Formal Maps and their Algebra

Valeria Fionda, Claudio Gutierrez Giuseppe Pirro

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 1 / 43

Outline

1 Introduction

2 Contribution

3 The notion of map

4 Related research

5 Preliminaries

6 Definition of Map

7 Maps as mathematical objects

8 Regions and maps on the Web

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 2 / 43

Outline

1 Introduction

2 Contribution

3 The notion of map

4 Related research

5 Preliminaries

6 Definition of Map

7 Maps as mathematical objects

8 Regions and maps on the Web

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 2 / 43

Outline

1 Introduction

2 Contribution

3 The notion of map

4 Related research

5 Preliminaries

6 Definition of Map

7 Maps as mathematical objects

8 Regions and maps on the Web

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 2 / 43

Outline

1 Introduction

2 Contribution

3 The notion of map

4 Related research

5 Preliminaries

6 Definition of Map

7 Maps as mathematical objects

8 Regions and maps on the Web

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 2 / 43

Outline

1 Introduction

2 Contribution

3 The notion of map

4 Related research

5 Preliminaries

6 Definition of Map

7 Maps as mathematical objects

8 Regions and maps on the Web

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 2 / 43

Outline

1 Introduction

2 Contribution

3 The notion of map

4 Related research

5 Preliminaries

6 Definition of Map

7 Maps as mathematical objects

8 Regions and maps on the Web

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 2 / 43

Outline

1 Introduction

2 Contribution

3 The notion of map

4 Related research

5 Preliminaries

6 Definition of Map

7 Maps as mathematical objects

8 Regions and maps on the Web

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 2 / 43

Outline

1 Introduction

2 Contribution

3 The notion of map

4 Related research

5 Preliminaries

6 Definition of Map

7 Maps as mathematical objects

8 Regions and maps on the Web

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 2 / 43

Introduction

Outline

1 Introduction

2 Contribution

3 The notion of map

4 Related research

5 Preliminaries

6 Definition of Map

7 Maps as mathematical objects

8 Regions and maps on the Web

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 3 / 43

Introduction

What is a map?

Maps are artifacts that orient users in information spaces

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 4 / 43

Introduction

What is a map?

Maps are artifacts that orient users in information spaces

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 4 / 43

Introduction

What is a map?

Maps are artifacts that orient users in information spaces

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 4 / 43

Introduction

Goals

Our broad goal is to investigate how cartographical principles can beapplied over the Web space

The web is a graph

In particular we are interested in the Web of linked data, that is ahuge, distributed, directed, labeled multigraph

Specific goals:

1) We want to build maps of a graph

2) Algebra of maps

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 5 / 43

Introduction

Goals

Our broad goal is to investigate how cartographical principles can beapplied over the Web space

The web is a graph

In particular we are interested in the Web of linked data, that is ahuge, distributed, directed, labeled multigraph

Specific goals:

1) We want to build maps of a graph

2) Algebra of maps

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 5 / 43

Introduction

Goals

Our broad goal is to investigate how cartographical principles can beapplied over the Web space

The web is a graph

In particular we are interested in the Web of linked data, that is ahuge, distributed, directed, labeled multigraph

Specific goals:

1) We want to build maps of a graph

2) Algebra of maps

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 5 / 43

Contribution

Outline

1 Introduction

2 Contribution

3 The notion of map

4 Related research

5 Preliminaries

6 Definition of Map

7 Maps as mathematical objects

8 Regions and maps on the Web

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 6 / 43

Contribution

Contribution

Our contributions are:1 We provide a formal general framework to cope with the notion of

map as a means to abstract graphs.

2 We studied the properties of different types of maps and developedefficient algorithms to compute them.

3 We introduced an algebra for maps.4 We tackle the problem of how to apply our framework to the Web:

1 By investigating how to specify regions of the Web - new semantics ofWeb navigational languages returning graphs

2 Obtaining maps from those regions

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 7 / 43

Contribution

Contribution

Our contributions are:1 We provide a formal general framework to cope with the notion of

map as a means to abstract graphs.2 We studied the properties of different types of maps and developed

efficient algorithms to compute them.

3 We introduced an algebra for maps.4 We tackle the problem of how to apply our framework to the Web:

1 By investigating how to specify regions of the Web - new semantics ofWeb navigational languages returning graphs

2 Obtaining maps from those regions

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 7 / 43

Contribution

Contribution

Our contributions are:1 We provide a formal general framework to cope with the notion of

map as a means to abstract graphs.2 We studied the properties of different types of maps and developed

efficient algorithms to compute them.3 We introduced an algebra for maps.

4 We tackle the problem of how to apply our framework to the Web:1 By investigating how to specify regions of the Web - new semantics of

Web navigational languages returning graphs2 Obtaining maps from those regions

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 7 / 43

Contribution

Contribution

Our contributions are:1 We provide a formal general framework to cope with the notion of

map as a means to abstract graphs.2 We studied the properties of different types of maps and developed

efficient algorithms to compute them.3 We introduced an algebra for maps.4 We tackle the problem of how to apply our framework to the Web:

1 By investigating how to specify regions of the Web - new semantics ofWeb navigational languages returning graphs

2 Obtaining maps from those regions

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 7 / 43

Contribution

Contribution

Our contributions are:1 We provide a formal general framework to cope with the notion of

map as a means to abstract graphs.2 We studied the properties of different types of maps and developed

efficient algorithms to compute them.3 We introduced an algebra for maps.4 We tackle the problem of how to apply our framework to the Web:

1 By investigating how to specify regions of the Web - new semantics ofWeb navigational languages returning graphs

2 Obtaining maps from those regions

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 7 / 43

Contribution

Contribution

Our contributions are:1 We provide a formal general framework to cope with the notion of

map as a means to abstract graphs.2 We studied the properties of different types of maps and developed

efficient algorithms to compute them.3 We introduced an algebra for maps.4 We tackle the problem of how to apply our framework to the Web:

1 By investigating how to specify regions of the Web - new semantics ofWeb navigational languages returning graphs

2 Obtaining maps from those regions

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 7 / 43

Contribution

Contribution

Our contributions are:1 We provide a formal general framework to cope with the notion of

map as a means to abstract graphs.2 We studied the properties of different types of maps and developed

efficient algorithms to compute them.3 We introduced an algebra for maps.4 We tackle the problem of how to apply our framework to the Web:

1 By investigating how to specify regions of the Web - new semantics ofWeb navigational languages returning graphs

2 Obtaining maps from those regions

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 7 / 43

The notion of map

Outline

1 Introduction

2 Contribution

3 The notion of map

4 Related research

5 Preliminaries

6 Definition of Map

7 Maps as mathematical objects

8 Regions and maps on the Web

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 8 / 43

The notion of map

A region

MartinScorsese

JohnFord

OrsonWelles

QuentinTarantino

StanleyKubrick

TimBurton

WoodyAllen

Lars vonTrier

DavidLynch

Influeces between directors

PeterJackson

TerryGilliam

DavidFincher

GregHarrison

Paul T.AndersonNicolas

Renf

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 9 / 43

The notion of map

Distinguished nodes

MartinScorsese

JohnFord

OrsonWelles

QuentinTarantino

StanleyKubrick

TimBurton

WoodyAllen

Lars vonTrier

DavidLynch

Influeces between directors

PeterJackson

TerryGilliam

DavidFincher

GregHarrison

Paul T.AndersonNicolas

Renf

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 10 / 43

The notion of map

Examples of maps

MartinScorsese

JohnFord

OrsonWelles

QuentinTarantino

StanleyKubrick

TimBurton

WoodyAllen

Lars vonTrier

DavidLynch

Influeces between directors

PeterJackson

TerryGilliam

DavidFincher

GregHarrison

Paul T.AndersonNicolas

Renf

MartinScorsese

JohnFord

QuentinTarantino

StanleyKubrick Woody

Allen

JohnFord

QuentinTarantino

StanleyKubrick Woody

Allen

Map 2Map 1

e1

e2

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 11 / 43

The notion of map

Abstraction level

A map should:

Represent the region

Be concise

Keep the connectivity

How much of the original region has to be included in the map?

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 12 / 43

The notion of map

Abstraction level

A map should:

Represent the region

Be concise

Keep the connectivity

How much of the original region has to be included in the map?

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 12 / 43

Related research

Outline

1 Introduction

2 Contribution

3 The notion of map

4 Related research

5 Preliminaries

6 Definition of Map

7 Maps as mathematical objects

8 Regions and maps on the Web

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 13 / 43

Related research

Graph summarization

Graph summarization [1,2,3]:

Goal: produce a compressed representation of an input graph G

Determine a function F in order to find a simplified structure Gs

satisfying some requirements

Removes some of the details from the graphs in order to reduce spaceconsumption by usually clustering nodes in partitions.

[1] C. Faloutsos, K.S. McCurley, and A. Tomkins. Fast Discovery of Connection Subgraphs. InKDD, pages 118-127. ACM, 2004.[2] J. Adibi, H. Chalupsky, E. Melz, A. Valente, et al. The KOJAK Group Finder: Connectingthe Dots via Integrated Knowledge-based and Statistical Reasoning. In AAAI, pages 800-807,2004.[3] F. Zhou, S. Malher, and H. Toivonen. Network Simplification with Minimal Loss ofConnectivity. In ICDM, pages 659-668. IEEE, 2010.

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 14 / 43

Related research

Graph indexing

Graph indexing [4]:

Goal: produce a list of graph substructures with references to theplace where they can be found

Make querying the graph faster

[4] X. Yan and J. Han. Graph Indexing. Managing and Mining Graph Data 2010, pages 161-180

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 15 / 43

Related research

Differences

1 These techniques work on the whole graphs, they do not providemeans to specify regions.

Benefit: the web cannot be mapped altogheter! Isolated portionsshould be considered (according to user interests)

2 We focus also on giving machine readable representations

structures and are represented in a standard format to be extended,reused and combined.

3 Our map frameworks allow to obtain maps that can be treated asmathematical objects by defining an algebra over maps.

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 16 / 43

Related research

Differences

1 These techniques work on the whole graphs, they do not providemeans to specify regions.

Benefit: the web cannot be mapped altogheter! Isolated portionsshould be considered (according to user interests)

2 We focus also on giving machine readable representations

structures and are represented in a standard format to be extended,reused and combined.

3 Our map frameworks allow to obtain maps that can be treated asmathematical objects by defining an algebra over maps.

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 16 / 43

Related research

Differences

1 These techniques work on the whole graphs, they do not providemeans to specify regions.

Benefit: the web cannot be mapped altogheter! Isolated portionsshould be considered (according to user interests)

2 We focus also on giving machine readable representations

structures and are represented in a standard format to be extended,reused and combined.

3 Our map frameworks allow to obtain maps that can be treated asmathematical objects by defining an algebra over maps.

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 16 / 43

Preliminaries

Outline

1 Introduction

2 Contribution

3 The notion of map

4 Related research

5 Preliminaries

6 Definition of Map

7 Maps as mathematical objects

8 Regions and maps on the Web

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 17 / 43

Preliminaries

Notation

Let G = (VG ,EG ) be a directed graph, VG the set of nodes, EG the set ofedges and u, v nodes in G and N ⊆ VG be a set of nodes:

u → v denotes an edge (u, v) ∈ EG

u � v denotes a path from u to v in G

u�Nv denotes a path from u to v in G not passing throughintermediate nodes in N

MartinScorsese

JohnFord

OrsonWelles

QuentinTarantino

StanleyKubrick

TimBurton

WoodyAllen

Lars vonTrier

DavidLynch

Influeces between directors

PeterJackson

TerryGilliam

DavidFincher

GregHarrison

Paul T.AndersonNicolas

Renf

John Ford → Martin ScorseseJohn Ford � Stanley Kubrick

N={Martin Scorsese, Woody Allen}

John Ford �N Stanley KubrickNOTJohn Ford �N Paul T. Anderson

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 18 / 43

Preliminaries

Notation

Let G = (VG ,EG ) be a directed graph, VG the set of nodes, EG the set ofedges and u, v nodes in G and N ⊆ VG be a set of nodes:

u → v denotes an edge (u, v) ∈ EG

u � v denotes a path from u to v in G

u�Nv denotes a path from u to v in G not passing throughintermediate nodes in N

MartinScorsese

JohnFord

OrsonWelles

QuentinTarantino

StanleyKubrick

TimBurton

WoodyAllen

Lars vonTrier

DavidLynch

Influeces between directors

PeterJackson

TerryGilliam

DavidFincher

GregHarrison

Paul T.AndersonNicolas

Renf

John Ford → Martin Scorsese

John Ford � Stanley Kubrick

N={Martin Scorsese, Woody Allen}

John Ford �N Stanley KubrickNOTJohn Ford �N Paul T. Anderson

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 18 / 43

Preliminaries

Notation

Let G = (VG ,EG ) be a directed graph, VG the set of nodes, EG the set ofedges and u, v nodes in G and N ⊆ VG be a set of nodes:

u → v denotes an edge (u, v) ∈ EG

u � v denotes a path from u to v in G

u�Nv denotes a path from u to v in G not passing throughintermediate nodes in N

MartinScorsese

JohnFord

OrsonWelles

QuentinTarantino

StanleyKubrick

TimBurton

WoodyAllen

Lars vonTrier

DavidLynch

Influeces between directors

PeterJackson

TerryGilliam

DavidFincher

GregHarrison

Paul T.AndersonNicolas

Renf

John Ford → Martin ScorseseJohn Ford � Stanley Kubrick

N={Martin Scorsese, Woody Allen}

John Ford �N Stanley KubrickNOTJohn Ford �N Paul T. Anderson

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 18 / 43

Preliminaries

Notation

Let G = (VG ,EG ) be a directed graph, VG the set of nodes, EG the set ofedges and u, v nodes in G and N ⊆ VG be a set of nodes:

u → v denotes an edge (u, v) ∈ EG

u � v denotes a path from u to v in G

u�Nv denotes a path from u to v in G not passing throughintermediate nodes in N

MartinScorsese

JohnFord

OrsonWelles

QuentinTarantino

StanleyKubrick

TimBurton

WoodyAllen

Lars vonTrier

DavidLynch

Influeces between directors

PeterJackson

TerryGilliam

DavidFincher

GregHarrison

Paul T.AndersonNicolas

Renf

John Ford → Martin ScorseseJohn Ford � Stanley Kubrick

N={Martin Scorsese, Woody Allen}John Ford �N Stanley KubrickNOTJohn Ford �N Paul T. Anderson

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 18 / 43

Definition of Map

Outline

1 Introduction

2 Contribution

3 The notion of map

4 Related research

5 Preliminaries

6 Definition of Map

7 Maps as mathematical objects

8 Regions and maps on the Web

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 19 / 43

Definition of Map

Map of a graph

Definition

A map M = (VM ,EM) of G is a graph such that VM ⊆ VG and each edge

(x , y) ∈ EM implies x � y in G

MartinScorsese

JohnFord

OrsonWelles

QuentinTarantino

StanleyKubrick

TimBurton

WoodyAllen

Lars vonTrier

DavidLynch

PeterJackson

TerryGilliam

DavidFincher

GregHarrison

Paul T.AndersonNicolas

Renf

MartinScorseseJohn

Ford

QuentinTarantino

StanleyKubrick

WoodyAllen

DavidLynch

MartinScorseseJohn

Ford

QuentinTarantino

StanleyKubrick

WoodyAllen

DavidLynch

MartinScorseseJohn

Ford

QuentinTarantino

StanleyKubrick

WoodyAllen

DavidLynch

MartinScorseseJohn

Ford

QuentinTarantino

StanleyKubrick

WoodyAllen

DavidLynch

MartinScorseseJohn

Ford

QuentinTarantino

StanleyKubrick

WoodyAllen

DavidLynch

MartinScorseseJohn

Ford

QuentinTarantino

StanleyKubrick

WoodyAllen

DavidLynch

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 20 / 43

Definition of Map

Map of a graph

Definition

A map M = (VM ,EM) of G is a graph such that VM ⊆ VG and each edge

(x , y) ∈ EM implies x � y in G

MartinScorsese

JohnFord

OrsonWelles

QuentinTarantino

StanleyKubrick

TimBurton

WoodyAllen

Lars vonTrier

DavidLynch

PeterJackson

TerryGilliam

DavidFincher

GregHarrison

Paul T.AndersonNicolas

Renf

MartinScorseseJohn

Ford

QuentinTarantino

StanleyKubrick

WoodyAllen

DavidLynch

MartinScorseseJohn

Ford

QuentinTarantino

StanleyKubrick

WoodyAllen

DavidLynch

MartinScorseseJohn

Ford

QuentinTarantino

StanleyKubrick

WoodyAllen

DavidLynch

MartinScorseseJohn

Ford

QuentinTarantino

StanleyKubrick

WoodyAllen

DavidLynch

MartinScorseseJohn

Ford

QuentinTarantino

StanleyKubrick

WoodyAllen

DavidLynch

MartinScorseseJohn

Ford

QuentinTarantino

StanleyKubrick

WoodyAllen

DavidLynch

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 20 / 43

Definition of Map

Complete map of a graph

Definition

A map M = (VM ,EM) of G is complete iff x � y in G implies x � y inM.

MartinScorsese

JohnFord

OrsonWelles

QuentinTarantino

StanleyKubrick

TimBurton

WoodyAllen

Lars vonTrier

DavidLynch

PeterJackson

TerryGilliam

DavidFincher

GregHarrison

Paul T.AndersonNicolas

Renf

MartinScorseseJohn

Ford

QuentinTarantino

StanleyKubrick

WoodyAllen

DavidLynch

MartinScorseseJohn

Ford

QuentinTarantino

StanleyKubrick

WoodyAllen

DavidLynch

MartinScorseseJohn

Ford

QuentinTarantino

StanleyKubrick

WoodyAllen

DavidLynch

MartinScorseseJohn

Ford

QuentinTarantino

StanleyKubrick

WoodyAllen

DavidLynch

MartinScorseseJohn

Ford

QuentinTarantino

StanleyKubrick

WoodyAllen

DavidLynch

MartinScorseseJohn

Ford

QuentinTarantino

StanleyKubrick

WoodyAllen

DavidLynch

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 21 / 43

Definition of Map

Complete map of a graph

Definition

A map M = (VM ,EM) of G is complete iff x � y in G implies x � y inM.

MartinScorsese

JohnFord

OrsonWelles

QuentinTarantino

StanleyKubrick

TimBurton

WoodyAllen

Lars vonTrier

DavidLynch

PeterJackson

TerryGilliam

DavidFincher

GregHarrison

Paul T.AndersonNicolas

Renf

MartinScorseseJohn

Ford

QuentinTarantino

StanleyKubrick

WoodyAllen

DavidLynch

MartinScorseseJohn

Ford

QuentinTarantino

StanleyKubrick

WoodyAllen

DavidLynch

MartinScorseseJohn

Ford

QuentinTarantino

StanleyKubrick

WoodyAllen

DavidLynch

MartinScorseseJohn

Ford

QuentinTarantino

StanleyKubrick

WoodyAllen

DavidLynch

MartinScorseseJohn

Ford

QuentinTarantino

StanleyKubrick

WoodyAllen

DavidLynch

MartinScorseseJohn

Ford

QuentinTarantino

StanleyKubrick

WoodyAllen

DavidLynch

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 21 / 43

Definition of Map

Complete map of a graph

Definition

A map M = (VM ,EM) of G is complete iff x � y in G implies x � y inM.

MartinScorsese

JohnFord

OrsonWelles

QuentinTarantino

StanleyKubrick

TimBurton

WoodyAllen

Lars vonTrier

DavidLynch

PeterJackson

TerryGilliam

DavidFincher

GregHarrison

Paul T.AndersonNicolas

RenfMartin

ScorseseJohnFord

QuentinTarantino

StanleyKubrick

WoodyAllen

DavidLynch

MartinScorseseJohn

Ford

QuentinTarantino

StanleyKubrick

WoodyAllen

DavidLynch

MartinScorseseJohn

Ford

QuentinTarantino

StanleyKubrick

WoodyAllen

DavidLynch

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 22 / 43

Definition of Map

Complete map of a graph

Completeness is not always enough to summarize information via maps

MartinScorsese

JohnFord

OrsonWelles

QuentinTarantino

StanleyKubrick

TimBurton

WoodyAllen

Lars vonTrier

DavidLynch

PeterJackson

TerryGilliam

DavidFincher

GregHarrison

Paul T.AndersonNicolas

Renf

MartinScorseseJohn

Ford

QuentinTarantino

StanleyKubrick

WoodyAllen

DavidLynch

MartinScorseseJohn

Ford

QuentinTarantino

StanleyKubrick

WoodyAllen

DavidLynch

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 23 / 43

Definition of Map

Complete map of a graph

Completeness is not always enough to summarize information via maps

MartinScorseseJohn

Ford

QuentinTarantino

StanleyKubrick

WoodyAllen

DavidLynch

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 24 / 43

Definition of Map

Complete map of a graph

Completeness is not always enough to summarize information via maps

MartinScorseseJohn

Ford

QuentinTarantino

StanleyKubrick

WoodyAllen

DavidLynch

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 25 / 43

Definition of Map

Complete map of a graph

Completeness is not always enough to summarize information via maps

MartinScorsese

JohnFord

OrsonWelles

QuentinTarantino

StanleyKubrick

TimBurton

WoodyAllen

Lars vonTrier

DavidLynch

PeterJackson

TerryGilliam

DavidFincher

GregHarrison

Paul T.AndersonNicolas

Renf

Redundant: paths from JohnFord to Quentin Tarantino in Gonly pass for some distinguished

nodes

MartinScorseseJohn

Ford

QuentinTarantino

StanleyKubrick

WoodyAllen

DavidLynch

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 26 / 43

Definition of Map

Route-Complete map of a graph

Definition

A map M = (VM ,EM) of G is route-complete iff x �VMy in G implies

x → y in M

MartinScorsese

JohnFord

OrsonWelles

QuentinTarantino

StanleyKubrick

TimBurton

WoodyAllen

Lars vonTrier

DavidLynch

PeterJackson

TerryGilliam

DavidFincher

GregHarrison

Paul T.AndersonNicolas

Renf

MartinScorseseJohn

Ford

QuentinTarantino

StanleyKubrick

WoodyAllen

DavidLynch

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 27 / 43

Definition of Map

Route-Complete map of a graph

Definition

A map M = (VM ,EM) of G is route-complete iff x �VMy in G implies

x → y in M

MartinScorsese

JohnFord

OrsonWelles

QuentinTarantino

StanleyKubrick

TimBurton

WoodyAllen

Lars vonTrier

DavidLynch

PeterJackson

TerryGilliam

DavidFincher

GregHarrison

Paul T.AndersonNicolas

Renf

MartinScorseseJohn

Ford

QuentinTarantino

StanleyKubrick

WoodyAllen

DavidLynch

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 27 / 43

Definition of Map

Non-Redundant map of a graph

Definition

A map M = (VM ,EM) of G is non-redundant iff x → y in M impliesx �VM

y in G .

MartinScorsese

JohnFord

OrsonWelles

QuentinTarantino

StanleyKubrick

TimBurton

WoodyAllen

Lars vonTrier

DavidLynch

PeterJackson

TerryGilliam

DavidFincher

GregHarrison

Paul T.AndersonNicolas

Renf

MartinScorseseJohn

Ford

QuentinTarantino

StanleyKubrick

WoodyAllen

DavidLynch

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 28 / 43

Definition of Map

Non-Redundant map of a graph

Definition

A map M = (VM ,EM) of G is non-redundant iff x → y in M impliesx �VM

y in G .

MartinScorsese

JohnFord

OrsonWelles

QuentinTarantino

StanleyKubrick

TimBurton

WoodyAllen

Lars vonTrier

DavidLynch

PeterJackson

TerryGilliam

DavidFincher

GregHarrison

Paul T.AndersonNicolas

Renf

MartinScorseseJohn

Ford

QuentinTarantino

StanleyKubrick

WoodyAllen

DavidLynch

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 28 / 43

Definition of Map

Good map of a graph

Definition

A map M = (VM ,EM) of G is good iff it is:

complete (x � y in G implies x � y in M)

route-complete (x �VMy in G implies x → y in M)

non-redundant (x → y in M implies x �VMy in G )

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 29 / 43

Definition of Map

Good map of a graph

Lemma

A map M = (VM ,EM) of G is good iff x → y in M ⇔ x�VMy in G ,

∀x , y ∈ VM

MartinScorsese

JohnFord

OrsonWelles

QuentinTarantino

StanleyKubrick

TimBurton

WoodyAllen

Lars vonTrier

DavidLynch

PeterJackson

TerryGilliam

DavidFincher

GregHarrison

Paul T.AndersonNicolas

Renf

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 30 / 43

Definition of Map

Good map of a graph

Lemma

A map M = (VM ,EM) of G is good iff x → y in M ⇔ x�VMy in G ,

∀x , y ∈ VM

MartinScorsese

JohnFord

OrsonWelles

QuentinTarantino

StanleyKubrick

TimBurton

WoodyAllen

Lars vonTrier

DavidLynch

PeterJackson

TerryGilliam

DavidFincher

GregHarrison

Paul T.AndersonNicolas

Renf

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 30 / 43

Definition of Map

Good map of a graph

Lemma

A map M = (VM ,EM) of G is good iff x → y in M ⇔ x�VMy in G ,

∀x , y ∈ VM

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 31 / 43

Definition of Map

Good map of a graph

Lemma

A map M = (VM ,EM) of G is good iff x → y in M ⇔ x�VMy in G ,

∀x , y ∈ VM

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 32 / 43

Definition of Map

Good map of a graph

Theorem

Let G = (VG ,EG ) be a graph. Given N ⊆ V , there is a unique good mapM over G.

(Sketch).

Existence and uniqueness follow from the previous lemma. The edgex → y in M is defined by the existence of a particular path in G .

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 33 / 43

Definition of Map

Good map of a graph

Theorem

Let G = (VG ,EG ) be a graph. Given N ⊆ V , there is a unique good mapM over G.

(Sketch).

Existence and uniqueness follow from the previous lemma. The edgex → y in M is defined by the existence of a particular path in G .

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 33 / 43

Definition of Map

Computing good maps of a graph

MartinScorsese

JohnFord

OrsonWelles

QuentinTarantino

StanleyKubrick

TimBurton

WoodyAllen

Lars vonTrier

DavidLynch

PeterJackson

TerryGilliam

DavidFincher

GregHarrison

Paul T.AndersonNicolas

Renf

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 34 / 43

Definition of Map

Computing good maps of a graph

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 35 / 43

Definition of Map

Computing good maps of a graph

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 36 / 43

Maps as mathematical objects

Outline

1 Introduction

2 Contribution

3 The notion of map

4 Related research

5 Preliminaries

6 Definition of Map

7 Maps as mathematical objects

8 Regions and maps on the Web

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 37 / 43

Maps as mathematical objects

Algebra of maps

Theorem

Let G = (VG ,EG ) be a graph and M(G ) the set of all good maps over G.Mi = (VMi

,EMi) ∈M(G ) is a map. Then:

1 The binary relation v over M(G ), defined by M1 v M2 iffVM1 ⊆ VM2 , is a partial order on M(G ).

2 The order v induces a Boolean algebra (M(G ),t,u,G , ∅), where:M1 tM2 is the unique good map of G over VM1 ∪ VM2 ; M1 uM2 isthe unique good map of G over VM1 ∩ VM2 .

3 There is an isomorphism of Boolean algebras from the algebra of sets(P(V ),∪,∩,V , ∅) to (M(G ),t,u,G , ∅), given by N � MN (theunique good map of N over G ).

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 38 / 43

Regions and maps on the Web

Outline

1 Introduction

2 Contribution

3 The notion of map

4 Related research

5 Preliminaries

6 Definition of Map

7 Maps as mathematical objects

8 Regions and maps on the Web

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 39 / 43

Regions and maps on the Web

Navigational language to specify regions

Navigational languages [5,6,7] declaratively specify nodes in a graphor sub-graph (possibly the Web).

No information about node connection is provided and then they arenot suitable for building maps.

We defined a general navigational language to deal with subgraphsbesides sets of nodes.

[5] W. W. W. Consortium. XML Path Language (Xpath) Recommendation., Nov. 1999.[6] V. Fionda, C. Gutierrez, and G. Pirro. Semantic Navigation on the Web of Data:Specification of Routes, Web Fragments and Actions. In WWW, pages 281290. ACM, 2012.[7] F. Alkhateeb, J.-F. Baget, and J. Euzenat. Extending SPARQL with Regular ExpressionPatterns (for querying RDF). JWS, 7(2):5773, 2009.

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 40 / 43

Regions and maps on the Web

Our framework

The high level specification for building maps of the Web is:

1 Specify the resources of interest (distinguished nodes): We leverage anavigational language.

2 Build the region R corresponding to this specification: We enhancedthe semantics of our navigational language to return subgraphsbesides sets of pairs of nodes.

3 Build a formal map corresponding to the region R: We build mapsfrom regions by using the map framework discussed in the talk.

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 41 / 43

Regions and maps on the Web

Our framework

The high level specification for building maps of the Web is:

1 Specify the resources of interest (distinguished nodes): We leverage anavigational language.

2 Build the region R corresponding to this specification: We enhancedthe semantics of our navigational language to return subgraphsbesides sets of pairs of nodes.

3 Build a formal map corresponding to the region R: We build mapsfrom regions by using the map framework discussed in the talk.

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 41 / 43

Regions and maps on the Web

Our framework

The high level specification for building maps of the Web is:

1 Specify the resources of interest (distinguished nodes): We leverage anavigational language.

2 Build the region R corresponding to this specification: We enhancedthe semantics of our navigational language to return subgraphsbesides sets of pairs of nodes.

3 Build a formal map corresponding to the region R: We build mapsfrom regions by using the map framework discussed in the talk.

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 41 / 43

Regions and maps on the Web

The implemented system

The map framework has been implemented in a tool, which can bedownloaded at the address http://mapsforweb.wordpress.com.

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 42 / 43

THANK YOU

V. Fionda, C. Gutierrez, G. Pirro Formal Maps and their Algebra 43 / 43