A Path Algebra for Mapping Multi-Relational Networks to Single-Relational Networks

A Path Algebra for Mapping Multi-Relational
Networks to Single-Relational Networks
Marko A. Rodriguez
marko@lanl.gov
T-7: Mathematical Modeling and Analysis Group
CNLS: Center for Nonlinear Studies
Los Alamos National Laboratory

June 26, 2008

Single- and Multi-Relational Networks
Human-D
Human-B

Human-F
Human-C

Human-A

Human-E

Publisher-A
Article-A

publishedBy

authored editorOf Human-B
Journal-A

containedIn authored

Human-A
authored
Article-B

Center for Non-Linear Studies Public Lecture - June 26, 2008

Presentation Article

Rodriguez M.A., Shinavier, J., “Exposing Multi-Relational Networks to
Single-Relational Network Analysis Algorithms”, LA-UR-08-03931, May
2008, http://arxiv.org/abs/0806.2274.

Acknowledgements:

• Ideas inspired by the MESUR problem space [Bollen et al., 2007].
• Vadas Gintautas aided in reviewing drafts of the article and presentation.
• Michael Ham aided in reviewing drafts of the presentation.
• Razvan Teodorescu taught me FoilTex and this is his template (I’m sorry,
I just don’t know how to change the color scheme!)


Problem Statement
Think about all of the known network analysis algorithms:

• geodesics: diameter, eccentricity [Harary & Hage, 1995], closeness
[Bavelas, 1950], betweenness [Freeman, 1977], ...
• spectral: PageRank [Brin & Page, 1998], eigenvector centrality
[Bonacich, 1987], ...
• community detection: leading eigenvector [Newman, 2006], edge
betweenness [Girvan & Newman, 2002], ...
• mixing pattens: scalar and discrete assortativity [Newman, 2003], ...
• on and on and on...

These algorithms have been developed for directed or undirected
single-relational networks. What do you do when you have a
multi-relational network?


Problem Statement

Now think about all of the known network analysis packages:

• Java Universal Network/Graph Framework (JUNG) [O’Madadhain et al., 2005]
• iGraph: Package for Complex Network Research [Csardi, 2006]
• Pajek
• NetworkX [Hagberg et al., n.d.]
• on and on and on...

These packages (for the most part) have been developed for directed or
undirected single-relational networks. What do you do when you have a
multi-relational network?


Problem Statement

• Do you reimplement all of the known algorithms to support a multi-
relational network?

• Even if you do, what do these algorithms look like?


Solution Statement

• You map your multi-relational network to a “meaningful” single-relational
network and re-use existing algorithms, packages, and theorems from the
single-relational domain.


Outline

• Formalizing Single- and Multi-Relational Networks

• Background on Multi-Relational Network Analysis

• The Elements of the Path Algebra

• The Operations of the Path Algebra

• Multi-Relational Network Analysis


An Undirected Single-Relational Network
Human-D
Human-B

Human-F
Human-C

Human-A

Human-E

All edges have a single homogenous meaning (e.g. co-author).

G = (V, E ⊆ {V × V })


A Directed Single-Relational Network
Article-D
Article-B

Article-F
Article-C

Article-A

Article-E

All edges have a single homogenous meaning (e.g. citation).

G = (V, E ⊆ (V × V ))


A Multi-Relational Network
Publisher-A
Article-A

publishedBy

authored editorOf Human-B
Journal-A


Human-A
authored
Article-B

Edges are heterogenous in meaning.

M = (V, E = {E0 ⊆ (V × V ), E1, . . . , Em})


Flatten the Multi-Relational Network

• Suppose you have a multi-relational network, where there exists only two
edge sets defined as coauthor and friend.

M = (V, E = {E0, E1})

and you want to determine the most central “scholar” in this network.

• It is not sufficient to simply ignore edge labels (flatten the multi-
relational network to a single-relational network) and execute a centrality
algorithm on the network. You will confuse central friendship with central
scholarship.


Extract a Single-Relational Network Component

• You could simply pull out the coauthor single relational network

G = (V, E0)

and calculate a centrality algorithm on that network to get your result.

• That works, but for more complex situations with “richer semantics”,
this mechanism will not work.


Execute a Grammar-Based Walker
• A walker obeys a “grammar” that speciﬁes the way in which the walker should move through the network
[Rodriguez, 2008].

coauthor primary
eigenvector grammar Publisher-A

while(true)
incr vertex counter publishedBy
go authored
go authored -1
but don't go back editorOf Human-B
to previous vertex Journal-A


Human-A
authored
Article-B

coauthor

• Problem – this solution mixes the analysis algorithm and the traversed implicit network.
• Solution – an algebra that is agnostic to the ﬁnal executing algorithm.


An Adjacency Matrix Representation of a
Single-Relational Network

n = |V |
0 1 0 0 1 Article-A

Article-D
0 0 0 0 0 Article-B

n = |V |
Article-B

0 0 0 1 1 Article-C

Article-F
Article-C
0 0 0 0 0 Article-D

0 0 0 0 0 Article-E

Article-A

Article-C

Article-D
Article-A

Article-B

Article-E
Article-E

* NOTE: Sorry about missing the vertex Article-F in the adjacency matrix. Too lazy to redo diagrams.


An Adjacency Matrix Representation of a
Single-Relational Network

A single-relational network 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.


A Three-Way Tensor Representation of a
Multi-Relational Network

0 1 1 0 0 Human-A

Article-A
0 0 0 0 0

n = |V |
Publisher-A
Article-A 0 0 0 0 0 Article-B

publishedBy
0 0 1 0 0 Human-B

In
authored Human-B

d
editorOf

ne
Journal-A
0 0 Journal-A

f
0 0 0

rO

y
ai

dB
ito
nt
m

ed
co

he

ed
is

Article-A

Article-B

Journal-A
Human-A

Human-B
=

or
bl
authored

th
pu
containedIn

au
Human-A
|E
authored
Article-B | n = |V |

* NOTE: Sorry about missing the vertex Publisher-A in the tensor. Too lazy to redo diagrams.


A Three-Way Tensor Representation of a
Multi-Relational Network

A three-way tensor can be used to represent a multi-relational network
[Kolda et al., 2005]. If

M = (V, E = {E0, E1, . . . , Em ⊆ (V × V )})

is a multi-relational network, then A ∈ {0, 1}n×n×m and

1 if (i, j) ∈ Em
Am
i,j =
0 otherwise.


The General Purpose of the Path Algebra

• Map a multi-relational tensor A ∈ {0, 1}n×n×m to a single-relational path matrix Z ∈ Rn×n .
+
• By performing operations on A, a single-relational path matrix is created whose “edges” are loaded
with meaning.
• For example, you can create a coauthorship network, a social science journal citation network, a
coauthorship network for scholars from the same university who have not been on the same project in
the last 10 years, but are in the same department, etc.
• The theorems of the algebra can be used to manipulate your mapping operation to a smaller/more
eﬃcient form (i.e. how a composition is spoken in words can diﬀer from its reduced form).

0 1 1 0 0 24 1 0 0 0

0 0 0 0 0 0 72 0 4 0

0 0 0 0 0 23 0 0 0 0

0 0 1 0 0 0 0 15.3 0 0

0 0 0 0 0 0 0 0 0 12

A ∈ {0, 1}n×n×m Z ∈ Rn×n
+


The Elements of the Path Algebra

• A ∈ {0, 1}n×n×m: a three-way tensor representation of a multi-relational
network.
• Z ∈ Rn×n: a path matrix derived by means of operations applied to A.
+
——————————————————————————————
• Cj ∈ {0, 1}n×n: “to” path filters.
• Ri ∈ {0, 1}n×n: “from” path filters.
• I ∈ {0, 1}n×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.


A1 : authored A2 : cites A3 : contains A4 : category A5 : developed
h ih ih ih ih i

Example 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 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 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.
• λA: scalar multiplication weights the entries of a matrix.
• A + B: matrix addition merges paths.


The Traverse Operation
• An interesting aspect of the single-relational adjacency matrix A ∈ {0, 1}n×n is that when it is raised
(k)
to the kth power, the entry Ai,j is equal to the number of paths of length k that connect vertex i to
vertex j [Chartrand, 1977].
(1)
• Given, by deﬁnition, that Ai,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,

(k) (k−1)
Ai,j = Ai,l · Al,j : k ≥ 2.
l∈V

a b c

a b c a b c a b c

a 0 1 0 a 0 1 0 a 0 0 1

b 0 0 1 · b 0 0 1 = b 0 0 0

c 0 0 0 c 0 0 0 c 0 0 0
there is a path of length 2
from a to c


h ih ih ih ih i

The 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.

A2
Article-A cites Article-B

A1 authored A1
authored

Human-A author-citation Human-B

Z

* NOTE: All diagrams are with respect to a “source” vertex (the blue vertex) in order to preserve clarity. In reality, the operations

operate on all vertices in parallel.


The Filter Operation
Various path ﬁlters can be deﬁned and applied using the entry-wise
Hadamard matrix product denoted ◦, where
 
A1,1 · B1,1 · · · A1,m · B1,m
A◦B= .
. ... .
. .
An,1 · Bn,1 · · · An,m · Bn,m

24 1 0 0 0 0 1 0 0 0 0 1 0 0 0

0 72 0 4 0 0 1 0 0 0 0 72 0 0 0

23 0 0 0 0 ◦ 1 0 0 0 0 = 23 0 0 0 0

0 0 15.3 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 12 0 0 0 0 0 0 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, 1}n×n → {0, 1}n×n

with a function rule of

1 if Ai,j = 0
n(A)i,j =
0 otherwise.

0 0 1 1 1 1 1 0 0 0

1 0 1 0 1 0 1 0 1 0

n 0 1 1 1 1 = 1 0 0 0 0

1 1 0 1 1 0 0 1 0 0

1 1 1 1 0 0 0 0 0 1


The Not Filter

If A ∈ {0, 1}n×n, then

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


h ih ih ih ih i

The Not Filter
A coauthorship path matrix is

Z = A1 · A1 ◦ n(I)

Article-A

A1 authored
A1
authored

Human-A coauthor Human-B

Z
n(I)
coauthor


The Clip Filter
The general purpose of clip is to take a path matrix and “clip”, or
normalize, it to a {0, 1}n×n matrix.

c : Rn×n → {0, 1}n×n
+

1 if Zi,j > 0
c(Z)i,j =
0 otherwise.

24 1 0 0 0 1 1 0 0 0

0 72 0 4 0 0 1 0 1 0

c 23 0 0 0 0 = 1 0 0 0 0

0 0 15.3 0 0 0 0 1 0 0

0 0 0 0 12 0 0 0 0 1


The Clip Filter

If A, B ∈ {0, 1}n×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)


h ih ih ih ih i

The Clip Filter
Suppose we want to create an author citation path matrix that does not allow self citation or coauthor
citations. „ « „ „ ««
1 2 1 1 1
Z= A ·A ·A ◦n c A · A ◦ n(I) ◦ n(I)
|{z}
| {z } | {z } no self
cites no coauthors

Z
author-citation Human-C

authored
2
A A1
Article-A cites Article-B

A 1 A1
authored authored authored

Human-A coauthor Human-B

n c A1 · A1 ◦ n(I)

self n(I)


h ih ih ih ih i

The Clip Filter

However, using various theorems of the algebra,

Z = A1 · A2 · A1 ◦ n c A1 · A1 ◦ n(I) ◦ n(I)
no self
cites no coauthors

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 a
vertex.
v − : Rn×n × N → {0, 1}n×n,
+

− 1 if k∈V Zi,k > 0
v (Z)i,j =
0 otherwise
turns a non-zero column into an all 1-column and

v + : Rn×n × N → {0, 1}n×n,
+

+ 1 if k∈V Zk,j > 0
v (Z)i,j =
0 otherwise
turns a non-zero row into an all 1-row.


The Vertex Filter
0 1 0 1 0 0 1 0 1 0

0 0 0 0 0 0 1 0 1 0

v− 0 2 0 32 0
= 0 1 0 1 0

0 23 0 0 0 0 1 0 1 0

0 0 0 0 0 0 1 0 1 0

v + not diagrammed, but acts the same except for makes 1-rows. Two import ﬁlters are the column and
row ﬁlters, C ∈ {0, 1}n×n and R ∈ {0, 1}n×n , respectively.

0 1 0 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0 0 0

C2 = 0 1 0 0 0 R3 = 1 1 1 1 1

0 1 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0 0


The Vertex Filter

• v −(Ci) = Ci
• v +(Rj ) = Rj
• v −(Z) = v +(Z )
• v +(Z) = v −(Z ) .


h ih ih ih ih i

The Vertex Filter
Assume that vertex 1 is the social science subject category vertex and we want to create a journal
citation network for social science journals only.
» „ «–
+ 4 3 2 3 − 4
h “ ” i
Z= v C1 ◦ A ◦ A ·A · A ◦v R1 ◦ A .
| {z } | {z }
soc.sci. journal articles articles in soc.sci. journals

social-science journal citation
Z 1 Social
Science

category

category
A2 Article-B contains Journal-B

A3 cites A3
v − R1 ◦ A4
Journal-A contains Article-A
cites
+ 4
v C1 ◦ A 2
Article-C contains Journal-C
A
A3


h ih ih ih ih i

The Vertex Filter
+ 4 3
h “ ” i
v C1 ◦ A ◦A
| {z }
soc.sci. journal articles

S J-A J-B J-C A-A A-B A-C S J-A J-B J-C A-A A-B A-C S J-A J-B J-C A-A A-B A-C

S 1 0 0 0 0 0 0 S 0 0 0 0 0 0 0 S 0 0 0 0 0 0 0
J-A
1 0 0 0 0 0 0 J-A
1 0 0 0 0 0 0 J-A
1 0 0 0 0 0 0
J-B
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
◦
0 J-B 0 J-B 0
=
1 1 1
J-C 1 0 0 0 0 0 0 J-C 0 0 0 0 0 0 0 J-C 0 0 0 0 0 0 0
A-A 1 0 0 0 0 0 0 A-A 0 0 0 0 0 0 0 A-A 0 0 0 0 0 0 0
A-B 1 0 0 0 0 0 0 A-B 0 0 0 0 0 0 0 A-B 0 0 0 0 0 0 0
A-C 1 0 0 0 0 0 0 A-C 0 0 0 0 0 0 0 A-C 0 0 0 0 0 0 0

C1 A4 C1 ◦ A4
S J-A J-B J-C A-A A-B A-C S J-A J-B J-C A-A A-B A-C S J-A J-B J-C A-A A-B A-C

S 0 0 0 0 0 0 0 S 0 0 0 0 0 0 0 S 0 0 0 0 0 0 0
J-A
1 1 1 1 1 1 1 J-A
0 0 0 0 1 0 0 J-A
0 0 0 0 1 0 0
J-B
1 1 1 1 1 1 1 J-B
0 0 0 0 0 1 0 J-B
0 0 0 0 0 1 0
J-C 0 0 0 0 0 0 0 ◦ J-C 0 0 0 0 0 0 1 = J-C 0 0 0 0 0 0 0
A-A 0 0 0 0 0 0 0 A-A 0 0 0 0 0 0 0 A-A 0 0 0 0 0 0 0
A-B 0 0 0 0 0 0 0 A-B 0 0 0 0 0 0 0 A-B 0 0 0 0 0 0 0
A-C 0 0 0 0 0 0 0 A-C 0 0 0 0 0 0 0 A-C 0 0 0 0 0 0 0

v + (C1 ◦ A4 ) A3 v + (C1 ◦ A4 ) ◦ A3


h ih ih ih ih i

The Vertex Filter

Z = v + C1 ◦ A4 ◦ A3 ·A2 · A3 ◦ v − R1 ◦ A4 .
soc.sci. journal articles articles in soc.sci. journals
However,

v − R1 ◦ A4 = v− C1 ◦ A4 Cx = Rx
= v + C1 ◦ A4 v +(Z) =
v −(Z ) .

Therefore, because A ◦ B = (A ◦ B) ,

Z = v + C1 ◦ A4 ◦ A3 ·A2 · v + C1 ◦ A4 ◦ A3 .
reused reused


The Weight and Merge Filter

• λZ: scalar multiplication weights paths.

• Y + Z: matrix addition merges paths.

24 1 0 0 0 0 1 0 0 0 24 2 0 0 0

0 72 0 4 0 0 10 0 0 0 0 82 0 4 0

23 0 0 0 0 + 1 0 34 0 0 = 24 0 34 0 0

0 0 15.3 0 0 0 0 0 0 0 0 0 15.3 0 0

0 0 0 0 12 0 0 0 0 2 0 0 0 0 14


h ih ih ih ih i

The Weight and Merge Filter

Z = 0.6 A1 · A1 ◦ n(I) + 0.4 A5 · A5 ◦ n(I)
coauthorship co-development
merges the article and software program collaboration path matrices as
speciﬁed by their respective weights of 0.6 and 0.4. The semantics of the
resultant is a software program and article collaboration path matrix that
favors article collaboration over software program collaboration. A
simpliﬁcation of the previous composition is

Z = 0.6 A1 · A1 + 0.4 A5 · A5 ◦ n(I).


Application to the Real-World

• A can be represented in a standard matrix manipulation package.

• Z can be constructed with the same matrix manipulation package.

• The path matrix Z has a weighted network representation.

Z = (V, E ⊆ (V × V ), λ), where λ : E → R+

• Z can be used in standard network analysis packages.


The Page Rank Tensor

In the matrix form of PageRank, there exist two adjacency matrices in
[0, 1]n×n denoted
1
1 |Γ+ (i)| if (i, j) ∈ E
Ai,j =
0 otherwise.
and
1
A2 =
i,j .
|V |
A1 is a row-stochastic adjacency matrix and A2 is a fully connected
adjacency matrix known as the teleportation matrix.


The Page Rank Tensor

The purpose of PageRank is to identify the primary eigenvector of a
weighted merged path matrix of the form

Z = δ · A1 + (1 − δ) · A2 .

Z is guaranteed to be a strongly connected single-relational path matrix
because there is some probability (deﬁned by 1 − δ) that every vertex is
reachable by every other vertex.


Conclusion

• Most of graph and network theory is concerned with the design of
theorems and algorithms for single-relational networks.

• Given a multi-relational network, you can manipulate a tensor
representation of it to yield a “semantically-rich” single-relational
network.

• Thus, a multi-relational network can be exposed to the concepts of the
single-relational domain.

Rodriguez M.A., Shinavier, J., “Exposing Multi-Relational Networks to
Single-Relational Network Analysis Algorithms”, LA-UR-08-03931, May
2008, http://arxiv.org/abs/0806.2274.


References

[Bavelas, 1950] Bavelas, A. 1950. Communication Patterns in Task
Oriented Groups. The Journal of the Acoustical Society of America,
22, 271–282.

[Bollen et al., 2007] Bollen, Johan, Rodriguez, Marko A., & Van de Sompel,
Herbert. 2007. MESUR: usage-based metrics of scholarly impact. In:
Joint Conference on Digital Libraries (JCDL07). Vancouver, Canada:
IEEE/ACM.

[Bonacich, 1987] Bonacich, Phillip. 1987. Power and centrality: A family
of measures. American Journal of Sociology, 92(5), 1170–1182.


[Brin & Page, 1998] Brin, Sergey, & Page, Lawrence. 1998. The anatomy
of a large-scale hypertextual Web search engine. Computer Networks and
ISDN Systems, 30(1–7), 107–117.

[Chartrand, 1977] Chartrand, Gary. 1977. Introductory Graph Theory.
Dover.

[Csardi, 2006] Csardi, Gabor. 2006. The igraph software package for
complex network research. InterJournal Complex Systems.

[Freeman, 1977] Freeman, L. C. 1977. A set of measures of centrality based
on betweenness. Sociometry, 40(35–41).

[Girvan & Newman, 2002] Girvan, Michelle, & Newman, M. E. J. 2002.
Community structure in social and biological networks. Proceedings of
the National Academy of Sciences, 99, 7821.


[Hagberg et al., n.d.] Hagberg, Aric, Schult, Daniel A., & Swart, Pieter J.
NetworkX. https://networkx.lanl.gov.

[Harary & Hage, 1995] Harary, Frank, & Hage, Per. 1995. Eccentricity and
centrality in networks. Social Networks, 17, 57–63.

[Kolda et al., 2005] Kolda, Tamara G., Bader, Brett W., & Kenny,
Joseph P. 2005. Higher-Order Web Link Analysis Using Multilinear
Algebra. In: Proceedings of the Fifth IEEE International Conference on
Data Mining ICDM’05. IEEE.

[Newman, 2003] Newman, M. E. J. 2003. Mixing patterns in networks.
Physical Review E, 67(2), 026126.

[Newman, 2006] Newman, M. E. J. 2006. Finding community structure in
networks using the eigenvectors of matrices. Physical Review E, 74(May).


[O’Madadhain et al., 2005] O’Madadhain, Joshua, Fisher, Danyel, Nelson,
Tom, & Krefeldt, Jens. 2005. JUNG: Java Universal Network/Graph
Framework.

[Rodriguez, 2008] Rodriguez, Marko A. 2008. Grammar-Based Random
Walkers in Semantic Networks. Knowledge-Based Systems, [in press].


A Path Algebra for Mapping Multi-Relational Networks to Single-Relational Networks

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