WoComToQC workshop lecture on Forman-Ricci curvature for applications in industry (social networks, disaster logistics, spatial data, and spatiotemporal goods pricing data).

1. APPLICATIONS
OF FORMAN-
RICCI
CURVATURE
Colleen M. Farrelly

2. Introduction
to Curvature

3. Types of Curvature:
Intrinsic
• Curvature at a point (local)
• Does not depend on the surface embedding
• Can be measured by:
• Defining a metric
• Measuring a geodesic
• Example:
• Gaussian curvature (product of principal
curvatures at a point)

4. Types of Curvature: Extrinsic
• Curvature of a surface
• Depends on the surface embedding
• Can be measured by:
• Examining rate of change from unit normal along surface
• Involves lifting manifold to higher-dimensional space,
measuring, and comparing to manifold measurements
• Example:
• Geodesic curvature
• Can have extrinsic curvature without intrinsic
curvature

5. Ricci Curvature
• Tensor defined locally to measure how a metric
tensor differs locally from Euclidean space
• Measures how a shape deforms as you move
along a geodesic line on the manifold
• Relates to the Laplacian tensor
• Symmetric properties
• Flow of Ricci curvature directly related to
solutions of partial differential equations defined
geometrically

6. Ricci Flow
• Partial differential equation for Riemannian
metric on a manifold
• Analogous to heat diffusion

7. Ricci Curvature: From
Continuous to Discrete
• Connection to Laplacian (and
Bochner formula)
• Allows for extensions of Ricci
curvature and flow to discrete
objects like graphs
• Includes Forman-Ricci curvature and
Forman-Ricci flow

8. Forman-Ricci Curvature
• Forman-Ricci curvature formula for weighted networks:
𝑅𝑖𝑐 𝑒 = 𝑤(𝑒)(
𝑤(𝑣1)
𝑤(𝑒)
+
𝑤(𝑣2)
𝑤(𝑒)
-
𝑒 𝑣2 ~𝑣
𝑒 𝑣1 ~𝑣
𝑤(𝑣1)
𝑤(𝑒)𝑤(𝑒𝑣1)
+
𝑤(𝑣2)
𝑤(𝑒)𝑤(𝑒𝑣2)
)
• e denotes an edge connecting two vertices, v1 and v2
• w denotes the weight of each vertex or edge
• Collapses to a simpler formula for unweighted networks
• Edge curvature related to degree
• Can sum edge curvatures to get a vertex centrality score based on curvature

9. Forman-Ricci
Flow
• To calculate flow:
• Subtract edge curvature
between time points
• Differences in Forman-Ricci
centrality of vertices related
to importance of vertex in
network over time

10. Applications
Systems biology
Sociology
Epidemiology
Finance
Engineering