Topological Data Analysis (Spring 2018) TDA on Networks · Topological Data Analysis (Spring 2018)...

Topological Data Analysis (Spring 2018)TDA on Networks

Instructor: Mehmet Aktas

March 27, 2018

1 / 20 Instructor: Mehmet Aktas TDA on Networks

Outline

1 Introduction

2 Filtrations on NetworksThe Rips Complex of a NetworkThe Dowker Filtration of a Network

3 Application

Introduction

Outline

1 Introduction

3 Application

Introduction

Graphs

Structured data representing relationship btw objects

Important in modeling sophisticated structures and their interaction

Formed by

A set of verticesA set of edges

Examples

Computer networks

Social networks

Protein interactionnetworks

Introduction

Graphs

Formed by

Examples

Computer networks

Social networks

Introduction

Graphs

Formed by

Examples

Computer networks

Social networks

Introduction

Graphs

Formed by

Examples

Computer networks

Social networks

Introduction

Graphs

Formed by

Examples

Computer networks

Social networks

Introduction

Graphs

Formed by

Examples

Computer networks

Social networks

Introduction

Attributed Graphs

Vertices have a set of attributes describing theproperties of them

Two source of data: Structure & Attribute

Everywhere

Social networks

Co-authorship networks

Introduction

Attributed Graphs

Everywhere

Social networks

Introduction

Attributed Graphs

Everywhere

Social networks

Introduction

Attributed Graphs

Everywhere

Social networks

Introduction

Attributed Graphs

Everywhere

Social networks

Introduction

Network Distances

Let G = (X ,E) be a weighted network where X is the set of vertices,E ⊂ X ×X is the set of edges, and ω ∶ X ×X → R is an edge weightfunction.

If there are two different edge weight functions ω,ω′ defined on G , wecan use the l∞ distance as a measure of network similarity between(G , ω) and (G , ω′):

∣∣ω − ω′∣∣l∞ ∶= maxe∈E

∣ω(e) − ω′(e)∣.

Given two vertex sets X and Y , we need to decide how to match uppoints of X with points of Y

Any such matching will yield a subset R ⊂ X ×Y , which is called acorrespondence.

Introduction

Network Distances

∣∣ω − ω′∣∣l∞ ∶= maxe∈E

∣ω(e) − ω′(e)∣.

Introduction

Network Distances

∣∣ω − ω′∣∣l∞ ∶= maxe∈E

∣ω(e) − ω′(e)∣.

Introduction

Network Distances

∣∣ω − ω′∣∣l∞ ∶= maxe∈E

∣ω(e) − ω′(e)∣.

Introduction

Network Distances

The distortion of a correspondence between X and Y is

dis(R) ∶= max(x ,y),(x ′,y ′)∈R

∣ωX (x , x ′) − ωY (y , y ′)∣.

We denote the set of all correspondences between X and Y byR(X ,Y ).

The network distance is defined as follows:

dN ((X , ωX ), (Y , ωY )) ∶= 1

R∈R(X ,Y )dis(R).

Introduction

Network Distances

The distortion of a correspondence between X and Y is

dis(R) ∶= max(x ,y),(x ′,y ′)∈R

∣ωX (x , x ′) − ωY (y , y ′)∣.

We denote the set of all correspondences between X and Y byR(X ,Y ).

The network distance is defined as follows:

dN ((X , ωX ), (Y , ωY )) ∶= 1

R∈R(X ,Y )dis(R).

Filtrations on Networks The Rips Complex of a Network

Outline

1 Introduction

3 Application

Rips Filtration

For a metric space (X ,dX ), the diameter of a subset σ ⊂ X is definedas diam(σ) ∶= maxx ,x ′∈σ dX (x , x ′).

The Rips complex of a metric space (X ,dX ) is defined for each r ∈ Ras

RδX ∶= {σ ∈ Pow(X ) ∶ diam(σ) ≤ δ}.

For any weigthed network (X , ωX ), define the weight of a subset as amap wgtX (σ) ∶ Pow(X )→ R given by:

wgtX (σ) ∶= maxx ,x ′∈σ

ωX (x , x ′)

Rips Filtration

ωX (x , x ′)

Rips Filtration

ωX (x , x ′)

Rips Filtration

The Rips complex of a network (X , ωX ) ∈ N is defined as

RδX ∶= {σ ∈ Pow(X ) ∶ wgtX (σ) ≤ δ}.

The Rips complex as defined above yields a valid simplicial complexon a network for each parameter δ ∈ R. Thus to any network(X , ωX ), we may associate the Rips filtration {Rδ

X ↪Rδ′

X}δ≤δ′ .For each k ∈ Z≥0, we denote the k-dimensional persistence diagram byDgmR

k (X ).

Proposition

Let (X , ωX ), (Y , ωY ) ∈ N . Then we have:

dB(DgmRk (X ),DgmR

k (Y )) ≤ 2dN (X ,Y ).

Rips Filtration

RδX ∶= {σ ∈ Pow(X ) ∶ wgtX (σ) ≤ δ}.

X ↪Rδ′

k (X ).

Proposition

dB(DgmRk (X ),DgmR

k (Y )) ≤ 2dN (X ,Y ).

Rips Filtration

RδX ∶= {σ ∈ Pow(X ) ∶ wgtX (σ) ≤ δ}.

X ↪Rδ′

k (X ).

Proposition

dB(DgmRk (X ),DgmR

k (Y )) ≤ 2dN (X ,Y ).

Rips Filtration

RδX ∶= {σ ∈ Pow(X ) ∶ wgtX (σ) ≤ δ}.

X ↪Rδ′

k (X ).

Proposition

dB(DgmRk (X ),DgmR

k (Y )) ≤ 2dN (X ,Y ).

Example

Drawbacks of Rips Filtration

It is blind to directed edge weights.

The Rips complex does not absorb information in dimensions higherthan one.

Simplices in a Rips complex are not formed with respect to any“central authority.” This could be undesirable in, for example, asmall-world network, where one would desire simplices to be formedwith respect to particular “hub” nodes.

Filtrations on Networks The Dowker Filtration of a Network

Outline

1 Introduction

3 Application

Dowker Filtration

Given a network (X , ωX ) ∈ N and δ ∈ R, Rd ,X ⊂ X ×X is defined as

Rδ,X ∶= {(x , x ′) ∶= {(x , x ′) ∶ ωX (x , x ′) ≤ δ}.

Dsiδ,X ∶= {σ = [x0, ..., xn] ∶ there exists x ′ ∈ X such that (xi , x ′) ∈

Rδ,X for each xi}.For δ′ ≥ δ, there is a natural inclusion Dsi

δ,X ↪Dsiδ′,X

This is called the Dowker sink filtration.

Dsoδ,X ∶= {σ = [x0, ..., xn] ∶ there exists x ′ ∈ X such that (x ′, xi) ∈

Rδ,X for each xi}.This is called the Dowker source filtration.

Dowker Filtration

Rδ,X ∶= {(x , x ′) ∶= {(x , x ′) ∶ ωX (x , x ′) ≤ δ}.

δ,X ↪Dsiδ′,X

Dowker Filtration

Rδ,X ∶= {(x , x ′) ∶= {(x , x ′) ∶ ωX (x , x ′) ≤ δ}.

δ,X ↪Dsiδ′,X

Dowker Filtration

Rδ,X ∶= {(x , x ′) ∶= {(x , x ′) ∶ ωX (x , x ′) ≤ δ}.

δ,X ↪Dsiδ′,X

Dowker Filtration

Rδ,X ∶= {(x , x ′) ∶= {(x , x ′) ∶ ωX (x , x ′) ≤ δ}.

δ,X ↪Dsiδ′,X

Dowker Filtration

Rδ,X ∶= {(x , x ′) ∶= {(x , x ′) ∶ ωX (x , x ′) ≤ δ}.

δ,X ↪Dsiδ′,X

Example

Sink vs Source

The sink and source filtrations are not equal in general. However ...

Theorem

For any k ∈ Z≥0 and (X , ωX ) ∈ N , we have

Dgmsik (X ) = Dgmso

k (X ).

Thus we may call either of the two diagrams above the k-dimensionalDowker diagram of X , denoted Dgm●k(X ).

Sink vs Source

Theorem

Dgmsik (X ) = Dgmso

k (X ).

Sink vs Source

Theorem

Dgmsik (X ) = Dgmso

k (X ).

Example

Application

Outline

1 Introduction

3 Application

Application

Simulated hippocampal networks

An animal explores a given environment or arena, specific “placecells” in the hippocampus show increased activity at specific spatialregions, called “place fields”.

Each place cell shows a spike in activity when the animal enters theplace field linked to this place cell, accompanied by a drop in activityas the animal moves far away from this place field.

Is the time series data of the place cell activity, referred to as “spiketrains”, enough to detect the structure of the arena?

Application

Experiment

Application

Experiment

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