Geometry of Discrete Graphs

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Geometry of Discrete Graphs

Brian Benson, Peter Ralli, Prasad Tetali

Kansas State University, Georgia Tech, Georgia Tech

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Optimal Transport Analogy

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Optimal Transport Analogy

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Ricci Curvature and Transportation Distance

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Ricci Curvature and Transportation Distance

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Ricci Curvature and Transportation Distance

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Ricci Curvature and Transportation Distance

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Ricci Curvature and Transportation Distance

Goal: Compare average distance between points in B(x ) and

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Positive Curvature and Transportation Distance

Parallel geodesics from x and y get closer together.

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Positive Curvature and Transportation Distance

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Negative Curvature and Transportation Distance

Parallel geodesics from x and y get farther apart.

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Negative Curvature and Transportation Distance

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Ricci Curvature and Transportation Distance

Ollivier (2009) made the following observations:

Positive Curvature: Transportation of points in B(x )

shorter, on average, to B(y ) compared to the distance

between x and y .

Negative Curvature: Must transport points in B(x ) farther,

on average, to B(y ) compared to the distance between x and

y .

Main Idea: For x , y points on the manifold, fix small > 0

and transport B(x ) to B(y ) using an optimal transportation

plan (cost is distance). Compare the average distance of the transport plan to the distance between x and y . Conclude a “curvature” between x and y (depending on ) based on this ratio.

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Ricci Curvature and Transportation Distance

between x and y .

y .

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Ricci Curvature and Transportation Distance

between x and y .

y .

plan (cost is distance). Compare the average distance of the transport plan to the distance between x and y . Conclude a “curvature” between x and y (depending on ) based on this ratio.

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Ricci Curvature and Transportation Distance

between x and y .

y .

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Ollivier Curvature

Let M be a Riemannian n-manifold. Create a probability measure at each point x ∈ M depending on small > 0:

dmx(w ) =

1

Vol(B(x ,))d Vol(w ), w ∈ B(x , ),

0, otherwise,

Recover the measure mx(A) of the subsets A of M by integrating

dmx over the set A.

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Ollivier Curvature

dmx(w ) =

1

Vol(B(x ,))d Vol(w ), w ∈ B(x , ),

0, otherwise,

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Ollivier Curvature

dmx(w ) =

1

Vol(B(x ,))d Vol(w ), w ∈ B(x , ),

0, otherwise,

dmx over the set A.

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Ollivier Curvature on Manifolds

dmx(w ) =

1

Vol(B(x ,))d Vol(w ), w ∈ B(x , ),

0, otherwise,

Define the Wasserstein distance between two measures as

W (mx, my) := inf

ξ∈Π(mx,my) Z

(w ,w0_)∈M×M

ρ(w , w0) d ξ(w , w0).

Here Π(mx, my) is the space of measures ξ on M × M such that

for any A ⊆ M, we have that ξ(A × M) = mx(A) and

ξ(M × A) = my(A).

The term ρ(w , w0) measures the distance from w to w0 and

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Ollivier Curvature on Manifolds

dmx(w ) =

1

Vol(B(x ,))d Vol(w ), w ∈ B(x , ),

0, otherwise,

Define the Wasserstein distance between two measures as

W (mx, my) := inf

ξ∈Π(mx,my) Z

(w ,w0_)∈M×M

ρ(w , w0) d ξ(w , w0).

Here Π(mx, my) is the space of measures ξ on M × M such that

for any A ⊆ M, we have that ξ(A × M) = mx(A) and

ξ(M × A) = my(A).

The term ρ(w , w0) measures the distance from w to w0 and

d ξ(w , w0) represents (infinitesimally) the mass which is traveling from w to w0.

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Ollivier Curvature on Manifolds

dmx(w ) =

1

Vol(B(x ,))d Vol(w ), w ∈ B(x , ),

0, otherwise,

Define the Wasserstein distance between two measures as W (mx, my) := Z ξ∈Π(mx,my) Z (w ,w0_)∈M×M ρ(w , w0) d ξ(w , w0). The Ollivier or coarse Ricci curvature of (x , y ) ∈ M × M is defined to be

κ(x , y ) := 1 −W (mx, my)

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Remarks on Ollivier Curvature on Manifolds

Ollivier’s curvature is defined for pairs of points in the manifold instead of at single points.

In spite of this, Ollivier proved that he could recover the classical Ricci curvature up to a factor using his definition! A major benefit of Ollivier’s definition is its applicability to graphs since it is defined using the metric and a measure, both of which arise naturally on graphs.

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Remarks on Ollivier Curvature on Manifolds

In spite of this, Ollivier proved that he could recover the classical Ricci curvature up to a factor using his definition!

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Remarks on Ollivier Curvature on Manifolds

In spite of this, Ollivier proved that he could recover the classical Ricci curvature up to a factor using his definition! A major benefit of Ollivier’s definition is its applicability to graphs since it is defined using the metric and a measure, both of which arise naturally on graphs.

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Ollivier Curvature on Graphs

Let G = (V , E ) be a finite, connected graph. Ollivier curvature for the graph G produces a quantity for curvature for each element in V × V . For each x ∈ V , the measure is the lazy random walk from the vertex x :

mx(v ) =    1 2, v = x , 1 2 deg(v ), v ∼ x , 0, otherwise.

Define the discrete Wasserstein distance between two measures as W (mx, my) := inf ξ∈Π(mx,my) X (w ,w0_{)∈V ×V} ρ(w , w0) · ξ(w , w0). The Ollivier or coarse Ricci curvature of (x , y ) ∈ V × V is defined to be

κ(x , y ) = 1 − W (mx, my)

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Ollivier Curvature on Graphs

Let G = (V , E ) be a finite, connected graph. Ollivier curvature for the graph G produces a quantity for curvature for each element in V × V . For each x ∈ V , the measure is the lazy random walk from the vertex x :

mx(v ) =    1 2, v = x , 1 2 deg(v ), v ∼ x , 0, otherwise.

Define the discrete Wasserstein distance between two measures as W (mx, my) := inf ξ∈Π(mx,my) X (w ,w0_{)∈V ×V} ρ(w , w0) · ξ(w , w0). The Ollivier or coarse Ricci curvature of (x , y ) ∈ V × V is defined to be

κ(x , y ) = 1 − W (mx, my)

ρ(x , y ) .

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Example: 3-Cube

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Example: 3-Cube

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Example: 3-Cube

W (mx, my) = 1 3 · 1 + 1 6 · 1 + 1 6· 1 = 2 3 κ(x , y ) = 1 − 2/3 1 = 1 3

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Example: A Very Simple Tree

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Example: A Very Simple Tree

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Example: A Very Simple Tree

W (mx, my) = 1

κ(x , y ) = 1 − 1

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Example: A Very Simple Tree

W (mx, my) = 1

κ(x , y ) = 1 − 1

1 = 0

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Example: A Cycle

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Example: A Cycle

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Example: A Cycle

W (mx, my) = 1

κ(x , y ) = 1 − 1

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Example: A Cycle

W (mx, my) = 1

κ(x , y ) = 1 − 1

1 = 0

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Example: A Very Simple Tree

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Example: A Very Simple Tree

W (mx, my) = 3

κ(x , y ) = 1 − 3

3 = 0

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Example: A Very Simple Tree

W (mx, my) = 3

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Example: A Cycle

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Example: A Cycle

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Example: A Cycle

W (mx, my) = 2 κ(x , y ) = 1 −2 3 = 1 3

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Volume Comparison Observations

Volume for graphs is counting measure on vertices.

Curvature more positive for vertices in the presence of cycles.

With cycles, ratio of edges to vertices is increased.

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Volume Comparison Observations

Goal: Predict size of subsets of the vertex sets in terms of the corresponding edge subsets.

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Volume Comparison Observations

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Volume Comparison Observations

Goal: Predict size of subsets of the vertex sets in terms of the corresponding edge subsets.

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Volume Comparisons on Manifolds

Bishop’s Volume Comparison Theorem (1963). Let M be a complete Riemannian n-manifold with Ricci curvature bounded below by (n − 1)k at every point. Then for every p ∈ M, we have

Voln Br(p) ≤ Voln B(k, r ).

where B(k, r ) is the ball of radius r in the space of constant sectional curvature k.

Heintze and Karcher (1979) generalize Bishop’s Theorem to a more arbitrary collection of subsets of M.

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Volume Comparisons on Manifolds

Goal: Find analogues of these theorems on graphs using Ollivier curvature.

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Volume Comparisons on Manifolds

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Application of Volume Comparison on Manifolds

Let M be a closed Riemannian n-manifold. Cheeger constant: h(M) = inf

Σ,A,B

Voln−1(Σ)

minVoln(A), Voln(B)

where Σ is a codimension-1 submanifold of M “splitting” M into A and B.

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Application of Volume Comparison on Manifolds

Σ,A,B

Voln−1(Σ)

minVoln(A), Voln(B)

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Application of Volume Comparison on Manifolds

Σ,A,B

Voln−1(Σ)

minVoln(A), Voln(B)

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Application of Volume Comparison on Manifolds

Σ,A,B

Voln−1(Σ)

minVoln(A), Voln(B)

Buser (1982): First non-zero eigenvalue of ∆ on M bounded by a quadratic in h(M) with coefficients depending on n and the Ricci curvature.

Agol (unpublished): Quantitative improvement of Buser. B. (2015): Analogue of Buser for all eigenvalues. Requires Heintze-Karcher volume comparison result.

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Application of Volume Comparison on Manifolds

Σ,A,B

Voln−1(Σ)

minVoln(A), Voln(B)

Agol (unpublished): Quantitative improvement of Buser.

B. (2015): Analogue of Buser for all eigenvalues. Requires Heintze-Karcher volume comparison result.

Buser-type results already known on graphs due to many authors using variants of CD curvature. Can we accomplish for Ollivier curvature?

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Application of Volume Comparison on Manifolds

Σ,A,B

Voln−1(Σ)

minVoln(A), Voln(B)

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Application of Volume Comparison on Manifolds

Σ,A,B

Voln−1(Σ)

minVoln(A), Voln(B)

Buser-type results already known on graphs due to many authors using variants of CD curvature. Can we accomplish for Ollivier curvature?