Many, if not most network analysis algorithms have been designed specifically for single-relational networks; that is, networks in which all edges are of the same type. For example, edges may either represent "friendship," "kinship," or "collaboration," but not all of them together. In contrast, a multi-relational network is a network with a heterogeneous set of edge labels which can represent relationships of various types in a single data structure. While multi-relational networks are more expressive in terms of the variety of relationships they can capture, there is a need for a general framework for transferring the many single-relational network analysis algorithms to the multi-relational domain. It is not sufficient to execute a single-relational network analysis algorithm on a multi-relational network by simply ignoring edge labels. This article presents an algebra for mapping multi-relational networks to single-relational networks, thereby exposing them to single-relational network analysis algorithms.
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A Path Algebra for Mapping Multi-Relational Networks to Single-Relational Networks
1. 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
2. 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
3. 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!)
Center for Non-Linear Studies Public Lecture - June 26, 2008
4. 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?
Center for Non-Linear Studies Public Lecture - June 26, 2008
5. 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?
Center for Non-Linear Studies Public Lecture - June 26, 2008
6. 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?
Center for Non-Linear Studies Public Lecture - June 26, 2008
7. 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.
Center for Non-Linear Studies Public Lecture - June 26, 2008
8. 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
Center for Non-Linear Studies Public Lecture - June 26, 2008
9. 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
Center for Non-Linear Studies Public Lecture - June 26, 2008
10. 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 })
Center for Non-Linear Studies Public Lecture - June 26, 2008
11. 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 ))
Center for Non-Linear Studies Public Lecture - June 26, 2008
12. A Multi-Relational Network
Publisher-A
Article-A
publishedBy
authored editorOf Human-B
Journal-A
containedIn authored
Human-A
authored
Article-B
Edges are heterogenous in meaning.
M = (V, E = {E0 ⊆ (V × V ), E1, . . . , Em})
Center for Non-Linear Studies Public Lecture - June 26, 2008
13. 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
Center for Non-Linear Studies Public Lecture - June 26, 2008
14. 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.
Center for Non-Linear Studies Public Lecture - June 26, 2008
15. 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.
Center for Non-Linear Studies Public Lecture - June 26, 2008
16. 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
containedIn authored
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.
Center for Non-Linear Studies Public Lecture - June 26, 2008
17. 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
Center for Non-Linear Studies Public Lecture - June 26, 2008
18. 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.
Center for Non-Linear Studies Public Lecture - June 26, 2008
19. 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.
Center for Non-Linear Studies Public Lecture - June 26, 2008
20. 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.
Center for Non-Linear Studies Public Lecture - June 26, 2008
21. 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.
Center for Non-Linear Studies Public Lecture - June 26, 2008
22. 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
+
Center for Non-Linear Studies Public Lecture - June 26, 2008
23. 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.
Center for Non-Linear Studies Public Lecture - June 26, 2008
24. 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.
Center for Non-Linear Studies Public Lecture - June 26, 2008
25. 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
Center for Non-Linear Studies Public Lecture - June 26, 2008
26. 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.
Center for Non-Linear Studies Public Lecture - June 26, 2008
27. 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
Center for Non-Linear Studies Public Lecture - June 26, 2008
28. A1 : authored A2 : cites A3 : contains A4 : category A5 : developed
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.
Center for Non-Linear Studies Public Lecture - June 26, 2008
30. The Filter Operation
• A◦1=A
• A◦0=0
• A◦B=B◦A
• A ◦ (B + C) = (A ◦ B) + (A ◦ C)
• A ◦ B = (A ◦ B) .
Center for Non-Linear Studies Public Lecture - June 26, 2008
31. 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
Center for Non-Linear Studies Public Lecture - June 26, 2008
32. The Not Filter
If A ∈ {0, 1}n×n, then
• n(n(A)) = A
• A ◦ n(A) = 0
• n(A) ◦ n(A) = n(A).
Center for Non-Linear Studies Public Lecture - June 26, 2008
33. A1 : authored A2 : cites A3 : contains A4 : category A5 : developed
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
Center for Non-Linear Studies Public Lecture - June 26, 2008
34. 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
Center for Non-Linear Studies Public Lecture - June 26, 2008
35. 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)
Center for Non-Linear Studies Public Lecture - June 26, 2008
36. A1 : authored A2 : cites A3 : contains A4 : category A5 : developed
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)
Center for Non-Linear Studies Public Lecture - June 26, 2008
37. A1 : authored A2 : cites A3 : contains A4 : category A5 : developed
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).
Center for Non-Linear Studies Public Lecture - June 26, 2008
38. 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.
Center for Non-Linear Studies Public Lecture - June 26, 2008
39. 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
Center for Non-Linear Studies Public Lecture - June 26, 2008
40. The Vertex Filter
• v −(Ci) = Ci
• v +(Rj ) = Rj
• v −(Z) = v +(Z )
• v +(Z) = v −(Z ) .
Center for Non-Linear Studies Public Lecture - June 26, 2008
41. A1 : authored A2 : cites A3 : contains A4 : category A5 : developed
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
Center for Non-Linear Studies Public Lecture - June 26, 2008
43. A1 : authored A2 : cites A3 : contains A4 : category A5 : developed
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
Center for Non-Linear Studies Public Lecture - June 26, 2008
45. A1 : authored A2 : cites A3 : contains A4 : category A5 : developed
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).
Center for Non-Linear Studies Public Lecture - June 26, 2008
46. 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
Center for Non-Linear Studies Public Lecture - June 26, 2008
47. 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.
Center for Non-Linear Studies Public Lecture - June 26, 2008
48. 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.
Center for Non-Linear Studies Public Lecture - June 26, 2008
49. 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.
Center for Non-Linear Studies Public Lecture - June 26, 2008
50. 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.
Center for Non-Linear Studies Public Lecture - June 26, 2008
51. 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.
Center for Non-Linear Studies Public Lecture - June 26, 2008
52. [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.
Center for Non-Linear Studies Public Lecture - June 26, 2008
53. [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).
Center for Non-Linear Studies Public Lecture - June 26, 2008
54. [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].
Center for Non-Linear Studies Public Lecture - June 26, 2008