# 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:**Dimensionality reduction is one of the basic operations in the toolbox of data analysts and designers of machine learning and pattern recognition systems. Given a large set of measured variables but few observations, an obvious idea is to reduce the degrees of freedom in the measurements by rep resenting them with a smaller set of more "condensed" variables. Another reason for reducing the dimensionality is to reduce computational load in further processing. A third reason is visualization.IEEE Signal Processing Magazine 04/2011; · 4.48 Impact Factor - SourceAvailable from: users.cis.fiu.edu[Show abstract] [Hide abstract]

**ABSTRACT:**Many developments have recently been made in mining dy-namic networks; however, effective visualization of dynamic networks remains a significant challenge. Dynamic networks are typically visualized via a sequence of static graph lay-outs. In addition to providing a visual representation of the network topology at each time step, the sequence should pre-serve the "mental map" between layouts of consecutive time steps to allow a human to interpret the temporal evolution of the network and gain valuable insights that are difficult to convey by summary statistics alone. We propose two regularized layout algorithms for visualizing dynamic net-works, namely dynamic multidimensional scaling (DMDS) and dynamic graph Laplacian layout (DGLL). These algo-rithms discourage node positions from moving drastically between time steps and encourage nodes to be positioned near other members of their group. We apply the proposed algorithms on several data sets to illustrate the benefit of the regularizers for producing interpretable visualizations.

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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

firstname.lastname@tkk.fi

ABSTRACT

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.

Keywords

Complex networks, graph visualization, latent variable model,

nonlinear dimensionality reduction

1. INTRODUCTION

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.

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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

link distributions.

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

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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

principles.

2. GRAPH DRAWING

In their classic paper [5], 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 [10]: even one of the fastest algorithms [4] has time

complexity O(n3) for graphs with n nodes. This has led to

development of so-called multi-level algorithms, introduced

by Walshaw [29] and Hadany and Harel [8]. 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 [7] 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 [7].

From this

2.2Spectral Methods

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 [9], it has only recently

become popular.

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 [9], following Koren’s ex-

position in [12]. A more detailed derivation can be found in

[13].

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

problem

X

min

x

E(x) =

(i,j)∈E

wi,j(x(i)−x(j))2

given: V ar(x) = 1,

(1)

where V ar(x) is the variance of x, that is,

V ar(x) =

n

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

1

Pn

i=1(x(i) − µ)2, where µ is the mean of x.

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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 [14], 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 [7]. Civril et al., who recently independently

rediscovered the algorithm, found that CMDS suffers less

from this tendency than other popular spectral methods[2].

2.3 VisualizingClusters: Edge-RepulsionLin-

Log

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) [20], 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

X

−

{u,v}∈V(2)

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

nodes.

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 [26], 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 [15].

2.4 Summary

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

UERLinLog =

{u,v}∈E

X

?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 [5]: 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 [20], which visualizes graph clusters, can be said to be

based on a clear visualization task.

3. MODEL-BASEDGRAPHVISUALIZATION

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

focus on.

A convenient, flexible choice is SSN-LDA (Simple Social

Network Latent Dirichlet Allocation) [30], 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

patterns.

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 [1],

v

i=1

d(p,q) =

u

u

t

n

X

(√pi−√qi)2, (3)

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

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posterior with Newton steps, then using Metropolis moves

with a Gaussian proposal distribution.

3.2 NonlinearDimensionalityReductionforVi-

sualization

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

[28], so it is a good choice for visualizing the graphs here.

In this paper we use the variant of NeRV introduced in [28]

which uses t-distributions to define the neighborhoods in the

output space; in [28] 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; [25]); 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).

4. EXPERIMENTS

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 [29]; Kruskal and Seery’s spectral method

[14], later independently rediscovered by Civril et al. as SDE

[2]; and Noack’s Edge-Repulsion Linlog [20]. Walshaw’s al-

gorithm is implemented as a Cytoscape [23] plugin [22]. Ali

Civril kindly provided his original implementation for SDE.

For ERLinLog, we used Noack’s publicly available imple-

mentation [19].

4.1 Datasets

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 [6]. 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.

Word-adjacency graphs.

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 [17]: 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 [18], which we call the Adjective-

noun graph. It has 112 words as nodes and 424 edges be-

tween them.

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 [21].

project [16] 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

The WordNet

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(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 [24] 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

four.

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.

4.3 Results

Graph visualizations. All graph layouts were visualized

with Cytoscape [23]. 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

graph.

5. DISCUSSION

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.

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6. ACKNOWLEDGMENTS

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.

The graph

Figure 2: Adjective-noun graph layouts. (A) Wal-

shaw, (B) SDE, (C) LinLog, (D) Our method. Col-

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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.

101