Augmentations of Forman's Ricci Curvature and their Applications in Community Detection

TL;DR

Evidence is provided that the AFRC frequently gives sufficient insight into the structure of a network to be used for community detection, and therefore provides a computationally cheaper alternative to previous ORC-based methods.

Abstract

The notion of curvature on graphs has recently gained traction in the networks community, with the Ollivier-Ricci curvature (ORC) in particular being used for several tasks in network analysis, such as community detection. In this work, we choose a different approach and study augmentations of the discretization of the Ricci curvature proposed by Forman (AFRC). We empirically and theoretically investigate its relation to the ORC and the un-augmented Forman-Ricci curvature. In particular, we provide evidence that the AFRC frequently gives sufficient insight into the structure of a network to be used for community detection, and therefore provides a computationally cheaper alternative to previous ORC-based methods. Our novel AFRC-based community detection algorithm is competitive with an ORC-based approach.

Figures (14)

• Figure 1: (a) Example of Forman-Ricci curvature. The edge $e=(1,2)$ has FRC equal to $\mathcal{F}(e)=4-\deg(1)-\deg(2)=-5$. (b) Example of augmented Forman-Ricci curvature. The edge $e$ is contained in the cycles $\gamma_1=1234$ and $\gamma_2=125$ and the AFRC of $e$ is equal to $\mathop{\mathrm{\mathcal{A} \mkern-3.65mu \mathcal{F}}}\nolimits(e)=2+2-3-1=0$. This augmentation arises from including faces corresponding to $\gamma_1$ and $\gamma_2$ and then applying Forman's formula to the resulting 2D complex. This example is worked out in detail in Appendix \ref{['appendix: detailed example']}. (c) Let the vertices be ordered as $1<2<3<4$ and thus $e=(1,2)<(3,4)=e'$. The edges $e,e'$ are contained in cycles $\gamma_3=1234$ and $\gamma_4=1243$. In $\gamma_3$ both edges are traversed from the small to large vertex ($1\rightarrow 2$ and $3\rightarrow 4$), so they are aligned and $\Gamma_{ee'}=1$. In $\gamma_2$ edge $e$ is traversed from small to large ($1\rightarrow 2$) while edge $e'$ is traversed from large to small ($4\rightarrow 3$), so the edges are not aligned in $\gamma_4$ and $\Gamma_{e'e}=1$.
• Figure 2: Distributions of the Augmented Forman-Ricci Curvature ($\mathcal{A} \mkern-3.65mu \mathcal{F}_{\space {3}}$) across edges in stochastic block models with weak ((a), SBM(2, 40, $0.5$, $0.2$)) and strong ((b), SBM(2, 40, $0.7$, $0.1$)) community structure. The curvature gap captures the (normalized) distance between the means of the edges between (orange) and within (blue) communities. (c) and (d) shows the corresponding networks with edges colored according to curvature.
• Figure 3: Distributions of $\mathcal{A} \mkern-3.65mu \mathcal{F}_{\space {4}}$(top) and the Ollivier-Ricci Curvature (bottom) for SBM(10, $k$, $0.7$, $0.05$) with $k\in\{5, 10, 15, 20\}$ (from left to right). As the community structure becomes more pronounced with larger community sizes, the curvature gaps increase. Note also the "switch" in the order of the within-community edges (blue) and the between-community edges (orange) in the AFRC histograms.
• Figure 4: Distributions of $\mathcal{A} \mkern-3.65mu \mathcal{F}_{\space {3}}$ in (a) and $\mathcal{A} \mkern-3.65mu \mathcal{F}_{\space {4}}$ in (b) for a HBG(40, 0.7, 0.05).
• Figure 5: AFRC community detection algorithm, here with $\mathcal{A} \mkern-3.65mu \mathcal{F}_{\space {3}}$.