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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Veröffentlicht in:	Algorithmica Jg. 87; H. 9; S. 1247 - 1273
Hauptverfasser:	Conte, Alessio, Grossi, Roberto, Kobayashi, Yasuaki, Kurita, Kazuhiro, Rucci, Davide, Uno, Takeaki, Wasa, Kunihiro
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	New York Springer US 01.09.2025 Springer Nature B.V
Schlagworte:	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
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Abstract	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.
AbstractList	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(k2) 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. 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. 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 $$G=(V,E)$$ 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 $$O(k^2)$$ 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 $$k = O(1)$$ k = O ( 1 ) , as it is often the case in practical settings, these algorithms are the first to achieve constant time per solution.
Author	Conte, Alessio Uno, Takeaki Grossi, Roberto Wasa, Kunihiro Kobayashi, Yasuaki Kurita, Kazuhiro Rucci, Davide
Author_xml	– sequence: 1 givenname: Alessio surname: Conte fullname: Conte, Alessio organization: University of Pisa – sequence: 2 givenname: Roberto surname: Grossi fullname: Grossi, Roberto organization: University of Pisa – sequence: 3 givenname: Yasuaki surname: Kobayashi fullname: Kobayashi, Yasuaki organization: Hokkaido University – sequence: 4 givenname: Kazuhiro surname: Kurita fullname: Kurita, Kazuhiro organization: Nagoya University – sequence: 5 givenname: Davide surname: Rucci fullname: Rucci, Davide email: davide.rucci@isti.cnr.it organization: ISTI-CNR – sequence: 6 givenname: Takeaki surname: Uno fullname: Uno, Takeaki organization: National Institute of Informatics – sequence: 7 givenname: Kunihiro surname: Wasa fullname: Wasa, Kunihiro organization: Hosei University
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Snippet	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... 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... 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...
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SubjectTerms	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
Title	Enumerating Graphlets with Amortized Time Complexity Independent of Graph Size
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