Introduction to Graphs Inhalt Non-Standard Database Systems Graph Databases 1 Introduction to Graphs Nikolaus Augsten [email protected] FB Computerwissenschaften Universit¨at Salzburg 2 Property Graph Model 3 Graph Database Implementations http://dbresearch.uni-salzburg.at Sommersemester 2016 Version 9. Juni 2016 Augsten (Univ. Salzburg) NSDB – Graph Databases Sommersemester 2016 1 / 32 Augsten (Univ. Salzburg) NSDB – Graph Databases Sommersemester 2016 2 / 32 Introduction to Graphs Introduction to Graphs Storing Data in Graphs – Examples Graph Terms Name: Bob Age: 27 knows knows Name: Alice Name: Clare graph G = (V , E) Age: 34 dislikes Age: 29 V : set of nodes (node = vertex) c Lena Wiese: Advanced Data Management, DeGruyter, 2015. E : set of edges adjacent nodes (=neighbors) are connected with an edge City: Hannover City: Braunschweig an edge is incident to a node if it is connected to the node Population: 522K 65km Population: 248K 35km 45km City: Hildesheim Population: 102K c Lena Wiese: Advanced Data Management, DeGruyter, 2015. Augsten (Univ. Salzburg) NSDB – Graph Databases Sommersemester 2016 3 / 32 Augsten (Univ. Salzburg) NSDB – Graph Databases Sommersemester 2016 4 / 32 Introduction to Graphs Introduction to Graphs Different Types of Graphs Simple Undirected Graphs v2 e1 e2 simple undirected graph v1 v3 e3 simple directed graph c Lena Wiese: Advanced Data Management, DeGruyter, 2015. undirectred multi-graph directed multi-graph edges are (unordered) two-element subsets of V , e.g., weighted graphs v , v = v , v E { 1 3} { 3 1} ∈ n(n 1) complete graph: maximum of − edges for n = V nodes 2 | | (without self-loops) Augsten (Univ. Salzburg) NSDB – Graph Databases Sommersemester 2016 5 / 32 Augsten (Univ. Salzburg) NSDB – Graph Databases Sommersemester 2016 6 / 32 Introduction to Graphs Introduction to Graphs Simple Directed Graphs Multigraphs v2 a pair of nodes may be connected by multiple edges (in the same e1 e2 direction) v1 v3 e3 undirected multigraph c Lena Wiese: Advanced Data Management, DeGruyter, 2015. v2 e1 e2 v1 v3 e3 edges E V V are (ordered) two-element tuples of V , e.g., e4 c Lena Wiese: Advanced Data Management, DeGruyter, 2015. ⊆ × (v1, v3) E, (v3, v1) / E ∈ ∈ directed multigraph source/tail node of an edge: outgoing (e.g., v1 in (v1, v3)) v2 e1 e2 target/head node of an edge: incoming (e.g., v3 in (v1, v3)) v1 v3 e3 complete graph: maximum of n(n 1) edges for n = V nodes e4 − | | c Lena Wiese: Advanced Data Management, DeGruyter, 2015. (without self-loops) Augsten (Univ. Salzburg) NSDB – Graph Databases Sommersemester 2016 7 / 32 Augsten (Univ. Salzburg) NSDB – Graph Databases Sommersemester 2016 8 / 32 Introduction to Graphs Introduction to Graphs Weighted Graph Graph Traversals depth-first: visit start node, recursively traverse all un-visited a weight (e.g., road distance) is assigned to edges neighbors in depth-first e : w v2 breath-first: visit start node (distance 0), visit all neighbors 1 1 e : w 2 2 (distance 1), then all other nodes in increasing distance order v1 v3 e3 : w3 Eulerian path/cycle: visit each edge exactly once e4 : w4 Hamiltonian path/cycle: visit each vertex exactly once c Lena Wiese: Advanced Data Management, DeGruyter, 2015. spanning tree: visit each vertex and a subset of edges such that visited vertices and edges form a tree Augsten (Univ. Salzburg) NSDB – Graph Databases Sommersemester 2016 9 / 32 Augsten (Univ. Salzburg) NSDB – Graph Databases Sommersemester 2016 10 / 32 Introduction to Graphs Introduction to Graphs Graph Data Structures Edge List edge list follows mathematical definition: store edges E and nodes V edge list as sets adjacency matrix add/delete edge/node are efficient incidence matrix small memory adjacency list most queries inefficient and require search among all edges: incidence list find all neighbors of a node find incident edges in directed graph traverse a specific path Augsten (Univ. Salzburg) NSDB – Graph Databases Sommersemester 2016 11 / 32 Augsten (Univ. Salzburg) NSDB – Graph Databases Sommersemester 2016 12 / 32 Introduction to Graphs Introduction to Graphs Adjacency Matrix Incidence Matrix matrix B of size V E matrix A of size V V | | × | | element b is 1 if edge e is incidentv2 to v (-1 for outgoing edge in | | × | | i,j i e i 1 e element ai,j is the number of (directed) edges between vi and vj directed graph) 2 v1 v3 adjacency matrix for undirected graphs is symmetric adding/deleting nodes/edges is problematice3 c Lena Wiese: Advanced Data Management, DeGruyter, 2015. adding/deleting nodes is problematic, adding/deleting edges is less memory than adjacency matrix for sparse graphs since no efficient zero-only columns storage size O( V 2), large overhead if graph is sparse (small average storage size may grow to O( V 3) (since E = O( V 2) in complete | | | | | | | | degree, i.e., few edges per node) graph) edge lookup by tail and head nodes is very efficient checking for the existence of an edge between vertex pair is expensive finding incident edges requires scanning matrix row or column finding incident edges requires searching matrix row finding the head for a given edge tail requires searching column Augsten (Univ. Salzburg) NSDB – Graph Databases Sommersemester 2016 13 / 32 Augsten (Univ. Salzburg) NSDB – Graph Databases Sommersemester 2016 14 / 32 Introduction to Graphs Introduction to Graphs Adjacency List Adjacency List – Examples each vertex stores linked list of incident edges (outgoing edges in simple, undirected graph directed graph) v1 v2 v3 edges are not stored explicitly v2 v1 v3 v3 v1 v2 adding/deleting nodes/edges is efficient c Lena Wiese: Advanced Data Management, DeGruyter, 2015. finding all neighbors is efficient directed multigraph small memory checking existence of edge between vertex pair requires search in v1 v2 v3 v3 v2 v2 v3 adjacency list e1 e2 v3 v1 v3 e3 finding incoming edges in directed graphs is inefficient (solution: c Lena Wiese: Advanced Data Management, DeGruyter, 2015. e4 forward and backward search adjacency list) c Lena Wiese: Advanced Data Management, DeGruyter, 2015. Augsten (Univ. Salzburg) NSDB – Graph Databases Sommersemester 2016 15 / 32 Augsten (Univ. Salzburg) NSDB – Graph Databases Sommersemester 2016 16 / 32 Introduction to Graphs Introduction to Graphs Incidence List Incidence List – Examples simple, undirected graph e1 e3 v1 each vertex stores linked list of incident edges (outgoing edges in v1, v2 v1, v3 { } { } v2 directed graph) e1 e2 e1 e2 v2 v1, v2 v2, v3 edges are listed explicitly such that information can be stored with v1 v3 { } { } e3 edges c Lena Wiese: Advanced Data Management, DeGruyter,e 2015.3 e2 v3 v1, v3 v2, v3 finding all neighbors is efficient { } { } c Lena Wiese: Advanced Data Management, DeGruyter, 2015. small memory directed multigraph checking existence of edge between vertex pair requires search in e1 e3 e4 v1 incidence list (v1, v2) (v1, v3) (v1, v3) finding incoming edges in directed graphs is inefficient (solution: e2 v2 forward and backward search incidence list) (v2, v3) v3 c Lena Wiese: Advanced Data Management, DeGruyter, 2015. Augsten (Univ. Salzburg) NSDB – Graph Databases Sommersemester 2016 17 / 32 Augsten (Univ. Salzburg) NSDB – Graph Databases Sommersemester 2016 18 / 32 Property Graph Model Property Graph Model Inhalt Property Graph Model directed, multi-relational, labeled multi-graph multi-relational single-relational graph: only one “kind” of nodes/edges multi-relational graph: nodes and edges have a type 1 Introduction to Graphs labels node label is the node type edge label is the edge type 2 Property Graph Model nodes and edges may have attributes name:value pairs name is the key (e.g., age) 3 v2 Graph Database Implementations e1 value has a domain (e.g., non-negativee2 integer) v1 v3 each node and each edge has an explicite3 ID e4 c Lena Wiese: Advanced Data Management, DeGruyter, 2015. only one edge of a specific type allowed between a given pair of nodes restrictions on edges can be defined (e.g., edges of type “likes” allowed only between nodes of type “person”) Augsten (Univ. Salzburg) NSDB – Graph Databases Sommersemester 2016 19 / 32 Augsten (Univ. Salzburg) NSDB – Graph Databases Sommersemester 2016 20 / 32 Property Graph Model Property Graph Model Property Graph – Social Network Example Property Graph – Social Network Example multiple edges between node pair only allowed if they differ by type Id: 2 Id: 2 Label: Person Label: Person Name: Bob Name: Bob Id: 4 Age: 27 Id: 5 Id: 4 Age: 27 Id: 5 Label: knows Label: knows Label: knows Label: knows since: 31-21-2009 since: 10-04-2011 since: 31-21-2009 since: 10-04-2011 Id: 1 Id: 3 Id: 1 Id: 3 Label: Person Label: Person Label: Person Label: Person Name: Alice Id: 6 Name: Charlene Name: Alice Id: 6 Name: Charlene Age: 34 Label: dislikes Age: 29 Age: 34 Label: dislikes Age: 29 c Lena Wiese: Advanced Data Management, DeGruyter, 2015. Id: 7 Label: dislikes (not allowed!) c Lena Wiese: Advanced Data Management, DeGruyter, 2015. Augsten (Univ. Salzburg) NSDB – Graph Databases Sommersemester 2016 21 / 32 Augsten (Univ. Salzburg) NSDB – Graph Databases Sommersemester 2016 22 / 32 Property Graph Model Graph Database Implementations Storing Property Graphs in Relations Inhalt Alternative 1: Nodes and their attributes: Node(NodeID, NodeLabel) Person(NodeID, Name, Age) π (Person) π (Node) 1 Introduction to Graphs NodeID ⊆ NodeID Edges and their attributes: Edge(EdgeID, EdgeLabel, Source, Target) Knows(EdgeID, Since) 2 Property Graph Model π (Knows) π (Edge) EdgeID ⊆ EdgeID π (Edge) π (Node) Source ⊆ NodeID π (Edge) π (Node)