multi-relational graph structures: from algebra to application

Multi-Relational Graph Structures:

From Algebra to Application

Marko A. RodriguezT-5, Center for Nonlinear StudiesLos Alamos National Laboratoryhttp://markorodriguez.com

October 27, 2009

Abstract

In a single-relational graph, all edges share the same meaning. In contrast,a multi-relational graph represents a heterogeneous set of edges, whereeach edge is labeled to denote the type of relationship that exists betweenthe two vertices it connects. While less prevalent than the single-relationalgraph, the multi-relational graph structure is beginning to see widespreadadoption in both academia and industry. An algebra for manipulatingmulti-relational graph structures and the realization of this algebra invarious application scenarios is presented in this talk.

MIT Lincoln Laboratory Lecture – Lexington, Massachusetts – October 27, 2009

My Computer Eco-System

• Articles/Lectures: LATEX, OmniGraffle, LATEX iT

• Software Development: Java, R Statistics

• Large-Scale Data Management: MySQL, Neo4j, Linked Process

• Graph/Network Analysis: iGraph, rPath, Confluence, JUNG

• Web of Data/Semantic Web: Open Sesame (SAIL), Protege

• 3D Modeling/Programming: Java Monkey Engine, Blender, Gimp

• Audio Synthesis/Processing: Max/MSP, ProTools


Outline

• Introduction to Graph Structures

? The Single-Relational Graph? The Multi-Relational Graph

• A Multi-Relational Path Algebra

• Application to Recommender Systems


A Single-Relational Graph Example

Article A

Article B

Article E

Article D

Article C Article F

An article citation graph. Each vertex is an article and each edge denotes that the tail

article cites the head article.


Single-Relational Graph Notation

• Homogenous set of vertex and edge types.1

• There are undirected and directed forms, where V is the set of verticesand E is an unordered or ordered set of edges, respectively.

? G = (V,E ⊆ V × V )? G = (V,E ⊆ (V × V )) (we will focus on directed graphs in this talk.)

• There is an adjacency matrix representation A ∈ 0, 1n×n, wheren = |V | and

Ai,j =

1 if (i, j) ∈ E0 otherwise.

1Unless the graph is bipartite.


The Use of Single-Relational Graphs in Research

• Most common graph structure used in 90’s and 00’s research.

? scholarly graphs: citations, coauthorship relationships, article/journalusage, acknowledgements, funding sources.

? technological graphs: software dependencies, Internet architecture,web citations.

? communication graphs: email correspondence, cell phone calls,micro-blog “following.”

• Numerous algorithms have been developed for analyzing such structures.

? geodesics: radius, diameter, eccentricity, closeness, betweenness.? spectral: eigenvector centrality, pagerank, spreading activation.? community detection: walktrap, edge betweenness, leading

eigenvector, spin-glass.


My Work with Single-Relational Graphs

• Articles of mine that make use of the single-relational graph structure.

? Bollen, J., Van de Sompel, H., Hagberg, A., Bettencourt, L.M.A, Chute, R., Rodriguez, M.A., Balakireva,L.L., “Clickstream Data Yields High-Resolution Maps of Science,” PLoS One, 4(3), e4803, 2009.

? Bollen, J., Van de Sompel, H., Rodriguez, M.A., “Towards Usage-Based Impact Metrics: First Results from the MESURProject,” Joint Conference on Digital Libraries (JCDL), 2008.

? Rodriguez, M.A., Pepe, A., “On the Relationship Between the Structural and Socioacademic Communities ofa Coauthorship Network,” Journal of Informetrics, 2(3), pp. 195–201, 2008.

? Rodriguez, M.A., Bollen, J., “An Algorithm to Determine Peer-Reviewers,” Conference on Information and KnowledgeManagement (CIKM), pp. 319–328, 2008.

? Rodriguez, M.A., Bollen, J., Van de Sompel, H., “Mapping the Bid Behavior of Conference Referees,” Journalof Informetrics, 1(1), pp. 62–82, 2007.

? Bollen, J., Rodriguez, M.A., Van de Sompel, H., “Journal Status,” Scientometrics, 69(3), pp. 669-687, 2006.? Rodriguez, M.A., Bollen, J., Van de Sompel, H., “The Convergence of Digital Libraries and the Peer-Review Process,”

Journal of Information Science, 32(2), pp. 149–159, 2006.? Rodriguez, M.A., Steinbock, D.J., “A Social Network for Societal-Scale Decision-Making Systems,” Proceedings of the

North American Association for Computational Social and Organizational Science Conference, 2004.

• They focus on supporting/analyzing/ranking/visualizing the scholarlycommunity and large-scale decision support systems (i.e. governancesystems).


Studying the Reading Behavior of Scholars

Bollen, J., Van de Sompel, H., Hagberg, A., Bettencourt, L.M.A, Chute, R., Rodriguez, M.A., Balakireva, L.L., “Clickstream

Data Yields High-Resolution Maps of Science,” PLoS One, 4(3), e4803, 2009.


Studying Characteristics that Lead to Coauthorship

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Rodriguez, M.A., Pepe, A., “On the Relationship Between the Structural and Socioacademic Communities of a Coauthorship

Network,” Journal of Informetrics, 2(3), pp. 195–201, 2008.


Predicting Referees Based on Coauthorship Patterns

SOMPEL

NELSON

LAGOZE

ARMS

MARCHIONINI

JESUROGA

FOO

LIM

SUGIMOTO

BORGMAN

FOX

MARSHALL

LEGGETTCHEN

GOLOVCHINSKYFURUTA

WITTEN

CUNNINGHAM

FUHR

NEUHOLD SOLVBERG

TAYLOR

SUMNER

FULKER

WRIGHT

JANEE

THANOS

KHOO

GIERSCH

ALLEN

SANCHEZRASMUSSENLYNCH BAKER

MOORE

RAY

CASSEL

TSE

CASTELLI

RECKER BISHOFF

Rodriguez, M.A., Bollen, J., Van de Sompel, H., “Mapping the Bid Behavior of Conference Referees,” Journal of Informetrics,

1(1), pp. 62–82, 2007.


A Multi-Relational Graph Example

Person A

Article B

Person E

Article D

Article C Article F

authored

cites

authored

peer-reviewed

cites

authored

acknowledges

A scholarly graph. Each vertex is a scholarly artifact and each edge denotes the type of

directed relationship that exists between the two scholarly artifacts it connects.


Multi-Relational Graph Notation

• Heterogeneous set of vertex types and a heterogeneous set of edge types.

• This data structure is becoming more prevalent due to both the SemanticWeb/Web of Data movement and the graph database movement.

• G = (V,E = E0, E1, . . . , Em ⊆ (V ×V )), where E is a family of typededge sets of length m. For example, E0 is the “authored” adjacencymatrix, E1 is the “cites” adjacency matrix, etc.

• There is a three-way tensor representation A ∈ 0, 1n×n×m, where

Aki,j =

1 if (i, j) ∈ Ek : k ≤ m0 otherwise.


A Three-Way Tensor Representation of aMulti-Relational Graph

0

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1

1

1

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0

0 0 0

0

0

0

0 0

0

0

0 0

0

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A ! 0, 1n!n!m

|E| =m |V | = n

|V|=n

authoredcite

s...


My Work with Multi-Relational Graphs

• Articles of mine that make use of the multi-relational graph structure.

? Rodriguez M.A., Shinavier, J., “Exposing Multi-Relational Networks to Single-Relational Network AnalysisAlgorithms,” Journal of Informetrics, in press, 2009. [Presented in the second part of this presentation.]

? Rodriguez, M.A., Geldart, J., “An Evidential Path Logic for Multi-Relational Networks,” Proceedings of the Associationfor the Advancement of Artificial Intelligence Spring Symposium: Technosocial Predictive Analytics Symposium, volumeSS-09-09, pp. 114–119, 2009.

? Rodriguez M.A., Bollen, J., Van de Sompel, H., “Automatic Metadata Generation using Associative Networks,” ACMTransactions on Information Systems, 27(2), pp. 1–20, 2009.

? Rodriguez, M.A., “Grammar-Based Random Walkers in Semantic Networks,” Knowledge-Based Systems,21(7), pp. 727–739, 2008. [Presented in the third part of this presentation.]

? Rodriguez, M.A., “Social Decision Making with Multi-Relational Networks and Grammar-Based Particle Swarms,” HawaiiInternational Conference on Systems Science (HICSS), pp. 39–49, 2007.

? Bollen, J., Rodriguez, M.A., Van de Sompel, H., Balakireva, L.L., Hagberg, A., “The Largest Scholarly SemanticNetwork...Ever.,” ACM World Wide Web Conference, 2007.

? Rodriguez, M.A., “A Multi-Relational Network to Support the Scholarly Communication Process,” International Journalof Public Information Systems, 2007(1), pp. 13–29, 2007.

• They focus on multi-relational graph algorithms, logic, informationretrieval, decision support systems, bibliometrics, recommender systems.


Resource Description Framework Graph

lanl:person_a

lanl:article_b

lanl:person_e

lanl:article_d

lanl:article_c lanl:article_f

lanl:authored

lanl:cites

lanl:authored

lanl:peer_reviewed

lanl:cites

lanl:authored

lanl:acknowledges

lanl: !" http://lanl.gov#

A scholarly graph. Each vertex and edge type is identified by a Uniform Resource

Identifier and thus, encoded in the address space of the World Wide Web.


Resource Description Framework Graph

• Vertices and edge labels are identified by Uniform Resource Identifiers(URI). Thus, there is a single address space where the world’s data canbe interrelated.

• G = (U ∪ B) × U × (U ∪ B ∪ L), where U is the set of all URIs, B isthe set of all blank nodes, and L is the set of all literals.

• There exist various implementations of this standard model.

? Open Sesame (http://openrdf.org/).? AllegroGraph (http://www.franz.com/agraph/allegrograph/).? OWLim (http://www.ontotext.com/owlim/).? Jena (http://jena.sourceforge.net/)


Linked Data and the Web of Data

http://dbpedia.org/resource/Albert Einstein

dbpedia:Albert_Einstein

dbpedia:United_States

flickr:Albert_Einstein

dbpedia:Alfred_Kleiner

dbpedia:citizenship dbpedia:doctoralAdvisor

dbpprop:hasPhotoCollection

http://farm4.static.flickr.com/3408/3547607847_65abfd03a5_m.jpg

foaf:depiction

http://farm1.static.flickr.com/60/170621225_661c705eb4_m.jpg

foaf:depiction

http://dbpedia.org/resource/Albert Einstein

http://www4.wiwiss.fu-berlin.de/flickrwrappr/photos/Albert_Einstein


My Work with Resource Description Framework Graphs

• Articles of mine that make use of RDF/Web of Data/Semantic Web.

? Rodriguez, M.A., “Interpretations of the Web of Data,” Data Management in the Semantic Web, eds. H. Jin and Z. Lv,Nova, in press, 2009.

? Rodriguez, M.A., “A Reflection on the Structure and Process of the Web of Data,” Bulletin of the American Society forInformation Science and Technology, 35(6), pp. 38–43, 2009.

? Rodriguez, M.A., “A Graph Analysis of the Linked Data Cloud,” http://arxiv.org/abs/0903.0194, February2009.

? Rodriguez, M.A., Allen, D.W., Shinavier, J., Ebersole, G., “A Recommender System to Support the ScholarlyCommunication Process,” KRS-2009-02, 2009. [Presented in the third part of this presentation.]

? Rodriguez, M.A., Watkins, J., “Faith in the Algorithm, Part 2: Computational Eudaemonics,” Lecture Notes in ArtificialIntelligence, eds. Velsquez, J.D., Howlett, R.J., and Jain, L.C., volume 5712, pp 813–820, 2009.

? Rodriguez, M.A., “General-Purpose Computing on a Semantic Network Substrate,” Emergent Web Intelligence,Advanced Information and Knowledge Processing series, Eds. R. Chbeir, A. Hassanien, A. Abraham, and Y. Badr, inpress, 2008.

? Rodriguez, M.A., Pepe, A., Shinavier, J., “The Dilated Triple,” Emergent Web Intelligence, Advanced Information andKnowledge Processing series, eds. R. Chbeir, A. Hassanien, A. Abraham, and Y. Badr, in press, 2008.

• They focus on graph algorithms, distributed computing, graph-basedcomputing, recommender systems.


The Web of Data as of March 2009

geospecies

freebase

dbpedia

libris

geneid

interpro

hgnc

symbol

pubmed

mgi

geneontology

uniprot

pubchem

unists

omim

homologene

pfam

pdb

reactome

chebi

uniparc

kegg

cas

uniref

prodomprosite

taxonomy

dailymed

linkedct

acm

dblprkbexplorer

laascnrs

newcastle

eprints

ecssouthampton

irittoulouseciteseer

pisa

resexibm

ieee

rae2001

budapestbme

eurecom

dblphannover

diseasome

drugbank

geonames

yago

opencyc

w3cwordnet

umbel

linkedmdb

rdfbookmashup

flickrwrappr

surgeradio

musicbrainz myspacewrapper

bbcplaycountdata

bbcprogrammes

semanticweborg

revyu

swconferencecorpus

lingvoj

pubguide

crunchbase

foafprofiles

riese

qdos

audioscrobbler

flickrexporter

bbcjohnpeel

wikicompany

govtrack

uscensusdata

openguides

doapspace

bbclatertotp

eurostat

semwebcentral

dblpberlin

siocsites

jamendo

magnatuneworldfactbook

projectgutenberg

opencalais

rdfohloh

virtuososponger

geospecies

freebase

dbpedia

libris

geneid

interpro

hgnc

symbol

pubmed

mgi

geneontology

uniprot

pubchem

unists

omim

homologene

pfam

pdb

reactome

chebi

uniparc

kegg

cas

uniref

prodomprosite

taxonomy

dailymed

linkedct

acm

dblprkbexplorer

laascnrs

newcastle

eprints

ecssouthampton

irittoulouseciteseer

pisa

resexibm

ieee

rae2001

budapestbme

eurecom

dblphannover

diseasome

drugbank

geonames

yago

opencyc

w3cwordnet

umbel

linkedmdb

rdfbookmashup

flickrwrappr

surgeradio

musicbrainz myspacewrapper

bbcplaycountdata

bbcprogrammes

semanticweborg

revyu

swconferencecorpus

lingvoj

pubguide

crunchbase

foafprofiles

riese

qdos

audioscrobbler

flickrexporter

bbcjohnpeel

wikicompany

govtrack

uscensusdata

openguides

doapspace

bbclatertotp

eurostat

semwebcentral

dblpberlin

siocsites

jamendo

magnatuneworldfactbook

projectgutenberg

opencalais

rdfohloh

virtuososponger

Rodriguez, M.A., “A Graph Analysis of the Linked Data Cloud,” http://arxiv.org/abs/0903.0194, February 2009.


The Web of Data as of March 2009data set domain data set domain data set domain

audioscrobbler music govtrack government pubguide booksbbclatertotp music homologene biology qdos socialbbcplaycountdata music ibm computer rae2001 computerbbcprogrammes media ieee computer rdfbookmashup booksbudapestbme computer interpro biology rdfohloh socialchebi biology jamendo music resex computercrunchbase business laascnrs computer riese governmentdailymed medical libris books semanticweborg computerdblpberlin computer lingvoj reference semwebcentral socialdblphannover computer linkedct medical siocsites socialdblprkbexplorer computer linkedmdb movie surgeradio musicdbpedia general magnatune music swconferencecorpus computerdoapspace social musicbrainz music taxonomy referencedrugbank medical myspacewrapper social umbel generaleurecom computer opencalais reference uniref biologyeurostat government opencyc general unists biologyflickrexporter images openguides reference uscensusdata governmentflickrwrappr images pdb biology virtuososponger referencefoafprofiles social pfam biology w3cwordnet referencefreebase general pisa computer wikicompany businessgeneid biology prodom biology worldfactbook governmentgeneontology biology projectgutenberg books yago generalgeonames geographic prosite biology . . .


Application Development on the Web of Data

Web of Data

127.0.0.1 127.0.0.2 127.0.0.3

Application 1 Application 2 Application 3

structures structuresstructures

processes processes processes

127.0.0.1 127.0.0.2 127.0.0.3

Application 1 Application 2 Application 3a. b.

structures structures structures

processes processes processes

a.) standard model b.) Web of Data model — public data changes the development

paradigm.


A Key/Value Graph Example

A

B

E

D

C F

type = personname = Markoage = 29

type = articlename = "Algori..."created = 1/1/09

type = articlename = "Network..."created = 2/1/08

type = citesweight = 1.0

type = authoredweight = 1.0

type = citesweight = 1.0

type = acknowledgesweight = 1.0

type = personname = Johanage = 37

type = peer-reviewedweight = -1.0

type = authoredweight = 0.5

type = authoredweight =1.0

type = articlename = "A Distributed..."created = 12/1/07

type = articlename = "Linked..."created = 1/30/09

A scholarly graph. Both vertices and edges maintain a key/value pair map that allows metadata to be

attached to them.


Key/Value Graph

• G = (V,E ⊆ (V ×V ), λ : (V ∪E)×Ω→ Σ), where Ω is the set of keysand Σ is the set of values.

• Has a convenient representation in object-oriented programminglanguages and used by various standards and graph packages.

? GraphML (http://graphml.graphdrawing.org/).? Neo4j (http://neo4j.org).? NetworkX (http://networkx.lanl.gov).? Confluence (http://markorodriguez.com/docs/conf/api/).? iGraph (http://igraph.sourceforge.net/).


Outline






Problem Statement

• There is a need to port all the known single-relational graph analysisalgorithms over to the multi-relational domain.

? Why?: There is a large body of algorithms in the domain of single-relational graph analysis.

? Why?: Multi-relational graph structures are becoming more prevalentand can be used to model more complex structures.

• The set of single-relational graph analysis algorithms should not be“blindly” applied to multi-relational graphs.

? Why?: For example, 〈marko, knows, johan〉 says more about socialcommunicaiton than 〈marko, livesInSameCityAs, bob〉.

? Why?: Multi-relational graph analysis algorithms must respect themeaning of the edges.


Solution Statement

• Provide an algebra to map a multi-relational graph to a“semantically-rich” single-relational graph that can be subjectedto all the known single-relational graph analysis algorithms.

Rodriguez M.A., Shinavier, J., “Exposing Multi-Relational Networks to

Single-Relational Network Analysis Algorithms,” Journal of Informetrics,

ISSN:1751-1577, Elsevier, doi:10.1016/j.joi.2009.06.004,

http://arxiv.org/abs/0806.2274, LA-UR-08-03931, in press, 2009.


A Three-Way Tensor Representation of aMulti-Relational Graph

As stated previously, a three-way tensor can be used to represent amulti-relational graph. If

G = (V,E = E0, E1, . . . , Em ⊆ (V × V ))

is a multi-relational graph, then A ∈ 0, 1n×n×m and

Aki,j =

1 if (i, j) ∈ Ek : k ≤ m0 otherwise.

A is the three-way tensor representation of the multi-relational graph.


The General Purpose of the Path Algebra

• Map a multi-relational tensor A ∈ 0, 1n×n×m to a single-relational path matrix

Z ∈ Rn×n+ — this path matrix is a weighted single-relational graph.

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A ! 0, 1n!n!m Z ! Rn!n+

1

3 4

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72

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15.3

12

!

• The created single-relational graph’s edges are loaded with meaning. For example,

given the right tensor, it is possible to create a coauthorship graph for scholars from

the same university who are not on the same project, but share a graduate student.

• The theorems of the algebra can be used to manipulate your operation to a more

efficient form.


The Elements of the Path Algebra

• A ∈ 0, 1n×n×m: a three-way tensor representation of a multi-relationalgraph.

• Z ∈ Rn×n+ : a path matrix derived by means of operations applied to A.

——————————————————————————————

• Cj ∈ 0, 1n×n: a “to” path filter.

• Ri ∈ 0, 1n×n: a “from” path filter.

• Ei,j ∈ 0, 1n×n: an entry path filter.

• I ∈ 0, 1n×n: the identity matrix as a self-loop filter.

• 1 ∈ 1n×n: a matrix in which all entries are equal to 1.

• 0 ∈ 0n×n: a matrix in which all entries are equal to 0.


The Operations of the 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 selectivelyexclude paths.

• n(A): not generates the complement of a 0, 1n×n matrix.

• c(A): clip generates a 0, 1n×n matrix from a Rn×n+ matrix.

• v±(A): vertex generates a 0, 1n×n matrix from a Rn×n+ matrix, where

only certain rows or columns contain non-zero values.

• λA: scalar multiplication weights the entries of a matrix.

• A + B: matrix addition merges paths.


hA1 : authored

i hA2 : cites

i hA3 : contains

i hA4 : category

i hA5 : developed

iExample Scholarly Tensor Used in the Remainder of the

Presentation

• A1 authored : human→ article

• A2 cites : article→ article

• A3 contains : journal→ article

• A4 category : journal→ subject category

• A5 developed : human→ program/software.


The Traverse Operation

• An interesting aspect of the single-relational adjacency matrix A ∈ 0, 1n×n is that when it is raised

to the kth power, the entry A(k)i,j is equal to the number of paths of length k that connect vertex i to

vertex j.

• Given, by definition, that A(1)i,j (i.e. Ai,j) represents the number of paths that go from i to j of length

1 (i.e. a single edge) and by the rules of ordinary matrix multiplication,

A(k)i,j =

∑l∈V

A(k−1)i,l ·Al,j : k ≥ 2.

0

0

1

0

0

0 0

1

0 0

0

1

0

0

0 0

1

0

·0

0

0

0

0

0 1

0

0

=

a b c

a b c

a

b

c

a b c a b c

a

b

c

a

b

c

there is a path of length 2 from a to c


hA1 : authored

i hA2 : cites

i hA3 : contains

i hA4 : category

i hA5 : developed

iThe Traverse Operation

Z = A1 · A2 · A1>,Zi,j defines the number of paths from vertex i to vertex j such that a path goes from author i to one the

articles he or she has authored, from that article to one of the articles it cites, and finally, from that cited

article to its author j. Semantically, Z is an author-citation single-relational path matrix.

Human A

authored

Article B

authored

Human D

Article Ccites

author-citation

A1

A2

A1!

Z

• NOTE: All diagrams are with respect to a “source” vertex (the blue vertex) in order to preserve clarity. In reality, theoperations operate on all vertices in parallel.


The Filter Operation

Various path filters can be defined and applied using the entry-wiseHadamard matrix product denoted , where

A B =

A1,1 ·B1,1 · · · A1,m ·B1,m... . . . ...

An,1 ·Bn,1 · · · An,m ·Bn,m

.

0

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0 0

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0! =

0

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23

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0 00

0

0

0

0 0

0

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0 0

0

0

Path Matrix Path Filter Filtered Path Matrix


The Filter Operation

• A 1 = A• A 0 = 0• A B = B A• A (B + C) = (A B) + (A C)• A> B> = (A B)>.


The Not Filter

The not filter is useful for excluding a set of paths to or from a vertex.

n : 0, 1n×n → 0, 1n×n

with a function rule of

n(A)i,j =

1 if Ai,j = 00 otherwise.

0

0

0

1

1

1

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1

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1 00

0

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1 0

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0 1

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0=n

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1

0 11

1

1

1

0 1

1

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1 0

1

1


The Not Filter

If A ∈ 0, 1n×n, then

• n(n(A)) = A• A n(A) = 0• n(A) n(A) = n(A).


hA1 : authored

i hA2 : cites

i hA3 : contains

i hA4 : category

i hA5 : developed

iThe Not Filter

A coauthorship path matrix is

Z = A1 · A1> n(I)

Human A

authored

Article B

Human Ccoauthor

A1 A1!

Z

authored

coauthor

n(I)


The Clip Filter

The general purpose of clip is to take a path matrix and “clip,” ornormalize, it to a 0, 1n×n matrix.

c : Rn×n+ → 0, 1n×n

c(Z)i,j =

1 if Zi,j > 00 otherwise.

0

0

0

72

1

15.3

0

0

0

23

0

24 00

0

0

0

4 0

0

0

0 12

0

0

0

0

0

1

1

1

0

0

0

1

0

1 00

0

0

0

1 0

0

0

0 1

0

0=c


The Clip Filter

If A,B ∈ 0, 1n×n and Y,Z ∈ Rn×n+ , then

• c(A) = A• c(n(A)) = n(c(A)) = n(A)• c(Y Z) = c(Y) c(Z)• n(A B) = c (n(A) + n(B))• n(A + B) = n(A) n(B)


hA1 : authored

i hA2 : cites

i hA3 : contains

i hA4 : category

i hA5 : developed

iThe Clip Filter

Suppose we want to create an author citation path matrix that does not allow self citation or coauthorcitations.

Z =

„A1 · A2 · A1>

«| z

cites

n

„c

„A1 · A1> n(I)

««| z

no coauthors

n(I)|zno self

Human A

authored

Article B

authored

Human E

Article Ccites

author-citation

A1

A2

A1!

Z

authored

Human D

A1!

authored

coauthor

self n(I)

n!c!A1 · A1! ! n(I)

""


hA1 : authored

i hA2 : cites

i hA3 : contains

i hA4 : category

i hA5 : developed

iThe Clip Filter

However, using various theorems of the algebra,

Z =(A1 · A2 · A1>

)︸︷︷︸

cites

n(c(A1 · A1> n(I)

))︸︷︷︸

no coauthors

n(I)︸︷︷︸no self

becomes

Z =(A1 · A2 · A1>

) n(c(A1 · A1>

)) n(I).


The Vertex Filter

In many cases, it is important to filter out particular paths to and from avertex.

v− : Rn×n+ × N→ 0, 1n×n,

v−(Z)i,j =

1 if

∑k∈V Zi,k > 0

0 otherwise

turns a non-zero column into an all 1-column and

v+ : Rn×n+ × N→ 0, 1n×n,

v+(Z)i,j =

1 if

∑k∈V Zk,j > 0

0 otherwise

turns a non-zero row into an all 1-row.


The Vertex Filter

0

23

2

0

1

0

0

0

0

0

0

0 10

0

0

0

0 0

32

0

0 0

0

0

1

1

1

1

1

0

0

0

0

0

0

0 10

0

0

0

1 0

1

1

1 0

0

0=v!

v+ not diagrammed, but acts the same except for makes 1-rows. Two import filters are the column and

row filters, C ∈ 0, 1n×n and R ∈ 0, 1n×n, respectively.

1

1

1

1

1

0

0

0

0

0

0

0 00

0

0

0

0 0

0

0

0 0

0

0

0

0

1

0

0

0

0

0

0

1

0

0 00

0

1

0

0 0

1

0

0 0

0

1C2 = R3 =


The Vertex Filter

• v−(Ci) = Ci

• v+(Rj) = Rj

• v−(Z) = v+(Z>)>• v+(Z) = v−(Z>)>.


hA1 : authored

i hA2 : cites

i hA3 : contains

i hA4 : category

i hA5 : developed

iThe Vertex Filter

Assume that vertex 1 is the social science subject category vertex and we want to create a journalcitation graph for social science journals only.

Z =hv+“C1 A4

” A3

i| z

soc.sci. journal articles

·A2 ·»A3> v

−„

R1 A4>«–

| z articles in soc.sci. journals

.

Social Science

Journal A

Journal E

Journal FArticle D

Article Ccategory

contains

contains

contains

Article B

cites

cites

category

v+!C1 !A4

"A3

A2

A2

A3!

A3!v!

!R1 !A4"

"

1social-science journal citation

Z


hA1 : authored

i hA2 : cites

i hA3 : contains

i hA4 : category

i hA5 : developed

iThe Vertex Filter

hv+“C1 A4

” A3

i| z


0000

0J-A

0

0

1111

1 00

00

0

0 0000 000

A-B

A-C

A-D

J-E

J-F

S

J-A A-B A-C A-DJ-E J-FS

0 000

0 0

0 01 0 0 0 00 01 0 0 0 0

0

00

0

C1

0000

0J-A

0

0

0011

0 00

00

0

0 0000 000

A-B

A-C

A-D

J-E

J-F

S


0 000

0 0

0 00 0 0 0 00 00 0 0 0 0

0

00

0

A4

0011

0J-A

0

0

0011

0 00

11

0

1 1100 001

A-B

A-C

A-D

J-E

J-F

S


0 011

0 0

0 00 0 0 0 00 00 0 0 0 0

0

11

0

v+(C1 !A4)

0000

0J-A

0

0

0000

0 00

00

0

0 1000 000

A-B

A-C

A-D

J-E

J-F

S


0 001

0 1

0 00 0 0 0 00 00 0 0 0 0

0

00

0! =

0000

0J-A

0

0

0000

0 00

00

0

0 1000 000

A-B

A-C

A-D

J-E

J-F

S


0 001

0 0

0 00 0 0 0 00 00 0 0 0 0

0

00

0

A3 v+(C1 !A4) !A3

! =0000

0J-A

0

0

0011

0 00

00

0

0 0000 000

A-B

A-C

A-D

J-E

J-F

S


0 000

0 0

0 00 0 0 0 00 00 0 0 0 0

0

00

0

C1 !A4


hA1 : authored

i hA2 : cites

i hA3 : contains

i hA4 : category

i hA5 : developed

iThe Vertex Filter

Z =[v+(C1 A4

) A3]︸︷︷︸


·A2 ·[A3> v−

(R1 A4>

)]︸︷︷︸

articles in soc.sci. journals

.

However,

v−(R1 A4>

)= v−

((C1 A4

)>)Cx = R>x

= v+(C1 A4

)>v+(Z) =v−(Z>)>.

Therefore, because A> B> = (A B)>,

Z =[v+(C1 A4

) A3]︸︷︷︸

reused

·A2 · [v+(C1 A4

) A3]︸︷︷︸

reused

>.


The Weight and Merge Operations

• λZ: scalar multiplication weights paths.

• Y + Z: matrix addition merges paths.

0

0

0

72

1

15.3

0

0

0

23

0

24 00

0

0

0

4 0

0

0

0 12

0

0

0

0

0

10

1

0

0

0

0

1

0

0 00

0

34

0

0 0

0

0

0 2

0

0+ =

0

0

0

2

15.3

0

0

0

24

0

24 00

0

34

0

4 0

0

0

0 14

0

0

82


hA1 : authored

i hA2 : cites

i hA3 : contains

i hA4 : category

i hA5 : developed

iThe Weight and Merge Operations

Z = 0.6(A1 · A1> n(I)

)︸︷︷︸

coauthorship

+ 0.4(A5 · A5> n(I)

)︸︷︷︸

co-development

merges the article and software program collaboration path matrices asspecified by their respective weights of 0.6 and 0.4. The semantics of theresultant is a software program and article collaboration path matrix thatfavors article collaboration over software program collaboration. Asimplification of the previous composition is

Z =[0.6(A1 · A1>

)+ 0.4

(A5 · A5>

)] n(I).


Outline






kReef: A Scholarly Recommendation Engine

1. The scholarly community is modeled using a multi-relational graph.

2. A “walker”-version of the path algebra is applied to the graph to support scholars.

Multi-Relational Graph Database

ontologyinstances

Grammar WalkerEngine TranslatorsAnalytics

Engine

Graphical User Interface

1

2

Rodriguez, M.A., Allen, D.W., Shinavier, J., Ebersole, G., “A Recommender System to Support the Scholarly Communication

Process,” KRS-2009-02, http://arxiv.org/abs/0905.1594, 2009.


kReef: Ontology Classes

core:Agent core:Item core:Event

core:Personcore:Group

core:Organization core:Project

core:Call

core:Collection

core:Book

core:Journal

core:Librarycore:Dataset

core:FundingOpportunity

core:Magazine

core:Document

core:Article

core:Viewgraph

core:Webpage

core:Media

core:Academic

core:Commerical

core:Government

core:Audio

core:Image

core:Videocore:CallForChapters

core:CallForPapers

core:CallForProposals

core:CallForTutorials

core:CallForWorkshops

core:Software

core:Conference

core:Course

core:Meeting

core:Panel

core:Presentation

core:Session

core:SocialEvent

core:Tutorial

core:Workshop

core:Keynotecore:Newspaper

core:Proceedings

core:Reefsource

Ag

Gr Pe

Or

Ac

Cm

Gv

Pj

It Ev

Do Co

Ar Bo

Jo

Lb

Mg

Np

Po

Vg

Wp

Md

Fu

Da

Sw

Ca Au

Im

Vi

Cc

Cp

Cl

Ct

Cw

Cs

Cf

Me

Kn

Pn

Wk

Tu

Se

Ss

Ps

• NOTE: All edges denote an rdf:subClassOf relationship (either directly or inferred).


kReef: Ontology Properties

Table 1: core:Reefsource rdf:Property relationsrdf:Property rdfs:domain rdfs:range

core:title core:Reefsource xsd:string

core:abstract core:Reefsource xsd:string

core:guid core:Reefsource xsd:string

Table 2: core:Agent rdf:Property relationsrdf:Property rdfs:domain rdfs:range

core:attends core:Agent core:Event

core:created core:Agent core:Item

core:member core:Group core:Person

core:subGroup core:Group core:Group

core:firstName core:Person xsd:string

core:lastName core:Person xsd:string

core:occupation core:Person xsd:string

core:sex core:Person core:Gender

Table 3: core:Item rdf:Property relationsrdf:Property rdfs:domain rdfs:range

core:cites core:Item core:Item

core:containedIn core:Item core:Collection

core:creationTime core:Item xsd:dateTime

core:doi core:Item xsd:anyURI

core:publisher core:Item core:Group

core:dueDate core:Call xsd:dateTime

core:callFor core:Call core:Reefsource

core:contains core:Collection core:Item

core:editor core:Collection core:Agent

core:isbn core:Collection xsd:anyURI

core:issn core:Collection xsd:anyURI

core:oaipmh core:Library xsd:anyURI

core:startPage core:Article xsd:int

core:endPage core:Article xsd:int

core:number core:Article xsd:int

core:volume core:Article xsd:int

Table 4: core:Event rdf:Property relationsrdf:Property rdfs:domain rdfs:range

core:startTime core:Event xsd:dateTime

core:endTime core:Event xsd:dateTime

core:presents core:Event core:Item

core:organizedBy core:Event core:Agent

core:subEvent core:Event core:Event


kReef: Instance Data Ingestion

Multi-Relational Graph Database

ontologyinstances

arXiv CiteULike

CiteSeerCrossRef

BibSonomy

CogPrintsCogPrints

Connotea

ACM, IEEE, IOP, Springer, Blackwell, Elsevier, etc.


kReef: Grammar Walker Engine Overview

• A walker-based implementation of the path algebra is applied to thescholarly model in order to support scholars in their professional lives.The path description is known as a “grammar” because it can be modeledas a finite state machine embedded in the walker.

? identify articles related to some interesting resource.? identify collaborators for a funding opportunity.? identify a publication venue for a newly created article.? identify referees to review an article.? identify resources of interest in one’s community.

Rodriguez, M.A., “Grammar-Based Random Walkers in Semantic Networks,” Knowledge-Based Systems, 21(7), pp. 727–739,

http://arxiv.org/abs/0803.4355, 2008.


kReef: Grammar Walker Engine Algorithm, Part 1

• First, when trying to solve a recommendation problem, determine whichabstract path should be searched to find a solution — this is usuallybased on hunch and then validated using real-world data.

? For example, what makes a good peer-reviewer/referee for an article:someone that is cited by the article and their respective coauthors.Moreover, a referee should not include the authors of the article ortheir coauthors one step away in the coauthorship network (conflict ofinterest).

• Let us denote the path description/grammar/contraint ψ.

Rodriguez, M.A., Bollen, J., “An Algorithm to Determine Peer-Reviewers,” Conference on Information and Knowledge

Management (CIKM), pp. 319–328, http://arxiv.org/abs/cs/0605112, 2008.



• Program a collection of discrete walkers to traverse the abstractpath defined by ψ. Each walker starts at some vertex i ∈ V and withan energy value ε ∈ R. As it walks the graph, its energy decays.

? Given the peer-review/referee example, the source vertex is the articlethat requires a set of referees.

!!t=1

t=2

t=3

i



• The solution to the problem is where the highest energy flow inthe network exists after k time steps.

? Given the peer-review example, the highest energy vertices are thosepeople most competent to review the article in question.

In short,Ψ× P(V )→ ω,

where Ψ is the set of all grammars, P(V ) is the set of all sets of sourcevertices, and ω : V → R is the resultant energy flow for each vertex in thegraph. Or,

Grammar︸︷︷︸path description

× Set<Vertex>︸︷︷︸source vertices

→ Map<Vertex, Double>︸︷︷︸ranked results

.


Other Application Scenarios

• Populating metadata poor resources with data propagated from metadatarich resources. Walkers take particular paths, pick up metadata fromrich resources, and attach metadata to atrophied resources.

? Rodriguez M.A., Bollen, J., Van de Sompel, H., “Automatic Metadata Generation using Associative Networks,” ACM

Transactions on Information Systems, 27(2), pp. 1–20, http://arxiv.org/abs/0807.0023, 2009.

• Generate a context-senstive representative decision-making structure thatreflects the voting behavior of the full population even as the actual votingpopulation wanes in size.

? Rodriguez, M.A., “Social Decision Making with Multi-Relational Networks and Grammar-Based Particle Swarms,” Hawaii

International Conference on Systems Science (HICSS), pp. 39–49, http://arxiv.org/abs/cs/0609034, 2007.


Future Work in this Area

• Further develop the path algebra. Explore other matrix and tensoroperations and determine if they are meaningful in the context ofmanipulating multi-relational graphs.

• Develop a programming language (Turing Complete?) to easilyrepresent path descriptions for walkers. Make it easier for developersto deploy swarms of walkers within a multi-relational network for variousapplication scenarios.

? Recommender systems? Vertex and edge ranking systems? Information retrieval systems? General graph analysis


Conclusion

• Thank you for your time...

? My homepage: http://markorodriguez.com? Linked Process: http://linkedprocess.org? Neno/Fhat: http://neno.lanl.gov? Collective Decision Making Systems: http://cdms.lanl.gov? Faith in the Algorithm: http://faithinthealgorithm.net? MESUR: http://www.mesur.org


http://markorodriguez.com

http://linkedprocess.org

http://neno.lanl.gov

http://cdms.lanl.gov

http://faithinthealgorithm.net

http://www.mesur.org

multi-relational graph structures: from algebra to application

Technology

article citation graph

common graph structure

use of single

v and1

protg e e

scholarly graphs

directed graphs

tail article