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Formal Maps and their Algebra

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Valeria Fionda
Valeria Fionda

Slides related to the papers: - Fionda V., Gutierrez C., Pirrò G. "Knowledge Maps of Web Graphs". n the proceedings of the “14th International Conference on Principles of Knowledge Representation and Reasoning (KR2014)”. - Fionda V., Pirrò G., Gutierrez C. "The Map Generator Tool". Demo session at the “13th International Semantic Web Conference (ISWC2014)”.

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Formal Maps and their Algebra Valeria Fionda, Claudio Gutierrez Giuseppe Pirr´o V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 1 / 43
Outline 1 Introduction V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 2 / 43
Outline 1 Introduction 2 Contribution V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 2 / 43
Outline 1 Introduction 2 Contribution 3 The notion of map V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 2 / 43
Outline 1 Introduction 2 Contribution 3 The notion of map 4 Related research V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 2 / 43
Outline 1 Introduction 2 Contribution 3 The notion of map 4 Related research 5 Preliminaries V. Fionda, C. Gutierrez, G. Pirr´o 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 V. Fionda, C. Gutierrez, G. Pirr´o 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 V. Fionda, C. Gutierrez, G. Pirr´o 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. Pirr´o 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. Pirr´o 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. Pirr´o 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. Pirr´o 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. Pirr´o Formal Maps and their Algebra 4 / 43
Introduction Goals Our broad goal is to investigate how cartographical principles can be applied over the Web space V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 5 / 43
Introduction Goals Our broad goal is to investigate how cartographical principles can be applied over the Web space The web is a graph In particular we are interested in the Web of linked data, that is a huge, distributed, directed, labeled multigraph V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 5 / 43
Introduction Goals Our broad goal is to investigate how cartographical principles can be applied over the Web space The web is a graph In particular we are interested in the Web of linked data, that is a huge, distributed, directed, labeled multigraph Specific goals: 1) We want to build maps of a graph 2) Algebra of maps V. Fionda, C. Gutierrez, G. Pirr´o 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. Pirr´o 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. V. Fionda, C. Gutierrez, G. Pirr´o 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. V. Fionda, C. Gutierrez, G. Pirr´o 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. V. Fionda, C. Gutierrez, G. Pirr´o 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 graphs 2 Obtaining maps from those regions V. Fionda, C. Gutierrez, G. Pirr´o 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 graphs 2 Obtaining maps from those regions V. Fionda, C. Gutierrez, G. Pirr´o 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 graphs 2 Obtaining maps from those regions V. Fionda, C. Gutierrez, G. Pirr´o 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 graphs 2 Obtaining maps from those regions V. Fionda, C. Gutierrez, G. Pirr´o 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. Pirr´o Formal Maps and their Algebra 8 / 43
The notion of map A region Martin Scorsese John Ford Orson Welles Quentin Tarantino Stanley Kubrick Tim Burton Woody Allen Lars von Trier David Lynch Influeces between directors Peter Jackson Terry Gilliam David Fincher Greg Harrison Paul T. Anderson Nicolas Renf V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 9 / 43
The notion of map Distinguished nodes Martin Scorsese John Ford Orson Welles Quentin Tarantino Stanley Kubrick Tim Burton Woody Allen Lars von Trier David Lynch Influeces between directors Peter Jackson Terry Gilliam David Fincher Greg Harrison Paul T. Anderson Nicolas Renf V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 10 / 43
The notion of map Examples of maps Martin Scorsese John Ford Orson Welles Quentin Tarantino Stanley Kubrick Tim Burton Woody Allen Lars von Trier David Lynch Influeces between directors Peter Jackson Terry Gilliam David Fincher Greg Harrison Paul T. Anderson Nicolas Renf Martin Scorsese John Ford Quentin Tarantino Stanley Kubrick Woody Allen John Ford Quentin Tarantino Stanley Kubrick Woody Allen Map 2Map 1 e1 e2 V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 11 / 43
The notion of map Abstraction level A map should: Represent the region Be concise Keep the connectivity V. Fionda, C. Gutierrez, G. Pirr´o 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. Pirr´o 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. Pirr´o 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 space consumption by usually clustering nodes in partitions. [1] C. Faloutsos, K.S. McCurley, and A. Tomkins. Fast Discovery of Connection Subgraphs. In KDD, pages 118-127. ACM, 2004. [2] J. Adibi, H. Chalupsky, E. Melz, A. Valente, et al. The KOJAK Group Finder: Connecting the 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 of Connectivity. In ICDM, pages 659-668. IEEE, 2010. V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 14 / 43
Related research Graph indexing Graph indexing [4]: Goal: produce a list of graph substructures with references to the place 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. Pirr´o Formal Maps and their Algebra 15 / 43
Related research Differences 1 These techniques work on the whole graphs, they do not provide means to specify regions. Benefit: the web cannot be mapped altogheter! Isolated portions should be considered (according to user interests) V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 16 / 43
Related research Differences 1 These techniques work on the whole graphs, they do not provide means to specify regions. Benefit: the web cannot be mapped altogheter! Isolated portions should 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. V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 16 / 43
Related research Differences 1 These techniques work on the whole graphs, they do not provide means to specify regions. Benefit: the web cannot be mapped altogheter! Isolated portions should 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 as mathematical objects by defining an algebra over maps. V. Fionda, C. Gutierrez, G. Pirr´o 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. Pirr´o 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 of edges and u, v nodes in G and N ⊆ VG be a set of nodes: Martin Scorsese John Ford Orson Welles Quentin Tarantino Stanley Kubrick Tim Burton Woody Allen Lars von Trier David Lynch Influeces between directors Peter Jackson Terry Gilliam David Fincher Greg Harrison Paul T. Anderson Nicolas Renf N={Martin Scorsese, Woody Allen} V. Fionda, C. Gutierrez, G. Pirr´o 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 of edges and u, v nodes in G and N ⊆ VG be a set of nodes: u → v denotes an edge (u, v) ∈ EG Martin Scorsese John Ford Orson Welles Quentin Tarantino Stanley Kubrick Tim Burton Woody Allen Lars von Trier David Lynch Influeces between directors Peter Jackson Terry Gilliam David Fincher Greg Harrison Paul T. Anderson Nicolas Renf John Ford → Martin Scorsese N={Martin Scorsese, Woody Allen} V. Fionda, C. Gutierrez, G. Pirr´o 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 of edges 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 Martin Scorsese John Ford Orson Welles Quentin Tarantino Stanley Kubrick Tim Burton Woody Allen Lars von Trier David Lynch Influeces between directors Peter Jackson Terry Gilliam David Fincher Greg Harrison Paul T. Anderson Nicolas Renf John Ford → Martin Scorsese John Ford Stanley Kubrick N={Martin Scorsese, Woody Allen} V. Fionda, C. Gutierrez, G. Pirr´o 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 of edges 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 through intermediate nodes in N Martin Scorsese John Ford Orson Welles Quentin Tarantino Stanley Kubrick Tim Burton Woody Allen Lars von Trier David Lynch Influeces between directors Peter Jackson Terry Gilliam David Fincher Greg Harrison Paul T. Anderson Nicolas Renf John Ford → Martin Scorsese John Ford Stanley Kubrick N={Martin Scorsese, Woody Allen} John Ford N Stanley Kubrick NOTJohn Ford N Paul T. Anderson V. Fionda, C. Gutierrez, G. Pirr´o 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. Pirr´o 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 V. Fionda, C. Gutierrez, G. Pirr´o 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 Martin Scorsese John Ford Orson Welles Quentin Tarantino Stanley Kubrick Tim Burton Woody Allen Lars von Trier David Lynch Peter Jackson Terry Gilliam David Fincher Greg Harrison Paul T. Anderson Nicolas Renf Martin ScorseseJohn Ford Quentin Tarantino Stanley Kubrick Woody Allen David Lynch Martin ScorseseJohn Ford Quentin Tarantino Stanley Kubrick Woody Allen David Lynch Martin ScorseseJohn Ford Quentin Tarantino Stanley Kubrick Woody Allen David Lynch Martin ScorseseJohn Ford Quentin Tarantino Stanley Kubrick Woody Allen David Lynch Martin ScorseseJohn Ford Quentin Tarantino Stanley Kubrick Woody Allen David Lynch Martin ScorseseJohn Ford Quentin Tarantino Stanley Kubrick Woody Allen David Lynch V. Fionda, C. Gutierrez, G. Pirr´o 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 in M. V. Fionda, C. Gutierrez, G. Pirr´o 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 in M. Martin Scorsese John Ford Orson Welles Quentin Tarantino Stanley Kubrick Tim Burton Woody Allen Lars von Trier David Lynch Peter Jackson Terry Gilliam David Fincher Greg Harrison Paul T. Anderson Nicolas Renf Martin ScorseseJohn Ford Quentin Tarantino Stanley Kubrick Woody Allen David Lynch Martin ScorseseJohn Ford Quentin Tarantino Stanley Kubrick Woody Allen David Lynch Martin ScorseseJohn Ford Quentin Tarantino Stanley Kubrick Woody Allen David Lynch Martin ScorseseJohn Ford Quentin Tarantino Stanley Kubrick Woody Allen David Lynch Martin ScorseseJohn Ford Quentin Tarantino Stanley Kubrick Woody Allen David Lynch Martin ScorseseJohn Ford Quentin Tarantino Stanley Kubrick Woody Allen David Lynch V. Fionda, C. Gutierrez, G. Pirr´o 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 in M. Martin Scorsese John Ford Orson Welles Quentin Tarantino Stanley Kubrick Tim Burton Woody Allen Lars von Trier David Lynch Peter Jackson Terry Gilliam David Fincher Greg Harrison Paul T. Anderson Nicolas Renf Martin ScorseseJohn Ford Quentin Tarantino Stanley Kubrick Woody Allen David Lynch Martin ScorseseJohn Ford Quentin Tarantino Stanley Kubrick Woody Allen David Lynch Martin ScorseseJohn Ford Quentin Tarantino Stanley Kubrick Woody Allen David Lynch V. Fionda, C. Gutierrez, G. Pirr´o 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 Martin Scorsese John Ford Orson Welles Quentin Tarantino Stanley Kubrick Tim Burton Woody Allen Lars von Trier David Lynch Peter Jackson Terry Gilliam David Fincher Greg Harrison Paul T. Anderson Nicolas Renf Martin ScorseseJohn Ford Quentin Tarantino Stanley Kubrick Woody Allen David Lynch Martin ScorseseJohn Ford Quentin Tarantino Stanley Kubrick Woody Allen David Lynch V. Fionda, C. Gutierrez, G. Pirr´o 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 Martin ScorseseJohn Ford Quentin Tarantino Stanley Kubrick Woody Allen David Lynch V. Fionda, C. Gutierrez, G. Pirr´o 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 Martin ScorseseJohn Ford Quentin Tarantino Stanley Kubrick Woody Allen David Lynch V. Fionda, C. Gutierrez, G. Pirr´o 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 Martin Scorsese John Ford Orson Welles Quentin Tarantino Stanley Kubrick Tim Burton Woody Allen Lars von Trier David Lynch Peter Jackson Terry Gilliam David Fincher Greg Harrison Paul T. Anderson Nicolas Renf Redundant: paths from John Ford to Quentin Tarantino in G only pass for some distinguished nodes Martin ScorseseJohn Ford Quentin Tarantino Stanley Kubrick Woody Allen David Lynch V. Fionda, C. Gutierrez, G. Pirr´o 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 VM y in G implies x → y in M V. Fionda, C. Gutierrez, G. Pirr´o 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 VM y in G implies x → y in M Martin Scorsese John Ford Orson Welles Quentin Tarantino Stanley Kubrick Tim Burton Woody Allen Lars von Trier David Lynch Peter Jackson Terry Gilliam David Fincher Greg Harrison Paul T. Anderson Nicolas Renf Martin ScorseseJohn Ford Quentin Tarantino Stanley Kubrick Woody Allen David Lynch V. Fionda, C. Gutierrez, G. Pirr´o 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 implies x VM y in G. V. Fionda, C. Gutierrez, G. Pirr´o 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 implies x VM y in G. Martin Scorsese John Ford Orson Welles Quentin Tarantino Stanley Kubrick Tim Burton Woody Allen Lars von Trier David Lynch Peter Jackson Terry Gilliam David Fincher Greg Harrison Paul T. Anderson Nicolas Renf Martin ScorseseJohn Ford Quentin Tarantino Stanley Kubrick Woody Allen David Lynch V. Fionda, C. Gutierrez, G. Pirr´o 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 VM y in G implies x → y in M) non-redundant (x → y in M implies x VM y in G) V. Fionda, C. Gutierrez, G. Pirr´o 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 VM y in G, ∀x, y ∈ VM V. Fionda, C. Gutierrez, G. Pirr´o 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 VM y in G, ∀x, y ∈ VM Martin Scorsese John Ford Orson Welles Quentin Tarantino Stanley Kubrick Tim Burton Woody Allen Lars von Trier David Lynch Peter Jackson Terry Gilliam David Fincher Greg Harrison Paul T. Anderson Nicolas Renf V. Fionda, C. Gutierrez, G. Pirr´o 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 VM y in G, ∀x, y ∈ VM V. Fionda, C. Gutierrez, G. Pirr´o 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 VM y in G, ∀x, y ∈ VM V. Fionda, C. Gutierrez, G. Pirr´o 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 map M over G. V. Fionda, C. Gutierrez, G. Pirr´o 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 map M over G. (Sketch). Existence and uniqueness follow from the previous lemma. The edge x → y in M is defined by the existence of a particular path in G. V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 33 / 43
Definition of Map Computing good maps of a graph Martin Scorsese John Ford Orson Welles Quentin Tarantino Stanley Kubrick Tim Burton Woody Allen Lars von Trier David Lynch Peter Jackson Terry Gilliam David Fincher Greg Harrison Paul T. Anderson Nicolas Renf V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 34 / 43
Definition of Map Computing good maps of a graph V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 35 / 43
Definition of Map Computing good maps of a graph V. Fionda, C. Gutierrez, G. Pirr´o 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. Pirr´o 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 over M(G), defined by M1 M2 iff VM1 ⊆ VM2 , is a partial order on M(G). 2 The order induces a Boolean algebra (M(G), , , G, ∅), where: M1 M2 is the unique good map of G over VM1 ∪ VM2 ; M1 M2 is the 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), , , G, ∅), given by N MN (the unique good map of N over G). V. Fionda, C. Gutierrez, G. Pirr´o 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. Pirr´o 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 graph or sub-graph (possibly the Web). No information about node connection is provided and then they are not suitable for building maps. We defined a general navigational language to deal with subgraphs besides sets of nodes. [5] W. W. W. Consortium. XML Path Language (Xpath) Recommendation., Nov. 1999. [6] V. Fionda, C. Gutierrez, and G. Pirr´o. 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 Expression Patterns (for querying RDF). JWS, 7(2):5773, 2009. V. Fionda, C. Gutierrez, G. Pirr´o 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 a navigational language. V. Fionda, C. Gutierrez, G. Pirr´o 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 a navigational language. 2 Build the region R corresponding to this specification: We enhanced the semantics of our navigational language to return subgraphs besides sets of pairs of nodes. V. Fionda, C. Gutierrez, G. Pirr´o 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 a navigational language. 2 Build the region R corresponding to this specification: We enhanced the semantics of our navigational language to return subgraphs besides sets of pairs of nodes. 3 Build a formal map corresponding to the region R: We build maps from regions by using the map framework discussed in the talk. V. Fionda, C. Gutierrez, G. Pirr´o 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 be downloaded at the address http://mapsforweb.wordpress.com. V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 42 / 43
THANK YOU V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 43 / 43

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Formal Maps and their Algebra

  • 1. Formal Maps and their Algebra Valeria Fionda, Claudio Gutierrez Giuseppe Pirr´o V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 1 / 43
  • 2. Outline 1 Introduction V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 2 / 43
  • 3. Outline 1 Introduction 2 Contribution V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 2 / 43
  • 4. Outline 1 Introduction 2 Contribution 3 The notion of map V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 2 / 43
  • 5. Outline 1 Introduction 2 Contribution 3 The notion of map 4 Related research V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 2 / 43
  • 6. Outline 1 Introduction 2 Contribution 3 The notion of map 4 Related research 5 Preliminaries V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 2 / 43
  • 7. Outline 1 Introduction 2 Contribution 3 The notion of map 4 Related research 5 Preliminaries 6 Definition of Map V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 2 / 43
  • 8. Outline 1 Introduction 2 Contribution 3 The notion of map 4 Related research 5 Preliminaries 6 Definition of Map 7 Maps as mathematical objects V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 2 / 43
  • 9. 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. Pirr´o Formal Maps and their Algebra 2 / 43
  • 10. 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. Pirr´o Formal Maps and their Algebra 3 / 43
  • 11. Introduction What is a map? Maps are artifacts that orient users in information spaces V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 4 / 43
  • 12. Introduction What is a map? Maps are artifacts that orient users in information spaces V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 4 / 43
  • 13. Introduction What is a map? Maps are artifacts that orient users in information spaces V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 4 / 43
  • 14. Introduction Goals Our broad goal is to investigate how cartographical principles can be applied over the Web space V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 5 / 43
  • 15. Introduction Goals Our broad goal is to investigate how cartographical principles can be applied over the Web space The web is a graph In particular we are interested in the Web of linked data, that is a huge, distributed, directed, labeled multigraph V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 5 / 43
  • 16. Introduction Goals Our broad goal is to investigate how cartographical principles can be applied over the Web space The web is a graph In particular we are interested in the Web of linked data, that is a huge, distributed, directed, labeled multigraph Specific goals: 1) We want to build maps of a graph 2) Algebra of maps V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 5 / 43
  • 17. 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. Pirr´o Formal Maps and their Algebra 6 / 43
  • 18. 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. V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 7 / 43
  • 19. 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. V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 7 / 43
  • 20. 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. V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 7 / 43
  • 21. 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 graphs 2 Obtaining maps from those regions V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 7 / 43
  • 22. 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 graphs 2 Obtaining maps from those regions V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 7 / 43
  • 23. 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 graphs 2 Obtaining maps from those regions V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 7 / 43
  • 24. 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 graphs 2 Obtaining maps from those regions V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 7 / 43
  • 25. 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. Pirr´o Formal Maps and their Algebra 8 / 43
  • 26. The notion of map A region Martin Scorsese John Ford Orson Welles Quentin Tarantino Stanley Kubrick Tim Burton Woody Allen Lars von Trier David Lynch Influeces between directors Peter Jackson Terry Gilliam David Fincher Greg Harrison Paul T. Anderson Nicolas Renf V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 9 / 43
  • 27. The notion of map Distinguished nodes Martin Scorsese John Ford Orson Welles Quentin Tarantino Stanley Kubrick Tim Burton Woody Allen Lars von Trier David Lynch Influeces between directors Peter Jackson Terry Gilliam David Fincher Greg Harrison Paul T. Anderson Nicolas Renf V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 10 / 43
  • 28. The notion of map Examples of maps Martin Scorsese John Ford Orson Welles Quentin Tarantino Stanley Kubrick Tim Burton Woody Allen Lars von Trier David Lynch Influeces between directors Peter Jackson Terry Gilliam David Fincher Greg Harrison Paul T. Anderson Nicolas Renf Martin Scorsese John Ford Quentin Tarantino Stanley Kubrick Woody Allen John Ford Quentin Tarantino Stanley Kubrick Woody Allen Map 2Map 1 e1 e2 V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 11 / 43
  • 29. The notion of map Abstraction level A map should: Represent the region Be concise Keep the connectivity V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 12 / 43
  • 30. 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. Pirr´o Formal Maps and their Algebra 12 / 43
  • 31. 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. Pirr´o Formal Maps and their Algebra 13 / 43
  • 32. 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 space consumption by usually clustering nodes in partitions. [1] C. Faloutsos, K.S. McCurley, and A. Tomkins. Fast Discovery of Connection Subgraphs. In KDD, pages 118-127. ACM, 2004. [2] J. Adibi, H. Chalupsky, E. Melz, A. Valente, et al. The KOJAK Group Finder: Connecting the 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 of Connectivity. In ICDM, pages 659-668. IEEE, 2010. V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 14 / 43
  • 33. Related research Graph indexing Graph indexing [4]: Goal: produce a list of graph substructures with references to the place 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. Pirr´o Formal Maps and their Algebra 15 / 43
  • 34. Related research Differences 1 These techniques work on the whole graphs, they do not provide means to specify regions. Benefit: the web cannot be mapped altogheter! Isolated portions should be considered (according to user interests) V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 16 / 43
  • 35. Related research Differences 1 These techniques work on the whole graphs, they do not provide means to specify regions. Benefit: the web cannot be mapped altogheter! Isolated portions should 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. V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 16 / 43
  • 36. Related research Differences 1 These techniques work on the whole graphs, they do not provide means to specify regions. Benefit: the web cannot be mapped altogheter! Isolated portions should 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 as mathematical objects by defining an algebra over maps. V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 16 / 43
  • 37. 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. Pirr´o Formal Maps and their Algebra 17 / 43
  • 38. Preliminaries Notation Let G = (VG , EG ) be a directed graph, VG the set of nodes, EG the set of edges and u, v nodes in G and N ⊆ VG be a set of nodes: Martin Scorsese John Ford Orson Welles Quentin Tarantino Stanley Kubrick Tim Burton Woody Allen Lars von Trier David Lynch Influeces between directors Peter Jackson Terry Gilliam David Fincher Greg Harrison Paul T. Anderson Nicolas Renf N={Martin Scorsese, Woody Allen} V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 18 / 43
  • 39. Preliminaries Notation Let G = (VG , EG ) be a directed graph, VG the set of nodes, EG the set of edges and u, v nodes in G and N ⊆ VG be a set of nodes: u → v denotes an edge (u, v) ∈ EG Martin Scorsese John Ford Orson Welles Quentin Tarantino Stanley Kubrick Tim Burton Woody Allen Lars von Trier David Lynch Influeces between directors Peter Jackson Terry Gilliam David Fincher Greg Harrison Paul T. Anderson Nicolas Renf John Ford → Martin Scorsese N={Martin Scorsese, Woody Allen} V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 18 / 43
  • 40. Preliminaries Notation Let G = (VG , EG ) be a directed graph, VG the set of nodes, EG the set of edges 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 Martin Scorsese John Ford Orson Welles Quentin Tarantino Stanley Kubrick Tim Burton Woody Allen Lars von Trier David Lynch Influeces between directors Peter Jackson Terry Gilliam David Fincher Greg Harrison Paul T. Anderson Nicolas Renf John Ford → Martin Scorsese John Ford Stanley Kubrick N={Martin Scorsese, Woody Allen} V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 18 / 43
  • 41. Preliminaries Notation Let G = (VG , EG ) be a directed graph, VG the set of nodes, EG the set of edges 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 through intermediate nodes in N Martin Scorsese John Ford Orson Welles Quentin Tarantino Stanley Kubrick Tim Burton Woody Allen Lars von Trier David Lynch Influeces between directors Peter Jackson Terry Gilliam David Fincher Greg Harrison Paul T. Anderson Nicolas Renf John Ford → Martin Scorsese John Ford Stanley Kubrick N={Martin Scorsese, Woody Allen} John Ford N Stanley Kubrick NOTJohn Ford N Paul T. Anderson V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 18 / 43
  • 42. 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. Pirr´o Formal Maps and their Algebra 19 / 43
  • 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 V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 20 / 43
  • 44. 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 Martin Scorsese John Ford Orson Welles Quentin Tarantino Stanley Kubrick Tim Burton Woody Allen Lars von Trier David Lynch Peter Jackson Terry Gilliam David Fincher Greg Harrison Paul T. Anderson Nicolas Renf Martin ScorseseJohn Ford Quentin Tarantino Stanley Kubrick Woody Allen David Lynch Martin ScorseseJohn Ford Quentin Tarantino Stanley Kubrick Woody Allen David Lynch Martin ScorseseJohn Ford Quentin Tarantino Stanley Kubrick Woody Allen David Lynch Martin ScorseseJohn Ford Quentin Tarantino Stanley Kubrick Woody Allen David Lynch Martin ScorseseJohn Ford Quentin Tarantino Stanley Kubrick Woody Allen David Lynch Martin ScorseseJohn Ford Quentin Tarantino Stanley Kubrick Woody Allen David Lynch V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 20 / 43
  • 45. 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 in M. V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 21 / 43
  • 46. 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 in M. Martin Scorsese John Ford Orson Welles Quentin Tarantino Stanley Kubrick Tim Burton Woody Allen Lars von Trier David Lynch Peter Jackson Terry Gilliam David Fincher Greg Harrison Paul T. Anderson Nicolas Renf Martin ScorseseJohn Ford Quentin Tarantino Stanley Kubrick Woody Allen David Lynch Martin ScorseseJohn Ford Quentin Tarantino Stanley Kubrick Woody Allen David Lynch Martin ScorseseJohn Ford Quentin Tarantino Stanley Kubrick Woody Allen David Lynch Martin ScorseseJohn Ford Quentin Tarantino Stanley Kubrick Woody Allen David Lynch Martin ScorseseJohn Ford Quentin Tarantino Stanley Kubrick Woody Allen David Lynch Martin ScorseseJohn Ford Quentin Tarantino Stanley Kubrick Woody Allen David Lynch V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 21 / 43
  • 47. 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 in M. Martin Scorsese John Ford Orson Welles Quentin Tarantino Stanley Kubrick Tim Burton Woody Allen Lars von Trier David Lynch Peter Jackson Terry Gilliam David Fincher Greg Harrison Paul T. Anderson Nicolas Renf Martin ScorseseJohn Ford Quentin Tarantino Stanley Kubrick Woody Allen David Lynch Martin ScorseseJohn Ford Quentin Tarantino Stanley Kubrick Woody Allen David Lynch Martin ScorseseJohn Ford Quentin Tarantino Stanley Kubrick Woody Allen David Lynch V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 22 / 43
  • 48. Definition of Map Complete map of a graph Completeness is not always enough to summarize information via maps Martin Scorsese John Ford Orson Welles Quentin Tarantino Stanley Kubrick Tim Burton Woody Allen Lars von Trier David Lynch Peter Jackson Terry Gilliam David Fincher Greg Harrison Paul T. Anderson Nicolas Renf Martin ScorseseJohn Ford Quentin Tarantino Stanley Kubrick Woody Allen David Lynch Martin ScorseseJohn Ford Quentin Tarantino Stanley Kubrick Woody Allen David Lynch V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 23 / 43
  • 49. Definition of Map Complete map of a graph Completeness is not always enough to summarize information via maps Martin ScorseseJohn Ford Quentin Tarantino Stanley Kubrick Woody Allen David Lynch V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 24 / 43
  • 50. Definition of Map Complete map of a graph Completeness is not always enough to summarize information via maps Martin ScorseseJohn Ford Quentin Tarantino Stanley Kubrick Woody Allen David Lynch V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 25 / 43
  • 51. Definition of Map Complete map of a graph Completeness is not always enough to summarize information via maps Martin Scorsese John Ford Orson Welles Quentin Tarantino Stanley Kubrick Tim Burton Woody Allen Lars von Trier David Lynch Peter Jackson Terry Gilliam David Fincher Greg Harrison Paul T. Anderson Nicolas Renf Redundant: paths from John Ford to Quentin Tarantino in G only pass for some distinguished nodes Martin ScorseseJohn Ford Quentin Tarantino Stanley Kubrick Woody Allen David Lynch V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 26 / 43
  • 52. Definition of Map Route-Complete map of a graph Definition A map M = (VM, EM) of G is route-complete iff x VM y in G implies x → y in M V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 27 / 43
  • 53. Definition of Map Route-Complete map of a graph Definition A map M = (VM, EM) of G is route-complete iff x VM y in G implies x → y in M Martin Scorsese John Ford Orson Welles Quentin Tarantino Stanley Kubrick Tim Burton Woody Allen Lars von Trier David Lynch Peter Jackson Terry Gilliam David Fincher Greg Harrison Paul T. Anderson Nicolas Renf Martin ScorseseJohn Ford Quentin Tarantino Stanley Kubrick Woody Allen David Lynch V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 27 / 43
  • 54. 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 implies x VM y in G. V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 28 / 43
  • 55. 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 implies x VM y in G. Martin Scorsese John Ford Orson Welles Quentin Tarantino Stanley Kubrick Tim Burton Woody Allen Lars von Trier David Lynch Peter Jackson Terry Gilliam David Fincher Greg Harrison Paul T. Anderson Nicolas Renf Martin ScorseseJohn Ford Quentin Tarantino Stanley Kubrick Woody Allen David Lynch V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 28 / 43
  • 56. 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 VM y in G implies x → y in M) non-redundant (x → y in M implies x VM y in G) V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 29 / 43
  • 57. Definition of Map Good map of a graph Lemma A map M = (VM, EM) of G is good iff x → y in M ⇔ x VM y in G, ∀x, y ∈ VM V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 30 / 43
  • 58. Definition of Map Good map of a graph Lemma A map M = (VM, EM) of G is good iff x → y in M ⇔ x VM y in G, ∀x, y ∈ VM Martin Scorsese John Ford Orson Welles Quentin Tarantino Stanley Kubrick Tim Burton Woody Allen Lars von Trier David Lynch Peter Jackson Terry Gilliam David Fincher Greg Harrison Paul T. Anderson Nicolas Renf V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 30 / 43
  • 59. Definition of Map Good map of a graph Lemma A map M = (VM, EM) of G is good iff x → y in M ⇔ x VM y in G, ∀x, y ∈ VM V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 31 / 43
  • 60. Definition of Map Good map of a graph Lemma A map M = (VM, EM) of G is good iff x → y in M ⇔ x VM y in G, ∀x, y ∈ VM V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 32 / 43
  • 61. Definition of Map Good map of a graph Theorem Let G = (VG , EG ) be a graph. Given N ⊆ V , there is a unique good map M over G. V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 33 / 43
  • 62. Definition of Map Good map of a graph Theorem Let G = (VG , EG ) be a graph. Given N ⊆ V , there is a unique good map M over G. (Sketch). Existence and uniqueness follow from the previous lemma. The edge x → y in M is defined by the existence of a particular path in G. V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 33 / 43
  • 63. Definition of Map Computing good maps of a graph Martin Scorsese John Ford Orson Welles Quentin Tarantino Stanley Kubrick Tim Burton Woody Allen Lars von Trier David Lynch Peter Jackson Terry Gilliam David Fincher Greg Harrison Paul T. Anderson Nicolas Renf V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 34 / 43
  • 64. Definition of Map Computing good maps of a graph V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 35 / 43
  • 65. Definition of Map Computing good maps of a graph V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 36 / 43
  • 66. 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. Pirr´o Formal Maps and their Algebra 37 / 43
  • 67. 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 over M(G), defined by M1 M2 iff VM1 ⊆ VM2 , is a partial order on M(G). 2 The order induces a Boolean algebra (M(G), , , G, ∅), where: M1 M2 is the unique good map of G over VM1 ∪ VM2 ; M1 M2 is the 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), , , G, ∅), given by N MN (the unique good map of N over G). V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 38 / 43
  • 68. 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. Pirr´o Formal Maps and their Algebra 39 / 43
  • 69. Regions and maps on the Web Navigational language to specify regions Navigational languages [5,6,7] declaratively specify nodes in a graph or sub-graph (possibly the Web). No information about node connection is provided and then they are not suitable for building maps. We defined a general navigational language to deal with subgraphs besides sets of nodes. [5] W. W. W. Consortium. XML Path Language (Xpath) Recommendation., Nov. 1999. [6] V. Fionda, C. Gutierrez, and G. Pirr´o. 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 Expression Patterns (for querying RDF). JWS, 7(2):5773, 2009. V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 40 / 43
  • 70. 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 a navigational language. V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 41 / 43
  • 71. 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 a navigational language. 2 Build the region R corresponding to this specification: We enhanced the semantics of our navigational language to return subgraphs besides sets of pairs of nodes. V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 41 / 43
  • 72. 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 a navigational language. 2 Build the region R corresponding to this specification: We enhanced the semantics of our navigational language to return subgraphs besides sets of pairs of nodes. 3 Build a formal map corresponding to the region R: We build maps from regions by using the map framework discussed in the talk. V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 41 / 43
  • 73. Regions and maps on the Web The implemented system The map framework has been implemented in a tool, which can be downloaded at the address http://mapsforweb.wordpress.com. V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 42 / 43
  • 74. THANK YOU V. Fionda, C. Gutierrez, G. Pirr´o Formal Maps and their Algebra 43 / 43