Modularity maximization and community detection in complex networks through recursive and hierarchical annealing in the D-Wave Advantage quantum processing units

TL;DR

This work presents a pure quantum annealing framework for modularity maximization in complex networks by hierarchically and recursively encoding binary problem instances, thereby avoiding one-hot constraints. It extends modularity optimization to weighted and directed networks and incorporates multiresolution analysis via the parameter $\gamma$, while delivering interpretable dendrograms that reveal hierarchical community structure. Through extensive benchmarking on various network models and real brain connectivity data, the approach yields competitive modularity scores against state-of-the-art classical methods with tractable running times and robust performance across topologies. The paper also analyzes hardware considerations, embedding strategies on D-Wave Pegasus, and provides the open-source Qommunity software to facilitate practical, reproducible experiments in network science and quantum computing.

Abstract

Quantum adiabatic optimization has long been expected to outperform classical methods in solving NP-type problems. While this has been proven in certain experiments, its main applications still reside in academic problems where the size of the system to be solved would not represent an obstacle to any modern desktop computer. Here we develop a systematic procedure to find the global optima of the modularity function to discover community structure in complex networks solely relying on pure annealers rather than hybrid solutions. We bypass the one-hot encoding constraints by hierarchically and recursively encoding binary instances of the problem that can be solved without the need to guess the exact penalties for the Lagrange multipliers. We study the variability, and robustness of the annealing process as a function of network size, directness of connections, topology, and the resolution of the communities. We show how our approach produces meaningful and at least equally optimal solutions to state-of-the-art community detection algorithms while maintaining tractable computing times. Lastly, due to its recursive nature, the annealing process returns intermediate subdivisions thus offering interpretable rather than black-box solutions. These \textit{dendrograms} can be used to unveil normal and pathological hidden hierarchies in brain networks hence opening the door to clinical workflows. Overall, this represents a first step towards an applicable practice-oriented usage of pure quantum annealing potentially bridging two segregated communities in modern science and engineering; that of network science and quantum computing.

Figures (31)

• Figure 1: Communities detected with Annealing and Louvain in chains of 3-cliques.a Modularity of the community structure found by the two algorithms measured as the relative increase (%) w.r.t. the Louvain solution. Individual points (in gray) and the mean increase (black diamonds). b Number of communities found with each algorithm. c, d Consensus matrices constructed with 5 different runs for each clique chain. e Percentage colorbar in c and d. f Output of the two algorithms and the generated graph (left) for a chain of 8 3-cliques. Both the Louvain and the hierarchical annealer solutions have the exact modularity.
• Figure 2: Hierarchical annealing solutions in random networks.a Maximum modularity ($\max Q$) of power-law–clustered networks as a function of network size. b Consensus matrices for the $N=100$ network in (a), comparing communities identified by the hierarchical annealing procedure with those obtained using alternative algorithms. Opacity is proportional to the Dice similarity score between pairs of communities. Black, blue, and red outlines indicate cases in which hierarchical annealing achieved equal, higher, or lower modularity, respectively, relative to the corresponding alternative solution. White entries denote zero Dice similarity. c Maximum modularity ($\max Q$) of Barabasi-Albert networks. d Consensus matrices of the Barabasi-Albert network ($N=100$) studied in (c). e Maximum modularity ($\max Q$) of Erdős-Renyi networks. f Consensus matrices of the Erdős-Renyi network ($N=100$) studied in (e). g Maximum modularity ($\max Q$) of directed scale-free. h Consensus matrices of the directed scale-free network ($N=70$) studied in (g). i-j Communities found the Hierarchical annealing (i) and Louvain (j) algorithms in the directed scale-free network in (h). See Fig. S5-S8 for the full set of consensus matrices.
• Figure 3: Hierarchical annealing as a function of inherent hierarchies in random networks.a Modularity obtained with the Leiden, Louvain, and Hierarchical Annealing algorithms across networks varying in size (panels: from left to right) and scale-freeness (marker shape: "Fraction" $f$), generated via preferential attachment Barabasi1999. b For networks of increasing size (panels: left to right), comparison of Hierarchical Annealing with Leiden (panels: top) and Louvain (panels: bottom). Black markers located on the diagonal indicate identical modularities; blue/red markers located on the upper/lower triangular parts denote superior/inferior annealing solutions. Marker shapes correspond to the legend in (a). c Relative improvement of annealing solutions in (a-b) as a function of $\alpha$ and network size. The green shaded region (top left) corresponds to scale-free networks. Bottom panels are identical but explicitly show cases excluded from the regression analyses (black markers).
• Figure 4: Hierarchical annealing in stochastic block models (SBM). Performance of benchmark methods relative to the hierarchical annealing procedure on an SBM with three communities ($p_{in}=0.9$, $p_{out}=0.05$). Recovery accuracy is assessed using four metrics against the ground truth, with the corresponding maximum modularity ($\max Q$) shown in the rightmost panel as a function of the mixing parameter $\lambda$.
• Figure 5: Hierarchical annealing in different clustered power-law networks for different resolution parameters. Each column depicts the results described below for a network of $N=10, 50, 80$ nodes, respectively. a Measure of the average overlap between the communities found by the Louvain and Hierarchical annealing algorithms (left axis, blue) and the number of communities (right axis, salmon and black) as a function of the resolution parameter $\gamma$. b Relative increase of the Hierarchical annealing measured w.r.t. the Louvain (top) and Leiden (bottom) solutions in (a). Black markers correspond to identical solutions, blue markers correspond to better solutions from the annealer, and red markers indicate worse performances than the Louvain alternative. c Maximum modularity per resolution value.