The Geometry of Graphs Paul Horn Department of Mathematics University of Denver May 21, 2016 P . Horn The Geometry of Graphs
Graphs Ultimately, I want to understand graphs: Collections of vertices and edges. P . Horn The Geometry of Graphs
Graphs Ultimately, I want to understand graphs: Collections of vertices and edges. Small graph: Easy to see what’s going on. P . Horn The Geometry of Graphs
Graphs Ultimately, I want to understand graphs: Collections of vertices and edges. Real-world graph: ??? P . Horn The Geometry of Graphs
Challenge Understanding large graphs leads to many challenges, especially as determining many graph properties are computationally hard. Goals: Be able to certify graph properties ‘cheaply’ computationally. Understand large-scale geometric properties of graphs (diameter, cuts, etc.) P . Horn The Geometry of Graphs
Challenge Understanding large graphs leads to many challenges, especially as determining many graph properties are computationally hard. Goals: Be able to certify graph properties ‘cheaply’ computationally. Understand large-scale geometric properties of graphs (diameter, cuts, etc.) One approach: via spectral graph theory. P . Horn The Geometry of Graphs
Spectral Graph Theory: Idea: Associate a matrix with a graph Matrix eigenvalues ⇔ Graph properties P . Horn The Geometry of Graphs
Spectral Graph Theory: Idea: Associate a matrix with a graph Matrix eigenvalues ⇔ Graph properties Normalized Laplace Operator ∆ = D − 1 A − I A = adjacency matrix D = diagonal degree matrix P . Horn The Geometry of Graphs
Spectral Graph Theory: Idea: Associate a matrix with a graph Matrix eigenvalues ⇔ Graph properties Normalized Laplace Operator ∆ = D − 1 A − I A = adjacency matrix D = diagonal degree matrix Unsymmetrized version of −L = D − 1 / 2 AD − 1 / 2 − I (normalized Laplacian ala Chung) Eigenvalues: 0 = − λ 0 ≤ − λ 1 ≤ · · · ≤ − λ n − 1 ≤ 2 . − λ 1 > 0 ⇐ ⇒ G connected ( # 0 ′ s = # connected components) − λ n − 1 < 2 ⇐ ⇒ no bipartite component. P . Horn The Geometry of Graphs
Geometry and Eigenvalues Cheeger’s Inequality If G is a graph, and λ 1 is the absolute value of second eigenvalue of ∆ , then 2 Φ ≥ λ 1 ≥ Φ 2 2 e ( X , ¯ X ) where Φ = min X ⊆ V ( G ) v �∈ X deg ( v ) } . min { � v ∈ X deg ( v ) , � P . Horn The Geometry of Graphs
Geometry and Eigenvalues Cheeger’s Inequality If G is a graph, and λ 1 is the absolute value of second eigenvalue of ∆ , then 2 Φ ≥ λ 1 ≥ Φ 2 2 e ( X , ¯ X ) where Φ = min X ⊆ V ( G ) v �∈ X deg ( v ) } . min { � v ∈ X deg ( v ) , � Quantitative version of statement that # 0 ′ s = # cc. P . Horn The Geometry of Graphs
Geometry and Eigenvalues Cheeger’s Inequality If G is a graph, and λ 1 is the absolute value of second eigenvalue of ∆ , then 2 Φ ≥ λ 1 ≥ Φ 2 2 e ( X , ¯ X ) where Φ = min X ⊆ V ( G ) v �∈ X deg ( v ) } . min { � v ∈ X deg ( v ) , � Quantitative version of statement that # 0 ′ s = # cc. Bound λ 1 ≥ Φ 2 2 : exact analogue of Cheeger’s inequality from differential geometry. Bound 2 Φ ≥ λ 1 : trivial for graphs, and not part of Cheeger’s inequality in geometry. P . Horn The Geometry of Graphs
Geometry and Eigenvalues Cheeger’s Inequality If G is a graph, and λ 1 is the absolute value of second eigenvalue of ∆ , then 2 Φ ≥ λ 1 ≥ Φ 2 2 e ( X , ¯ X ) where Φ = min X ⊆ V ( G ) v �∈ X deg ( v ) } . min { � v ∈ X deg ( v ) , � Quantitative version of statement that # 0 ′ s = # cc. Bound λ 1 ≥ Φ 2 2 : exact analogue of Cheeger’s inequality from differential geometry. Bound 2 Φ ≥ λ 1 : trivial for graphs, and not part of Cheeger’s inequality in geometry. Buser’s inequality: Non-negatively curved manifolds satisfy λ 1 = O (Φ 2 ) . P . Horn The Geometry of Graphs
Spectral Graph Theory Graph eigenvalues control many geometric properties of graphs: Isoperimetric constant (via Cheeger’s inequality) Diameter/distance between subsets Diffusion of random walks/heat dispersion. ... Eigenvalues provide a certificate of many graph properties, but are global in nature. P . Horn The Geometry of Graphs
Spectral Graph Theory Graph eigenvalues control many geometric properties of graphs: Isoperimetric constant (via Cheeger’s inequality) Diameter/distance between subsets Diffusion of random walks/heat dispersion. ... Eigenvalues provide a certificate of many graph properties, but are global in nature. Goal: Find local way to certify similar geometric properties. P . Horn The Geometry of Graphs
Curvature on Graphs Curvature of manifold: Local quantity measuring how fast manifold expands near a point Zero curvature: (Locally) expands like R n Positive curvature: (Locally) expands slower than R n (like on sphere) Negative curvature: (Locally) expands faster than R n P . Horn The Geometry of Graphs
Curvature on Graphs Curvature of manifold: Local quantity measuring how fast manifold expands near a point Zero curvature: (Locally) expands like R n Positive curvature: (Locally) expands slower than R n (like on sphere) Negative curvature: (Locally) expands faster than R n Curvature lower bounds have strong geometric and analytic consequences. eg. Bochner Formula: If M is an n -dim’l manifold, curvature ≥ − K , then for all smooth f : M → R ∆ |∇ f | 2 ≥ 2 n (∆ f ) 2 − 2 �∇ f , ∇ ∆ f � − K |∇ f | 2 P . Horn The Geometry of Graphs
Curvature on Graphs Curvature of manifold: Local quantity measuring how fast manifold expands near a point Zero curvature: (Locally) expands like R n Positive curvature: (Locally) expands slower than R n (like on sphere) Negative curvature: (Locally) expands faster than R n Curvature lower bounds have strong geometric and analytic consequences. eg. Bochner Formula: If M is an n -dim’l manifold, curvature ≥ − K , then for all smooth f : M → R ∆ |∇ f | 2 ≥ 2 n (∆ f ) 2 − 2 �∇ f , ∇ ∆ f � − K |∇ f | 2 Definitions of curvature for graphs take a consequences and make it a definition in the graph case. eg. based on: Degree (natural measure of local expansion) ‘Transportation distance’ (Ollivier/Lott, Villani/Lin, Lu, Yau) Satisfying analytic condition like Bochner Formula. P . Horn The Geometry of Graphs
Road to understanding One major route to understanding geometric properties from curvature: Curvature lower bound ⇓ Control of solutions to heat equation (solns to ∂ ∂ t u = ∆ u ) ⇓ Geometric information on graph/manifold P . Horn The Geometry of Graphs
Road to understanding One major route to understanding geometric properties from curvature: Curvature lower bound ⇓ Control of solutions to heat equation (solns to ∂ ∂ t u = ∆ u ) ⇓ Geometric information on graph/manifold Grigor’yan and Saloff-Coste (for manifolds) and Delmotte (for graphs) show: Strong control of solutions (Harnack inequalities) ⇔ Eigenvalue condition on balls (Poincaré inequality) plus volume growth condition (volume doubling) ⇔ Gaussian behavior for fundamental solutions. P . Horn The Geometry of Graphs
Road to understanding One major route to understanding geometric properties from curvature: Curvature lower bound ⇓ Control of solutions to heat equation (solns to ∂ ∂ t u = ∆ u ) ⇓ Geometric information on graph/manifold Manifold case: A curvature lower bound implies the Li-Yau inequality, a local estimate of how heat diffuses. Li-Yau Inequality: If u positive solution to heat equation on n -dim’l non-negatively curved compact manifold, |∇ u | 2 − u t u ≤ n 2 t . u 2 Can be integrated to derive Harnack inequality, and imply three (equivalent) conditions. P . Horn The Geometry of Graphs
Road to understanding One major route to understanding geometric properties from curvature: Curvature lower bound ⇓ Control of solutions to heat equation (solns to ∂ ∂ t u = ∆ u ) ⇓ Geometric information on graph/manifold Version of Li-Yau inequality for graphs derived by Bauer, H., Lippner, Lin, Mangoubi, Yau Introduce new version of graph curvature, the exponential curvature dimension inequality Can derive a Harnack inequality, but not quite strong enough to imply three conditions. H., Lin, Liu, Yau: Use different methods to prove non-negatively curved graphs satisfy equivalent conditions. P . Horn The Geometry of Graphs
Road to understanding One major route to understanding geometric properties from curvature: Curvature lower bound ⇓ Control of solutions to heat equation (solns to ∂ ∂ t u = ∆ u ) ⇓ Geometric information on graph/manifold Remark: Understanding solutions to the heat equation themselves is interesting on graphs. Related to diffusion of continuous time random walk. P . Horn The Geometry of Graphs
Road to understanding One major route to understanding geometric properties from curvature: Curvature lower bound ⇓ Control of solutions to heat equation (solns to ∂ ∂ t u = ∆ u ) ⇓ Geometric information on graph/manifold To continue, want to revisit Key step: Proving Li-Yau inequality for graphs. P . Horn The Geometry of Graphs
Goal To prove Li-Yau inequality for graphs, and its further geometric consequences need to understand how curvature plays a role in the proofs. Challenges that arise dictate how to proceed. P . Horn The Geometry of Graphs
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