Enumerating Graphlets with Amortized Time Complexity Independent of Graph Size

Graphlets of order  k in a graph G are connected subgraphs induced by k  nodes (called k -graphlets) or by k  edges (called edge k -graphlets). They are among the interesting subgraphs in network analysis to get insights on both the local and global structure of a network. While several algorithms e...

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Bibliographic Details
Published in:Algorithmica Vol. 87; no. 9; pp. 1247 - 1273
Main Authors: Conte, Alessio, Grossi, Roberto, Kobayashi, Yasuaki, Kurita, Kazuhiro, Rucci, Davide, Uno, Takeaki, Wasa, Kunihiro
Format: Journal Article
Language:English
Published: New York Springer US 01.09.2025
Springer Nature B.V
Subjects:
Algorithm Analysis and Problem Complexity
Algorithms
Complexity
Computer Science
Computer Systems Organization and Communication Networks
Data Structures and Information Theory
Graph theory
Mathematics of Computing
Network analysis
Theory of Computation
Amortization
Enumeration
Graph algorithms
Subgraphs
Graphlets
ISSN:0178-4617, 1432-0541
Online Access:Get full text
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Summary:Graphlets of order  k in a graph G are connected subgraphs induced by k  nodes (called k -graphlets) or by k  edges (called edge k -graphlets). They are among the interesting subgraphs in network analysis to get insights on both the local and global structure of a network. While several algorithms exist for discovering and enumerating graphlets, the amortized time complexity of such algorithms typically depends on the size of the graph G , or its maximum degree. In real networks, even the latter can be in the order of millions, whereas k is typically required to be a small value. In this paper we provide the first algorithm to list all graphlets of order  k in a graph G = ( V , E ) with an amortized time complexity depending solely on the order k , contrarily to previous approaches where the cost depends also on the size of G or its maximum degree. Specifically, we show that it is possible to list k -graphlets in O ( k 2 ) time per solution, and to list edge k -graphlets in O ( k ) time per solution. Furthermore we show that, if the input graph has bounded degree, then the amortized time for listing k -graphlets is reduced to O ( k ). Whenever k = O ( 1 ) , as it is often the case in practical settings, these algorithms are the first to achieve constant time per solution.
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ISSN:0178-4617
1432-0541
DOI:10.1007/s00453-025-01312-0