Null-adjusted persistence function for high-resolution community detection

TL;DR

The paper tackles the limitations of modularity (notably the resolution limit) and persistence probability (tending to increase with cluster size) in community detection. It introduces null-adjusted persistence ${\cal P}^{\star}$, which compares observed persistence to its null-model baseline derived from a configuration model, yielding a partition objective ${\cal P}^{\star}_{\Pi}$ that satisfies ${\cal P}^{\star}_{\Pi} = {\cal P}_{\Pi} - 1$. The authors prove key analytical properties, show scale invariance and absence of a resolution limit, derive a size-independent merging threshold, and implement a Louvain-style heuristic along with MILP formulations. Empirical results on caveman and LFR benchmarks, as well as a real Facebook network, demonstrate improved high-resolution detection and richer mesoscale structure compared to modularity, particularly in networks with degree heterogeneity. Overall, null-adjusted persistence offers a principled, interpretable, and practically effective objective for community detection across diverse network types and sizes.

Abstract

Modularity and persistence probability are two widely used quality functions for detecting communities in complex networks. In this paper, we introduce a new objective function called null-adjusted persistence, which incorporates features from both modularity and persistence probability, as it implies a comparison of persistence probability with the same null model of modularity. We prove key analytic properties of this new function. We show that the null-adjusted persistence overcomes the limitations of modularity, such as scaling behavior and resolution limits, and the limitation of the persistence probability, which is an increasing function with respect to the cluster size. We propose to find the partition that maximizes the null-adjusted persistence with a variation of the Louvain method and we tested its effectiveness on benchmark and real networks. We found out that maximizing null-adjusted persistence outperforms modularity maximization, as it detects higher resolution partitions in dense and large networks.

Figures (11)

• Figure 2.1: Panel (a): caveman graph discussed in the text; panel (b): comparison between persistence probability (black line) and null-adjusted persistence (blue line).
• Figure 3.1: $Q_{\mathcal{C}}$ (red line) and ${{\cal P}^{\star}_{\cal C}}\xspace$ (blue line) as functions of $m_i$. For the sake of representation, $m_{e}$ and $m$ are set to values $m_{e}=2$ and $m=12$. Both functions vanish at the same values $m_{i}=5\pm 2\sqrt{6}$. Conversely, the modularity has a maximum for $m_{i}=5$, whereas the null-adjusted persistence attains the maximum at $m_{i}=2\sqrt{3}-1=2.4641$.
• Figure 3.2: Connected network $G$ formed by the inner clique $K_{3}$ and three leaves (panels (a) and (c)). Optimal partition of the network $G$ according to modularity and null-adjusted persistence (panel (b)).
• Figure 3.3: Disconnected network $G$ formed by two cliques $K_{3}$ with leaves.
• Figure 3.4: Optimal partition $\Pi$ of the disconnected network $G$ formed by two cliques $K_{3}$ with leaves, according to modularity $Q(\Pi)$).

Theorems & Definitions (11)

• Definition 1
• Proposition 1
• proof
• Proposition 2
• proof
• Proposition 3
• proof
• Proposition 4
• proof
• Proposition 5