Katz Centrality

SQL function: cugraph_katz_centrality

Official cuGraph reference: C API

Score vertices from the number of walks that reach them, attenuating longer walks and adding a baseline contribution.

Signature

cugraph_katz_centrality(table_name [, src_col, dst_col [, weight_col [, options_json]]])

Relation inputs

The first positional argument names a registered edge table or view (the edges role). Parenthesized relation subqueries are not accepted; metadata validation uses the same registered name.

Vertex ID types

The edges relation declares the accepted vertex-ID domains. Numeric calls preserve the existing numeric schema. When logical string support is declared, Utf8, LargeUtf8, and Utf8View endpoint columns share one logical domain; their vertex-identity outputs are canonicalized to Utf8.

Domain	Accepted endpoint inputs	Output contract
Numeric edge endpoints	Int32, Int64	The numeric output schema is used for numeric calls.
Logical string edge endpoints	Utf8, LargeUtf8, Utf8View	Vertex identity columns are canonicalized to Utf8; scores, distances, counts, coordinates, and opaque labels remain numeric.

The native mapping type is Int64. Call-specific output schemas come from gpu_validate_call.

Logical string side-input limitations:

• edge ID columns and edge-ID predicate side inputs are not supported for logical string graphs

Scalar arguments & JSON options

Positional scalar arguments

src_col and dst_col name the edge endpoint columns; both are optional and default to src and dst.

Argument	Type	Required	Default	Notes
weight_col	Utf8|null	no		accepted as an edge-column binding; native algorithm execution does not consume weights; semantic effect: none for this algorithm

JSON options

Option	Type	Default	Constraints	Description
alpha	Float64	0.01	max 1; > 0
beta	Float64	1	> 0
epsilon	Float64	0.000001	> 0
max_iterations	UInt32	1000	min 1

Graph construction options

Graph construction follows the shared defaults (directed=true, renumbering, python_cugraph policy) documented in Graph Construction Options.

Output schema

Column	Type	Nullable	Description
vertex	Int64|Utf8	no	Vertex receiving the Katz centrality score.
value	Float64	no	Katz centrality score for the vertex.

These are generic descriptor schemas; validate the call to get the concrete, table-specific output schema.

Examples

This example runs on the citation network demo dataset.

alpha decides how deep credit flows

Katz centrality counts all incoming walks, discounting a walk of length n by alpha^n. At the default alpha (0.01) long chains contribute almost nothing and the ranking is nearly in-degree. Raising alpha to 0.05 shifts credit to papers whose citers are themselves heavily cited, transitively — window functions over the two results make the shift visible as rank movement:

WITH katz AS (
  SELECT vertex, ROW_NUMBER() OVER (ORDER BY value DESC) AS katz_rank
  FROM cugraph_katz_centrality('citation_edges', 'src', 'dst', NULL,
                               '{"alpha": 0.05}')),
deg AS (
  SELECT vertex, in_degree, ROW_NUMBER() OVER (ORDER BY in_degree DESC) AS degree_rank
  FROM cugraph_in_degrees_all('citation_edges', 'src', 'dst'))
SELECT p.title, p.year, k.katz_rank, d.degree_rank, d.in_degree
FROM katz k
JOIN deg d ON d.vertex = k.vertex
JOIN papers p ON p.paper_id = k.vertex
WHERE k.katz_rank <= 8
ORDER BY k.katz_rank;

title	year	katz_rank	degree_rank	in_degree
Distinctive Image Features from Scale-Invariant Keypoints	2004	1	1	22,892
The Nature of Statistical Learning Theory	1995	2	6	14,508
Histograms of oriented gradients for human detection	2005	3	9	13,768
Object recognition from local scale-invariant features	1999	4	58	6,300
New Directions in Cryptography	1976	5	69	6,007
The Design and Analysis of Computer Algorithms	1974	6	103	5,168
A Computational Approach to Edge Detection	1986	7	24	9,069
A method for obtaining digital signatures and public-key cryptosystems	1978	8	50	6,580

The risers are the foundations: New Directions in Cryptography climbs from in-degree rank #69 to Katz rank #5 and the RSA paper from #50 to #8, because the papers citing them anchor entire downstream literatures. Katz is the natural choice on citation-style graphs — near-acyclic structure starves eigenvector centrality, while the beta base score keeps every vertex non-zero here. Convergence requires alpha below the reciprocal of the graph's largest eigenvalue: pushing it too high fails with a convergence error rather than returning partial scores.

Limitations & lifecycle

No algorithm-specific limitations.

Validate before running

Dry-run validation checks registered relation metadata, column presence, static dtypes, and options only; it does not scan edge data, construct a graph, or prove source-vertex existence:

SELECT * FROM gpu_validate_call(
  'cugraph_katz_centrality',
  '{"schema_version":1,"relations":{"edges":{"table":"target_edges"}},"options":{"src_col":"src","dst_col":"dst"}}'
);

See GPU Function Catalog API for the full gpu_validate_call contract.