Bakry-Émery-Ricci curvature: An alternative network geometry measure in the expanding toolbox of graph Ricci curvatures

TL;DR

This work introduces Bakry-Émery-Ricci curvature as a vertex-based discrete curvature for graphs, grounded in the curvature-dimension framework and Bochner calculus, and compares it to Forman-Ricci and Ollivier-Ricci curvatures across diverse model and real networks. The authors formulate BEK via a semidefinite program in the limit $n\to\infty$ and analyze its distribution, correlations with other graph measures, and role in network robustness, highlighting strong ties to Augmented Forman and Ollivier curvature and consistently negative BEK values across vertices. Key findings show BEK distributions are broader in scale-free and hyperbolic graphs, BEK is highly informative about vertex centrality and robustness, and BEK behaves differently from scalar curvature in denser networks. The results position BEK as a computationally efficient, interpretable tool for understanding network cohesion and suggest avenues for extending the framework to directed/weighted networks and community detection tasks.

Abstract

The characterization of complex networks with tools originating in geometry, for instance through the statistics of so-called Ricci curvatures, is a well established tool of network science. There exist various types of such Ricci curvatures, capturing different aspects of network geometry. In the present work, we investigate Bakry-Émery-Ricci curvature, a notion of discrete Ricci curvature that has been studied much in geometry, but so far has not been applied to networks. We explore on standard classes of artificial networks as well as on selected empirical ones to what the statistics of that curvature are similar to or different from that of other curvatures, how it is correlated to other important network measures, and what it tells us about the underlying network. We observe that most vertices typically have negative curvature. Random and small-world networks exhibit a narrow curvature distribution whereas other classes and most of the real-world networks possess a wide curvature distribution. When we compare Bakry-Émery-Ricci curvature with two other discrete notions of Ricci-curvature, Forman-Ricci and Ollivier-Ricci curvature for both model and real-world networks, we observe a high positive correlation between Bakry-Émery-Ricci and both Forman-Ricci and Ollivier-Ricci curvature, and in particular with the augmented version of Forman-Ricci curvature. Bakry-Émery-Ricci curvature also exhibits a high negative correlation with the vertex centrality measure and degree for most of the model and real-world networks. However, it does not correlate with the clustering coefficient. Also, we investigate the importance of vertices with highly negative curvature values to maintain communication in the network. The computational time for Bakry-Émery-Ricci curvature is shorter than that required for Ollivier-Ricci curvature but higher than for Augmented Forman-Ricci curvature.

Figures (10)

• Figure 1: (a) Bakry-Émery-Ricci curvature of a vertex $x$. $S_1(x)$ and $S_2(x)$ contain the vertices present in the 1-sphere and the 2-sphere respectively. All the vertices included in the 2-ball $B_2(x)$ are taken into account for the computation of Bakry-Émery-Ricci curvature at vertex $x$. The colors indicate the connected components of the punctured ball $B_2(x)\backslash \{x\}$. The illustration is inspired by cushing2020bakry. (b,c) Forman-Ricci curvature of an edge $e$ containing the vertices $x$ and $y$. Edges that share a child (a lower dimensional face) or a parent (a higher dimensional face) to the edge $e$ contribute to Forman-Ricci curvature. (b) Edges $e_{11}$, $e_{12}$, $e_{13}$, $e_{14}$, $e_{15}$ share a vertex $x$ with $e$ and edges $e_{21}$, $e_{22}$, $e_{23}$, $e_{24}$ share a vertex $y$ with $e$. Therefore all these edges are taken into account for the Forman-Ricci curvature of edge $e$. (c) For Augmented Forman-Ricci curvature of edge $e$, apart from all the edges mentioned above, the two-dimensional simplices $f_1$ and $f_2$ also contribute. (d,e) Illustration of Ollivier-Ricci curvature. (d) For a given vertex $x$, neighbors belong to the sphere $S_1(x)$ centered at $x$. Points on the sphere $S_1(y)$ centered at $x$ are transported to the points of the sphere centered at $y$. $\mu_{x, y}(x_2, y_1)$ is a probability measure from the vertex $x_2$ to the vertex $y_1$ through the edge $e$ with marginals equal to the measures $m_x$ and $m_y$ on $S_1(x)$ and $S_1(y)$, respectively. (e) An illustration of the measures $m_x$ and $m_y$. Points will be transported with average distance $(1-\kappa(x,y))d(x,y)$, where $\kappa(x,y)$ denotes the Ollivier-Ricci curvature between the nodes $x$ and $y$.
• Figure 2: Distribution of Bakry-Émery-Ricci curvature for different model networks with the number of vertices, $n$ = 1000. (a) ER model with $p$ = 0.005, (b) WS model with $k$ = 6, $p$ = 0.5, (c) BA model with $m$ = 3 and (d) HGG model with $k$ = 3, $\gamma$ = 2, $T$ = 0.
• Figure 3: Distribution of Bakry-Émery-Ricci curvature for different real-world networks (a) Email communication, (b) Chicago road network, (c) US Power Grid, (d) Euro road network, (e) Human protein interactions, and (f) Hamsterster friendships networks.
• Figure 4: Spearman correlation (rounded to two decimal places) of Bakry-Émery-Ricci curvature with Forman-Ricci curvature, Augmented Forman-Ricci curvature, Ollivier-Ricci curvature, degree, clustering coefficient, betweenness centrality, eigenvector centrality, and closeness centrality in model networks. The columns correspond to the eight different vertex measures. The rows correspond to different model networks.
• Figure 5: Spearman correlation (rounded to two decimal places) of Bakry-Émery-Ricci curvature with Forman-Ricci curvature, Augmented Forman-Ricci curvature, Ollivier-Ricci curvature, degree, clustering coefficient, betweenness centrality, eigenvector centrality, and closeness centrality in six real-world networks. The columns correspond to the eight different vertex measures. The rows correspond to six different real-world networks.