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Distance & Measures Operator Set

Operator Category: Distance & Measures

Number of Algorithms: 24

Applicable Stages: Network health check, topological span assessment, center/periphery identification, global compactness analysis, graph resistance and random‑walk analysis, propagation/tracing cost evaluation, assortativity analysis, connection preference analysis, mixing matrix modeling, structural preference diagnosis

Product Positioning: Provides a unified capability base for answering “how large is the network span / which nodes are in the center or on the periphery / what is the overall distance cost / whether the graph is compact / whether nodes tend to connect to similar nodes / whether high‑degree nodes connect to each other / how different attributes mix in connections / how neighbor‑degree structure is distributed”.

I. Operator Set Overview​

The Distance & Measures operator set focuses on global distance structure, center‑periphery positioning, overall compactness, random‑walk stability, graph resistance characteristics, assortative mixing preferences, and mixing patterns, addressing the following questions:

  1. Distance and Span Structure

    • How far is a node from its farthest reachable node?
    • What is the distance between the two farthest nodes in the network?
    • Where is the optimal center of the network?
    • Which nodes lie on the structural periphery?
    • Is the network compact, and what is its overall distance cost?

  2. Center and Periphery Identification

    • Which nodes are best suited as monitoring entries, service centers, or coordination hubs?
    • Which nodes are most remote or hardest to cover?
    • How are the radius, diameter, and center sets distributed?

  3. Graph Resistance and Random‑Walk Measures

    • What is the resistance distance between two nodes?
    • What is the effective graph resistance of the whole graph?
    • How efficient is long‑term random‑walk access on the graph?
    • Does the network exhibit high diffusion or reachability cost?

  4. Assortativity and Connection Preferences

    • Do high‑degree nodes tend to connect to high‑degree nodes?
    • Do nodes with the same attribute tend to connect to each other?
    • Are nodes with similar numeric attributes more likely to be connected?
    • Is the network assortative, disassortative, or neutral?

  5. Mixing Matrices and Edge‑Endpoint Relationships

    • How frequently do different attribute classes connect to each other?
    • How do nodes of different degrees connect?
    • How are attribute values or degrees paired across each edge?
    • Can connection preferences be turned into matrices or dictionaries for downstream modeling?

II. Operator Capability Classification​

Capability TypeCorresponding OperatorDescription
Node farthest distanceeccentricityDistance to farthest reachable node
Graph radiusradiusMinimum eccentricity in the graph
Graph diameterdiameterMaximum eccentricity in the graph
Graph centercenterSet of nodes with minimum eccentricity
Graph peripheryperipherySet of nodes with eccentricity equal to diameter
Graph barycenterbarycenterSet of nodes with minimum sum of shortest‑path distances
Harmonic diameterharmonic_diameterEffective distance scale using harmonic mean
Wiener indexwiener_indexSum of all‑pairs shortest‑path distances
Resistance distanceresistance_distanceResistance distance between node pairs
Effective graph resistanceeffective_graph_resistanceGlobal effective resistance of the graph
Kemeny constantkemeny_constantMean hitting time to stationarity for random walks
Degree assortativity coefficientdegree_assortativity_coefficientCorrelation of degrees between connected nodes
Degree Pearson correlationdegree_pearson_correlation_coefficientPearson correlation of degrees across edges
Attribute assortativity coefficientattribute_assortativity_coefficientHomophily/heterophily for categorical attributes
Numeric assortativity coefficientnumeric_assortativity_coefficientCorrelation of numeric attributes across edges
Average neighbor degreeaverage_neighbor_degreeAverage degree of neighbors for each node
Average degree connectivityaverage_degree_connectivityFor each degree, average neighbor degree
Attribute mixing matrixattribute_mixing_matrixMatrix of connections between attribute classes
Degree mixing matrixdegree_mixing_matrixMatrix of connections between degree values
Attribute mixing dictionaryattribute_mixing_dictDictionary version of attribute mixing
Degree mixing dictionarydegree_mixing_dictDictionary version of degree mixing
Generic mixing dictionarymixing_dictBuild mixing dictionary from arbitrary endpoint value pairs
Node attribute XY generatornode_attribute_xyGenerate (attribute value, attribute value) pairs for each edge
Node degree XY generatornode_degree_xyGenerate (degree, degree) pairs for each edge

III. General Input/Output Conventions​

  • Input G: NetworkX Graph / DiGraph
  • Common Input Parameters:
    • weight: Edge weight field name, function, or None
    • e: Pre‑computed eccentricity dictionary (reused by radius, diameter, center, periphery)
    • sp: Pre‑computed shortest‑path distance dictionary (for eccentricity, barycenter, harmonic_diameter)
    • usebounds: Acceleration for undirected graphs using bounds
    • attribute: Node attribute name (for attribute‑based assortativity and mixing)
    • nodes: Subset of nodes to restrict computation
    • x / y: Degree type for source/target in directed graphs (in / out)
    • source / target: Direction or degree type for neighbor‑degree algorithms
    • normalized: Whether to normalize mixing matrices or dictionaries
  • Common Output Types:
    • Distance measures: numeric, list of nodes, or dict[node → value]
    • Resistance / random‑walk: numeric or node‑pair measures
    • Assortativity: float correlation coefficient
    • Neighbor degree: dict[node → average_degree] or dict[degree → average_neighbor_degree]
    • Mixing relations: matrix, nested dictionary, or edge‑endpoint value iterator

IV. Detailed Operator Descriptions​

1. eccentricity – Eccentricity​

Description
For each node, computes the maximum shortest‑path distance to any reachable node. A larger eccentricity means the node is more peripheral.

Product Value

  • Measures worst‑case reachability distance
  • Identifies remote nodes and coverage blind spots
  • Foundation for radius, diameter, center, periphery

Typical Scenarios

  • Network coverage analysis
  • Service entry point selection
  • Tracking cost estimation
  • Remote node identification
  • Topology health check

Applicability & Characteristics

  • Graph type: Directed / Undirected
  • Output: dict[node → eccentricity] or single value
  • Note: For disconnected graphs, interpret per component or reachable subgraph

2. radius – Graph Radius​

Description
The minimum eccentricity among all nodes. It is the smallest possible worst‑case distance from an optimal center.

Product Value

  • Measures the best‑center worst‑case coverage distance
  • Indicates global compactness
  • Useful for hub location and coordination ability assessment

Typical Scenarios

  • Service center placement
  • Monitoring entry deployment
  • Emergency coordination hub selection
  • Organisational coordination cost analysis
  • Network compactness evaluation

Applicability & Characteristics

  • Graph type: Directed / Undirected
  • Output: numeric
  • Can reuse pre‑computed eccentricity for speed

3. diameter – Graph Diameter​

Description
The maximum eccentricity among all nodes. It is the longest shortest‑path distance between any two nodes.

Product Value

  • Measures the maximum span of the network
  • Indicates worst‑case propagation, tracking, or collaboration cost
  • Allows structural comparison across networks

Typical Scenarios

  • Social network span analysis
  • Maximum detour distance in transportation
  • Maximum hierarchical distance in supply chains
  • Upper bound of communication latency
  • Topology health check

Applicability & Characteristics

  • Graph type: Directed / Undirected
  • Output: numeric
  • Interpret carefully for disconnected graphs

4. center – Graph Center​

Description
Returns the set of nodes whose eccentricity equals the graph radius. These are the structural centers.

Product Value

  • Identifies best global center nodes
  • Supports selection of service, monitoring, scheduling, and coordination hubs
  • Helps understand central locations in the network

Typical Scenarios

  • Monitoring node selection
  • Service facility layout
  • Hub node identification
  • Emergency response centre selection
  • Network centrality interpretation

Applicability & Characteristics

  • Graph type: Directed / Undirected
  • Output: list of center nodes
  • Multiple centers may exist

5. periphery – Graph Periphery​

Description
Returns the set of nodes whose eccentricity equals the graph diameter. These nodes lie on the structural boundary.

Product Value

  • Identifies coverage blind spots and remote positions
  • Supports edge node, isolated risk point, and structural boundary analysis
  • Useful for network optimisation and adding edges

Typical Scenarios

  • Remote device identification
  • Peripheral user analysis
  • Transportation terminal analysis
  • Supply chain endpoint risk identification
  • Network coverage optimisation

Applicability & Characteristics

  • Graph type: Directed / Undirected
  • Output: list of periphery nodes
  • Often used together with diameter and eccentricity

6. barycenter – Graph Barycenter​

Description
Returns the set of nodes that minimise the sum of shortest‑path distances to all other nodes. Unlike center (which minimises worst‑case distance), barycenter minimises total distance cost.

Product Value

  • Selects nodes with smallest average access cost
  • Suitable for logistics hubs, coordination centres, or service placement
  • Complements center by considering total cost instead of worst case

Typical Scenarios

  • Warehouse / logistics center location
  • Organisational coordination hub analysis
  • Network service node selection
  • Average access cost optimisation
  • Road network center identification

Applicability & Characteristics

  • Graph type: Typically for connected graphs
  • Output: list of barycenter nodes
  • Can use weight for weighted shortest paths

7. harmonic_diameter – Harmonic Diameter​

Description
Uses the harmonic mean of pairwise distances to measure the effective distance scale of the graph. More robust to extreme distances or disconnected components than ordinary diameter.

Product Value

  • Provides a robust global distance measure
  • Suitable for graphs that are not fully connected or have local fractures
  • Allows comparison of overall reachability efficiency

Typical Scenarios

  • Large‑scale social network distance analysis
  • Effective distance in disconnected graphs
  • Compactness comparison before/after network evolution
  • Propagation efficiency assessment
  • Topology optimisation evaluation

Applicability & Characteristics

  • Graph type: Directed / Undirected
  • Output: numeric
  • Focuses on overall effective distance rather than the single farthest pair

8. wiener_index – Wiener Index​

Description
Sum of shortest‑path distances over all unordered node pairs. Measures the global distance cost of the graph.

Product Value

  • Quantifies overall compactness of the network
  • Supports structural comparison between different networks
  • Useful for propagation, collaboration, transportation cost evaluation

Typical Scenarios

  • Total path cost analysis in transportation networks
  • Collaboration cost in supply chains
  • Compactness comparison of social networks
  • Molecular graph structure analysis
  • Before/after topology optimisation

Applicability & Characteristics

  • Graph type: Directed / Undirected
  • Output: numeric
  • Can be computed unweighted or weighted

9. resistance_distance – Resistance Distance​

Description
Treats the graph as an electrical network and computes the equivalent resistance distance between two nodes. It accounts for multiple parallel paths (redundancy) in addition to the shortest path.

Product Value

  • Measures global connection strength between nodes
  • Reflects path redundancy influence on proximity
  • More nuanced than simple shortest‑path distance

Typical Scenarios

  • Robustness analysis
  • Power grid / communication network structure analysis
  • Node similarity measurement
  • Multi‑path redundancy assessment
  • Graph kernel feature construction

Applicability & Characteristics

  • Graph type: Typically undirected, connected
  • Output: numeric resistance distance
  • Smaller distance means stronger overall connectivity

10. effective_graph_resistance – Effective Graph Resistance​

Description
Computes the total effective resistance of the whole graph, a global measure of the graph’s resistance (inverse of robustness).

Product Value

  • Measures overall connectivity resilience
  • Reflects path redundancy and structural tightness
  • Allows robustness comparison across networks

Typical Scenarios

  • Power network resilience analysis
  • Communication network robustness assessment
  • Transportation network redundancy analysis
  • Graph structure optimisation comparison
  • Network vulnerability analysis

Applicability & Characteristics

  • Graph type: Typically undirected, connected
  • Output: numeric
  • Lower values indicate tighter connectivity and stronger redundancy

11. kemeny_constant – Kemeny Constant​

Description
The Kemeny constant of a random walk on the graph. It measures the expected time to reach stationarity, averaged over starting nodes.

Product Value

  • Quantifies mixing efficiency of random walks
  • Supports diffusion, access, and convergence analysis
  • Serves as a global reachability and stability measure

Typical Scenarios

  • Random walk model analysis
  • Web page access modelling
  • Random walk‑based recommendation recall
  • Diffusion process stability analysis
  • Markov chain network evaluation

Applicability & Characteristics

  • Graph type: Typically connected graphs (or suitable directed graphs)
  • Output: numeric
  • Related to random walks, transition matrices, and stationary analysis

12. degree_assortativity_coefficient – Degree Assortativity Coefficient​

Description
Measures the correlation of degrees between connected nodes.

  • r > 0: Assortative (high‑degree nodes connect to high‑degree nodes)
  • r < 0: Disassortative (high‑degree nodes connect to low‑degree nodes)
  • r ≈ 0: No linear degree preference

Product Value

  • Determines whether high‑degree nodes cluster together or form a core‑periphery
  • Supports connection preference analysis for active accounts
  • Identifies core‑periphery structure

Typical Scenarios

  • Social network assortativity analysis
  • Core‑periphery diagnosis
  • High‑activity account connection preference
  • Biological network connection patterns
  • Transaction network hierarchical structure analysis

Applicability & Characteristics

  • Graph type: Directed / Undirected
  • Output: float correlation coefficient
  • For directed graphs, can specify in‑ or out‑degree for source/target

13. degree_pearson_correlation_coefficient – Degree Pearson Correlation Coefficient​

Description
Pearson correlation of degrees across edges. Equivalent to degree assortativity in the undirected case.

Product Value

  • Standard correlation view of degree mixing
  • Suitable for statistical analysis and cross‑network comparison
  • Can be cross‑checked with degree assortativity

Typical Scenarios

  • Degree correlation analysis
  • Statistical modelling of network structure
  • High‑degree node connection preference
  • Core‑periphery identification
  • Assortativity comparison across networks

Applicability & Characteristics

  • Graph type: Directed / Undirected
  • Output: float
  • Interpretation similar to degree assortativity coefficient

14. attribute_assortativity_coefficient – Attribute Assortativity Coefficient​

Description
Measures whether nodes with the same categorical attribute value are more likely to be connected.

  • r > 0: Homophily (same attribute connects)
  • r < 0: Heterophily (different attributes connect)
  • r ≈ 0: Weak preference

Product Value

  • Diagnoses homophily/heterophily in the network
  • Supports analysis of connection preferences for departments, types, risk levels
  • Useful for community, organisation, and risk structure analysis

Typical Scenarios

  • Intra‑ / inter‑department collaboration analysis
  • Risk‑level homophily analysis
  • User type connection preference
  • Institutional hierarchical connectivity
  • Community homophily evaluation

Applicability & Characteristics

  • Graph type: Directed / Undirected
  • Required parameter: attribute
  • Output: float

15. numeric_assortativity_coefficient – Numeric Assortativity Coefficient​

Description
Measures whether the numeric attribute values (e.g., age, score, risk level) of connected nodes are correlated (Pearson correlation).

Product Value

  • Supports connection preference analysis for continuous or ordinal attributes
  • Determines whether similar numeric values tend to connect
  • Suitable for risk, grade, activity, size, etc.

Typical Scenarios

  • Do accounts with similar risk scores connect?
  • Do users of similar ages follow each other?
  • Do nodes with similar activity levels cluster?
  • Do companies of similar size trade?
  • Node score vs connection preference analysis

Applicability & Characteristics

  • Graph type: Directed / Undirected
  • Required parameter: attribute
  • Output: float correlation

16. average_neighbor_degree – Average Neighbor Degree​

Description
For each node, computes the average degree of its neighbors. Indicates whether a node is connected to high‑activity or low‑activity neighbors.

Product Value

  • Assesses the quality of neighbors a node connects to
  • Identifies “core‑attached” or “peripheral” nodes
  • A useful local structural feature

Typical Scenarios

  • Auxiliary node influence assessment
  • Social network neighbor quality analysis
  • Fraud network peripheral node identification
  • Transaction partner activity analysis
  • Network hierarchical structure modelling

Applicability & Characteristics

  • Graph type: Directed / Undirected
  • Output: dict[node → average_neighbor_degree]
  • Can use source, target, weight parameters

17. average_degree_connectivity – Average Degree Connectivity​

Description
Groups nodes by their degree and computes, for each degree value, the average degree of their neighbors. Shows what degrees nodes of a given degree tend to connect to.

Product Value

  • Reveals mixing patterns between degree layers
  • Indicates assortativity or disassortativity at the degree‑level
  • Suitable for plotting degree connectivity curves

Typical Scenarios

  • Analysis of whether high‑degree nodes connect to high‑degree nodes
  • Core‑periphery pattern identification
  • Network hierarchical structure analysis
  • Activity connection preference in social networks
  • Degree‑related patterns in biological networks

Applicability & Characteristics

  • Graph type: Directed / Undirected
  • Output: dict[degree → average_neighbor_degree]
  • Often interpreted together with assortativity coefficients

18. attribute_mixing_matrix – Attribute Mixing Matrix​

Description
Constructs a matrix where rows and columns correspond to attribute values, and entries count (or proportion) of edges between attribute classes.

Product Value

  • Converts attribute connection preference into a matrix
  • Supports quantitative analysis of inter‑class connection frequencies
  • Foundational for assortativity and homophily analysis

Typical Scenarios

  • Inter‑department collaboration matrix
  • Risk level connection matrix
  • User type interaction matrix
  • Industry transaction relation matrix
  • Community connection preference analysis

Applicability & Characteristics

  • Graph type: Directed / Undirected
  • Required parameter: attribute
  • Output: matrix (list of lists or numpy array)
  • Can be normalized

19. degree_mixing_matrix – Degree Mixing Matrix​

Description
Constructs a matrix where rows and columns correspond to degree values, and entries count (or proportion) of edges between nodes of those degrees.

Product Value

  • Shows connection patterns across degree layers
  • Supports assortativity and core‑periphery analysis
  • Useful for statistical modelling of network structure

Typical Scenarios

  • High‑degree node connection preference analysis
  • Core‑periphery connectivity pattern analysis
  • Visualisation of assortativity/disassortativity
  • Hierarchical degree structure modelling
  • Topology comparison

Applicability & Characteristics

  • Graph type: Directed / Undirected
  • Output: matrix
  • For directed graphs, can specify x and y degree types

20. attribute_mixing_dict – Attribute Mixing Dictionary​

Description
Dictionary representation of attribute mixing, with keys as (attr_u, attr_v) and values as counts or proportions. Sparse version of the mixing matrix.

Product Value

  • Easy inspection of specific attribute pair connections
  • Suitable for sparse categories or rule extraction
  • Lightweight alternative to mixing matrix

Typical Scenarios

  • Category connection frequency counting
  • Department‑department connection lookup
  • Risk type interaction analysis
  • Multi‑label node relation analysis
  • Attribute preference rule extraction

Applicability & Characteristics

  • Graph type: Directed / Undirected
  • Required parameter: attribute
  • Output: nested dictionary
  • Can be normalized

21. degree_mixing_dict – Degree Mixing Dictionary​

Description
Dictionary representation of degree mixing, keys as (deg_u, deg_v) with counts or proportions.

Product Value

  • Direct view of degree‑pair connection distribution
  • Supports assortativity, disassortativity, and hierarchical analysis
  • Convenient for custom statistics or rule extraction

Typical Scenarios

  • Degree‑layer connection statistics
  • High‑low degree connection analysis
  • Core‑periphery structural interpretation
  • Network generation model parameter analysis
  • Topological connection preference modelling

Applicability & Characteristics

  • Graph type: Directed / Undirected
  • Output: nested dictionary
  • For directed graphs, can specify degree types for source/target

22. mixing_dict – Generic Mixing Dictionary​

Description
Given a sequence of endpoint value pairs (generated from edges), builds a mixing dictionary. A generic tool underlying attribute and degree mixing.

Product Value

  • Supports custom mixing statistics beyond attributes/degrees
  • Extensible for other node properties
  • Builds custom structural indicators

Typical Scenarios

  • Custom category mixing statistics
  • Edge endpoint label pair counting
  • Connection preference rule generation
  • Graph feature engineering
  • Relational pattern modelling

Applicability & Characteristics

  • Input: Iterable of (value_u, value_v) pairs
  • Output: mixing dictionary
  • Optional normalisation

23. node_attribute_xy – Node Attribute XY Generator​

Description
Generates, for each edge, the pair of attribute values of its two endpoints. Used as input for attribute mixing matrix/dictionary or custom assortativity calculations.

Product Value

  • Provides attribute pairing data for further analysis
  • Supports custom connection preference analysis
  • Data preparation step for downstream statistics

Typical Scenarios

  • Generating department‑department pairs
  • Risk level edge pairs
  • User type interaction pairs
  • Custom attribute assortativity computation
  • Preprocessing for mixing matrix construction

Applicability & Characteristics

  • Graph type: Directed / Undirected
  • Required parameter: attribute
  • Output: iterator of (attr_u, attr_v) pairs

24. node_degree_xy – Node Degree XY Generator​

Description
Generates, for each edge, the pair of degrees of its two endpoints. Used as input for degree mixing matrix/dictionary or custom degree correlation analysis.

Product Value

  • Provides degree pairing data for analysis
  • Supports custom degree‑based statistics
  • Foundation for degree mixing and assortativity

Typical Scenarios

  • High‑degree node connection pattern analysis
  • Custom degree assortativity calculation
  • Degree mixing matrix construction
  • Core‑periphery structure analysis
  • Graph statistical feature engineering

Applicability & Characteristics

  • Graph type: Directed / Undirected
  • Output: iterator of (deg_u, deg_v) pairs
  • For directed graphs, can specify x and y degree types

1. Distance and Span Analysis​

  • Node farthest distance: eccentricity
  • Maximum network span: diameter
  • Best‑center worst‑case distance: radius
  • Structural center nodes: center
  • Structural periphery nodes: periphery
  • Overall distance cost: wiener_index
  • Robust effective distance: harmonic_diameter

2. Centrality and Overall Cost​

  • Nodes with smallest total distance: barycenter
  • Nodes with smallest worst‑case distance: center
  • Compare global distance cost: wiener_index
  • Average‑sense reachability efficiency: harmonic_diameter

3. Graph Resistance and Random Walk​

  • Pairwise resistance distance: resistance_distance
  • Global effective resistance: effective_graph_resistance
  • Random‑walk mixing efficiency: kemeny_constant

4. Assortativity and Connection Preference​

  • High‑degree clustering: degree_assortativity_coefficient
  • Degree Pearson correlation: degree_pearson_correlation_coefficient
  • Categorical homophily: attribute_assortativity_coefficient
  • Numeric attribute correlation: numeric_assortativity_coefficient

5. Neighbor Degree Structure​

  • Node‑level average neighbor degree: average_neighbor_degree
  • Degree‑level average neighbor degree: average_degree_connectivity

6. Mixing Matrices and Dictionaries​

  • Attribute mixing matrix: attribute_mixing_matrix
  • Degree mixing matrix: degree_mixing_matrix
  • Attribute mixing dictionary: attribute_mixing_dict
  • Degree mixing dictionary: degree_mixing_dict
  • Custom mixing: mixing_dict

7. Edge Endpoint Pair Generation​

  • Attribute value pairs: node_attribute_xy
  • Degree value pairs: node_degree_xy

VI. Typical Questions That Can Be Directly Answered​

  • “What is the maximum span of this network?”
  • “Which nodes are at the network center?”
  • “Which nodes are most remote and hardest to cover?”
  • “From the optimal center, what is the worst case distance?”
  • “What is the total distance cost of the whole network?”
  • “Which node has the smallest sum of distances to all others?”
  • “Is this network compact?”
  • “What is the resistance distance between two nodes?”
  • “Is the effective graph resistance high or low?”
  • “How efficient is random walk mixing on this graph?”
  • “Do high‑degree nodes tend to connect to high‑degree nodes?”
  • “Does the network exhibit core‑periphery structure?”
  • “Do nodes of the same department/type/risk level connect more?”
  • “Are nodes with similar numeric attributes more likely to connect?”
  • “What is the average neighbor degree for each node?”
  • “What degrees do nodes of a given degree connect to?”
  • “What is the mixing matrix between attribute classes?”
  • “What is the mixing matrix between degree layers?”
  • “For each edge, how are the attribute values paired?”
  • “For each edge, how are the degrees paired?”

VII. Engineering and Usage Notes​

  1. Connectivity Requirements Distance metrics (eccentricity, radius, diameter, center, periphery, barycenter, wiener_index) require careful interpretation for disconnected graphs.

    • Undirected: compute per connected component or take the largest component.
    • Directed: consider strongly connected components or interpret within reachable subgraphs.

  2. Weighted Distance Semantics weight should represent distance, cost, time, or impedance. If edge weights represent capacity, similarity, or strength (larger = better), transform them appropriately (e.g., reciprocal or negative log, ensuring positivity).

  3. Interpreting Assortativity Assortativity coefficients show correlation, not causation.

    • r > 0: like connects to like.
    • r < 0: opposite connects.
    • r ≈ 0: no linear preference.

  4. Mixing Matrices and Dictionaries

    • Matrices are good for visualisation and numerical computations.
    • Dictionaries are good for sparse categories, rule extraction, and direct lookup.
    • Normalisation changes interpretation (frequency vs. probability).

  5. Performance Recommendations

    • When computing eccentricity, radius, diameter, center, periphery together, pre‑compute and reuse eccentricity.
    • All‑pairs shortest‑path based metrics are expensive on large graphs; consider sampling, component analysis, or approximation.
    • Assortativity and mixing measures are generally much lighter.

VIII. Operator List​

No.Operator NameDescription
1eccentricityCompute node eccentricity
2radiusCompute graph radius
3diameterCompute graph diameter
4centerGet graph center nodes
5peripheryGet graph periphery nodes
6barycenterGet graph barycenter nodes
7harmonic_diameterCompute harmonic diameter
8wiener_indexCompute Wiener index
9resistance_distanceCompute resistance distance between nodes
10effective_graph_resistanceCompute effective graph resistance
11kemeny_constantCompute Kemeny constant
12degree_assortativity_coefficientCompute degree assortativity coefficient
13degree_pearson_correlation_coefficientCompute degree Pearson correlation
14attribute_assortativity_coefficientCompute attribute assortativity coefficient
15numeric_assortativity_coefficientCompute numeric assortativity coefficient
16average_neighbor_degreeCompute average neighbor degree
17average_degree_connectivityCompute average degree connectivity
18attribute_mixing_matrixBuild attribute mixing matrix
19degree_mixing_matrixBuild degree mixing matrix
20attribute_mixing_dictBuild attribute mixing dictionary
21degree_mixing_dictBuild degree mixing dictionary
22mixing_dictBuild generic mixing dictionary
23node_attribute_xyGenerate node attribute XY pairs
24node_degree_xyGenerate node degree XY pairs