Problem-Solving using Graph Traversals: Searching, Scoring, Ranking, and Recommendation

Problem-Solving using Graph Traversals Searching, Scoring, Ranking, and Recommendation Marko A. Rodriguez Graph Systems Architect http://markorodriguez.com http://twitter.com/twarko AT&Ti Technical Talk - Glendale, California – July 27, 2010 July 26, 2010

Abstract A graph is a data structure that links a set of vertices by a set of edges. Modern graph databases support multi-relational graph structures, where there exist different types of vertices (e.g. people, places, items) and different types of edges (e.g. friend, lives at, purchased). By means of index-free adjacency, graph databases are optimized for graph traversals and are interacted with through a graph traversal engine. A graph traversal is defined as an abstract path whose instance is realized on a graph dataset. Graph databases and traversals can be used for searching, scoring, ranking, and in concert, recommendation. This presentation will explore graph structures, algorithms, traversal algebras, graph-related software suites, and a host of examples demonstrating how to solve real-world problems, in real-time, with graphs. This is a whirlwind tour of the theory and application of graphs.

Outline • Graph Structures, Algorithms, and Algebras • Graph Databases and the Property Graph • TinkerPop Open-Source Graph Product Suite • Real-Time, Real-World Use Cases for Graphs

difﬁculty graphs algebra databases indices time data models Diﬃculty Chart software algorithms real-world conclusion

G = (V, E)

A Vertex There once was a vertex i ∈ V named tenderlove.

Two Vertices And then came along another vertex j ∈ V named sixwing. Thus, i, j ∈ V .

A Directed Edge Our tenderlove extended a relationship to sixwing. Thus, (i, j) ∈ E.

The Single-Relational, Directed Graph More vertices join, create edges and, in turn, the graph grows...

The Single-Relational, Directed Graph as a Matrix A single-relational graph deﬁned as G = (V, E ⊆ (V × V )) can be represented as the adjacency matrix A ∈ {0, 1}n×n, where 1 if (i, j) ∈ E Ai,j = 0 otherwise.

The Single-Relational, Directed Graph as a Matrix 0 1 1 0 1 0 0 1 1 0 0 0 0 1 0 0 G A

The Single-Relational, Directed Graph • All vertices are homogenous in meaning—all vertices denote the same type of object (e.g. people, webpages, etc.).1 • All edges are homogenous in meaning—all edges denote the same type of relationships (e.g. friendship, works with, etc.).2 1 This is not completely true. All n-partite single-relational graphs allow for the division of the vertex set into n subsets, where V = n Ai : Ai ∩ Aj = ∅. Thus, its possible to implicitly type the vertices. i 2 This is not completely true. There exists an injective, information-preserving function that maps any multi-relational graph to a single-relational graph, where edge types are denoted by topological structures. Thus, at a “higher-level,” it is possible to create a heterogenous set of relationships. Rodriguez, M.A., “Mapping Semantic Networks to Undirected Networks,” International Journal of Applied Mathematics and Computer Sciences, 5(1), pp. 39–42, 2009. [http://arxiv.org/abs/0804.0277]

Applications of Single-Relational Graphs • Social: define how people interact (collaborators, friends, kins). • Biological: define how biological components interact (protein, food chains, gene regulation). • Transportation: define how cities are joined by air and road routes. • Dependency: define how software modules, data sets, functions depend on each other. • Technology: define the connectivity of Internet routers, web pages, etc. • Language: define the relationships between words.

The Limitations of Single-Relational Graph Modeling Friendship Graph Favorite Graph Works-For Graph Unfortunately, single-relational graphs are independent of each other. This is because G = (V, E)—there is only a single edge set E (i.e. a single type of relation).

Numerous Algorithms for Single-Relational Graphs We would like a more flexible graph modeling construct, but unfortunately, most of our graph algorithms were designed for single-relational graphs.3 • Geodesic: diameter, radius, eccentricity, closeness, betweenness, etc. • Spectral: random walks, PageRank, eigenvector centrality, spreading activation, etc. • Assortativity: scalar, categorical, hierarchal, etc. • Others: ...4 We can solve this with multi-relational graphs and a path algebra. 3 For a fine book on graph analysis algorithms, please see: Brandes, U., Erlebach T., “Network Analysis: Methodolgical Foundations,” edited book, Springer, 2005. 4 One of the purposes of this presentation is advocate for local graph analysis algorithms (i.e. priors-based, relative) vs. global graph analysis algorithms. Most popular graph analysis algorithms are global in that they require an analysis of the whole graph (or a large portion of a graph) to yield results. Local analysis algorithms are dependent on sub-graphs of the whole and in effect, can boast faster running times.

G = (V, E)

A Directed Edge

A Directed, Labeled Edge friend Lets specify the type of relationship that exists between tenderlove and sixwing. Thus, (i, j) ∈ Efriend.

Growing a Multi-Relational Graph friend friend Lets make the friendship relationship symmetric. Thus, (j, i) ∈ Efriend.

Growing a Multi-Relational Graph friend friend friend friend Lets add marko to the mix: k ∈ V . This graph is still single-relational. There is only one type of relation.

Growing a Multi-Relational Graph friend friend favorite friend friend Lets add an (i, l) ∈ Efavorite. Now there are multiple types of relationships: Efriend and Efavorite (2 edge sets).

The Multi-Relational, Directed Graph • At this point, there is a multi-relational, directed graph: G = (V, E), where E = (E0, E1, . . . , Em ⊆ (V × V )).5 • Vertices can denote different types of objects (e.g. people, places).6 • Edge can denote different types of relationships (e.g. friend, favorite).7 • This is the data model of the Web of Data—the RDF data model.8 5 Another representation is G ⊆ (V × Ω × V ), where Ω ⊆ Σ∗ is the set of legal edge labels. 6 Vertex types can be determined by the domain and range specification of the respective edge relation/label/predicate. Or, another way, by means of an explicit typing relation such as a, type, b . 7 Edge types are determined by the label that accompanies the edge. 8 This is not completely true. The vertex set is split into URIs (U ), literals (L), and blank/anonymous nodes (B ), such that G ⊆ ((U × B) × U × (U × B × L)). [http://www.w3.org/RDF/]

The Multi-Relational, Directed Graph as a Tensor A three-way tensor can be used to represent a multi-relational graph. If G = (V, E = {E0, E1, . . . , Em ⊆ (V × V )}) is a multi-relational graph, then A ∈ {0, 1}n×n×m and 1 if (i, j) ∈ Em : 1 ≤ k ≤ m Ak i,j = 0 otherwise. Thus, each edge set in E represents an adjacency matrix and the combination of m adjacency matrices forms a 3-way tensor.

The Multi-Relational, Directed Graph as a Tensor friend 0 0 0 0 0 0 0 1 friend favorite 0 0 0 0 0 0 0 0 s er sw nd an e ite fri or G A v fa

Multi-Relational Graph Algorithms “Can we evaluate single-relational graph analysis algorithms on a multi-relational graph?”

The Meaning of Edge Meanings loves loves loves hates hates hates loves loves hates hates • Multi-relationally: tenderlove is more liked than marko. • Single-relationally: tenderlove and marko simply have the same in-degree. Given, lets say, degree-centrality, tenderlove and marko are equal as they have the same number of relationships. The edge labels do not eﬀect the output of the degree-centrality algorithm.

What Do You Mean By “Central?” answer ... answer_for ite or v What is your favorite fa answer_by bookstore? favorite question_by ... friend friend friend Lets focus speciﬁcally on centrality. What is the most central vertex in a multi-relational graph? Who is the most central friend in the graph—by friendship, by question answering, by favorites, etc?

Primary Eigenvector “What does the primary eigenvector of a multi-relational graph mean?”91011 9 We will use the primary eigenvector for the following argument. Note that the same argument applies for all known single-relational graph algorithms (i.e. geodesic, spectral, community detection, etc.). 10 Technical details are left aside such as outgoing edge probability distributions and the irreducibility of the graph. 11 The popular PageRank vector is deﬁned as the primary eigenvector of a low-probability fully connected graph combined with the original graph (i.e. both graphs maintain the same V ).

Primary Eigenvector: Ignoring Edge Labels |V |×|V | • If π = Bπ, where B ∈ N+ is the adjacency matrix formed by merging the edge sets in E, then edge labels are ignored—all edges are treated equally. • In this “ignoring labels”-model, there is only one primary eigenvector for the graph—one deﬁnition of centrality. • With a heterogenous set of vertices connected by a heterogenous set of edges, what does this type of centrality mean?

Primary Eigenvector: Isolating Subgraphs • Are there other primary eigenvectors in the multi-relational graph? • You can ignore certain edge sets and calculate the primary eigenvector (e.g. pull out the single-relational “friend”-graph.) π = Afriendπ, where Afriend ∈ {0, 1}|V |×|V | is the adjacency matrix formed by the edge set Efriend. • Thus, you can isolate subgraphs (i.e. adjacency matrices) of the multi-relational graph and calculate the primary eigenvector for those subgraphs. • In this “isolation”-model, there are m deﬁnitions of centrality—one for each isolated subgraph.12 12 Remember, A ∈ {0, 1}n×n×m .

Ultimately what we want is...

Primary Eigenvector: Turing Completeness • What about using paths through the graph—not simply explicit one-step edges? • What about determining centrality for a relation that isn’t explicit in E (i.e. Ak ∈ A)? In general, what about π = Xπ, where X is a derived adjacency matrix of the multi-relational graph. For example, if I know who everyone’s friends are, then I know (i.e. can infer, derive, compute) who everyone’s friends-of-a-friends (FOAF) are. What about the primary eigenvector of the derived FOAF graph? • In the end, you want a Turing-complete framework—you want complete control (universal computability) over how π moves through the multi-relational graph structure.13 13 These ideas are expounded upon at great length throughout this presentation.

A Path Algebra for Evaluating Single-Relational Algorithms on Multi-Relational Graphs • There exists a multi-relational graph algebra for mapping single-relational graph analysis algorithms to the multi-relational domain.14 • The algebra works on a tensor representation of a multi-relational graph. • In this framework and given the running example, there are as many primary eigenvectors as there are abstract path deﬁnitions. 14 * Rodriguez M.A., Shinavier, J., “Exposing Multi-Relational Networks to Single-Relational Network Analysis Algorithms,” Journal of Informetrics, 4(1), pp. 29–41, doi:10.1016/j.joi.2009.06.004, 2009. [http://arxiv.org/abs/0806.2274] * Rodriguez, M.A., “Grammar-Based Random Walkers in Semantic Networks,” Knowledge-Based Systems, 21(7), pp. 727–739, doi:10.1016/j.knosys.2008.03.030, 2008. [http://arxiv.org/abs/0803.4355] * Rodriguez, M.A., Watkins, J.,“Grammar-Based Geodesics in Semantic Networks,” Knowledge-Based Systems, in press, doi:10.1016/j.knosys.2010.05.009, 2010.

The Operations of the Multi-Relational Path Algebra • A · B: ordinary matrix multiplication determines the number of (A, B)- paths between vertices. • A : matrix transpose inverts path directionality. • A ◦ B: Hadamard, entry-wise multiplication applies a ﬁlter to selectively exclude paths. • n(A): not generates the complement of a {0, 1}n×n matrix. • c(A): clip generates a {0, 1}n×n matrix from a Rn×n matrix. + • v ±(A): vertex generates a {0, 1}n×n matrix from a Rn×n matrix, where + only certain rows or columns contain non-zero values. • xA: scalar multiplication weights the entries of a matrix. • A + B: matrix addition merges paths.

Primary Eigenvectors in a Multi-Relational Graph • Friend: Afriend π 2 • FOAF: Afriend · Afriend π ≡ Afriend π 2 • FOAF (no self): Afriend ◦ n(I) π 15 2 • FOAF (no friends nor self): Afriend ◦ n Afriend ◦ n(I) π • Co-Worker: Aworks at · Aworks at ◦ n (I) π • Friend-or-CoWorker: 0.65Afriend + 0.35 Aworks at · Aworks at ◦ n ( I) π • ...and more.16 15 I ∈ {0, 1}|V |×|V | : Ii,i = 1—the identity matrix. 16 Note, again, that the examples are with respect to determining the primary eigenvector of the derived adjacency matrix. The same argument holds for all other single-relational graph analysis algorithms. In general, the path algebra provides a means of creating “higher-order” (i.e. semantically-rich) single-relational graphs from a single multi-relational graph. Thus, these derived matrices can be subjected to standard single-relational graph analysis algorithms.

Deriving “Semantically Rich” Adjacency Matrices 0 0 0 0 0 0 0 0 = 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 ∪ 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 s an f) d er n 0 0 0 0 sw se ie nd fri rs o -fr an e e ite fri (n -of d sw l en A Afriend · A friend or nd ◦ n(I) A v e e fa rit fri vo fa 2 Afriend ◦ n(I) friend-of-a-friend (no self) Use the multi-relational graph to generate explicit edges that were implicitly deﬁned as paths. Those new explicit edges can then be memoized17 and re-used (time vs. space tradeoﬀ)—aka path reuse. 17 Memoization Wikipedia entry: http://en.wikipedia.org/wiki/Memoization.

Benefits, Drawbacks, and Future of the Path Algebra • Benefit: Provides a set of theorems for deriving equivalences and thus, provides the foundation for graph traversal engine optimizers.18 Serves a similar purpose as the relational algebra for relational databases.19 • Drawback: The algebra is represented in matrix form and thus, operationally, works globally over the graph.20 • Future: A non-matrix-based, ring theoretic model of graph traversal that supports +, −, and · on individual vertices and edges. The Gremlin [http://gremlin.tinkerpop.com] graph traversal engine presented later provides the implementation before a fully-developed theory. 18 Rodriguez M.A., Shinavier, J., “Exposing Multi-Relational Networks to Single-Relational Network Analysis Algorithms,” Journal of Informetrics, 4(1), pp. 29–41, 2009. [http://arxiv.org/abs/0806.2274] 19 Codd, E.F., “A Relational Model of Data for Large Shared Data Banks,” Communications of the ACM, 13(6), pp. 377–387, doi:10.1145/362384.362685, 1970. 20 It is possible to represent local traversals using vertex filters at the expense of clumsy notation.

The Simplicity of a Graph • A graph is a simple data structure. • A graph states that something is related to something else (the foundation of any other data structure).21 • It is possible to model a graph in various types of databases.22 Relational database: MySQL, Oracle, PostgreSQL JSON document database: MongoDB, CouchDB XML document database: MarkLogic, eXist-db etc. 21 A graph can be used to represent other data structures. This point becomes convenient when looking beyond using graphs for typical, real-world domain models (e.g. friends, favorites, etc.), and seeing their applicability in other areas such as modeling code (e.g. http://arxiv.org/abs/0802.3492), indices, etc. 22 For the sake of diagram clarity, the examples to follow are with respect to a single-relational, directed graph. Note that it is possible to model multi-relational graphs in these types of database as well.

Representing a Graph in a Relational Database outV | inV ------------ A A | B A | C C | D B C D | A D

Representing a Graph in a JSON Database { A : { outE : [B, C] A } B : { outE : [] } B C C : { outE : [D] } D : { outE : [A] D } }

Representing a Graph in an XML Database graphml graph node id=A / A node id=B / node id=C / node id=D / edge source=A target=B / edge source=A target=C / B C edge source=C target=D / edge source=D target=A / /graph /graphml D

Deﬁning a Graph Database “If any database can represent a graph, then what is a graph database?”

Defining a Graph Database A graph database is any storage system that provides index-free adjacency.2324 23 There is no “official” definition of what makes a database a graph database. The one provided is my definition (respective of the influence of my collaborators in this area). However, hopefully the following argument will convince you that this is a necessary definition. Given that any database can model a graph, such a definition would not provide strict enough bounds to yield a formal concept (i.e. ). 24 There is adjacency between the elements of an index, but if the index is not the primary data structure of concern (to the developer), then there is indirect/implicit adjacency, not direct/explicit adjacency. A graph database exposes the graph as an explicit data structure (not an implicit data structure).

Deﬁning a Graph Database by Example Toy Graph Gremlin (stuntman) B E A C D

Graph Databases and Index-Free Adjacency B E A C D • Our gremlin is at vertex A. • In a graph database, vertex A has direct references to its adjacent vertices. • Constant time cost to move from A to B and C . It is dependent upon the number of edges emanating from vertex A (local).

Graph Databases and Index-Free Adjacency B E A C D The Graph (explicit)

Non-Graph Databases and Index-Based Adjacency B E A B C A B,C E D,E D E C D • Our gremlin is at vertex A.

Non-Graph Databases and Index-Based Adjacency B E A B C A B,C E D,E D E C D • In a non-graph database, the gremlin needs to look at an index to determine what is adjacent to A. • log2(n) time cost to move to B and C . It is dependent upon the total number of vertices and edges in the database (global).

Non-Graph Databases and Index-Based Adjacency B E A B C A B,C E D,E D E C D The Index (explicit) The Graph (implicit)

Index-Free Adjacency • While any database can implicitly represent a graph, only a graph database makes the graph structure explicit.25 • In a graph database, each vertex serves as a “mini index” of its adjacent elements.26 • Thus, as the graph grows in size, the cost of a local step remains the same.27 25 Please see http://markorodriguez.com/Blarko/Entries/2010/3/29_MySQL_vs._Neo4j_on_a_ Large-Scale_Graph_Traversal.html for some performance characteristics of graph traversals in a relational database (MySQL) and a graph database (Neo4j). 26 Each vertex can be intepreted as a “parent node” in an index with its children being its adjacent elements. In this sense, traversing a graph is analogous in many ways to traversing an index—albeit the graph is not an acyclic connected graph (tree). (a vision espoused by Craig Taverner) 27 A graph, in many ways, is like a distributed index.

Graph Databases Do Make Use of Indices A B C } Index of Vertices (by id) D E } The Graph • There is more to the graph than the explicit graph structure. • Indices index the vertices by their properties (e.g. ids, name, latitude).28 28 Graph databases can be used to create index structures. In fact, in the early days of Neo4j, Neo4j used its own graph structure to index the properties of its vertices—a graph indexing a graph. A thought iterated many times over by Craig Taverner who is interested in graph databases for geo-spatial indexing/analysis.

The Patterns of Relational and Graph Databases • In a relational database, operations are conceptualized set-theoretically with the joining of tuple structures being the means by which normalized/separated data is associated. • In a graph database, operations are conceptualized graph-theoretically with paths over edges being the means by which non-adjacent/separated vertices are associated.29 In theory and ignoring performance, both models have the same expressivity and allow for the same manipulations. But such theory does not determine intention and the mental ruts that any approach engrains. The graph database provides a novel perspective on the ancient necessity to manipulate information. 29 Rodriguez, M.A., Neubauer, P., “The Graph Traversal Pattern,” ATTi and NeoTechnology Technical Report, currently in review, 2010. [http://arxiv.org/abs/1004.1001]

Property Graphs and Graph Databases • Most graph databases support a graph data model known as a property graph. • A property graph is a directed, attributed, multi-relational graph. In other words, vertices and edges are equipped with a collection of key/value pairs.30 30 Rodriguez, M.A., Neubauer, P., “Constructions from Dots and Lines,” Bulletin of the American Society for Information Science and Technology, American Society for Information Science and Technology, 2010. [http://arxiv.org/abs/1006.2361]

From a Multi-Relational Graph... friend friend favorite friend friend

...to a Property Graph name=marko location=Santa Fe lat=11111 gender=male long=22222 created_at=123456 friend friend favorite name=sixwing location=West Hollywood gender=male created_at=234567 friend friend created_at=234567

Why the Property Graph Model? • Standard single-relational graphs do not provide enough modeling ﬂexibility for use in real-world situations.31 • Multi-relational graphs do and the Web of Data (RDF) world demonstrates this to be the case in practice. • Property graphs are perhaps more practical because not every datum needs to be “related” (e.g. age, name, etc.). Thus, the edge and key/value model is a convenient dichotomy.32 • Property graphs provide ﬁner-granularity on the meaning of an edge as the key/values of an edge add extra information beyond the edge label. 31 This is not completely true—researchers use the single-relational graph all the time. However, in most data rich applications, its limiting to work with a single edge type and a homogenous population of vertices. 32 RDF has a similar argument in that literals can only be the object of a triple. However, in practice, when represented in a graph database, there is a single literal vertex denoting that literal and thus, is traversable like any other vertex.

Graph Type Morphisms weighted graph add weight attribute property graph remove attributes remove attributes no op labeled graph no op semantic graph no op directed graph remove edge labels remove edge labels make labels URIs no op remove directionality rdf graph multi-graph remove loops, directionality, and multiple edges simple graph no op undirected graph

TinkerPop: Making Stuﬀ for the Fun of It • Open source software group started in 2008 focusing on graph data structures, graph query engines, graph-based programming languages, and, in general, tools and techniques for working with graphs. [http://tinkerpop.com] [http://github.com/tinkerpop] Current members: Marko A. Rodriguez (ATTi), Peter Neubauer (NeoTechnology), Joshua Shinavier (Rensselaer Polytechnic Institute), and Pavel Yaskevich (“I am no one from nowhere”).

TinkerPop Productions • Blueprints: Data Models and their Implementations [http://blueprints.tinkerpop.com] • Pipes: A Data Flow Framework using Process Graphs [http://pipes.tinkerpop.com] • Gremlin: A Graph-Based Programming Language [http://gremlin.tinkerpop.com] • Rexster: A RESTful Graph Shell [http://rexster.tinkerpop.com] Wreckster: A Ruby API for Rexster [http://github.com/tenderlove/wreckster] There are other TinkerPop products (e.g. Ripple, LoPSideD, TwitLogic, etc.), but for the purpose of this presentation, only the above will be discussed.

Blueprints: Data Models and their Implementations Blueprints • Blueprints is the like the JDBC of the graph database community. • Provides a Java-based interface API for the property graph data model. Graph, Vertex, Edge, Index. • Provides implementations of the interfaces for TinkerGraph, Neo4j, Sails (e.g. AllegroGraph, HyperSail, etc.), and soon (hopefully) others such as InﬁniteGraph, InfoGrid, Sones, DEX, and HyperGraphDB.33 33 HyperGraphDB makes use of an n-ary graph structure known as a hypergraph. Blueprints, in its current form, only supports the more common binary graph.

Pipes: A Data Flow Framework using Process Graphs Pipes • A dataﬂow framework with support for Blueprints-based graph processing. • Provides a collection of “pipes” (implement Iterable and Iterator) that are connected together to form processing pipelines. Filters: ComparisonFilterPipe, RandomFilterPipe, etc. Traversal: VertexEdgePipe, EdgeVertexPipe, PropertyPipe, etc. Splitting/Merging: CopySplitPipe, RobinMergePipe, etc. Logic: OrPipe, AndPipe, etc.

Gremlin: A Graph-Based Programming Language Gremlin G = (V, E) • A Turing-complete, graph-based programming language that compiles Gremlin syntax down to Pipes (implements JSR 223). • Support various language constructs: :=, foreach, while, repeat, if/else, function and path deﬁnitions, etc. ./outE[@label=‘friend’]/inV ./outE[@label=‘friend’]/inV/outE[@label=‘friend’]/inV[g:except($ , .)] g:key(‘name’,‘Aaron Patterson’)[0]/outE[@label=‘favorite’]/inV/@name

Rexster: A RESTful Graph Shell reXster • Allows Blueprints graphs to be exposed through a RESTful API (HTTP). • Supports stored traversals written in raw Pipes or Gremlin. • Supports adhoc traversals represented in Gremlin. • Provides “helper classes” for performing search-, score-, and rank-based traversal algorithms—in concert, support for recommendation. • Aaron Patterson (ATTi) maintains the Ruby connector Wreckster.

Typical TinkerPop Graph Stack GET http://{host}/{resource} Neo4j NativeStore TinkerGraph

Using Graphs in Real-Time Systems • Most popular graph algorithms require global graph analysis. Such algorithms compute a score, a vector, etc. given the structure of the whole graph. Moreover, many of these algorithms have large running times: O(|V | + |E|), O(|V | log |V |), O(|V |2), etc. • Many real-world situations can make use of local graph analysis.34 Search for x starting from y. Score x given its local neighborhood. Rank x relative to y. Recommend vertices to user x. 34 Many web applications are “ego-centric” in that they are with respect to a particular user (the user logged in). In such scenarios, local graph analysis algorithms are not only prudent to use, but also, beneﬁcial in that they are faster than global graph analysis algorithms. Many of the local analysis algorithms discussed run in the sub-second range (for graphs with “natural” statistics).

Applications of Graph Databases and Traversal Engines: Searching, Scoring, and Ranking ˆ • Searching: given a power multi-set of vertices (P(V )) and a path description (Ψ), return the vertices at the end of that path.35 ˆ ˆ P(V ) × Ψ → P(V ) • Scoring: given some vertices and a path description, return a score. ˆ P(V ) × Ψ → R • Ranking: given some vertices and a path description, return a map of scored vertices. ˆ P(V ) × Ψ → (V × R) 35 Use cases need not be with respect to vertices only. Edges can be searched, scored, and ranked as well. However, in order to express the ideas as simply as possible, all discussion is with respect to vertices.

Applications of Graph Databases and Traversal Engines: Recommendation • Recommendation: searching, scoring, and ranking can all be used as components of a recommendation. Thus, recommendation is founded on these more basic ideas. Recommendation aids the user by allowing them to make “jumps” through the data. Items that are not explicitly connected, are connected implicitly through recommendation (through some abstract path Ψ). • The act of recommending can be seen as an attempt to increase the density of the graph around a user’s vertex. For example, recommending user i ∈ V places to visit U ⊂ V , will hopefully lead to edges of the form i, visited, j : ∀j ∈ U .36 36 A standard metric for recommendation quality is seen as how well it predicts the user’s future behavior. That is, does it predict an edge.

There Is More Than “People Who Like X Also Like Y .” • A system need not be limited to one type of recommendation. With graph-based methods, there are as many recommendations as there are abstract paths. • Use recommendation to aid the user in solving problems (i.e. computationally derive solutions for which your data set is primed for). Examples below are with respect to problem-solving in the scholarly community.37 Recommend articles to read. (articles) Recommend collaborators to work on an idea/article with. (people) Recommend a venue to submit the article to. (venues) Recommend an editor referees to review the article. (people)38 Recommend scholars to talk to and concepts to talk to them about at the venue. (people and tags) 37 Rodriguez, M.A., Allen, D.W., Shinavier, J., Ebersole, G., “A Recommender System to Support the Scholarly Communication Process,” KRS-2009-02, 2009. [http://arxiv.org/abs/0905.1594] 38 Rodriguez, M.A., Bollen, J., “An Algorithm to Determine Peer-Reviewers,” Conference on Information and Knowledge Management (CIKM), pp. 319–328, doi:10.1145/1458082.1458127, 2008. [http: //arxiv.org/abs/cs/0605112]

Real-Time, Domain-Speciﬁc, Graph-Based, Problem-Solving Engine Ψ5 Ψ1 Real-Time + Ψ4 Ψn Ψ2 Ψ3 = Domain-Speciﬁc Graph-Based Problem-Solving Engine Library of Path/Traversal Expressions Graph Data Set Your domain model (i.e. graph dataset) determines what traversals you can design, develop, and deploy. Together, these determine which types of problems you can solve automatically/computationally for yourself, your users.

Applicable in Various, Seemingly Diverse Areas • Applications to a techno-social government (i.e. collective decision making systems).39 0.20 correct decisions 0.00 0.05 0.10 0.15 0.95 direct democracy dynamically distributed democracy 0.80 proportion oferror 0.65 dynamically distributed democracy direct democracy 0.50 100 90 80 70 60 50 40 30 20 10 100 90 80 70 60 50 40 30 20 10 0 0 percentage of active citizens percentage of active citizens (n) 39 Fig. 5. The relationship between k and evote for direct democracy (gray * Rodriguez, M.A., Watkins, J.H., “Revisiting the Age of Enlightenment from a Collective The plot provides line) and dynamically distributed democracy (black line). Decision Making Systems k the proportion of identical, correct decisions over a simulation that was run Perspective,” First Monday, 14(8), 2009. [http://arxiv.org/abs/0901.3929] with 1000 artificially generated networks composed of 100 citizens each. Fig. 6. A visualization of a network of t * Rodriguez, M.A., “Social Decision Making with Multi-Relational Networks and Grammar-Based Particle Swarms,” color denotes their “political tenden citizen’s Hawaii International Conference on Systems Science (HICSS), pp. 39–49, 2007. [http://arxiv.org/abs/cs/0609034] is 1, and layout. is 0.5. purple The layout algori As previously stated, let x ∈ [0, 1]n denote the political Reingold * Rodriguez, M.A., Steinbock, D.J., “A Social Network for Societal-Scale each citizen in this population, where xi is the of the North tendency of Decision-Making Systems,” Proceedings tendency of citizen i and, for the purpose of simulation, is American Association for Computational Social and Organizational Science Conference, 2004. [http://arxiv.org/abs/cs/ determined from a uniform distribution. Assume that every 1 n “vote power” and this is represe 0412047] citizen in a population of n citizens uses some social network- such that the total amount of vote based system to create links to those individuals that they 1. Let y ∈ Rn denote the total amo + believe reflect their tendency the best. In practice, these links flowed to each citizen over the cours may point to a close friend, a relative, or some public figure a ∈ {0, 1}n denotes whether citizen whose political tendencies resonate with the individual. In in the current decision making pro other words, representatives are any citizens, not political values of a are biased by an unfair candidates that serve in public office. Let A ∈ [0, 1]n×n denote of making the citizen an active parti the link matrix representing the network, where the weight of the citizen inactive. The iterative alg an edge, for the purpose of simulation, is denoted where ◦ denotes entry-wise multip 1 − |xi − xj | if link exists

Toy Graph Dataset lat=11111 long=22222 name=marko created_at=123456 4 name=sixwing location=West Hollywood location=Santa Fe gender=male favorite gender=male friend friend 1 2 3 favorite created_at=234567 friend favorite 6 name=Bryce Canyon favorite 5 name=charlie We will use the toy-graph above to demonstrate Gremlin (to introduce the syntax). However, in parallel, we will also use a large graph of the same schema to demonstrate how SQL/MySQL compares relative to Gremlin/Neo4j on traversal-based queries (i.e. for relational databases, queries with table joins).

Dataset Schema in Neo4j Neo4j [http://neo4j.org] is a “schema-less” database. However, ultimately, data is represented according to some schema whether that schema be explicit in the database, in the code interacting with the database, or in the developer’s head.40 Please note the schema diagrammed below is a non-standard convention.41 name=string name=string location=string lat=double gender=string long=double type=Person type=Place Person Place friend favorite 40 A better term for “schema-less” might have been “dynamic schema.” 41 For expressive, standardized graph-based schema languages, refer to RDFS [http://www.w3.org/TR/ rdf-schema/] and OWL [http://www.w3.org/TR/owl-features/] of the Web of Data community.

Dataset Schema in MySQL CREATE TABLE friend ( outV INT NOT NULL, inV INT NOT NULL); CREATE INDEX friend_outV_index USING BTREE ON friend (outV); CREATE INDEX friend_inV_index USING BTREE ON friend (inV); CREATE TABLE favorite ( outV INT NOT NULL, inV INT NOT NULL); CREATE INDEX favorite_outV_index USING BTREE ON favorite (outV); CREATE INDEX favorite_inV_index USING BTREE ON favorite (inV); CREATE TABLE metadata ( vertex INT NOT NULL, _key VARCHAR(100) NOT NULL, _value VARCHAR(100), PRIMARY KEY (vertex, _key)); CREATE INDEX metadata_vertex_index USING BTREE ON metadata (vertex); CREATE INDEX metadata_key_index USING BTREE ON metadata (_key); CREATE INDEX metadata_value_index USING BTREE ON metadata (_value);

Experiment Discussion • First, for each experiment, no cache is used. For each query (or run of queries), caches are reset/ﬂushed and the query is performed.42 • Second, for each experiment, a “stable point” (i.e. performance with full caching) is found through the repeated evaluation of the same query. • Evaluations are done on my laptop using SQL/MySQL(5.1.45) and Gremlin(0.5-alpha)/Neo4j(1.1).43 • I am not an expert in relational databases. Be aware of all of my choices (table design, indexes used, query representation, etc.).44 42 I believe, from looking at the behavior of MySQL, MySQL caches maintain joined structure in main memory for subsequent queries. Neo4j caches by maintaining active portions of the graph in main memory. 43 Note that Gremlin 0.5-alpha is much more performant than Gremlin 0.2.2. Also, running times presented are likely to change with optimizations (discussed later)—consider all times in passing only. 44 For the more interested, please do experiments yourself with your particular domain models and queries.

Loading Identical Data into MySQL and Neo4j For the ﬁrst half of the examples, we will use a small data set. Later we will increase this data set by 10,000,000 edges and compare again. The reason is to test how indices eﬀect the performance of standard queries. As indices grow, log2(n) becomes costly. mysql (SELECT * FROM friend) UNION (SELECT * FROM favorite) 71100 rows in set (0.47 sec) gremlin g:count($_g/E) ==71100 results returned in 145.427ms (0.145 sec) First thing to note—graph databases don’t have a notion of “tables,” the entire graph is one atomic entity.

Basic Gremlin gremlin (1 + 2) * 4 div 5 ==2.4 gremlin marko + a. + rodriguez ==marko a. rodriguez gremlin func ex:add-one($x) $x + 1 end gremlin foreach $y in g:list(1,2,3,4) g:print(ex:add-one($y)) end 2 3 4 5

Searching Example: Friends gremlin $_g := neo4j:open(‘/data/mygraph’) name=marko location=Santa Fe lat=11111 gremlin $_ := g:id(1) gender=male long=22222 ==v[1] gremlin . ==v[1] 3 4 gremlin ./outE created_at=123456 ==e[10][1-friend-2] friend favorite name=sixwing ==e[11][1-friend-3] location=West Hollywood ==e[12][1-favorite-4] gender=male gremlin ./outE[@label=‘friend’]/inV/@name friend ==sixwing 1 2 ==marko gremlin ./outE[@label=‘friend’]/inV/@gender favorite favorite ==male created_at=234567 friend ==male gremin ./outE[@label=‘friend’] 6 /inV[@location=‘Santa Fe’]/@name name=Bryce Canyon favorite ==marko 5 name=charlie

Searching Friends SQL/MySQL vs. Gremlin/Neo4j What are the names of Rand Fitzpatrick’s friends?45 mysql SELECT friend.inV, b._value FROM friend, metadata as a, metadata as b WHERE a._key=‘name’ AND a._value=‘Rand Fitzpatrick’ AND a.vertex=friend.outV AND b.vertex=friend.inV AND b._key=‘name’; 97 rows in set (0.32 sec -- 320.0 ms) gremlin g:key(‘name’,‘Rand Fitzpatrick’)/outE[@label=‘friend’]/inV/@name 97 results returned (0.00258 sec -- 25.88 ms) 45 When in cache (through repeated, identical querying), SQL/MySQL evaluates in ∼0.005 seconds (5ms) and Gremlin/Neo4j evaluates in ∼0.0002 seconds (0.2ms).

Searching Example: FOAF (No Friends, No Self) gremlin . name=marko location=Santa Fe lat=11111 ==v[1] gender=male long=22222 gremlin ./outE[@label=‘friend’]/inV /outE[@label=‘friend’]/inV ==v[1] 3 4 ==v[1] created_at=123456 ==v[5] friend favorite name=sixwing gremlin (./outE[@label=‘friend’] location=West Hollywood /inV)[g:assign(‘$x’)] gender=male /outE[@label=‘friend’] friend /inV[g:except(.,$_)][g:except(.,$x)] 1 2 /@name ==charlie favorite favorite created_at=234567 friend 6 name=Bryce Canyon favorite 5 name=charlie

Searching FOAF (Not Self) SQL/MySQL vs. Gremlin/Neo4j What are the names of Rand Fitzpatrick’s friends friends who are not Rand (note: this may include Rand’s friends)?46 mysql SELECT mb._value FROM friend as a, friend as b, metadata as ma, metadata as mb WHERE ma._key=‘name’ AND ma._value=‘Rand Fitzpatrick’ AND ma.vertex=a.outV AND a.inV=b.outV AND b.outV != ma.vertex AND b.inV = mb.vertex AND mb._key=‘name’ 8985 rows in set (0.47 sec -- 470.00 ms) gremlin g:key(‘name’,‘Rand Fitzpatrick’)/outE[@label=‘friend’] /inV/outE[@label=‘friend’]/inV[g:except(.,$_)]/@name 8985 results returned (0.053 sec -- 53.879 ms) 46 When in cache (through repeated, identical querying), SQL/MySQL evaluates in ∼0.03 seconds (30ms) and Gremlin/Neo4j evaluates in ∼0.015 seconds (15ms).

Searching Example: Friend’s Favorites gremlin . name=marko location=Santa Fe lat=11111 ==v[1] gender=male long=22222 gremlin ./outE[@label=‘friend’]/inV /outE[@label=‘favorite’]/inV ==v[6] 3 4 ==v[6] created_at=123456 gremlin ./outE[@label=‘friend’]/inV friend favorite name=sixwing /outE[@label=‘favorite’ and @created_at234500] location=West Hollywood /inV/@name gender=male ==Bryce Canyon friend 1 2 favorite favorite created_at=234567 friend 6 name=Bryce Canyon favorite 5 name=charlie

Searching FOAF (No Self) Favorites SQL/MySQL vs. Gremlin/Neo4j What do Rand’s friends friends (who are not Rand) favorite?47 mysql SELECT mb._value FROM friend as fa, friend as fb, favorite, metadata as ma, metadata as mb WHERE ma._key=‘name’ AND ma._value=‘Rand Fitzpatrick’ AND ma.vertex=fa.outV AND fa.inV=fb.outV AND fb.inV != ma.vertex AND fb.inV=favorite.outV AND mb.vertex=favorite.inV AND mb._key=‘name’; 364905 rows in set (11.17 sec -- 11170.0 ms) gremlin g:key(‘name’,‘Rand Fitzpatrick’)/outE[@label=‘friend’] /inV/outE[@label=‘friend’]/inV[g:except(.,$_)] /outE[@label=’favorite’]/inV/@name 364905 results returned (2.278 sec -- 2278.59 ms) 47 When in cache (through repeated, identical querying), SQL/MySQL evaluates in ∼6.25 seconds (6250ms) and Gremlin/Neo4j evaluates in ∼1.0 second (1000ms).

A Traversal Detour Through the Web of Data ECS South- Sem- Wiki- BBC Surge ampton LIBRIS Web- company Playcount Radio Central RDF Data ohloh Resex Doap- Buda- Music- space Semantic ReSIST brainz Audio- pest Eurécom Project Flickr Web.org MySpace Scrobbler QDOS SW BME Wiki exporter Wrapper Conference IRIT Corpus Toulouse RAE National BBC BBC Crunch 2001 Science FOAF SIOC ACM BBC Music Later + John Base Revyu Foundation Jamendo Peel profiles Sites TOTP Open- Guides DBLP flickr RKB Project Pub Geo- Euro- wrappr Explorer Guten- Virtuoso Guide names stat Pisa CORDIS berg Sponger eprints BBC Programmes Open Calais RKB riese World Linked ECS Magna- Fact- MDB IEEE New- South- tune book ampton castle RDF Book DBpedia Mashup Linked GeoData lingvoj Freebase LAAS- US CiteSeer Census CNRS W3C DBLP Data IBM WordNet Hannover UniRef GEO UMBEL Species DBLP Gov- Track Berlin Reactome LinkedCT UniParc Open Taxonomy Cyc Yago Drug PROSITE Daily Bank Med Pub GeneID Chem Homolo KEGG UniProt Gene Pfam ProDom Disea- CAS Gene some ChEBI Ontology Symbol OMIM Inter Pro UniSTS PDB HGNC MGI PubMed As of July 2009 Image produced by Richard Cyganiak and Anja Jentzsch. [http://linkeddata.org/]

Deﬁning the Web of Data • The Web of Data is similar to the Web of Documents (of common knowledge), but instead of referencing documents (e.g. HTML, images, etc.) with the URI address space, individual datum are referenced.4849 http://markorodriguez.com, foaf:fundedBy, http://atti.com http://markorodriguez.com, foaf:name, Marko Rodriguez http://markorodriguez.com, foaf:age, 30 http://markorodriguez.com, foaf:knows, http://tenderlovemaking.com • In graph theoretic terms, the Web of Data is a multi-relational graph deﬁned as G ⊆ (U ∪ B) × U × (U ∪ B ∪ L), where U is the set of all URIs, B is the set of all blank/anonymous nodes, and L is the set of all literals. 48 The Web of Data is also known as the Linked Data Web, the Giant Global Graph, the Semantic Web, the RDF graph, etc. 49 * Rodriguez, M.A., “Interpretations of the Web of Data, Data Management in the Semantic Web, eds. H. Jin and Z. Lv, Nova Publishing, in press, 2010. [http://arxiv.org/abs/0905.3378] * Rodriguez, M.A., “A Graph Analysis of the Linked Data Cloud,” Technical Report, KRS-2009-01, 2009. [http://arxiv.org/abs/0903.0194]

Some of the Datasets on the Web of Data data set domain data set domain data set domain audioscrobbler music govtrack government pubguide books bbclatertotp music homologene biology qdos social bbcplaycountdata music ibm computer rae2001 computer bbcprogrammes media ieee computer rdfbookmashup books budapestbme computer interpro biology rdfohloh social chebi biology jamendo music resex computer crunchbase business laascnrs computer riese government dailymed medical libris books semanticweborg computer dblpberlin computer lingvoj reference semwebcentral social dblphannover computer linkedct medical siocsites social dblprkbexplorer computer linkedmdb movie surgeradio music dbpedia general magnatune music swconferencecorpus computer doapspace social musicbrainz music taxonomy reference drugbank medical myspacewrapper social umbel general eurecom computer opencalais reference uniref biology eurostat government opencyc general unists biology flickrexporter images openguides reference uscensusdata government flickrwrappr images pdb biology virtuososponger reference foafprofiles social pfam biology w3cwordnet reference freebase general pisa computer wikicompany business geneid biology prodom biology worldfactbook government geneontology biology projectgutenberg books yago general geonames geographic prosite biology ...

Web of Data Dataset Dependencies homologenekegg projectgutenberg symbol libris cas bbcjohnpeel unists diseasome dailymed w3cwordnet chebi hgnc pubchem eurostat mgi omim wikicompany geospecies geneid reactome drugbank worldfactbook magnatune pubmed opencyc uniparc freebase linkedct uniprot taxonomy interpro uniref geneontologypdb umbel yago pfam dbpedia bbclatertotp govtrack prosite prodom flickrwrappropencalais uscensusdata surgeradio lingvoj linkedmdb virtuososponger homologenekegg projectgutenberg rdfbookmashup symbol libris swconferencecorpus geonames musicbrainz myspacewrapper dblpberlin pubguide cas bbcjohnpeel revyu unists jamendo diseasome dailymed w3cwordnet chebi rdfohloh hgnc bbcplaycountdata pubchem eurostat mgi omim wikicompany geospecies semanticweborg siocsites riese geneid foafprofiles reactome drugbank worldfactbook audioscrobbler bbcprogrammes magnatune dblphannover openguides pubmed opencyc uniparc crunchbase freebase linkedct uniprot taxonomy doapspace interpro uniref geneontology pdb umbel yago pfam dbpedia bbclatertotp govtrack flickrexporter budapestbme qdos prosite prodom flickrwrappropencalais semwebcentral uscensusdata eurecom ecssouthampton dblprkbexplorer surgeradio newcastle lingvoj linkedmdb pisa rae2001 virtuososponger acm eprints irittoulouse rdfbookmashup laascnrs citeseer swconferencecorpus geonames musicbrainz myspacewrapper ieee resex dblpberlin pubguide ibm revyu jamendo rdfohloh bbcplaycountdata semanticweborg siocsites riese foafprofiles openguides audioscrobbler bbcprogrammes dblphannover crunchbase doapspace flickrexporter

Web of Data Transforms Development Paradigm A new application development paradigm emerges. No longer do data and application providers need to be the same entity (left). With the Web of Data, its possible for developers to write applications that utilize data that they do not maintain (right).50 Application 1 Application 2 Application 3 Application 1 Application 2 Application 3 processes processes processes processes processes processes Web of Data structures structures structures structures structures structures 127.0.0.1 127.0.0.2 127.0.0.3 127.0.0.1 127.0.0.2 127.0.0.3 50 Rodriguez, M.A., “A Reﬂection on the Structure and Process of the Web of Data,” Bulletin of the American Society for Information Science and Technology, 35(6), pp. 38–43, doi:10.1002/bult.2009.1720350611, 2009. [http://arxiv.org/abs/0908.0373]

Extending our Knowledge of Bryce Canyon National Park gremlin $h := lds:open() gremlin $_ := g:add-v($h, ‘http://dbpedia.org/resource/Bryce_Canyon_National_Park’) ==v[http://dbpedia.org/resource/Bryce_Canyon_National_Park] gremlin ./outE ==e[dbpedia:Bryce_Canyon_National_Park - dbpprop:reference - http://www.nps.gov/brca/] ==e[dbpedia:Bryce_Canyon_National_Park - dbpprop:iucnCategory - II@en] ==e[dbpedia:Bryce_Canyon_National_Park - dbpedia-owl:numberOfVisitors - 1012563^^xsd:integer] ==e[dbpedia:Bryce_Canyon_National_Park - skos:subject - dbpedia:Category:Colorado_Plateau] ==e[dbpedia:Bryce_Canyon_National_Park - dbpprop:visitationNum - 1012563^^xsd:int] ==e[dbpedia:Bryce_Canyon_National_Park - dbpedia-owl:abstract - Bryce Canyon National Park is a national park located in southwestern Utah in the United States...@en] ==e[dbpedia:Bryce_Canyon_National_Park - dbpprop:area - 35835.0^^http://dbpedia.org/datatype/acre] ==e[dbpedia:Bryce_Canyon_National_Park - rdf:type - dbpedia-owl:ProtectedArea] ==e[dbpedia:Bryce_Canyon_National_Park - dbpedia-owl:location - dbpedia:Garfield_County%2C_Utah] ==e[dbpedia:Bryce_Canyon_National_Park - dbpprop:nearestCity - dbpedia:Panguitch%2C_Utah] ==e[dbpedia:Bryce_Canyon_National_Park - dbpprop:established - 1928-09-15^^xsd:date] ... 51 51 Linked Data Sail (LDS) was developed by Joshua Shinavier (RPI and TinkerPop) and connects to Gremlin through Gremlin’s native support for Sail (i.e. for RDF graphs). LDS caches the traversed aspects of the Web of Data into any quad-store (e.g. MemoryStore, AllegroGraph, HyperGraphSail, Neo4jSail, etc.).

Augmenting Traversals with the Web of Data Lets extend our query over the Web of Data. Perhaps incorporate that into our searching, scoring, ranking, and recommendation. gremlin $visits := ./outE[@label=‘dbpprop:visitationNum’]/inV/@value ==1012563 gremlin $acreage := ./outE[@label=‘dbpprop:area’]/inV/@value ==35835.0 ### imagine wrapping traversals in Gremlin functions: ### func lds:acreage($h, $v) and func lds:visitors($h, $v) gremlin ./outE[@label=‘friend’]/inV/outE[@label=‘favorite’] /inV[lds:acreage($h, .) 1000000 and lds:visitors($h, .) 2000000]/@name ==Bryce Canyon Thus, what do tenderlove’s friends favorite that are small in acreage and visitation?52 52 In Gremlin, its possible to have multiple graphs open in parallel and thus, mix and match data from each graph as desired. Hence, demonstrated by the example above, its possible to mix Web of Data RDF graph data and Blueprints property graph data.

Using the Web of Data for Music Recommendation Yet another aside: Using only the Web of Data data to recommend musicians/bands with a simplistic, edge-boolean spreading activation algorithm.53 gremlin $_ := ==The Tubes g:id(‘http://dbpedia.../Grateful_Dead’) ==Bob Dylan ==v[http://dbpedia.../Grateful_Dead] ==New Riders of the Purple Sage gremlin lds:spreading-activation(.) ==Bruce Hornsby ==Jerry Garcia Acoustic Band ==Donna Jean Godchaux ==BK3 ==Kingfish ==Phil Lesh and Friends ==Jerry Garcia Band ==Old and In the Way ==Donna Jean Godchaux Band ==RatDog ==The Other Ones ==The Dead ==Bobby and the Midnites ==Heart of Gold Band ==Furthur ==Legion of Mary ==Rhythm Devils 53 Please read the following for interesting, deeper ideas in this space: Clark, A., “Associative Engines: Connectionism, Concepts, and Representational Change,” MIT Press, 1993.

Another View of the TinkerPop Stack GET http://{host}/{resource} Local Dataset Web of Data owl:sameAs

Scoring Example: How Many of My Friends Favorite X? gremlin . name=marko location=Santa Fe lat=11111 ==v[1] gender=male long=22222 gremlin ./outE[@label=‘friend’]/inV ==v[3] ==v[2] 3 4 gremlin g:count(./outE[@label=‘friend’]/inV created_at=123456 /outE[@label=‘favorite’] friend favorite name=sixwing /inV[@id=6]) location=West Hollywood ==2 gender=male friend 1 2 favorite favorite created_at=234567 friend 6 name=Bryce Canyon favorite 5 name=charlie