11institutetext: University of Perugia, Perugia, Italy 11email:
<EMAIL_ADDRESS>22institutetext: University of
Tübingen, Tübingen, Germany 22email: {henry.foerster,julia.katheder}@uni-
tuebingen.de

#  Evaluating Animation Parameters
for Morphing Edge Drawings††thanks: Research partially supported by: $(i)$
University of Perugia, Ricerca di Base 2021, Proj. “AIDMIX — Artificial
Intelligence for Decision Making: Methods for Interpretability and
eXplainability”; $(ii)$ MUR PRIN Proj. 2022TS4Y3N - “EXPAND: scalable
algorithms for EXPloratory Analyses of heterogeneous and dynamic Networked
Data”; $(iii)$ MUR PRIN Proj. 2022ME9Z78 - “NextGRAAL: Next-generation
algorithms for constrained GRAph visuALization”; $(iv)$ University of Perugia,
Ricerca di Base, grant RICBA22CB; $(v)$ DFG grant Ka 812-18/2.

Carla Binucci 11 0000-0002-5320-9110 Henry Förster 22 0000-0002-1441-4189
Julia Katheder 22 0000-0002-7545-0730 Alessandra Tappini 22
0000-0001-9192-2067

###### Abstract

Partial edge drawings (ped) of graphs avoid edge crossings by subdividing each
edge into three parts and representing only its stubs, i.e., the parts
incident to the end-nodes. The morphing edge drawing model (med) extends the
ped drawing style by animations that smoothly morph each edge between its
representation as stubs and the one as a fully drawn segment while avoiding
new crossings. Participants of a previous study on med (Misue and Akasaka,
GD19) reported eye straining caused by the animation. We conducted a user
study to evaluate how this effect is influenced by varying animation speed and
animation dynamic by considering an easing technique that is commonly used in
web design. Our results provide indications that the easing technique may help
users in executing topology-based tasks accurately. The participants also
expressed appreciation for the easing and a preference for a slow animation
speed.

###### Keywords:

morphing edge drawings readability user study easing function

##  1 Introduction

Edge crossings are well-known to reduce the readability and the perceived
aesthetics of graph drawings; see, e.g., [24, 26]. Bruckdorfer and Kaufmann
[5] suggested a quite rigorous solution that avoids crossings in straight-line
drawings by drawing edges only partially. More precisely, in a _partial edge
drawing_ (ped) of a graph, each edge is subdivided into three parts and the
middle part is not drawn; the drawn parts are called _stubs_. While the model
has received some attention in follow-up studies (see, e.g., [2, 4, 6, 7, 8,
10, 15, 28]), its practical applicability seems to be hindered by the fact
that peds cannot be read as quickly as traditional node-link diagrams [6]. On
the other hand, there are indications that they can be interpreted more
accurately than drawings where edges are fully drawn [6].

Recently, Misue and Akasaka [20] suggested to enhance the ped drawing style by
a sequence of animations that smoothly morph each edge between its partial
representation and its representation as a fully drawn segment. With the
resulting drawing style, known as _morphing edge drawing_ (med), the goal is
to help the users read drawings more quickly while maintaining the readability
offered by ped. A user study by Misue and Akasaka [20] indicates that the
model can achieve these goals.

An unfortunate side effect of the addition of the animation appears to be eye
fatigue experienced by the users: Namely, while the users of Misue’s and
Akasaka’s study [20] reported that in fact “[morphing] made it easy to confirm
the exact adjacency”, they also stated that “[their] eyes [were] strained.”
The users also indicated potential reasons for this negative effect stating
that “[it was] messy and difficult to [focus on]”, that “the stubs [changed]
too fast”, and that “the time for stubs to connect [was] too short.” The
latter two aspects have been considered in Misue’s follow-up study [19], where
the speed of the morphing animation was substantially reduced while there was
a small delay added to the animation at the time that the two stubs meet. It
is worth noting that – aside from these two changes – both previous studies on
med [19, 20] have mainly focused on minimizing the total animation duration,
which is proposed to be a main factor in user response time. However, the
existing user study’s focus [20] was not to investigate different animation
speeds, but to establish the validity of med in comparison to ped and
traditional straight-line drawings.

We also remark that meds may be regarded as _staggered_ animations between
peds and traditional straight-line drawings. These types of animations have
been previously investigated for morphs between two 2D scatterplots of the
same data points [11]. Curiously, in contrast to Misue and Akasaka’s findings
for med [20], Chevalier et al. [11] did not observe indications for benefits
of staggered morphing animations between two scatterplots.

#### Our contribution.

We study meds from a user-centric point of view. To this end, we investigate
in a user study to which extent animation speed influences user response time
and accuracy in the execution of tasks. Moreover, to counteract the eye
straining reported by the participants of the previous user study [20], we
smoothen the animation by an approach called _easing_ that is widely used in
webdesign [14, 17] and experimentally evaluate its impact111Research data and
all stimuli available at https://github.com/tuna-
pizza/MEDSupplementaryMaterial.. We remark that to the best of our knowledge,
easing has not been evaluated in the context of graph animations before. Our
results indicate that fast animation and easing can help users in performing
topology-based tasks. Also, while participants appreciated the easing
technique, they preferred a slower speed.

The paper is structured as follows. Section 2 introduces meds and the
animation parameters we consider. Section 3 describes the design of our user
study. Section 4 discusses the quantitative and the subjective results of our
experiment, as well as its limitations. Section 5 lists some future research
directions.

##  2 Morphing Edge Drawings

#### Model Description.

A med of a graph $G=(V,E)$ is a _dynamic_ drawing, i.e., a function
$\Gamma:G\times T\rightarrow\mathbb{R}^{2}$ where $T$ is the time interval of
the drawing animation. In this paper, we study straight-line meds, where each
vertex $v\in V$ is time-independently mapped to a point in the plane and each
edge $(u,v)\in E$ is mapped to two fragments of the segment $\overline{uv}$,
one incident to $u$ and one incident to $v$. We call these two subsegments
_stubs_ , which change length smoothly over time.

Depending on the length changing process, meds are _symmetric_ if at any time
the two stubs of any edge have the same length. Previous studies show that
symmetry allows the user to efficiently match stubs belonging to the same edge
[2]. We can describe the stub length of an edge $(u,v)$ in a symmetric med by
a _stub length ratio_ function $\delta_{(u,v)}(t)$ that describes the ratio
between the stub length and the length of the segment $\overline{uv}$. If the
minimum of the stub length ratio function of all edges is the same value
$\delta_{0}$, we call the drawing _homogeneous_. We consider only _symmetric_
and _homogeneous_ meds, also known as $\delta_{0}$-shmeds, as it is known that
symmetric and homogeneous peds are the easiest to read [2].

Similar to previous studies [19, 20], we consider only stub length ratio
functions that morph stub lengths between two distinct states (namely
$\delta_{0}$ and $1/2$) such that the transition between both states is
_strictly monotone_. Namely, we allow such a smooth transition to happen
several times for each edge, starting from specific time frames
$T_{start}(u,v)\subset T$ during which $\delta_{(u,v)}(t)=\delta_{0}$. After
some predefined elapsed time $\tau_{(u,v)}$ the edge is represented as a
straight-line segment, i.e., $\delta_{(u,v)}(t)=1/2$. Afterwards, for some
duration $\tau_{1/2}$, edge $(u,v)$ is completely visible ensuring that the
user can see the full edge for some frames in a rendered animation as
suggested by Misue [19]. Following this small delay, the animation reverts and
$\delta_{(u,v)}=\delta_{0}$ again after time $\tau_{(u,v)}$. Due to the
relation of both parts of the animation, in the following we only describe the
first part of the animation. As in previous studies [19], we use the same type
of animation for all morphs of an edge, i.e., $\tau_{(u,v)}$ is assumed to be
constant. To facilitate distinguishing between several runs of an animation,
we allow a new animation to start only after $\tau_{distinct}$ time has
elapsed.

If we consider the animations of two edges $e_{1}$ and $e_{2}$, we may observe
that an _avoidable_ crossing (between two stubs of length ratio strictly
greater than $\delta_{0}$) can occur if their animations are not scheduled
accordingly. We want to avoid that the stubs of $e_{1}$ and $e_{2}$ ever meet
at their avoidable crossing as otherwise they can be perceived as the same
object according to Gestalt principles [27]. To counteract this from
happening, we also allow $e_{1}$ to pass through the same point $e_{2}$ has
passed through (and vice-versa) only after $\tau_{distinct}$ time has elapsed.

#### Animation Parameters.

We want to describe the morph from $\delta_{(u,v)}(t)=\delta_{0}$ to
$\delta_{(u,v)}(t+\tau_{(u,v)})=1/2$ with two easier to understand parameters
that we can fix for the _entire_ med so that the aesthetic effect of all edge
morphs is similar.

The first parameter is the _speed_ of the stub length morphing animation. In
the previous studies on med [19, 20], the speed was a constant $\sigma$
holding for all edges. We can extend the speed parameter to morphings with
_non-constant speed_ by bounding the _average morphing speed_ $\sigma_{a}$ or
the _maximum morphing speed_ $\sigma_{m}$. It is straight-forward to keep
$\sigma_{a}$ fixed as for fixed $\sigma_{a}$, $\tau_{(u,v)}$ simply depends on
the length of segment $\overline{uv}$. Bounding $\sigma_{m}$ is more tricky as
it requires $\delta_{(u,v)}$ to be differentiable to be efficient to compute.
Thus, in this paper, we only consider the average morphing speed $\sigma_{a}$
as a parameter.

In addition, we may want to consider animations with a non-constant speed.
This may be desirable as humans are used to movements in nature that usually
are at first gradually accelerating before reaching a maximum speed before
slowing again before coming to a halt. This observation has been considered in
multimedia design for a long time [23] and gave rise to so-called _easing_ of
animations [14, 23]. Namely, an _easing function_ $\eta:[0,1]\rightarrow[0,1]$
expresses for a given time elapsed ratio $\rho_{t}$ the progress of the
animation $\rho_{a}$. In this regard, $\rho_{t}=0$ means that at the current
time-frame the animation starts (in our case, $t=t_{s}$ for a $t_{s}\in
T_{start}(u,v)$) whereas $\rho_{t}=1$ means that at the current time-frame the
animation finishes (in our case, $t=t_{s}+\tau_{(u,v)}$). Meanwhile
$\rho_{a}=0$ means that the animation has just started (in our case, the stub
length ratio is $\delta_{0}$) whereas $\rho_{a}=1$ means that it has been
fully displayed (in our case, the stub length ratio has become $1/2$). For our
purposes, we want that $\eta$ is _strictly monotone_ and _invertible_ _on the
interval $[0,1]$_ to be able to apply the algorithm of Misue and Akasaka
[20](refer to Appendix 0.A for a description of the algorithm).

##  3 User Study

###  3.1 Experimental Conditions

Figure 1: Easing functions.

Table 1: Parameters used in our experiments. Model: | SlowLin | SlowEas | FastLin | FastEas
---|---|---|---|---
$\sigma_{a}$ | 100 px/s | 100 px/s | 200 px/s | 200 px/s
$\eta$ | Linear | Easing | Linear | Easing
$\tau_{1/2}$ |  100 ms
$\tau_{distinct}$ |  50 ms
$\delta_{0}$ |  1/4

We investigate how morphing speed and easing function influence the
readability of meds. In the first study on med, $\sigma_{a}$ was chosen at
$10^{\circ}$/s to be as fast as possible222 The metric ∘/s refers to the
temporal change of the visual angle, which is the size of the image of the
object on the observer’s retina. Naturally, this metric requires a controlled
environment which we believe to conflict with a variance in user preferences.
while remaining human-tractable [20] whereas in the follow-up study [19] it
was significantly reduced to 100 px/s. We intend to evaluate med as a
generally applicable tool for graph visualizations in practical applications,
where the concrete usage depends on user preferences. As a result, we believe
that px/s is a more useful metric than ∘/s as the latter metric requires to
control the relative positioning of the user towards the screen333The concrete
effect of a speed measured in px/s vastly depends on the screen resolutions.
Hence, we fixed the speed assuming a resolution of 1920 px $\times$ 1080 px
and scaled the drawings so to cover the same fraction of the screen size on
any resolution.. We investigate two speeds:

  * Slow.

Here, we set $\sigma_{a}=100$ px/s as in [19].

  * Fast.

Here, we set $\sigma_{a}=200$ px/s so to have a significantly faster speed as
comparison which allows to check how a shorter animation duration at the cost
of higher speed influences readability.

In previous studies on med [19, 20], as an easing function, the function
$\rho_{a}=\eta(\rho_{t})=\rho_{t}$, which is also known as the Linear easing
function, has been used. We believe that the aesthetic appeal of meds can be
improved by having a non-constant speed allowing for a more natural appearance
of the movement of the stubs. As a side effect, edges can be displayed _almost
fully drawn_ for a longer duration which may allow users to more efficiently
observe them; see Fig. 1. There is a plethora of different easing functions in
the literature [23]. Currently, functions described by _cubic Bézier curves_
appear as the web-design standard [14, 17] with a particular focus on the
function described by points $P_{0}=(0,0)$, $P_{1}=(0.25,0.1)$,
$P_{2}=(0.25,1)$ and $P_{3}=(1,1)$ which is commonly known also as Easing; see
Fig. 1. It is worth noting that cubic Bézier curves are defined as a function
that maps a parameter $p$ to the tuple $(\rho_{t},\rho_{a})$. As a result, it
is not straight-forward to compute $\rho_{a}$ in terms of $\rho_{t}$ [17] and
implementations typically solve this problem numerically [16]. For our
purposes, we oriented ourselves at the implementation used in the Firefox web
browser [12].

The combination of speed (Slow/Fast) and easing function (Linear/Easing) gives
rise to the four models, i.e., the experimental conditions, that we compare in
our study; see Table 1. Note that in all models we set $\tau_{1/2}=100$ ms (as
in [19]) and $\tau_{distinct}=50$ ms. Further, we used $\delta_{0}=1/4$ as in
[20].

###  3.2 Tasks and Hypotheses

We defined five tasks, reported in Table 2, for which we also provide the
classification according to the taxonomy by Lee et al. [18]. Fig. 2 shows some
examples of trials. We designed the tasks so that they require to explore the
drawing locally and globally, are easy to explain, can be executed in a
reasonably short time, and can be easily measured. Also, most of them have
already been used in previous graph visualization user studies (e.g., [1, 13,
22, 25]). The main purpose of our study is to evaluate the differences between
our experimental conditions in terms of readability, understandability, and
effectiveness. For this reason we defined _interpretation tasks_ , according
to the top-level classification by Burch et al. [9].

Table 2: Tasks used in our experiment. Task | Classification | Description |  |
---|---|---|---|---
| T1
---
(Adjacency)
| topology-based
---
(adjacency)
| Is there an edge that connects
---
the two blue nodes?
|
| T2
---
(NeighborhoodSize)
| topology-based
---
(adjacency)
| How many blue nodes are connected
---
with the orange node?
|
| T3
---
(PathLength)
| topology-based
---
(connectivity)
| Is there a path of length at most $k$
---
that connects the blue node with the
orange node?
|
| T4
---
(CommonNeighbors)
| topology-based
---
(common connection)
| How many nodes are connected
---
with both the blue nodes?
|
| T5
---
(InterRegionEdges)
overview |  | How many edges directly connect
---
the two highlighted parts?
|

Figure 2: Examples of trials: (a) T1 (Adjacency), (b) T2 (NeighborhoodSize),
(c) T3 (PathLength) with $k=2$, (d) T4 (CommonNeighbors), (e) T5
(InterRegionEdges).

We formulated three experimental hypotheses about the effectiveness of the
different conditions to support users in the execution of analysis tasks:

  1. H1.

Fast speed and cubic Bézier curve Easing give lower response time than Slow
speed and Linear easing.

  2. H2.

Slow speed and cubic Bézier curve Easing give lower error rate than Fast speed
and Linear easing.

  3. H3.

Slow speed and cubic Bézier curve Easing are preferred by the users rather
than Fast speed and Linear easing.

Our rationale behind H1 is that the time used for morphing all edges with Fast
speed is shorter than with Slow speed, and cubic Bézier curve Easing makes
stubs touch for a longer time than with Linear easing. About H2, we think that
the user can benefit from cubic Bézier curve Easing and Slow speed in terms of
accuracy, since the movement of the stubs is more gradual and connections are
visualized for a longer time than with Linear easing and Fast speed. Finally,
about H3 we think that users may find a Fast animation more irritating and
distracting than a Slow one, and the movement of stubs with Linear easing less
smooth than with cubic Bézier curve Easing.

###  3.3 Stimuli

We investigated our hypotheses on visualizations of real-world networks, more
specifically drawn from the annual GD Contest444See
https://mozart.diei.unipg.it/gdcontest/ for further information.:

boardgames. This graph from the GD Contest 2023 contains the 40 most popular
boardgames as nodes which are connected if fans of one game also liked the
other one (data according to https://boardgamegeek.com/).

marvel. This bipartite graph from the GD Contest 2019 represents which
characters from the Marvel Cinematic Universe are featured in which film.

kpop. This graph consists of a subset of the data from the GD Contest 2020 and
shows the nodes with betweenness centrality at least 0.3 from the largest
connected component of the subgraph induced by K-Pop bands and labels.
Connections in this network showcase cooperations between the entities.

We chose the specific networks as they all have a moderate size reasonably
displayable on a standard screen (40 to 58 nodes) and offer various densities
(ranging from 2.12 to 5.35). We embedded the stimuli graphs using a force
directed algorithm available in the D3.js library [3]. The resulting drawings
roughly required the same area; see Table 3.

In our drawings, nodes are represented as disks of radius 7 px filled gray,
blue or orange (where the colored variants are used to highlight nodes of
interest for the tasks), while the boundaries of the disks and edges are drawn
black and 2 px wide. The animations were scheduled using the algorithm of
Misue and Akasaka [20] with an improvement of [19] (see Appendix 0.A for
details) and rendered at 30 fps. The resulting animations for SlowEas
(FastEas) were on average $20\%$ ($11\%$, resp.) slower than the corresponding
animations for SlowLin (FastLin, resp.). Comparing the different speeds, the
animations for SlowLin (SlowEas) were on average $74\%$ ($87\%$, resp.) slower
than the ones for FastLin (FastEas, resp.); see Table 3. We produced a trial
for each combination of network and task, so to obtain a total of $15$ trials
for each experimental condition.

Table 3: Data about the stimuli used in our experiment. Stimulus | Nodes | Edges | Density | Resolution | SlowLin | SlowEas | FastLin | FastEas
---|---|---|---|---|---|---|---|---
boardgames | 40 | 214 | 5.35 | 1030$\times$820 px | 7.79s | 8.74s | 4.25s | 4.87s
marvel | 52 | 152 | 2.92 | 846$\times$974 px | 4.92s | 5.61s | 2.66s | 3.08s
kpop | 58 | 123 | 2.12 | 1142$\times$858 px | 5.31s | 7.09s | 3.43s | 3.55s

###  3.4 Experimental Process

For our experiment, we opted for a between-subject design where each
participant is exposed to one of the four conditions and hence to $15$ trials.
Indeed, a within-subject design would imply that each user sees the same
experimental object $20$ times, which makes it difficult to avoid the learning
and the fatigue  effect. In addition, the between-subject study design allowed
us to query subjective feedback from the users about a specific condition
after a series of trials. The users performed an online test prepared with the
LimeSurvey tool (https://www.limesurvey.org/).

The phases of the survey are the following:

  1. 1.

we collect some information about the user;

  2. 2.

we assign the condition to the user by adopting a round robin approach;

  3. 3.

we present a video tutorial and some training questions;

  4. 4.

we present the $15$ trials in random order;

  5. 5.

we ask for some subjective feedback: $(i)$ Two 5-point Likert scale questions
about the beauty of the drawings and the easiness of the questions, $(ii)$ one
question about the perceived speed of the movements, $(iii)$ a question asking
users if they found the tasks tiring, and $(iv)$ an optional free-form
feedback.

We used the gdnet, ieee_vis, and infovis mailing lists to recruit
participants. Also, we involved our engineering and CS students of the
universities of Perugia and Tübingen.

##  4 Results and Discussion

We collected questionnaires from 84 participants (21 per condition); see Table
4 for some details.

Table 4: Data about the participants. Gender | Age | Screen size
---|---|---
Woman | Man | Transgender | Prefer not to say | 18-24 | 21% | <13” | 2%
31% | 63% | 1% | 5% | 25-29 | 26% | 13” | 13%
Educational level | 30-34 | 21% | 14” | 21%
High school | Bachelor | Master | Doctoral degree | 35-39 | 8% | 15” | 20%
14% | 27% | 30% | 29% | 40-44 | 7% | >15” | 38%
Expertise | 45-49 | 5% | No answer | 5%
None | Low | Medium | High | 50-59 | 5% |  |
12% | 31% | 26% | 31% | 60+ | 6% |  |

###  4.1 Quantitative Results

For each question, we recorded answers and response times of all participants.
For T1 (Adjacency) and T3 (PathLength), the error rate of a user is computed
as the ratio between the number of wrong answers and the total number of
questions, that is $3$ for each task. For T2 (NeighborhoodSize), T4
(CommonNeighbors), and T5 (InterRegionEdges), the error on a question is
computed as $1-\frac{1}{1+|v_{u}-v_{c}|}$, where $v_{u}$ is the value given by
the user and $v_{c}$ is the correct value.

We performed a Shapiro-Wilk test [29, 30] and we found that the populations
were normally distributed for tasks T2 (NeighborhoodSize), T4
(CommonNeighbors), and T5 (InterRegionEdges). We thus performed a one-way
ANOVA test and post-hoc pairwise comparisons of the considered placement
strategies by using Tukey’s HSD test [21]. Regarding T1 (Adjacency) and T3
(PathLength), we performed the non-parametric Kruskal-Wallis test. For all the
above tests, the significance level was set to $\alpha=0.05$. From the tests,
we did not observe any statistically significant difference between the four
experimental conditions. However, by looking at the box-plots of the response
time and error rate, which are reported in Figs. 3 and 4, one can observe
interesting phenomena555 Observe that the scales on the y-axes in Figs. 3 and
4 differ between subfigures. .

Regarding response time, the plots in Fig. 3 show that for all tasks but T5
(InterRegionEdges), Fast speed and cubic Bézier curve Easing perform better
than Slow speed and Linear easing, respectively. This behavior can also be
observed over all tasks, but cubic Bézier curve Easing seems to perform better
than Linear easing only if combined with Fast speed (FastEas model). A similar
trend is exhibited by the error rate reported in Fig. 4. In particular, for
tasks T2 (NeighborhoodSize), T3 (PathLength), and T4 (CommonNeighbors) cubic
Bézier curve Easing performs better than Linear easing in the case of equal
speed. Although average values are very low for T1 (Adjacency), they show
slightly better performance with Fast speed and cubic Bézier curve Easing than
with Slow speed and Linear easing, respectively. This behavior can also be
observed over all tasks.

Figure 3: Response time aggregated by task.

Figure 4: Error rate aggregated by task.

###  4.2 Subjective Results

The percentage of answers to the Likert scale questions are reported in Figs.
5 and 5 (the answer distributions as box-plots can be found in Appendix 0.B);
we assigned a score from 1 (lowest) to 5 (highest) to each answer. While there
is no statistically significant difference among the conditions, we can
observe the following. Regarding beauty, users seem to prefer Slow speed and
cubic Bézier curve Easing rather than Fast speed and Linear easing. Indeed,
the models with Slow speed received a percentage of positive appreciations (4
and 5) that is higher than the one obtained by the models with Fast speed, and
the models with cubic Bézier curve Easing received a higher percentage of
positive appreciations than the models with Linear easing. Also, the average
ratings suggest that the preferred model is SlowEas. About easiness, on
average FastEas and SlowLin are the models that were found to be most
difficult and easiest, respectively.

The percentage of users who considered the speed adequate is on average higher
for the models with Slow speed than for the ones with Fast speed; see Fig. 5.
The percentage of users who found it tiring to work with Fast speed is higher
than the one of users who worked with Slow speed; see Fig. 5. Among the slow
models, Linear easing seems to produce visualizations that are rated as less
tiring than cubic Bézier curve Easing.

Figure 5: Subjective results. In (a) and (b), 1=lowest and 5=highest.

Additional user feedback. Overall $42\%$ of users submitted additional
feedback amounting to 35 submissions, while the proportion of submissions in
each group was rather balanced, with SlowEas and FastLin giving the most
comments ($10$ each); see Fig. 6. We performed a thematic analysis on the
users’ feedback and we identified four recurring themes, namely _subjective
impressions and emotional impact_ ($30$), closely followed by _challenges and
obstructions_ when performing the study ($29$), _suggestions for improvements_
($17$), and _positive feedback to the proposed model_ ($11$); refer to Fig. 6.
The different groups have largely made similar points, mostly with no clear
trend to be noted. For a breakdown of each theme by model, see Figs. 6–6.

Figure 6: Subjective results derived from additional user feedback.

#### Subjective impressions and emotional impact.

Negative emotions ($5$ for Easing, $19$ for Linear) were reported more often
than positive ones ($6$). The dominant factor was a sense of
difficulty/frustration ($10$), followed by exhaustion/nausea ($5$), and
doubting practicality ($4$, Slow only). Other negative sentiments are
irritation/confusion ($3$, Slow only) and stress/overwhelm ($2$, Fast only).
On the positive side, users reported a liking ($3$), a sense of learning
($2$), and fun ($1$).

#### Challenges and obstructions.

Users mostly described that waiting for the animation of an edge hindered them
when performing the tasks ($7$, mostly Easing). Second most reported were
technical issues ($5$, mostly delay or glitches in the animation), followed by
a distraction due to the simultaneous movement ($4$). Other challenges were
the general difficulty to find graph structures requested by the tasks ($3$),
miscounting or losing progress ($3$, Fast only), discerning adjacency due to
nodes placement ($3$), finding coping mechanisms to alleviate difficulty ($2$,
Fast only) and not understanding task descriptions ($2$).

#### Suggestions for improvements.

Users most frequently requested control over the animations ($7$, e.g.,
through hovering or clicking). Further, it was suggested to use more colors
and different line-widths to highlight relevant structures ($5$). Concerning
network layout, two users in the FastEas group suggested that a better angular
resolution might improve task performance. The second requested interactive
feature was the possibility to mark nodes for tracking progress ($2$, Fast
only). Lastly, there was one call for better animation scheduling.

#### Positive feedback to proposed model.

Most often it was mentioned that the animation made the recognition of the
graph topology clearer ($4$), followed by the potential of the model in
restricted scenarios, i.e., for certain graphs or tasks ($3$, Easing only).
For the Slow speed, two users found the speed to be appropriate. Finally, two
users experienced technical issues with animation: They continued solving the
tasks as for peds, which they described as much harder than meds.

###  4.3 Discussion of our Results

We did not observe statistically significant differences between the four
models in terms of response time and error rate, which suggests that none of
the models clearly outperforms the others in terms of readability. However, an
analysis of the aggregate data provides some initial indications that cubic
Bézier curve Easing may help users in executing topology-based tasks in an
accurate manner. We do not have any indications for the overview task (i.e.,
T5 (InterRegionEdges)), whose behavior seems not to be in line with the other
tasks and the error rate is very high for all models (the percentages of wrong
answers for the marvel, boardgames, and kpop networks are $100\%$, $99\%$, and
$75\%$, respectively). Our interpretation is that this task is not suitable
for meds, since the connections between the boundaries of the highlighted
parts of the network appear and disappear repeatedly, thus distracting the
user and making it hard to keep track of the previously considered edges.
Moreover, while we observed slightly lower response times for the Fast
animation speed and the cubic Bézier curve Easing the collected data does not
suggest a model clearly outperforming the others in terms of response time.

Referring to our hypotheses, the data thus provide indications that only
partially support H1 and H2. In particular, Slow speed does not give a lower
error rate than Fast speed, while cubic Bézier curve Easing behaves better
than Linear easing. Our interpretation for this phenomenon is that, when
analyzing a connection between two nodes, the waiting time for the edge to
appear might be long, and this is more evident with a slow animation. Further,
the differences between Linear and cubic Bézier curve Easing may be caused by
the circumstance that the longer time when stubs are drawn almost fully in
cubic Bézier curve Easing helped the users when performing the tasks. Our
subjective results support H3. The analysis of the users’ free-form feedback
suggests that the cubic Bézier curve Easing is preferred over Linear easing.
Concerning the speed, the subjective feedback favors the Slow speed for
beauty, speed appropriateness and tiredness, while the easiness of the tasks
shows no clear trend. We believe that there are two different key effects at
play here: First, the higher waiting times for the Slow speed are perceived by
the users as a distracting factor as also mentioned in the free-form answers.
Second, the Fast speed shows animations in different parts of the drawing in
rapid succession, which may be overwhelming and lead to miscounting or losing
progress. Thus, we think that an intermediate speed between 100 and 200 px/s
would be most appropriate. This seems to be also partially supported by the
answers about speed; see Fig. 5. Also, note that FastEas has the highest
maximum animation speed and the lowest ratings as an _Okay_ speed.

###  4.4 Limitations of our Experiment.

First, we did not observe statistically significant differences between the
four models in terms of response time and error rate, which indicates that
none of the models outperforms the others in terms of readability. Possible
differences between the models could emerge with a higher number of
participants or by considering a different study design, i.e., a within-
subject experiment.

Second, we noticed that the only overview task that we considered (T5
(InterRegionEdges)) turned out to be very challenging for the users, which
made us conclude that meds are not a good paradigm to perform such a task.
Perhaps, the choice of different overview tasks would have allowed us a better
evaluation.

Moreover, the choice of not allowing interaction implied to use networks that
fit into a standard screen, thus facilitating the execution of an online test.
We think that enabling interaction may allow the evaluation of our models on
larger networks and could facilitate the users in the execution of tasks, e.g.
by employing selective animation and marking. On the other hand, we believe
that this would preferably require a controlled experiment study design.

Finally, the readability of meds may be sensitive to the scheduling algorithm
used for the animation of edges and to the one used to produce the layout.
This justifies further investigation with different scheduling and layout
algorithms.

##  5 Conclusions and Future Research Directions

We presented a user study that evaluates the readability of meds by comparing
different speeds and easing functions for the animation of edges. Since our
results do not exhibit statistically significant differences from a task
performance point of view, we suggest to follow user preferences. Namely,
users tend to favor the cubic Bézier curve Easing and rate slower animation
speed as more beautiful and appropriate. Thus, choosing an intermediate
animation speed may be the preferred option in practical applications.

Our study has some limitations and cannot be generalized to settings
significantly different from ours. This motivates further experiments with
larger networks, additional tasks, different scheduling algorithms for the
animation of edges, and interaction features. Regarding the last aspect, we
believe that the incorporation of user interaction can significantly improve
the model by only showing animations of edges or nodes that are selected by
the users. We emphasize that this has also been suggested by some participants
of our experiment.

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##  Appendix 0.A Algorithms for Computing MEDs

Misue and Akasaka [20] suggested a greedy algorithm to compute meds of short
total animation duration. Namely, they sorted edges in decreasing order of
$\tau_{(u,v)}$ and started the animation of the next edge at the earliest time
frame that do not introduce avoidable crossings. Misue [19] suggested three
variants of the algorithm that aim to minimize the average time between two
morphs of the same edge:  1. a variant that greedily picks edges to be
animated again as long as there it is possible within $T$,  2. a variant that
allows animations to begin before the start of $T$ and end after the end of
$T$ yielding a cyclic overlap,  3. a variant that allows a certain number of
avoidable crossings.  For the purpose of this study we decided to implement
Variant 1 and ignore the latter two variants. Namely, Variant 2 removes the
initial time frame during which all edges are at stub length ratio
$\delta_{0}$, which may serve as a useful anchor point for a viewer, while
only yielding minor improvements [19]. Variant 3 introduces the number of
crossings as an additional variable that may interfere with our experiments.
While the algorithmic descriptions of Misue and Akasaka [19, 20] do not
include $\tau_{1/2}$ and $\tau_{distinct}$, it is straight-forward to add
them. Meanwhile, the algorithms work for any invertible function
$\delta_{(u,v)}$ which for our purposes require that $\eta$ is invertible.

##  Appendix 0.B Box-plots for Subjective Analysis

Figure 7: Box-plots concerning subjective results.