Graph Visualization with Latent Variable Models
ABSTRACT Large graph layout design by choosing locations for the vertices on the plane, such that the drawn set of edges is understandable, is a tough problem. The goal is ill-defined and usually both optimization and evaluation criteria are only very indirectly related to the goal. We suggest a new and surprisingly effective visualization principle: Position nodes such that nearby nodes have similar link distributions. Since their edges are similar by definition, the edges will become visually bundled and do not interfere. For the definition of similarity we use latent variable models which incorporate the user's assumption of what is important in the graph, and given the similarity construct the visualization with a suitable nonlinear projection method capable of maximizing the precision of the display. We finally show that the method outperforms alternative graph visualization methods empirically, and that at least in the special case of clustered data the method is able to properly abstract and visualize the links. TKK reports in information and computer science, ISSN 1797-5042; 20
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ABSTRACT: Correction to Annals of Applied Statistics 1 (2007) 17--35 [doi:10.1214/07-AOAS114] Comment: Published in at http://dx.doi.org/10.1214/07-AOAS136 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org)12/2007;
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ABSTRACT: Graphs have been widely used to model relationships among data. For large graphs, excessive edge crossings make the display visually cluttered and thus difficult to explore. In this paper, we propose a novel geometry-based edge-clustering framework that can group edges into bundles to reduce the overall edge crossings. Our method uses a control mesh to guide the edge-clustering process; edge bundles can be formed by forcing all edges to pass through some control points on the mesh. The control mesh can be generated at different levels of detail either manually or automatically based on underlying graph patterns. Users can further interact with the edge-clustering results through several advanced visualization techniques such as color and opacity enhancement. Compared with other edge-clustering methods, our approach is intuitive, flexible, and efficient. The experiments on some large graphs demonstrate the effectiveness of our method.IEEE Transactions on Visualization and Computer Graphics 01/2008; 14(6):1277-84. · 1.90 Impact Factor
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ABSTRACT: A number of recent studies have focused on the statistical properties of networked systems such as social networks and the Worldwide Web. Researchers have concentrated particularly on a few properties that seem to be common to many networks: the small-world property, power-law degree distributions, and network transitivity. In this article, we highlight another property that is found in many networks, the property of community structure, in which network nodes are joined together in tightly knit groups, between which there are only looser connections. We propose a method for detecting such communities, built around the idea of using centrality indices to find community boundaries. We test our method on computer-generated and real-world graphs whose community structure is already known and find that the method detects this known structure with high sensitivity and reliability. We also apply the method to two networks whose community structure is not well known--a collaboration network and a food web--and find that it detects significant and informative community divisions in both cases.Proceedings of the National Academy of Sciences 07/2002; 99(12):7821-6. · 9.74 Impact Factor
Graph Visualization With Latent Variable Models
Juuso Parkkinen, Kristian Nybo, Jaakko Peltonen, and SamuelKaski
Aalto University Schoold of Science and Technology,
Department of Information and Computer Science,
Helsinki Institute for Information Technology HIIT
P.O. Box 15400, FI-00076 Aalto, Finland
Graphs are central representations of information in many
domains including biological and social networks. Graph vi-
sualization is needed for discovering underlying structures
or patterns within the data, for example communities in a
social network, or interaction patterns between protein com-
plexes. Existing graph visualization methods, however, of-
ten fail to visualize such structures, because they focus on lo-
cal details rather than global structural properties of graphs.
We suggest a novel modeling-driven approach to graph vi-
sualization: As usually in modeling, choose the (generative)
model such that it captures what is important in the data.
Then visualize similarity of the graph nodes with a suitable
multidimensional scaling method, with similarity given by
the model; we use a multidimensional scaling method opti-
mized for a rigorous visual information retrieval task. We
show experimentally that the resulting method outperforms
existing graph visualization methods in finding and visual-
izing global structures in graphs.
Complex networks, graph visualization, latent variable model,
nonlinear dimensionality reduction
Complex networks are actively studied in many fields of
science. For example, epidemiologists analyze social net-
works to understand and predict how epidemics spread, and
biologists study protein–protein interaction networks to gain
insight into biological functions and diseases. An established
way of exploring the structure of any complicated data set
is to visualize it. The most common approach to visualiz-
ing networks is straight-line graph drawing: each node in
the network is drawn as a glyph on the plane, and a link
between two nodes is drawn as a straight line between the
glyphs. The task of the visualization algorithm is then to
arrange the nodes so that a good visualization is produced.
Permission to make digital or hard copies of all or part of this work for
personal or classroom use is granted without fee provided that copies are
not made or distributed for profit or commercial advantage and that copies
bear this notice and the full citation on the first page. To copy otherwise, to
republish, to post on servers or to redistribute to lists, requires prior specific
permission and/or a fee.
MLG ’10 Washington DC, USA
Copyright 2010 ACM 978-1-4503-0214-2/10/07 ...$10.00.
Because it is in general impossible to reveal every aspect
of a complex and large data set in a single visualization,
every visualization is necessarily a compromise. The best
we can do is to choose what we consider to be the most
important aspect of the graph and then design a visualiza-
tion that shows that aspect as effectively as possible. We
will see in Section 2 that most existing graph drawing meth-
ods are not based on an explicit choice of what to visualize.
For example, force-directed algorithms, the oldest and most
popular class of methods, are formulated purely in terms of
local properties of the graph: two nodes connected by an
edge should be drawn close to each other, but nodes should
not be allowed to overlap. It is hard to say what a visual-
ization based on this local principle will reveal about global
structural properties, unless the structure of the graph is
very simple and regular, as in the case of a grid or a mesh.
Perhaps because of this, authors of graph drawing methods
have traditionally tested their methods on graphs with sim-
ple and regular structures (see, e.g., [5, 7, 29]). We will see
that conventional graph drawing methods do fail to reveal
global structure in more complex graphs.
To meet the challenges posed by complex graphs, we pro-
pose a new approach to graph visualization. If we were to
visualize only the nodes of a graph, a natural principle would
be that nodes placed nearby in the layout should be ‘sim-
ilar’ in some respect. For graphs where no separate node
features are available, the only information about a node is
what other nodes it links to; therefore, it is natural to as-
sume that nodes nearby on the display should have similar
Directly comparing the observed links from two nodes
would yield only a noisy measure of similarity; in many do-
mains the observed links represent stochastic measurements,
such as measurements of gene interactions, and an observed
graph contains only a sample of possible links. It is then
reasonable to assume that the links arise from underlying
link distributions; more generally, the distributions them-
selves can arise from several latent processes that generate
links between nodes. If the activities of such latent processes
were known for each node, they could be used to rigorously
compute node similarities, which could then be used to op-
timize a graph layout.
We will estimate the link distributions of the nodes, and
the underlying latent processes, by learning a generative la-
tent variable model for the links in a graph. The similarities
of nodes can then be compared by using any of several com-
mon distance measures between distributions.
Given a good estimate of node similarities, we want to
produce a rigorous visualization that will place nodes with
similar link distributions close-by on the display; in addition
to yielding an easily interpretable layout of the nodes, a
side benefit is that the links from such groups of nodes form
intuitive bundles of links that start and end at similar nodes.
To produce such a visualization, we will use a novel rigorous
framework for visualization which we introduced recently.
As far as we know, this principle for visualizing graphs
is new. We will start by discussing earlier graph drawing
2. GRAPH DRAWING
In their classic paper , Fruchterman and Reingold for-
mulate the problem of graph drawing as producing a draw-
ing “according to some generally accepted aesthetic crite-
ria”: distribute the vertices evenly in the image, make edge
lengths uniform, reflect inherent symmetry, minimize edge
crossings, and conform to the drawing frame. Fruchterman
and Reingold note that their algorithm does not explicitly
strive for these goals, but that it does well in terms of the
three first. They state their drawing method is based on
two principles: that vertices connected by an edge should be
drawn close to each other; and that vertices should not be
drawn too close to each other.
This is the task definition that is implicitly used in both
classical and very recent papers on force-based and spectral
graph drawing algorithms. We say ‘implicitly’ because most
papers that we cite in these categories do not explicitly for-
mulate a task, but the task can be inferred from their choices
of comparison methods and evaluation criteria.
2.1 Force-Based Methods
Force-based algorithms are arguably the most established
and wide-spread class of graph drawing algorithms. Simple
force-based algorithms are commonly used also in graph vi-
sualization papers such as [3, 11] where the graph layout is
only a starting point.
The principle can be explained by a physical analogy:
imagine each edge is a spring with an equilibrium length
k, and each node is a steel ring to which its inbound edges
are attached. Place the nodes at arbitrary initial positions
in the plane, and then release them; they move according
to forces exerted on them by the extending and contract-
ing springs, until the system finds an equilibrium, which is
the layout. Due to this analogy, force-based methods are
sometimes called spring embedders.
The classical Fruchterman-Reingold (FR) algorithm fol-
lows this analogy, but instead of simulating physical forces,
it uses an iterative computation with similar properties. In
brief, at each iteration, each node x is displaced by con-
tributions from other nodes: each of the other nodes z re-
pulses x by a displacement k2/d(x,z) where d(x,z) is the
distance between x and z; on the other hand, each node
y connected to x by an edge attracts x by a displacement
d(x,y)/k. Similar ideas are used in other classical force-
based methods. Fruchterman and Reingold also introduce a
faster grid-variant of their algorithm.
The classical force-based algorithms are slow for large
graphs : even one of the fastest algorithms  has time
complexity O(n3) for graphs with n nodes. This has led to
development of so-called multi-level algorithms, introduced
by Walshaw  and Hadany and Harel . The core idea
of both algorithms is the same; we describe Walshaw’s algo-
rithm, which we use as a comparison method in Section 4.
Walshaw’s algorithm starts by creating a sequence of in-
creasingly coarse approximations of the graph, call it G0, by
finding a maximal independent subset S of edges. A set of
edges is independent if no two edges in the set are incident
on the same node; it is maximal if no more edges can be
added to the set without breaking the independence crite-
rion. Walshaw finds such subsets with an approximate algo-
rithm. Once an (approximate) maximal independent subset
of edges S is found, a coarse graph G1 approximating G0
is created, by collapsing the edges in S: all edges in S are
deleted, and any pair of nodes connected by a deleted edge
are replaced by a single node. Next, an even coarser ap-
proximation G2 is created by applying the coarsening step
to G1, and so on. Once there is an approximation Gn with
sufficiently few nodes, we compute a layout for it with the
Fruchterman-Reingold grid-variant algorithm.
layout, an initial layout is interpolated for Gn−1: each node
in Gn that represents a pair of nodes in Gn−1 is replaced
by two nodes connected by an edge. This initial layout is
refined by the Fruchterman-Reingold algorithm, and is then
used to interpolate a layout for Gn−2, and so on until we
have a layout for the original graph.
Since Walshaw published his algorithm, several other multi-
level methods have been developed; see  for a compar-
ison.They are all at least an order of magnitude faster
than Fruchterman and Reingold’s grid-variant algorithm,
and generally produce much better layouts .
Spectral graph drawing algorithms offer another solution
to the problem of computational complexity in traditional
graph drawing algorithms. Although the spectral approach
was introduced as early as 1970 , it has only recently
The term ‘spectral method’ typically refers to any graph
layout algorithm that bases its layout on the eigenvectors
(i.e., the spectral decomposition) of some matrix derived
from the graph. As an example of this class of algorithms,
we describe Hall’s original method , following Koren’s ex-
position in . A more detailed derivation can be found in
Hall’s method computes layouts of a weighted graph G(V,E)
with n nodes, into m < n dimensions. Consider first the case
m = 1. The one-dimensional graph drawing is formulated
as finding an x ∈ Rn, where x(i) is the coordinate of the
ith node, such that x solves the constrained minimization
given: V ar(x) = 1,
where V ar(x) is the variance of x, that is,
V ar(x) =
In words, the task is to find a layout x that minimizes the
distances between nodes that are connected by an edge; the
constraint V ar(x) = 1 prevents nodes from being mapped
too close to each other. Thus we can interpret the mini-
mization problem (1) as a (one-dimensional) mathematical
formulation of Fruchterman and Reingold’s two graph draw-
ing principles: nodes connected by an edge should be drawn
close to each other; and nodes should not be drawn too close
to each other. To produce a two-dimensional drawing, the
i=1(x(i) − µ)2, where µ is the mean of x.
method simply computes another coordinate vector y that
satisfies (1), but it is additionally required that there be no
correlation between x and y. It is easily shown that these
vectors x and y are the second and third smallest eigenvec-
tors of the Laplacian matrix of the graph.
Other spectral methods generally follow the same struc-
ture; mainly the matrix whose eigenvectors are calculated
varies. The classical MDS (CMDS) algorithm by Kruskal
and Seery , one of our comparison methods, uses the
shortest path pairwise distance matrix. Spectral methods
are significantly faster than the fastest force-based methods,
but they have a tendency to produce layouts with many over-
lapping nodes . Civril et al., who recently independently
rediscovered the algorithm, found that CMDS suffers less
from this tendency than other popular spectral methods.
2.3 VisualizingClusters: Edge-RepulsionLin-
Although multi-level force-based algorithms and spectral
methods offer great improvements over classical drawing al-
gorithms, these improvements are mainly technical inno-
vations that allow the algorithms to scale to much larger
graphs than their predecessors could handle; the graph draw-
ing task is still the same. Noack recently proposed ERLin-
Log (Edge-Repulsion LinLog) , a graph drawing method
specifically designed to show graph clusters, a task that
Noack shows is not only different from, but actually in con-
flict with the classical Fruchterman-Reingold task.
Informally, Noack calls a subgraph a cluster if it has many
internal edges and few edges to the remaining graph; he in-
troduces various clustering measures with which the notion
can be formalized. The goal of ERLinLog is to show clusters
in the layout by grouping densely connected nodes (nodes in
the same cluster) and separating sparsely connected nodes
(nodes not in the same cluster). This is accomplished by
minimizing the cost function
where p(u) and p(v) are the coordinates of nodes u and v in
the layout. Noack presents a theorem stating that a layout
with minimal cost minimizes the ratio of the mean distance
between connected nodes to the mean distance between all
There have been some other papers with tasks similar to
Noack’s. Van Ham and Van Wijk modified an earlier variant
of LinLog to create an interactive visualization system for
small-world graphs such as social networks , Lehmann
and Kottler also proposed an algorithm for drawing clus-
tered graphs, but based on the idea of computing a special
kind of spanning tree .
We have seen that most papers that introduce a new graph
drawing method do not explicitly state what exactly their
method tries to visualize. Because it is generally impossible
for a single visualization to capture every possible aspect
of a graph, however, every graph layout method necessary
favors some aspects over others, and hence has at least an
implicit objective. We saw that methods without explicitly
?p(u) − p(v)?
degudegv log?p(u) − p(v)?, (2)
stated objectives, which includes most force-based and spec-
tral methods, tend to share implicitly the goals of Fruchter-
man and Reingold : to place nodes connected by an edge
close to each other, to keep edge lengths uniform, and to
minimize the number of edge crossings. Of all the methods
reviewed in this chapter, only Noack’s Edge-Repulsion Lin-
Log , which visualizes graph clusters, can be said to be
based on a clear visualization task.
Our proposed graph visualization principle, which was
motivated in the previous sections, includes the following
steps: (1) Devise a latent variable model for capturing es-
sential structure from a graph; (2) Compute distances be-
tween the graph nodes in the latent space; and (3) Apply
a non-linear dimensionality reduction to visualize the nodes
(and links) in two dimensions. In the following subsections
we describe each step.
3.1 Generative Model for Graphs
We assume that the links have been generated from a
latent variable model, where the latent variables capture
what is central in the graph, and the rest is noise. We should,
as usual in modeling, build our assumptions about the data
into the model, and those assumptions will then determine
what kinds of properties of the graphs the visualization will
A convenient, flexible choice is SSN-LDA (Simple Social
Network Latent Dirichlet Allocation) , a generative topic-
type model for graphs. In SSN-LDA each node is associated
with a membership vector over a set of latent components.
Each component is in turn associated with a distribution
over the nodes in the graph. Edges are generated by first
drawing a component for the starting node, and then draw-
ing the receiving node from the component-specific distri-
bution. The assumption behind this generative process is
that the graph can be decomposed into overlapping latent
components, that is, groups of nodes with similar edge dis-
tributions. Hence we assume that the components are more
important for graph visualization than are details in link
For SSN-LDA the latent space takes the form of compo-
nent probabilities given the node, and thus the distances
between link distributions should be evaluated in terms of
those probabilities.We will use the quickly computable
Hellinger distance which has been proven useful for topic
models earlier ,
where p and q are the probability distributions over the com-
ponents. The distances could alternatively be computed in
the link space (that is, based on the link distributions from
each node rather than the higher-level component distribu-
tions), using information-geometric formulations.
SSN-LDA has two hyperparameters, α and β, controlling
the node-wise distributions over the components and the
component-wise distributions over the nodes, respectively.
We learn these parameters from the data, by putting vague
gamma priors for both parameters and sampling their values
from their posteriors. Our sampling approach for the hy-
perparameters consists of first finding the maximum of the
posterior with Newton steps, then using Metropolis moves
with a Gaussian proposal distribution.
The final step is to position the nodes on the display so
that nearby nodes will have similar link distributions. This
is a task of nonlinear dimensionality reduction (multidimen-
sional scaling) for visualization.
We have recently introduced a rigorous formalization of
nonlinear dimensionality reduction for visualization as an
information retrieval task [27, 28], which solves the prob-
lem that such visualization has typically had no well-defined
goal. Here the goal of visualization is to produce a low-
dimensional display from which the analyst can retrieve which
points are neighbors. Since a two-dimensional display can-
not perfectly represent a high-dimensional data set, all meth-
ods make errors: some original neighbors are missed in the
visualization, and some points are falsely retrieved as neigh-
bors. The total cost of these errors is equivalent to the
tradeoff between precision and recall of the information re-
trieval; the tradeoff is determined by the relative cost of a
miss versus a false neighbor, as set by the analyst. Both
precision and recall are valid goals for visualization.
The information retrieval criterion is a rigorous objective
for visualization; we have introduced the Neighbor Retrieval
Visualizer (NeRV) method [27, 28] which optimizes this cri-
terion to produce optimal visualizations for information re-
trieval. In addition to being based on a rigorous formalism,
NeRV has performed very well in quantitative comparisons
, so it is a good choice for visualizing the graphs here.
In this paper we use the variant of NeRV introduced in 
which uses t-distributions to define the neighborhoods in the
output space; in  this variant was called t-NeRV but here
we simply call it NeRV.
In NeRV, the tradeoff between optimizing precision and
recall is controlled by a parameter λ ∈ [0,1], defining the
relative cost to the analyst of a miss versus a false neighbor.
This parameter is set according to the needs of the analyst;
both λ = 0 (maximizing precision; minimizing false neigh-
bors) and λ = 1 (maximizing recall; minimizing misses) are
useful goals. When the tradeoff parameter is set to λ = 1
(maximizing recall), NeRV with the t-distributions corre-
sponds to the recent“t-Distributed Stochastic Neighbor Em-
bedding”method (t-SNE; ); more generally, NeRV yields
a flexible tradeoff between precision and recall. In exper-
iments, we will use both λ = 1 (maximizing recall) and
λ = 0.1 (maximizing mostly precision).
We next show empirically how our proposed graph visu-
alization method can successfully find both assortative and
disassortative structures, whereas existing methods fail in
the latter task. We also show this quantitatively based on
external ground truth.
We compare our method against three existing graph draw-
ing methods discussed in Section 2: Walshaw’s multi-level
force-based algorithm ; Kruskal and Seery’s spectral method
, later independently rediscovered by Civril et al. as SDE
; and Noack’s Edge-Repulsion Linlog . Walshaw’s al-
gorithm is implemented as a Cytoscape  plugin . Ali
Civril kindly provided his original implementation for SDE.
For ERLinLog, we used Noack’s publicly available imple-
The Football graph.
We first test the methods on a graph representing foot-
ball teams and their games, with 115 nodes and 613 edges.
Each node represents a college football team in the United
States, and an edge between two teams implies that the
teams played each other during the Division I games of the
2000 season . Each team is known to belong to one of
12 conferences. The conference structure is highly assorta-
tive: The teams in each conference played heavily against
each other. In addition, there is some structure in games
between the conferences which is not as obvious.
To verify the ability of our proposed method to find dis-
assortative structure from graphs we apply it on word ad-
jacency graphs. These graphs represent the relationships of
words within text: A link is assigned between two directly
adjacent words in a text. Word-adjacency graphs have been
shown to exhibit disassortative structure : Words from
the same word classes (e.g., adjectives, nouns), tend to ap-
pear next to words from different classes more often that
words from the same class.
As a simple demonstration we use the adjacency network
of common adjectives and nouns in the novel David Copper-
field by Charles Dickens , which we call the Adjective-
noun graph. It has 112 words as nodes and 424 edges be-
We also constructed a larger word-adjacency graph based
on seven novels by Jane Austen (Emma, Lady Susan, Mans-
field Park, Northanger Abbey, Persuasion, Pride and Preju-
dice, and Sense and Sensibility). The novels are obtained
from the Project Gutenberg website .
project  offers a set of commonly used adjectives, nouns,
and verbs, called Core WordNet, which we use here as the
nodes of the graph. The graph was constructed by assigning
a directed link between two Core WordNet words if they ap-
pear next to each other in the text of the Jane Austen novels
at least two times. The graph was then binarized (so that
multiple links between the same two words are counted only
once), self links were removed, and the largest connected
component was taken, resulting in a binary directed graph
with 879 nodes and 2284 links between them. This graph is
later denoted as the Jane Austen graph. The word classes
(adjective, noun, or verb) are available for each graph node
(Core WordNet word); note that some words may belong to
multiple word classes, for example, ‘open’ can be used both
as an adjective and as a verb.
4.2 Experiment Setup
We compare our proposed method to three existing graph
layout methods, representing the three main approaches to
graph visualization: Walshaw, SDE, and LinLog (described
in Section 2). All comparison methods were run with their
default settings, except that for LinLog the number of iter-
ations was raised to 1000 to assure convergence.
We optimize SSN-LDA with a collapsed Gibbs sampler.
We first run 10,000 burn-in iterations and then take 50 sam-
ples with 50 iteration interval. We also set the number of
components beforehand, estimated based on external data
(and in the small demonstrations simply by setting it man-
ually). In case such external data is not available, there are
a couple of other means; a natural choice for a generative
model would be to evaluate the predictive likelihood of left-
out data for different numbers of components, but in the
case of graphs this is not straightforward, as the data points
are interdependent. Another possibility is to use a Hier-
archical Dirichlet Process (HDP) prior  for the number
of components, but due to the strick-breaking construction
HDP has the unwanted property of producing only few large
components and a long tail of tiny ones.
For the Football graph the number of components was
simply set to 12, to match to the number of ground truth
classes. For the Adjective-noun graph the number of ground
truth classes is two, but setting the number of components
to two would result in essentially one-dimensional visualiza-
tion (each node would be positioned between the two com-
ponents). We thus double the number of components to
In the Jane Austen graph we used external data, the
known classes. We ran LDA with several numbers of com-
ponents, ranging from 3 to 30, and evaluated the result-
ing Hellinger distance matrices by computing the k near-
est neighbors classification performance with respect to the
ground truth classes, with k set to 5. The best performance
was obtained with 5 components.
For the first demonstrations, Adjective-noun and Football
graphs, the NeRV parameter λ was simply set to 0.1. In
the Jane Austen graph we demonstrate the tradeoff between
maximizing precision and recall by using both λ = 0.1 and
λ = 1.0, respectively.
In addition, we also need to set the neighborhood size for
NeRV. Here we follow the rule-of-thumb that the number of
components times neighborhood size should roughly match
the number of nodes in the graph. This way the optimized
neighbors would represent the latent components found by
the model, and which the user is interested in. Finally, we
ran each NeRV ten times and chose the best run according
to the NeRV cost function.
Graph visualizations. All graph layouts were visualized
with Cytoscape . From the Football graph visualiza-
tions (Fig. 1) we see that LinLog and our method are able
to detect clear clusters, most of which match exactly to the
known conferences of the teams. This was expected, as Lin-
Log is designed to find exactly this kind of assortative clus-
ter structure from graphs, and our method is designed to be
able to represent both assortative and disassortative struc-
ture. We note that some teams do not follow the pattern
and are displaced from their conference members. Walshaw
and SDE do not show clear clusters, but also in their layouts
the nodes with same colors are somewhat grouped together,
in Walshaw more clearly than in SDE.
In the case of the Adjective-noun graph (Fig. 2) the ability
of our method to detect disassortative structures becomes
evident. While the three other methods fail to find any
structure and completely mix the adjectives and nouns, our
method is able to separate the classes well into four clus-
ters. Similar behavior can be seen in the Jane Austen graphs
(Fig. 3, subfigures A-E): again, our method can separate the
word classes (here adjectives, nouns, and verbs) into several
clusters while the others do not find the structure. The other
methods do show slight separation of verbs from others, but
even this faint structure would be practically impossible to
see without plotting the ground truth class colors onto the
layout. In contrast, our method finds a very clear grouping
of the nodes (words) and bundling of the links.
Subfigures D and E in Figure 3 demonstrate the interest-
ing visual tradeoff between maximizing precision and recall.
In both visualizations the word classes are separated nicely,
but in the precision end the clusters are more distinct from
each other. This highlights the formation of edge bundles
between certain node clusters, which allow the analyst to
discover the underlying linking patterns in the graph. On
the other hand, the other end of the tradeoff (maximizing
recall) yields a layout with less separation between nodes
having moderate similarity. Both graph layouts are useful
but for different needs of the analyst.
Quantitative validation. To verify that the visual inspec-
tion gave the right impression about how well the word
classes are separated, we perform a simple quantitative eval-
uation of visualization quality, by evaluating class purity on
the display with respect to the word classes of the nodes:
we perform a leave-one-out classification of the nodes on the
display, using k-nearest neighbor (KNN) classification with
k = 5. In case a word (node) to be classified belongs to mul-
tiple word classes, a neighbor sharing any of those classes is
counted as a positive outcome for the classification.
Subfigure F in Figure 3 shows KNN classification results
for the Jane Austen graph. It is clear that our proposed
methods are able to detect the word classes from the graph
significantly better than the other three methods. Note that
although the number of components for our method was
chosen using the classes, the effect of the number of com-
ponents on classification performance (about 3%) was small
compared to the difference to other methods; thus the ad-
vantage of our method did not depend on that choice. The
classification accuracy difference between the precision and
recall ends of the tradeoff in our method is minimal com-
pared to the difference to other methods, showing that both
ends of the tradeoff can capture essential properties of the
We have introduced a new principle for graph visualiza-
tion, and shown how it can be taken into use in a method
which outperforms existing graph visualization algorithms in
discovering and visualizing global structures from complex
graphs. What is particularly attractive is that the method is
model-based. A generative model of the graph is assumed,
and the visualization focuses on those properties in the link
distribution the generative model models well, that is, con-
siders important. The visualization can be made to focus on
different properties of the graph by changing the model. In
effect, the principle turns graph visualization, for which only
mostly heuristic solutions have existed so far, into a genera-
tive modeling problem. With our method it is also possible
to control the unavoidable tradeoff between precision and re-
call in the visualization, depending on the particular needs
of the analyst.
The obvious disadvantage is longer running time, but the
computation of the graphs in this paper only took some tens
of minutes on a standard PC for all of the algorithms.
The authors belong to AIRC. Ju.P. and K.N. are funded
by the HeCSE and FICS graduate schools respectively and
Ja.P. is supported by the Academy of Finland, decision num-
ber 123983. This work was also supported in part by the
PASCAL2 Network of Excellence, ICT 216886.
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Figure 1: Football graph layouts. (A) Walshaw, (B)
SDE, (C) LinLog, (D) Our method. Colors corre-
spond to the 12 football conferences.
layouts by LinLog and our method reveal the con-
ference structure the most clearly.
Figure 2: Adjective-noun graph layouts. (A) Wal-
shaw, (B) SDE, (C) LinLog, (D) Our method. Col-
ors: blue nodes are adjectives, red nodes are nouns.
Only our method is able to reveal the structure
where nouns mostly link to adjectives and vice versa.
Figure 3: Jane Austen graph layouts. (A) Walshaw, (B) SDE, (C) LinLog, (D) Our method, λ = 1.0 (maxi-
mizing recall), (E) Our method, λ = 0.1 (maximizing mostly precision). Blue nodes are adjectives, red nodes
are nouns, green nodes are verbs, yellow nodes have multiple word classes. Our method again reveals the
linking structure the most clearly. Subfigures D-E demonstrate the precision-recall tradeoff that the analyst
can control in our method. Maximizing precision and recall yield different useful visualizations of the same
structure: maximizing precision divides data into smaller, tighter groups and edges into tighter bundles,
whereas maximizing recall yields less separation between nodes having moderate similarity. (E) Quantitative
validation. The table shows the 5 nearest neighbour classification accuracy for each method, with respect to
the known word classes. Our method performs best.