load balancing, and task splitting
• Can be used for two-region partition, AB-group experiments, etc.
• More oriented toward balanced cuts or reducing cross-group connections than generic community detection

Typical Scenarios

• Computational task graph bipartition scheduling
• Network partitioning and load balancing
• Structured splitting for A/B testing
• Circuit or module partitioning

Key Parameters

• partition: optional initial bipartition
• max_iter: maximum iterations
• weight: edge weight
• seed: random seed

Applicability & Characteristics

• Graph type: undirected
• Output: tuple[set, set]
• Characteristics: heuristic, not guaranteed to be globally optimal

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4.3 Cycle Structures & Loop Detection

17. simple_cycles — Simple Cycle Enumeration

Description
Enumerate all simple cycles in the graph. A simple cycle is a closed walk where no node (except start/end) repeats.

Product Value

• Discover fund recycling, circular transactions, cyclic dependencies
• Identify feedback loops and cyclic propagation structures
• Useful for closed-loop risk investigation in directed networks

Typical Scenarios

• Financial network: circular transaction chains
• Software dependency: cyclic dependency chains
• Biological network: feedback regulation loops
• Process network: abnormal closed-loop processes

Key Parameters

• length_bound: limit cycle length to avoid output explosion

Applicability & Characteristics

• Graph type: primarily directed
• Output: cycle path iterator or list
• Note: Number of cycles can be exponential; set length bound or filter subgraph for large graphs

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18. cycle_basis — Cycle Basis

Description
Return a cycle basis for an undirected graph – a set of independent cycles that can generate all cycles.

Product Value

• Decompose complex cycle structures into basic components
• Suitable for circuit, road, pipeline network cycle analysis
• Better for structural summarization than enumerating all cycles

Typical Scenarios

• Circuit networks: fundamental loop analysis
• Road networks: basic closed block identification
• Pipeline systems: basic loop structure identification
• Relationship networks: fundamental closed structure summary

Key Parameters

• root: optional starting node

Applicability & Characteristics

• Graph type: undirected
• Output: list[list[node]]

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19. minimum_cycle_basis — Minimum Cycle Basis

Description
Return a minimum cycle basis for an undirected graph – a set of independent cycles with minimal total length or cost.

Product Value

• Prefers lower cost, smaller, or better-weighted cycle bases compared to a generic basis
• Suitable for scenarios requiring the most compact closed-loop explanation
• Useful in circuits, roads, pipelines for low-cost cycle analysis

Typical Scenarios

• Circuit networks: minimal fundamental loop analysis
• Road networks: shortest closed block identification
• Pipeline systems: low-cost fundamental loop analysis
• Chemical structures: minimal ring system identification

Key Parameters

• weight: edge weight attribute (distance, cost, length)

Applicability & Characteristics

• Graph type: undirected
• Output: list[list[node]]
• Suitable for: small to medium graphs or scenarios needing minimal closed-loop summary

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20. triadic_census — Triadic Census

Description
Count the number of each type of 3-node subgraph (triad) in a directed graph. Different triad patterns represent reciprocity, transitivity, cyclicity, null connections, etc.

Product Value

• Characterize directed network structure at the triad level
• Determine whether the network favors transitive, cyclic, or reciprocal relationships
• Suitable for microstructure analysis of social, transaction, communication, organization, and ecological networks

Typical Scenarios

• Social networks: mutual following, one-way following, three-person closed loop statistics
• Transaction networks: three-account cyclic transaction structure analysis
• Organization networks: hierarchical transitive vs. feedback relationships
• Communication networks: three-node message flow pattern analysis

Applicability & Characteristics

• Graph type: directed
• Output: dictionary of triad type counts
• Suitable for: microstructure pattern statistics and network fingerprinting

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4.4 Community Evaluation & Validation

21. modularity — Modularity Score

Description
Compute the modularity Q value for a given community partition, measuring whether within-community edges are more abundant than random expectation and between-community edges relatively few.

Product Value

• Provides a unified quality score for community partitions
• Useful for comparing different community detection algorithms or parameter settings
• The objective function optimized by Louvain, Leiden, greedy modularity, etc.

Typical Scenarios

• Compare Louvain vs. Leiden partition quality
• Select better result after adjusting resolution parameter
• Evaluate whether a manual grouping fits the network structure
• Quality check before deploying community detection results

Key Parameters

• communities: list of communities (should be a strict partition normally)
• weight: edge weight
• resolution: resolution parameter

Applicability & Characteristics

• Output: float
• Note: Higher modularity is generally better, but must be interpreted in business context and community scale

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22. partition_quality — Coverage and Performance

Description
Compute two quality metrics for a community partition:

• coverage: proportion of edges that are inside communities
• performance: proportion of node pairs that are either inside the same community or not connected across communities

Product Value

• Evaluate partition quality from different angles
• Coverage emphasizes whether within-group edges are abundant
• Performance considers both within-group edges and between-group separation

Typical Scenarios

• Quality assessment of community detection results
• Comparison between manual grouping and algorithmic grouping
• Judging clarity of community boundaries
• Evaluating gang splitting effectiveness in risk control

Applicability & Characteristics

• Input: community partition
• Output: (coverage, performance)
• Note: Typically requires a strict partition

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23. is_partition — Partition Validity Check

Description
Check whether a given list of communities forms a strict partition, i.e., covers all nodes and no overlap between communities.

Product Value

• Basic validation before community evaluation
• Prevent missing or duplicate nodes that could distort metrics
• Quality check before passing community results downstream

Typical Scenarios

• Verify that algorithm output covers all users
• Check manual grouping for duplicate members
• Validate community results before computing modularity
• Legality check before writing community results to database

Applicability & Characteristics

• Input: graph and community list
• Output: bool
• Note: Overlapping results like k_clique_communities typically do not satisfy strict partition

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V. Recommended Usage Guide (Selection Advice)

• First, measure whether the network is cliquish

  • Global cliquishness: average_clustering / transitivity
  • Local cliquishness: clustering
  • Triangle closure: triangles
  • 4-cycle redundancy: square_clustering

• Quickly obtain communities

  • Default for large graphs: louvain_communities / leiden_communities
  • Fast coarse grouping: label_propagation_communities / asyn_lpa_communities
  • Fixed number of communities: asyn_fluidc

• Require interpretability or hierarchical splits

  • Hierarchical communities: girvan_newman
  • Bipartition: kernighan_lin_bisection
  • Small graph greedy validation: naive_greedy_modularity_communities

• Find overlapping tight circles

  • Overlapping dense communities: k_clique_communities

• Handle bipartite graph communities

  • Bipartite spectral modularity bipartition: spectral_modularity_bipartition

• Detect closed loops, cyclic dependencies, or fund recycling

  • All simple cycles: simple_cycles
  • Cycle basis: cycle_basis
  • Minimum cycle basis: minimum_cycle_basis

• Analyze triadic patterns

  • Directed triad statistics: triadic_census

• Evaluate community results

  • Modularity: modularity
  • Coverage and performance: partition_quality
  • Validity check: is_partition

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VI. Typical Questions That Can Be Answered Directly (Examples)

• "Is the network overall loose random or small-world? Give me an overall metric."
• "Which nodes have the tightest friend circles? Top-20 clustering coefficients."
• "Which nodes participate in the most triangles?"
• "Does this bipartite network have significant 4-cycle redundancy?"
• "Automatically partition the network into natural communities and output each community's members."
• "Use Louvain and Leiden to detect communities and compare modularity."
• "I want a fixed partition into 5 communities. How should I do it?"
• "Bisect the graph into two parts while minimizing cross-group connections."
• "Find overlapping core circles."
• "Is this community result a strict partition? Are any nodes missing or duplicated?"
• "I have two community partitions; which is better? Give modularity and coverage/performance."
• "Find all fund closed loops or cyclic dependency chains, with cycle length no more than 6."
• "Output the minimum cycle basis of the undirected graph for basic closed-loop structure analysis."
• "Count triadic patterns in the directed graph and see how many transitive and cyclic triads exist."

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VII. Engineering Implementation Considerations

1. Clarify analysis objectives first

   • Clustering coefficients answer "Is local relationship dense?"
   • Community detection answers "How do nodes naturally group?"
   • Cycle algorithms answer "Are there closed loops or cyclic paths?"
   • Evaluation metrics answer "Is this partition good?"

2. No single correct answer for community detection

   • Louvain, Leiden, LPA, Girvan-Newman have different mechanisms and may yield different results.
   • Combine modularity, partition_quality, and business interpretation for selection.
   • For stochastic algorithms, set seed for reproducibility.

3. Prefer efficient algorithms for large graphs

   • Large graphs: louvain_communities, leiden_communities, asyn_lpa_communities.
   • girvan_newman is more interpretable but suited for small graphs.
   • Clique and cycle enumeration may be very slow on large graphs.

4. Distinguish overlapping vs. strict partitions

   • k_clique_communities allows node to belong to multiple communities.
   • modularity, partition_quality, is_partition are typically for strict partitions.
   • Do not directly evaluate overlapping communities with strict-partition metrics unless transformed.

5. Do not arbitrarily project bipartite graphs

   • Projecting user-item, author-paper can create many spurious edges.
   • Prefer bipartite methods like spectral_modularity_bipartition that preserve the bipartite structure.
   • If projection is necessary, define the projection rule and edge weights clearly.

6. Control output size for cycle algorithms

   • simple_cycles can have exponentially many results.
   • Use length_bound or filter subgraph in production.
   • For undirected graphs, prefer cycle_basis or minimum_cycle_basis for structural summary.

7. Triangles, squares, triads as structural features

   • triangles, square_clustering, triadic_census can serve as ML or rule-based features.
   • They do not directly produce communities but capture local patterns and structural anomalies.

8. Modularity has resolution limits

   • resolution affects community size.
   • Higher resolution typically yields smaller communities, lower resolution larger communities.
   • Choose scale based on business scenario, not only the highest modularity.

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VIII. Operator List

No.	Operator Name	Description (Chinese)
1	clustering	Node clustering coefficient
2	average_clustering	Average clustering coefficient
3	transitivity	Global transitivity
4	triangles	Triangle count
5	square_clustering	Square clustering coefficient
6	greedy_modularity_communities	Greedy modularity communities
7	girvan_newman	Girvan-Newman hierarchical communities
8	label_propagation_communities	Synchronous label propagation communities
9	asyn_lpa_communities	Asynchronous label propagation communities
10	k_clique_communities	k-clique percolation overlapping communities
11	louvain_communities	Louvain communities
12	leiden_communities	Leiden communities
13	simple_cycles	Simple cycle enumeration
14	cycle_basis	Cycle basis
15	modularity	Modularity score
16	partition_quality	Coverage and performance
17	is_partition	Partition validity check
18	naive_greedy_modularity_communities	Naive greedy modularity communities
19	spectral_modularity_bipartition	Spectral modularity bipartition (bipartite)
20	minimum_cycle_basis	Minimum cycle basis
21	triadic_census	Triadic census (directed)
22	asyn_fluidc	Asynchronous fluid communities
23	kernighan_lin_bisection	Kernighan-Lin bisection