Visualizing Dynamic Weighted Digraphs with Partial Links

Hansj

org Schmauder, Michael Burch and Daniel Weiskopf

VISUS, University of Stuttgart, Stuttgart, Germany

Keywords:

Partial Links, Dynamic Graph Visualization, Time-varying Data.

Abstract:

Graphs are traditionally represented as node-link diagrams, but these typically suffer from visual clutter when

they become denser, i.e. more vertices and edges are present in the data set. Partial link drawings have been

introduced for node-link diagrams aiming at reducing visual clutter caused by link crossings. Although this

concept was shown to perform well for some parameter settings, it has not been used for visually encoding

dynamic weighted digraphs. In this paper we investigate the problem of visualizing time-varying graphs as

one node-link diagram in a speciﬁc layout by exploiting the links as timelines. Partially drawn links are used

to show the graph dynamics by splitting each link into as many segments as time steps have to be represented.

Conventional 2D layout algorithms can be applied while simultaneously showing the evolution over time.

Color-coded links represent the changing weights. We use tapered links to reduce possible overlaps at the link

target nodes that would occur when using traditional arrow-based directed links. We experiment with different

graph layouts and different numbers of data dimensions, i.e. number of vertices, edges, and time steps. We

illustrate the usefulness of the technique in a case study investigating dynamic migration data.

1 INTRODUCTION

The visualization of dynamic weighted graphs is of

interest in many application domains. For example,

call graphs in software development, contacts among

people in a social network, or protein-protein interac-

tions in the ﬁeld of bioinformatics have a relational

structure which is changing over time.

Node-link diagrams are the most convenient vi-

sual metaphor to visually encode relationships among

objects. The relations are graphically depicted as

straight links connecting related objects, where the

objects are displayed as circular, rectangular, or tri-

angular shapes to mention the most important ones.

Although node-link diagrams are intuitive representa-

tions, they typically suffer from visual clutter (Rosen-

holtz et al., 2005) caused by many link crossings

when the graphs become denser and denser. In con-

trast, an adjacency matrix representation is useful for

dense graphs but suffers from bad performance of

path-related tasks (Ghoniem et al., 2004).

The visualization of dynamic graphs (Beck et al.,

2014) makes this problem even more challenging.

Using animated node-link diagrams is one solution

towards solving this problem, but static displays of

dynamic data beneﬁt from preserving a viewer’s men-

tal map (Purchase et al., 2006), thus supporting com-

parison tasks and the visual exploration of such time-

varying data for trends. The drawback of static di-

agrams is the reduced visual scalability, i.e., only a

limited number of graphs of a sequence can be dis-

played demanding for a suitable visualization which

allows graph comparisons and scales for longer graph

sequences.

In this work we propose a compromise repre-

sentation beneﬁting from the strengths of node-link

diagrams as well as of static displays of dynamic

graph data combined into a single graph representa-

tion which is advantageous for visual scalability. To

achieve this goal we map a timeline to each directed

edge which begins at the node where the edge starts

and points to the target node. Each link is split into

as many segments as time steps have to be displayed.

The static representation also makes it easier to ap-

ply interaction techniques than in the animated case.

Moreover, an additional hierarchical organization of

the graph vertices can easily be attached to a static

diagram.

Applying this concept to complete links soon

leads to a situation where visual clutter occurs, mak-

ing the diagram unreadable and useless. To miti-

gate this situation we apply the concept of partial

links (Burch et al., 2011b; Bruckdorfer and Kauf-

mann, 2012), i.e., we allow the viewer to interactively

123

Schmauder H., Burch M. and Weiskopf D..

Visualizing DynamicWeighted Digraphs with Partial Links.

DOI: 10.5220/0005303801230130

In Proceedings of the 6th International Conference on Information Visualization Theory and Applications (IVAPP-2015), pages 123-130

ISBN: 978-989-758-088-8

 2015 SCITEPRESS (Science and Technology Publications, Lda.)

reduce the link lengths until the clutter is reduced

and visual patterns can be derived. The presented ap-

proach is also dependent on the generated graph lay-

out. We illustrate the usefulness of our technique by

showing real-world graph data from people migration

behavior. In this scenario we demonstrate interaction

techniques implemented in the visualization tool and

ﬁnally, use our technique to gain insights from such

time-varying weighted relational data.

2 RELATED WORK

There are many application domains dealing with dy-

namic relational data. Consequently, related literature

in this ﬁeld comes in a variety of forms.

There are two camps of researchers in the ﬁeld

of dynamic graph visualization. Time-to-time map-

ping as it is used in animation is one way to show the

time dependency, whereas time-to-space mapping is

another way to visually depict time-varying graphs.

Several comparative user studies focus on the ques-

tion which of the two general visualization principles

of dynamic data, in particular dynamic graph data,

leads to better user performances (Archambault et al.,

2011; Ghani et al., 2012; Tversky et al., 2002).

When animating a graph, a node-link diagram

is generally laid out and is smoothly transformed

into the sequence of layouts one after the other.

This process demands for a good layout for both,

each single graph in the sequence as well as for

the whole sequence in order to preserve a viewers’

mental map (Purchase et al., 2006) guaranteed by a

high degree of dynamic stability. Ofﬂine (Diehl and

org, 2002) and online (Frishman and Tal, 2008) ap-

proaches are investigated for their suitability to repre-

senting dynamic graphs.

Animation has some general drawbacks apart

from high algorithmic complexities. The viewer can

only see one graph at a time, leading to problems

when comparing several graphs in the sequence to de-

rive time-varying visual patterns and insights in the

data such as trends, counter-trends, or anomalies. For

this reason time-to-space mappings (Burch and Diehl,

2008; Stein et al., 2010; Brandes and Nick, 2011)

have been developed which present a subsequence of

the evolving graph in one view. This concept allows

one to visually analyze a dynamic graph by having a

look at all the graphs side by side similar to a small

multiples representation (Tufte, 1983; Tufte, 1990).

One drawback of such a small multiples diagram

is its poor visual scalability. For example, in the paral-

lel edge splatting technique (Burch et al., 2011a) only

one representation row is used for a graph sequence

which is already enhanced by the authors by apply-

ing the concept of Rapid Serial Visual Presentation

(RSVP) (Beck et al., 2012). However, in their RSVP

variant the graph sequence is animated and only one

time window containing a subsequence of graphs is

displayed. The parallel edge splatting idea was there-

fore extended to a grid-based mapping of the graph

sequence while also supporting a ﬂip-book metaphor

in order to make the visualization more scalable in the

time dimension (Burch and Weiskopf, 2014).

A recent article surveyed existing research in the

ﬁeld of dynamic graph visualization (Beck et al.,

2014). The paper states the observation that to-

day more and more static representations of dynamic

graph data are designed and less animated diagrams.

But negatively, the designed graph visualization typi-

cally use a small multiples representation which does

not allow us to integrate the time-varying weights of

the graph edges into a single static graph view.

As an enhancement in our work, instead, we do

not use small multiples representations; nor do we

use an animated sequence of graph diagrams. We, in-

stead, show one graph in a convenient layout, but we

additionally use each directed link as a timeline start-

ing at the origin node. To avoid visual clutter (Rosen-

holtz et al., 2005) we do not draw the link completely

but allow the user to interactively vary the link length

similar to the originally proposed (Becker et al., 1995)

and later evaluated (Rusu et al., 2011; Burch et al.,

2011b) partially drawn links. Moreover, we use ta-

pered links which do not use explicit arrow heads at

the target vertices and consequently, further unclutter

the node-link diagram (Holten et al., 2011). A similar

concept has already been used in the TimeSpiderTrees

visualization (Burch et al., 2010) in which the graph

sequence is mapped to growing circles. The edges

are visually encoded as straight links but only drawn

partially with the goal to reduce overlaps and visual

clutter.

Partial edges were mathematically modeled as a

graph drawing problem (Bruckdorfer and Kaufmann,

2012) and integrated it as an interaction technique into

a graph visualization tool (Burch et al., 2014). How-

ever, in their work, they did not investigate the visual

encoding of dynamic weighted and directed graphs by

using partially drawn links, which we illustrate in this

paper.

3 DATA MODEL

We model a directed weighted graph mathematically

G = (V, E)

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Figure 1: A directed graph containing 16 vertices and 26 edges displayed as node-link diagram using tapered links. Lengths

are varied: (a) 100 % link length, (b) 60 %, (c) 20 %.

Figure 2: A dynamic weight displayed as a partial link divided into color-coded segments: (a) 4 weights and 50 percent link

length. (b) 8 weights and 50 percent link length. (c) 8 time steps where at 2 of them, no edges are available, i.e. 6 weights

and 50 percent link length. (d)–(f) The scenarios from (a) to (c) with 100 percent link length.

where

V = {v

, . . . , v

}

denotes the set of n ∈ N vertices and

E ⊆ V ×V

the set of directed edges. Each edge e ∈ E is associ-

ated with a weight w(e) ∈ R given by a weight func-

tion w : E → R.

In the context of this work, a layout L of a graph

G is a function

L : G 7−→

{

, y

), . . . , (x

, y

)

}

⊆ N

which takes an abstract graph data set modeled as G

and maps all vertices contained in G to (x, y)-positions

in the two-dimensional space. In this visual encod-

ing strategy, we follow aesthetic graph drawing cri-

teria (Ware et al., 2002) which are responsible for

making a generated node-link diagram readable, ex-

plorable, and understandable. The partial link visual-

ization tool is implemented in the C# programming

language, which supports an easy extension of the

functionality by additional source code for more lay-

out techniques.

A dynamic graph

Γ = (G

, . . . , G

)

consists of a sequence of k ∈ N static graphs (single

graphs). We deﬁne the union of all graphs as G

, E

) with

[

i=1

, E

[

i=1

We use this union graph G

in our visualization ap-

proach to compute a 2D layout.

4 VISUALIZATION TECHNIQUE

Our visualization technique is based on node-link dia-

grams. Each link represents a directed weighted edge

and a timeline is visually encoded starting at the ori-

gin vertex and pointing to the target vertex. Color

coding is used to represent the time-varying weights.

We use tapered links to reduce overlaps at the target

vertices when many edges point to the same vertex.

Moreover, the tapered edges are perceptually useful to

directly derive the edge direction (always from thicker

to thinner end).

4.1 Partial Links

Partially drawn links have been introduced to reduce

explicit link crossings in node-link diagrams. By

shortening the links, the exact connections become

more difﬁcult to trace, leading to ambiguities which

node is the edge’s actual target node. We allow the

viewer to interactively vary those link lengths until a

balance between reduction of visual clutter and target

node ambiguities is achieved.

Figure 1 illustrates the impact of link length re-

duction on the visual appearance of the node-link di-

agram. One can directly see how the number of link

crossings is reduced but also how much more difﬁ-

cult it becomes to solve path-related tasks the less of

the link is displayed. For example, in scenario (c), no

explicit link crossings remain for the 20 percent link

lengths, but it also gets harder to trace paths in the

graph.

VisualizingDynamic WeightedDigraphswithPartialLinks

125

4.2 Time-varying Partial Links

Each graph edge is visually encoded by a straight par-

tial link. To each of these links a timeline is attached

starting at the origin vertex and pointing to the tar-

get vertex. The link is equally divided into as many

segments as graphs have to be displayed. When more

graphs have to be displayed as pixels on the link can

be color-coded, we use a weight aggregation tech-

nique.

Figure 2 illustrates some partial link scenarios

with dynamic weights in either 50 percent link lengths

(a)–(c) or 100 percent link lengths (d)–(f). Here we

can see that the weight is oscillating between low and

high values and that at some time steps, the edge may

not be available at all (c), (f).

Figure 3 illustrates how dynamic weights can be

visualized in a directed dynamic graph. A planar

graph without link crossings in the layout is used for

illustrative purposes.

Figure 3: A small, directed planar graph with dynamic

weights and the weight color mapping.

4.3 Interaction Techniques

Figure 4 illustrates the GUI of our visualization tool.

On the left part of the GUI the user can interactively

change parameters while the dynamic graph view on

the right hand side is directly updated.

Apart from generating a static overview of the dy-

namic weighted graph data, we support an analyst by

several interaction techniques to explore the data. In

the following we give a short list of the most impor-

tant ones of these techniques. These can be classi-

ﬁed into techniques that allow graph layout changes,

node/link/time interval selections, visual appearance

changes to nodes and links, ﬁltering techniques, and

details-on-demand.

Graph Layout. We take the union graph G

of the

complete graph sequence into account when comput-

ing a general graph layout. The user can interactively

decide which graph layout is applied to the graph data

set. To this end we support force-directed, circular,

and random layouts. After that, the user is also able

to drag and drop single nodes. All adjacent edges

are then also moved. This helps the user to make

small layout changes and to see to which other nodes a

dragged node is connected because the adjacent links

are smoothly moved around.

Selection. The user can interactively select either

single nodes and links or node and link groups by

clicking on them one after the other. By using rubber-

banding, a connected region can be deﬁned in which

all directly selected links and the selected nodes’ out-

going links are marked for link length manipulation.

Visual Appearance. The diameter of nodes and the

width of links can be changed on demand. Tapered

and traditional link representations are supported.

Showing the links all in their complete length

(100 percent link length) soon leads to a situation pro-

ducing vast amounts of visual clutter. The addition-

ally attached timeline to each link makes the readabil-

ity and pattern detection even worse. For this reason,

the user is able to interactively and smoothly adjust

the link lengths.

We support several color codings which can be se-

lected from a given menu. The most important ones

are linear optimal, vegetation, topographic, blue-to-

red, or heated object color scales. Also logarithmic

and double-logarithmic mappings are supported. Al-

pha blending can be applied on demand to transpar-

ent links, which is useful when these are crossing. A

color gradient is used for the color coding, in which

the lowest value is fully transparent and the links be-

come more and more opaque with increasing values.

Filtering. Since we are dealing with weighted

graphs, a user can interactively ﬁlter edges for spe-

ciﬁc weights. The ﬁltered out edges are shown in gray

color, i.e. they are still displayed for context infor-

mation, all other edge weights are still color-coded.

Also, some time periods might be uninteresting, be-

cause of a stability pattern for example. In this case,

the user can ﬁlter out such time intervals in all of

the links. The ﬁltered out time periods can either be

grayed out or removed completely from the visualiza-

tion, which leaves more space for the remaining time

steps. If vertices are attached with descriptions in tex-

tual form, a text ﬁlter can be applied. All vertices

matching a given substring can either be ﬁltered out

or only those can be shown.

Details-on-Demand. Hovering the mouse over a

node gives additional information about this node in

textual form, either in a separate panel or in form of

tool tips. Also the weight values can be displayed.

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Figure 4: The graphical user interface of our visualization tool: On the left hand side the user can make parameter settings,

whereas on the right hand side the currently selected dynamic weighted graph is displayed.

5 MIGRATION GRAPHS

In this scenario we visually explore the dynamic

country-to-country migration of people in the world.

The corresponding graph contains 226 vertices,

34,968 edges per time frame on average, and spans

over 5 time periods (measured every 10 years between

1960–2010). The edge weights are problematic since

they range from 1 to 9,367,910. Our tool can solve

this by applying a logarithmic weight mapping before

color coding. Moreover, to reduce visual clutter, we

are able to use transparent links for the lower weight

values.

Figure 5 shows the dynamic migration graph for

all countries in the world over ﬁve decades. For il-

lustrative purposes, we ﬁrst generated a random graph

layout, which has the beneﬁt that the vertices are more

or less equally distributed in the 2D plane. From this

ﬁgure, we can draw various conclusions, of which

some are illustrated below:

• China and Hong Kong: If we have a look at the

dynamic edge pointing from China to Hong Kong,

we can see that there seems to be a missing data

point, i.e. for the decade from 1970 until 1980, no

migration data is recorded.

• Ukraine and Russia: There was an increase of

people migrating between Ukraine and the Rus-

sian Federation and vice versa, but after the year

2000, the number of people migrating in either di-

rection has dramatically reduced.

• Pakistan and India: From 1960 to 1970, many

people immigrated from Pakistan to India, but

not that many in the other direction. The num-

bers rapidly decreased decade by decade, but still

many people immigrate between both countries,

which can be seen by the green colored peaks of

the links pointing to each other.

• Poland and Germany: There is much immi-

gration from Poland to Germany in nearly every

decade, but not from Germany to Poland.

• Mexico and United States: The number of peo-

ple immigrating from Mexico to the United States

is strongly increasing, which can be seen by the

red color-coded link peak.

Since we also have a hierarchical organization of

the vertices (the hierarchy of the countries based on

the continents, regions, etc.) we can use the geo-

graphic information to place the vertices as well (see

Figure 6). The vertices in this ﬁgure are placed close

to the centroid of the area representing each country.

As this is probably the most familiar way of visualiz-

ing geographic positions, the viewer can readily spot

the countries of interest. In this ﬁgure we use 45 per-

VisualizingDynamic WeightedDigraphswithPartialLinks

127

Figure 5: Migration data visualized with 40 percent link lengths and opaque color coding for low values.

cent link lengths and additionally adapt the color scale

to show clearer differences to the world map in the

background. Thanks to the color coding with high

transparency for low migration values, we can still see

differences. For example, France and Lesotho, both

on the right side on the very top in ﬁgure 5, have a

quite different number of target countries: For France,

there are 987 outgoing edges, for Lesotho there are

only 234 edges.

6 LIMITATIONS AND

SCALABILITY

Although we designed a useful and interactive visu-

alization technique which combines the evolution of

a graph into one single graph representation in a spe-

ciﬁc layout, we are aware of the fact that there are also

many limitations of our approach.

• Layout Dependency: The interpretation strongly

depends on the layout of the graph as it is also the

case for static graph visualization. Visual clutter

might be reduced and graph patterns might be bet-

ter perceivable if a suitable layout is generated.

• Link Length Dependency: The link length can

be reduced, but we are aware of the fact that

if the links become too short, many more target

vertex ambiguities will occur, leading to misin-

terpretations of the data. Moreover, differently

long links (depending on the distance of two

nodes) cause differently stretched timelines which

may cause problems when comparing their time-

varying weights.

• Color Codings: As with any visualization tech-

nique, the applied color coding has a strong im-

pact on the strength of the visualization and its in-

terpretability for visual patterns. For this reason,

we leave the selection of a suitable color coding

to the user, but to achieve basic expressiveness,

the colors’ intensities should be proportional to

the encoded weight.

• Path-related Tasks: Our approach has beneﬁts

when solving path-related tasks since a node-link

diagram is used and the dynamics of the graph is

integrated into one single static diagram. Such

path tracing is problematic in matrix representa-

tions, graph animations, or small multiple repre-

sentations.

• Scalability in the Time Dimension: An increase

of the number of time steps sooner or later leads

to an aggregation of the color-coded weights en-

coded in each link which may lead to a loss of

visible time-varying patterns.

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Figure 6: Migration data visualized with 45 percent link lengths and semi-transparent color coding in a world map layout.

• Graph Planarity: In particular, our approach is

suitable for planar graphs where no link cross-

ings occur. In such a graph scenario the ques-

tion arises if we need partially links at all since

those are speciﬁcally designed for minimizing vi-

sual clutter mainly caused by link crossings. But

if a directed edge occurs in both directions be-

tween the same vertices, a 40 percent link length

makes sense even in a planar graph layout.

• Decreasing Thickness: The tapered links are

beneﬁcial at the target vertices since they do not

use arrow heads which would cause additional

visual clutter. On the negative side, the time-

dependent weights are visualized with different

link thickness (large ones at the beginning and

smaller ones towards the end). Whereas it is

suitable for the visualization of node relations,

this may cause perceptual problems when inves-

tigating edge weight variations over time (Stone,

2011).

7 CONCLUSION AND FUTURE

WORK

In this paper we introduced a dynamic graph visual-

ization technique which makes use of partially drawn

links. The links are exploited as timelines starting

at the adjacent vertices and pointing to the target

vertices. The user can interactively change the link

lengths in order to reduce visual clutter caused by

many explicit link crossings. We experimented with

different graph layouts and illustrated our novel idea

in a case study investigating dynamic migration data.

We described interaction techniques and their impact

on the perceived graphs, in particular on the partially

drawn links.

For future work, we plan to also visually integrate

an existing or computed hierarchical organization of

the vertices. This is helpful to further navigate in the

data and to ﬁlter it on different levels of hierarchi-

cal granularity. Our novel approach should be eval-

uated in a comparative user study which would give

insights into the readability and usability. Also data

sets from different application domains might be of

interest such as dynamic social network data along

with domain expert feedback for our use cases.

ACKNOWLEDGEMENTS

The authors would like to thank the German Research

Foundation (DFG) for ﬁnancial support of the project

within the Cluster of Excellence in Simulation Tech-

nology (EXC 310/2) at the University of Stuttgart.

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