Fast maximal clique enumeration in weighted temporal networks

TL;DR

This work tackles the scalable enumeration of $(\delta,\gamma)$-maximal cliques in weighted temporal networks by reformulating the temporal clique definitions to bound time intervals by actual edge occurrences and to use a cumulative weight threshold $\gamma$. It introduces a two-phase algorithm with a linear-time stretching phase and a pruning-enhanced bulking phase, along with a custom node labeling that reduces duplicate exploration and memory usage. The method achieves substantial speedups over prior approaches on large real-world datasets, often by orders of magnitude, and extends applicability to weighted networks. The weighted extension enables more nuanced group connectivity analyses, capturing both temporal and strength-based constraints on clique formation. Overall, the approach improves interpretability, scalability, and practical relevance for dynamic network analysis.

Abstract

Cliques, groups of fully connected nodes in a network, are often used to study group dynamics of complex systems. In real-world settings, group dynamics often have a temporal component. For example, conference attendees moving from one group conversation to another. Recently, maximal clique enumeration methods have been introduced that add temporal (and frequency) constraints, to account for such phenomena. These methods enumerate so called (delta,gamma)-maximal cliques. In this work, we introduce an efficient (delta,gamma)-maximal clique enumeration algorithm, that extends gamma from a frequency constraint to a more versatile weighting constraint. Additionally, we introduce a definition of (delta,gamma)-cliques, that resolves a problem of existing definitions in the temporal domain. Our approach, which was inspired by a state-of-the-art two-phase approach, introduces a more efficient initial (stretching) phase. Specifically, we reduce the time complexity of this phase to be linear with respect to the number of temporal edges. Furthermore, we introduce a new approach to the second (bulking) phase, which allows us to efficiently prune search tree branches. Consequently, in experiments we observe speed-ups, often by several order of magnitude, on various (large) real-world datasets. Our algorithm vastly outperforms the existing state-of-the-art methods for temporal networks, while also extending applicability to weighted networks.

Figures (13)

• Figure 1: Given $\delta = 4$ and $\gamma = 2$ and temporal network (a) shown as a link stream model, then (b) highlights the three maximal ($\delta, \gamma$)-cliques $C_1 = (\{A,B\}, [-1,4])$ in orange (with backslash pattern), $C_2 = (\{B,C\}, [0,5])$ in green (with cross pattern), and $C_3 = (\{A,B,C\}, [0,4])$ in blue (with forward slash pattern), that are by existing definitions ($\delta, \gamma$)-maximal. Note that the edges included in $C_1$ and $C_2$ are also all included in $C_3$. Therefore, $C_1$ and $C_2$ are effectively subgraphs of $C_3$. Finally, (c) shows the only remaining ($\delta, \gamma$)-maximal clique, when the time interval is bounded by the first and last temporal edge occurrences included. This clique ($C_4 = (\{A,B,C\}, [1,4])$) is found by our proposed definition (Def. \ref{['def:dgnew-clique']}). Notice, that $C_4$ is the bounded form of $C_3$ from (b) and that the bounded forms of cliques $C_1$ and $C_2$ are no longer maximal.
• Figure 2: $\delta = 5, \gamma = 3$
• Figure 3: $\delta = 5, \gamma = 4$
• Figure 5: A visual representation of the clique time interval and properties on a timeline
• Figure 6: Inner-recursion extension visualization

Theorems & Definitions (5)

• Definition 1: $\delta$-clique
• Definition 2: $\delta$-maximal clique
• Definition 3: ($\delta, \gamma$)-clique
• Definition 4: ($\delta, \gamma$)-maximal clique
• proof