Analyzing Modularity Maximization in Approximation, Heuristic, and Graph Neural Network Algorithms for Community Detection

Community detection, a fundamental problem in computational sciences, finds applications in various domains. Heuristics are often employed to detect communities through maximizing an objective function, modularity, over partitions of network nodes. Our research delves into the performance of different modularity maximization algorithms in achieving optimal partitions. We use 104 networks, comprising real-world instances from diverse contexts and synthetic graphs with modular structures. We analyze ten inexact modularity-based algorithms against an exact baseline which is an exact integer programming method that globally optimizes modularity. The ten algorithms analyzed include eight heuristics, two variations of a graph neural network algorithm, and several variations of the Bayan approximation algorithm. Our analysis uncovers substantial dissimilarities between the partitions obtained by most commonly used modularity-based methods and any optimal partition of the networks, as indicated by both adjusted and reduced mutual information metrics. Importantly, our results show that near-optimal partitions are often disproportionately dissimilar to any optimal partition. Taken together, our analysis points to a crucial limitation of the commonly used unguaranteed modularity-based methods for discovering communities: they rarely produce an optimal partition or a partition resembling an optimal partition even on networks with modular structures. If modularity is to be used for detecting communities, approximate optimization algorithms are recommendable for a more methodologically sound usage of modularity within its applicability limits.

Heuristic Modularity Maximization Algorithms for Community Detection Rarely Return an Optimal Partition or Anything Similar

Community detection is a classic problem in network science with extensive applications in various fields. The most commonly used methods are the algorithms designed to maximize modularity over different partitions of the network nodes into communities. Using 80 real and random networks from a wide range of contexts, we investigate the extent to which current heuristic modularity maximization algorithms succeed in returning modularity-maximum (optimal) partitions. We evaluate (1) the ratio of their output modularity to the maximum modularity for each input graph and (2) the maximum similarity between their output partition and any optimal partition of that graph. Our computational experiments involve eight existing heuristic algorithms which we compare against an exact integer programming method that globally maximizes modularity. The average modularity-based heuristic algorithm returns optimal partitions for only 16.9% of the 80 graphs considered. Results on adjusted mutual information show considerable dissimilarity between the sub-optimal partitions and any optimal partitions of the graphs in our experiments. More importantly, our results show that near-optimal partitions tend to be disproportionally dissimilar to any optimal partition. Taken together, our analysis points to a crucial limitation of commonly used modularity-based algorithms for discovering communities: they rarely return an optimal partition or a partition resembling an optimal partition. Given this finding, developing an exact or approximate algorithm for modularity maximization is recommendable for a more methodologically sound usage of modularity in community detection.

The Bayan Algorithm: Detecting Communities in Networks Through Exact and Approximate Optimization of Modularity

Community detection is a classic problem in network science with extensive applications in various fields. The most commonly used methods are the algorithms designed to maximize a utility function, modularity, across different ways that a network can be partitioned into communities. Despite their name and design philosophy, current modularity maximization algorithms generally fail to maximize modularity or guarantee any proximity to an optimal solution. We propose the Bayan algorithm which, unlike the existing methods, returns network partitions with a guarantee of either optimality or proximity to an optimal solution. At the core of the Bayan algorithm is a branch-and-cut scheme that solves a sparse integer programming formulation of the modularity maximization problem to optimality or approximate it within a factor. We analyze the performance of Bayan against 22 existing algorithms using synthetic and real networks. Through extensive experiments, we demonstrate Bayan's distinctive capabilities not only in maximizing modularity, but more importantly in accurate retrieval of ground-truth communities. Bayan's comparative level of performance remains stable over variations in the amount of noise in the data (graph) generation process. The performance of Bayan as an exact modularity maximization algorithm also reveals the theoretical capability limits of maximum-modularity partitions in accurate retrieval of communities. Overall our analysis points to Bayan as a suitable choice for a methodologically grounded detection of communities through exact (approximate) maximization of modularity in networks with up to $\sim10^3$ edges (and larger networks). Prospective advances in graph optimization and integer programming can push these limits further.

Return migration of German-affiliated researchers: Analyzing departure and return by gender, cohort, and discipline using Scopus bibliometric data 1996-2020

The international migration of researchers is a highly prized dimension of scientific mobility and motivates considerable policy debate. However, tracking migration life courses of researchers is challenging due to data limitations. In this study, we use Scopus bibliometric data on 8 million publications from 1.1 million researchers who have published at least once with an affiliation address from Germany in 1996-2020. We describe several key steps and algorithms we develop that enable us to construct the partial life histories of published researchers in this period. These tools allow us to explore both the out-migration of researchers with German affiliations as well as the subsequent return of a share of this group - the returnees. Our analyses shed light on important career stages and gender disparities between researchers who remain in Germany and those who both migrate out and those who eventually return. Return migration streams are even more gender imbalanced and point to the importance of additional efforts to attract female researchers back to Germany. We document a slightly declining trend in return migration with cohorts which, for most disciplines, is associated with decreasing German collaboration ties among cohorts of researchers who leave Germany. Also, gender disparities for the most gender imbalanced disciplines are unlikely to be mitigated by return migration given the gender compositions in cohorts of researchers who leave Germany and those who return. This analysis reveals new dimensions of scholarly migration by investigating the return migration of published researchers which is critical for science policy development.