Maximal Clique Enumeration (MCE) plays a pivotal role in analyzing large-scale graphs across diverse domains, yet improving its practical efficiency remains challenging. In this work, we introduce EkC (Extended k-Clique), a novel iterative algorithm that identifies maximal cliques initialized by a specified set of k-seed nodes. Unlike traditional combinatorial approaches that explore the entire graph space recursively, EkC operates by first identifying common neighbors of k-seed nodes and then iteratively expanding to form larger tuples (2-tuples, 3-tuples, up to n-tuples) while ensuring connectivity constraints. This targeted expansion strategy, which focuses only on neighbors in the seed set and their interconnections, significantly reduces the search space while preserving contextual relevance for real-world applications. The overall worst case time complexity of EkC is observed to be <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="42979_2025_4271_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="108" /> </InlineMediaObject> <EquationSource Format="TEX">$O(n \times V \times d^2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>×</mo> <mi>V</mi> <mo>×</mo> <msup> <mi>d</mi> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Experimental evaluation across four real-world network datasets demonstrates EkC’s superior performance against five state-of-the-art methods in terms of in terms of time. EkC cuts the time needed for extracting cliques by <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="42979_2025_4271_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="31" /> </InlineMediaObject> <EquationSource Format="TEX">$25\%$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>25</mn> <mo>%</mo> </mrow> </math></EquationSource> </InlineEquation> in smaller networks and also by <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="42979_2025_4271_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="31" /> </InlineMediaObject> <EquationSource Format="TEX">$50\%$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>50</mn> <mo>%</mo> </mrow> </math></EquationSource> </InlineEquation> in larger networks when compared with combinatorial versions of the same. To validate the algorithm’s practical utility, we implemented EkC in a recipe recommendation system, where its ability to efficiently identify ingredient clusters through seed-node-based clique enumeration proved particularly valuable. These comprehensive results, spanning both benchmark comparisons and real-world application, establish EkC as an effective solution for scenarios requiring focused clique enumeration with improved computational performance.

An Efficient Seed-Initialized Algorithm for Maximal Clique Enumeration and Its Applications

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发表日期：

2025年8月13日

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Sandeep Khanna,
Chiranjoy Chattopadhyay,
Suman Kundu

摘要

Maximal Clique Enumeration (MCE) plays a pivotal role in analyzing large-scale graphs across diverse domains, yet improving its practical efficiency remains challenging. In this work, we introduce EkC (Extended k-Clique), a novel iterative algorithm that identifies maximal cliques initialized by a specified set of k-seed nodes. Unlike traditional combinatorial approaches that explore the entire graph space recursively, EkC operates by first identifying common neighbors of k-seed nodes and then iteratively expanding to form larger tuples (2-tuples, 3-tuples, up to n-tuples) while ensuring connectivity constraints. This targeted expansion strategy, which focuses only on neighbors in the seed set and their interconnections, significantly reduces the search space while preserving contextual relevance for real-world applications. The overall worst case time complexity of EkC is observed to be $O(n \times V \times d^2)$ $O (n \times V \times d^{2})$ . Experimental evaluation across four real-world network datasets demonstrates EkC’s superior performance against five state-of-the-art methods in terms of in terms of time. EkC cuts the time needed for extracting cliques by $25\%$ $25 %$ in smaller networks and also by $50\%$ $50 %$ in larger networks when compared with combinatorial versions of the same. To validate the algorithm’s practical utility, we implemented EkC in a recipe recommendation system, where its ability to efficiently identify ingredient clusters through seed-node-based clique enumeration proved particularly valuable. These comprehensive results, spanning both benchmark comparisons and real-world application, establish EkC as an effective solution for scenarios requiring focused clique enumeration with improved computational performance.