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Abstract

We present sequence of time-aligned edge plots (STEP): a sequence- and edge-scalable visualization of dynamic networks and, more broadly, graph ensembles. We construct the graph sequence by ordering the individual graphs based on specific criteria, such as time for dynamic networks. To achieve scalability with respect to long sequences, we partition the sequence into equal-sized subsequences. Each subsequence is represented by a horizontal axis placed between two vertical axes. The horizontal axis depicts the order within the subsequence, while the two vertical axes depict the source and destination vertices. Edges within each subsequence are depicted as segmented lines extending from the source vertices on the left to the destination vertices on the right throughout the entire subsequence, and only the segments corresponding to the sequence members where the edges occur are drawn. By partitioning the sequence, STEP provides an overview of the graphs’ structural changes and avoids aspect ratio distortion. We showcase the utility of STEP for two realistic datasets. Additionally, we evaluate our approach by qualitatively comparing it against three state-of-the-art techniques using synthetic graphs with varying complexities. Furthermore, we evaluate the generalizability of STEP by applying it to a graph ensemble dataset from the architecture domain.

Keywords

Visualization dynamic networks graph ensembles sequence time scalability partial edge architecture

Introduction

When visualizing static networks, a technique is deemed effective if it accurately depicts the networks’structural properties and preserves them as network complexity increases in size and density. In visualization literature, there has been a lot of work on visualizing^1,2 and evaluating^3–5 techniques for static networks. Node-link diagrams and adjacency matrices are two of the most used and studied techniques. Node-link diagrams depict connectivity information in an intuitive way, making them a favored choice for path analysis tasks when network density is low. In contrast, adjacency matrices can reliably depict clusters in small and dense networks. Both techniques, however, reach their limits as network complexity increases and the task at hand changes,^6,7 eliciting other techniques for static network visualization.^8–10

In the real world, networks are rarely static but rather evolving (a.k.a. dynamic networks). Having time as an additional dimension besides network size and density makes the visualization task even more challenging. That is, a visualization technique does not only need to depict the networks’ structural properties and preserve them as networks scale in size and density but also show temporal patterns such as stability, periodicity, and trends associated with these properties, and preserve them as networks scale in time. Small multiples approaches^11,12 are often used to visualize dynamic networks. The networks from different points in time are placed adjacent to each other, providing an intuitive way to track the evolution of networks’ structural properties. However, such an approach does not scale in time.

In this work, we tackle that challenge by introducing STEP, a two-dimensional visualization technique for dynamic networks that is scalable with respect to the density and time dimensions. Time is depicted along the x-axis, while network nodes are represented on the y-axis. To achieve temporal scalability, STEP partitions the timeline into equally spaced intervals (see Figure 1(b)). Each interval is visually represented using a horizontal axis positioned between two parallel vertical axes. Network links are then visualized within each interval by connecting the source and destination nodes with segmented lines throughout the entire interval, and only the segments corresponding to the timepoints when the links occur are actually drawn. By partitioning the timeline, STEP offers a comprehensive overview of the evolving network structure and avoids aspect ratio distortion, particularly when dealing with a large number of timepoints (see Figure 1). Additionally, by partially drawing lines, STEP promises to scale better than the state-of-the-art with respect to network density.

Figure 1.

A visual comparison between (a) TEP and (b) STEP using the SIPRI Arms Transfers dataset. By partitioning the timeline into equidistant intervals, STEP avoids aspect ratio distortion and offers a comprehensive overview of the evolving network structure, which is hard to obtain in TEP.

While we originally designed STEP for visualizing dynamic networks, the technique is also applicable for visualizing a collection of graphs that are generated using probabilistic or random models (i.e., graph ensembles). The ensemble members, however, do not necessarily have temporal information associated with them. In that case, an order is established by sorting them according to one or more data attributes. In that sense, dynamic networks can be viewed as graph ensembles ordered by the time attribute. Such a generalization allows us to apply STEP to different datasets and problem domains.

This paper extends and reuses materials from our previous work¹³. Our earlier paper introduced the concept of time-aligned edge plots (TEP) for drawing edges as partial lines aligned in time (see Figure 1 (a)). We extend our previous work in the following ways:

We extend the time-aligned edge plots (TEP) technique by introducing timeline partitioning, resulting in a sequence of TEP. Hence, the name STEP. This extension allows us to avoid aspect ratio distortion with a large number of timepoints and provides a good overview of the evolving graph structural patterns.

We include another dataset (SIPRI Arms Transfer) in addition to the software call graph dataset for the use case-based evaluation.

We add another visualization technique to the comparative parameter study: stacked edge splatting (SES).

We present a case study with one domain expert from the architecture domain to validate the generalizability of STEP for visualizing graph ensembles.

Related work

Static graphs build a complex data structure composed of vertices and edges. To visualize them, node-link diagrams or adjacency matrices are often used. Dynamic graphs add time as a third data dimension, posing a visualization challenge. In recent decades, dynamic graph visualization became an active area of research. Beck et al.¹⁴ discuss two major concepts to encode the time dimension into the graphs: animation (time-to-time mapping)^15,16 and timeline (time-to-space mapping).^17,18 Animation bears a close resemblance to the real world. According to the Congruence Principle, it is an intuitive choice for conveying concepts of change.¹⁹ However, when used for analysis tasks, animation may be too fast to be accurately apprehended and less effective than its static counterpart.²⁰ This is due to the limited capacity of humans to preserve information^21–23 and detect changes (change blindness).²⁴

Alternatively, time-to-space mapping approaches use the screen space to encode the time dimension. A simple approach is to juxtapose node-link diagrams,^25,26 adjacency matrices,^27,28 or bipartite graph layouts²⁹ in a small multiples fashion. These approaches, while providing a high-level overview of the network structural changes, do not scale with respect to time. Additionally, it requires the viewers to revisit the graphs back and forth to obtain an overview of the temporal changes, which is a challenging task.

To address these problems, other approaches adopt a stacking representation to overlay small multiples on top of each other, resulting in a more condensed visualization where the changes between different timepoints can be instantly perceived, thus, reducing the users’ cognitive load. Bach et al.^27,30 stack the adjacency matrices on top of each other, resulting in a three-dimensional cube or a pile of adjacency matrices. However, since the adjacency matrix layout is a two-dimensional representation, adding the time dimension results in a three-dimensional representation, making it difficult to obtain an overview of the dynamic graph without using interaction techniques. Moreover, the screen space consumed by the adjacency matrix grows quadratically with respect to the number of vertices, making these techniques only suitable for visualizing small graphs.

To preserve both networks’ structural properties and temporal patterns, other approaches^31,32 attempt to reduce the complexity of dynamic graphs by modeling the individual graphs as points in high-dimensional space and projecting them to a two-dimensional node-link graph layout. However, in this case, the original graph’s structure is not visible anymore, and the meaning of the resulting projection can be hard to grasp. In our work, we aim to obtain a direct representation of dynamic graphs rather than indirectly through the use of dimensionality reduction.

We situate our work close to massive sequence views,¹⁸ interleaved edge splatting,¹⁷ and stacked edge splatting.³³ The three approaches result in a two-dimensional visualization, where the x-axis encodes time, and the y-axis encodes the positions of vertices. Additionally, they employ vertex ordering to reduce the amount of visual clutter. Nevertheless, the three approaches fail to maintain the appropriate balance between preserving the networks’ structural properties and the temporal patterns. Massive sequence views¹⁸ and interleaved edge splatting¹⁷ emphasize the structural properties by plotting the entire graph structure at each timepoint, resulting in overdrawing problems and obscuring the detection of both the structural properties and the temporal patterns. Stacked edge splatting,³³ in contrast, reveals the temporal patterns by using a separate canvas area to plot the edges’ temporal events. However, obtaining an overview of the graph structure at a certain time point becomes challenging since the graph structure is aggregated and plotted in a different canvas area.

In our work, we aim to find a better balance between preserving the graphs’ structural properties and temporal patterns. We do that by partitioning the timeline into equidistant intervals and partially drawing the graph edges within each interval aligned with the timepoints in which they occur. The timeline partitioning facilitates tracking structural changes in the networks by comparing these intervals, similar to small multiple approaches. Simultaneously, the partial drawing of edges reveals the temporal patterns.

Method

In this section, we present STEP, which is an evolution of our earlier introduced time-aligned edge plots (TEP).¹³ We begin by introducing our data model and subsequently delve into the details of TEP. We then introduce STEP, discussing technical aspects related to edge depiction, vertex ordering, and visual patterns.

Data model

We model a dynamic graph as an ordered sequence $S = (G_{O (1)}, G_{O (2)}, \dots, G_{O (N)})$ , where $G_{O (i)}$ is an instance of the graph at index $i$ after temporally ordering the sequence using the function $O (\cdot)$ , such that $1 \leq i \leq N$ . Each $G_{O (i)}$ consists of a set of vertices $V$ and a set of edges $E_{O (i)} \subseteq E$ . This definition generalizes to graph ensembles by allowing the function $O (\cdot)$ to be based on other relevant data attributes.

With respect to the graph density, we adopt a linear density definition³⁴ as $d_{ℓ} = | E | / | V |$ , which resembles the complexity of the real-world networks.³⁵

Time-aligned edge plots

TEP consists of one horizontal axis placed in-between two vertical parallel axes (see Figure 2 (middle)). The horizontal axis represents an interval of time, while the vertical axes represent the graph vertices. Vertices are replicated and placed on both vertical axes in the same order (i.e., axis duplication). The left vertical axis represents the source vertices, whereas the right one represents the destination vertices, adopting a layout otherwise known from visualizations of bipartite graphs. To visualize edges, we connect the source and destination vertices with segmented straight lines, and only the segments that correspond to the timepoints where the edges occur are drawn. This reduces the number of depicted lines significantly, resulting in a less cluttered visual representation and revealing the temporal patterns associated with the edges.

Figure 2.

Comparison between small multiples of node-link diagrams (left), TEP (middle), and STEP (right). Edges over time are depicted by a single line in TEP, resulting in a less cluttered representation and revealing temporal patterns. To scale TEP in time, we partition the timeline into equidistant intervals, each represented by a TEP, resulting in a sequence of TEP or STEP.

Figure 2 shows a comparison between small multiples of a node-link diagram (left) and TEP (middle). One can perceive several temporal patterns by looking at TEP. For example, the periodic pattern of edges C→D and D→E, the stability of edges A → C and B → A, and the gaps in edges E → B and A → D. These patterns are difficult to obtain by looking at the small multiples, regardless of the underlying representation (i.e., node-link diagram, adjacency matrix, or bipartite layout). Typically, it requires the viewers to revisit the individual graphs back and forth in order to see these patterns. Additionally, it is even more difficult to spot temporal synchronization, such as the alternating behavior between the two edges C → D and D → E in the small multiples view. In contrast, in TEP, the alignment of edges over time allows the temporal synchronization to appear as spatial synchronization and, therefore, is easier to see.

Sequence of TEP

While TEP, in principle, can reveal the temporal patterns associated with the edges, the technique has limited scalability with respect to the time dimension and is prone to aspect ratio distortion. Figure 1(a) shows an example of such an issue. As the network scales in time, it becomes challenging to recognize the edges, especially in the middle region of the timeline. To avoid that issue, we partition the entire timeline into equidistant time intervals, each represented by a TEP, resulting in a sequence of TEP (see Figure 1(b)). By partitioning the timeline, we not only ensure a well-proportioned ratio between the width and height but also provide a comprehensive overview of the evolving graph structure akin to the small multiples approaches. However, in the case of STEP, each multiple (i.e., TEP) is a composition of several timepoints.

Finding a balance between width and height is crucial to line-based plots.^36,37 As seen in Figure 3, opting for a wide aspect ratio while emphasizing the time dimension distorts the graphs’ structural patterns. Conversely, narrow aspect ratios may compromise the clarity of the time dimension. While a 1:1 aspect ratio seems an intuitive choice, our experimentation showed that an aspect ratio around the golden ratio (1:1.618) yields visually appealing outcomes while preserving the right balance between the two axes. The length of the time intervals (i.e., the number of timepoints per TEP) is determined based on the aspect ratio and the width of a timepoint segment, a user-defined parameter.

Figure 3.

The influence of different aspect ratios on the time and vertex dimensions. While wide aspect ratios distort the structural patterns, narrow ones impair the time dimension.

Edge depiction

In TEP, the depiction of edges depends on their occurrence frequency over time. While stable edges will be depicted as solid lines, periodic edges will appear as stippled or partially drawn lines (see Section “Visual patterns”). As a result, the visibility of an edge and, therefore, the ability to identify its source and target nodes correlates with its occurrence frequency.

Previous work^38,39 has shown that breaking the edges at crossing points³⁸ or drawing them at 75% of their full length³⁹ can reduce visual clutter without compromising the recognition of edges in cluttered areas. Since edges can occur for short periods (i.e., outliers), it becomes challenging to see them, let alone trace the source and target nodes. To address this problem, we draw reference lines for edges in the background based on the occurrence frequency, a user-defined parameter. For example, the users can select to draw the reference lines for edges that have occurrence frequency between the minimum and the maximum of a defined interval. To distinguish between the reference lines in the background and the actual edges in the foreground, we increase the reference lines’ transparency (see Figure 2 (middle)).

To further amplify the recognition of edges in highly cluttered areas, we use color to encode the slopes of the lines. Additionally, we blend the lines by applying alpha compositing. This is similar to previous work,⁴⁰ where color has been used to encode a combination of lines’ distance, angle, and crossings.

As shown in Figure 4(c), encoding the slope by color and applying alpha compositing increases the visibility of edges in crossing areas. Furthermore, we implemented an interactive, nonlinear time-zooming lens that allows the users to magnify individual graphs within TEP by moving the mouse cursor over them.

Figure 4.

By encoding the slope by color and applying alpha compositing, we amplify the recognition of edges in highly cluttered areas: (a) edges are drawn as opaque lines, (b) edges are blended by applying alpha compositing, and (c) edges are blended, and the slope is encoded by color.

We experimented with two ways of depicting the edges as straight lines or Bezier curves. While straight lines are preferable for accurately depicting slopes, Bezier curves are esthetically pleasing (see Figure 5). Additionally, we experimented with three options to encode a time-varying attribute of the graph (i.e., edge weight): (a) constructing a bi-variate color map from a uni-variate isoluminant color map;⁴¹ (b) using the line width as an additional variable to encode the edge weight; (c) a combination of both. We found that options (b) and (c) provide a more accurate representation of edge weight, with option (b) being more esthetically pleasing due to the flexibility it offers when it comes to choosing the color map (see Figure 6). Nevertheless, the three options reach their limits as the density of the graph increases.

Figure 5.

Two options for depicting edges: (a) straight lines and (b) Bezier curves. While straight lines are preferable for accurately depicting slopes, Bezier curves are more visually pleasing.

Figure 6.

Three options to encode a time-varying attribute while using the color to encode the line slope: (a) color encoding using bi-variate color map, (b) line width encoding, and (c) a combination of both.

Vertex ordering

Since vertices are placed on the vertical axis, vertex-ordering algorithms play a critical role in obtaining a less cluttered visual representation. In graph theory, the problem of linearly ordering the vertices to optimize a cost function is an NP-hard problem, known as minimum linear arrangement,⁴² or MinLA problem. In our work, we combine hierarchical clustering with simulated annealing to obtain an efficient arrangement of vertices. Hierarchical agglomerative clustering is a deterministic algorithm except for the tied distances. The algorithm proceeds in a bottom-up approach by merging the two most similar clusters, until all clusters are hierarchically merged into one single cluster containing all graph vertices. The similarity between two graph vertices $v_{i}$ and $v_{j}$ is obtained by calculating the Jaccard coefficient:

$J (\bar{V_{i}}, \bar{V_{j}}) = \frac{| \bar{V_{i}} \cap \bar{V_{j}} |}{| \bar{V_{i}} \cup \bar{V_{j}} |} \in [0, 1],$

where $\bar{V_{i}}$ and $\bar{V_{j}}$ are the sets of direct neighbors for vertices $v_{i}$ and $v_{j}$ , respectively.

In other words, this similarity metric describes a measure for a common neighborhood of vertices. Vertices that have similar out- and in-connections will belong to the same cluster, and therefore, placed close to each other on the vertical axes. The resulting hierarchy is further reordered to minimize a cost function according to:

$minimize : \bar{l} (π_{V}) = \frac{1}{| E |} \sum_{\forall e \in E} l (e),$

where $\bar{l} (π_{V})$ is the average edge length in the graph, $π_{V}$ is a permutation on the set of vertices V, and $l (e)$ is the edge length defined as:

$l (v_{i}, v_{i}) = \frac{| i - j |}{| V | - 1},$

where $i$ and $j$ are the positions of $v_{i}$ and $v_{j}$ on the vertical axis in permutation $π_{V}$ .

This allows us to further optimize the global connections between clusters without losing the hierarchical relationship between them. To do this, the hierarchy is traversed from top to bottom (breadth-first approach). For each cluster, the left and right children are swapped, and the cost function is computed before and after the swapping so that we keep the configuration that achieves minimum cost.

We further use the flexibility of simulated annealing to optimize the hierarchy resulting from the clustering algorithm, resulting in an even more optimized order of vertices. Similar to previous work,¹⁸ we define a proportional cooling schedule, that is, $T_{i + 1} = α T_{i}$ , where $α$ is a constant close to but smaller than 1, known as the cooling factor. We chose a low value for the initial temperature $(T_{0} = 100)$ and cooling rate $(α = 0.999)$ .

Visual patterns

Applying a proper vertex-ordering technique and aligning the edge over time allows STEP and TEP to reveal some of the graph topological features and temporal patterns. In the following, we describe graph structural features and temporal patterns that can be recognized in a single TEP. STEP relies on the combination of such features and patterns, that is, directly building on the following discussion.

Structural patterns

The fan pattern describes a single graph node that has a lot of incoming (fan-in) or outgoing (fan-out) connections. The connections usually spread in all directions, forming a fan-like shape (see Figure 7(a) and (b)). If the same node has both fan-in and fan-out patterns, the overlapping between both patterns will form a diamond-like shape. The cross pattern describes a symmetrical relationship between two nodes, where the connections between them exist in both directions. This pattern forms a cross-like shape where the intersection point occurs exactly at the center of the plot (see Figure 7(c)). The cluster pattern describes a group of graph nodes that are strongly connected. In an ordered graph, these connections appear as horizontal lines (see Figure 7(d)). The cross-cluster pattern is a combination of both the cross and the cluster patterns. It resembles connections that exist in both directions between two clusters of nodes, resulting in an hourglass-like shape (see Figure 7(e)).

Figure 7.

Structural patterns in TEP: (a) fan-out, (b) fan-in, (c) cross, (d) cluster, and (e) cross-cluster.

Temporal patterns

The stability pattern describes a constant presence of certain connection(s) over time. For a single connection, this pattern appears as a solid line going from the source to the destination (see Figure 8(a)). The gap pattern describes a break in the continuity of a stable connection(s) for extended periods of time. For a single connection, this pattern appears as a solid line with a gap occurring at the beginning, the middle, or the end of it (see Figure 8(b)). The periodicity pattern describes a regularly repeated presence and absence of certain connection(s) over time. For a single connection, this pattern appears as a dotted or dashed line. The lengths of the dashes or the gaps between the dots are not necessarily uniform over time (see Figure 8(c)). A trend describes a gradual increase or decrease of certain connection(s) over time. For a single connection, this pattern appears as a dotted or a dashed line, where the lengths of the dashes or the gaps between the dots are gradually increasing or decreasing over time (see Figure 8(d)). The phase shift pattern describes a series of two or more periodic connections that are regularly repeated in the same order over time. It appears as two or more dotted or dashed lines synchronized together over time (see Figure 8(e)). This pattern is an analytical insight that cannot be obtained easily using other visualization approaches. The anomaly pattern describes an unusual presence of certain connections at specific points in time. It appears as an unusual presence of a dot or a dash in a temporally coherent structure (see Figure 8(f)). An anomaly timepoint is a point in time where the graph density remarkably deviates from the average density.

Figure 8.

Temporal patterns in TEP: (a) stability, (b) gap, (c) periodicity, (d) trend, (e) phase shift, and (f) anomaly.

Sequential ordering and graph ensembles

Since TEP and STEP model a dynamic network as a sequence of graphs, they assume that these graphs are inherently ordered, typically based on temporal criteria. We extend our approach to scenarios where no temporal information associated with the graphs could be used to provide a natural ordering. In particular, our goal is to make TEP and STEP applicable to ensembles of graphs. Here, we assume that we have a set of individual graphs that share a set of vertices. Then, we need a way to construct an ordered sequence of graphs from the unordered set. One viable approach is to arrange the sequence based on the topological similarity between the graphs. For example, by hierarchically clustering the sequence using pairwise Euclidean distances calculated from the connectivity matrices. However, other ordering approaches might also be useful as long as some coherence between neighboring sequence members is achieved. It should be noted that (re)-ordering could even be useful for dynamic graph visualization if the original time ordering fails to provide sufficient structural coherence.

Use cases

We evaluate the versatility of STEP by applying it to two real-world datasets. The software call graph dataset assesses the ability of STEP to reveal temporal patterns, while the SIPRI Arms Transfers dataset tests the effectiveness in visualizing structural changes. Please see the supplemental materials for the datasets files.

Software call graph

The software call graph dataset contains the dynamic method calls of an open-source Java system called JHotDraw.⁴³ We chose to visualize the dynamic method calls resulting from a drawing scenario done by Beck et al.⁴⁴ using the JHotDraw graphical editor (JavaDrawApp). The test scenario was to start the program, create a new file, draw a rectangle, draw a circle, and write a text into the circle. The dynamic method calls from this scenario were used to evaluate different dynamic graph visualization techniques before.^17,33,44 The dataset contains 787 vertices and 1077 timepoints, and has an average edge density of approximately $d_{ℓ} = 0.91$ .

Figure 9 shows the dynamic method calls of the drawing scenario visualized using STEP over one interval. At the top of the figure, we annotated the timeline with the corresponding user actions. At the bottom, we marked four temporal patterns that we could identify. Each depicts function calls that are triggered as a reaction to a mouse movement. Beck et al.⁴⁴ provided some interpretations of these patterns. For example, Pattern reflects the function calls when the user moves the mouse over the toolbar. Pattern and Pattern represent the mouse movement over an empty canvas area and the drawn objects, respectively. Pattern reflects the function calls when the user draws (drags) the object. In this way, one can tell the drawing scenario by visually following the user actions and function call patterns, as depicted in Figure 9.

Figure 9.

Software call graph dataset visualized using STEP over one interval. Four temporal patterns can be identified (annotated at the bottom).

Each of the user actions annotated in Figure 9 at the top results in a dense timepoint, where many function calls occur. A typical example is the first timepoint when the program starts, as shown in Figure 10(a). These are probably function calls related to the program’s initialization process. Another example is when the user creates a new file (Figure 10(b)). These timepoints are considered outliers in terms of density compared to the whole graph sequence.

Figure 10.

Dense call graphs resulting from user actions: (a) starting the program and (b) creating a new file.

SIPRI Arms Transfers

The second dataset is the SIPRI Arms Transfers dataset published by Stockholm International Peace Research Institute.⁴⁵ The dataset is aggregated on a yearly basis from 1950 to 2017 (i.e., 68 timepoints), contains 259 vertices, and has an average edge density of approximately $d_{ℓ} = 1.11$ .

Figure 11(a) shows the 68 years visualized using STEP over four intervals. In the first interval (1950–1966), the arms transfer landscape appears to be dominated by three countries: the United States , the United Kingdom , and the Soviet Union , as the three biggest exporters. and appear to share the same importing countries and have more spread than , in contrast, exported to a more limited group of countries such as East Germany (GDR) and Czechoslovakia . In the second interval (1967–1984), we notice new players joining the scene. Notably, France takes a prominent position at the top of the chart, whereas Germany and Switzerland appear toward the bottom of the plot. We also notice an increasing volume of trade from the three major players (, , and ). This trend continues to the third interval (1985–2000), with new players such as Canada , Sweden , Brazil , and Italy joining the scene. In the fourth interval (2001–2017), we notice the disappearance of and and the decline in the and trade activities. Overall, the network exhibits the preferential attachment property of the Barabási network model, where few countries serve as primary hubs and attract connections over time. (i.e., fan-out pattern).

Figure 11.

The SIPRI Arms Transfers dataset visualized using STEP at four different time intervals. (a) Vertices are randomly ordered. The emergence of new players and networks’ structural changes can be noticed as the network evolves. (b) Vertices are ordered using hierarchical clustering. Ordering the network vertices reveals countries that share similar profiles.

To gain further insights into countries with similar trade profiles, we applied hierarchical clustering to the graph vertices. As annotated in Figure 11(b), the ordering organizes the countries into four distinct sets or clusters along the y-axis: {}, {, China , Russia }, {, , , , }, {}. The European cluster seems to have a lot of trade within itself. However, stands out with a slightly different profile than the rest of the cluster with more exports to African countries. The superpower cluster {, , } has more trade volume not only toward Middle Eastern countries such as Saudi Arabia, India, Pakistan, Jordan, UAE, and Egypt but also toward countries in the European cluster, as well as Asian and African countries such as Congo, Zambia, Guinea, Ghana, Sri Lanka, Nepal, Vietnam, among others. The aforementioned Middle Eastern countries engage in transactions with both the superpower and European clusters. The two singular clusters {} and {} engage in a bidirectional exchange, which is visible in the first two intervals (see Figure 11(b)).

Regarding the temporal patterns, there is an overall increasing trend in arms trade over the years, with being an exception. Exports from exhibit a non-constant pattern, displaying periodic fluctuations over time. Similar periodic patterns are observed in the first two intervals between and East Germany (GDR). However, detecting these patterns proved challenging due to visual clutter. While the ordering of vertices aimed to reduce this clutter, it resulted in a densely packed plot of edges toward the middle of the graph, which did not contribute to recognizing these patterns effectively.

Parameter study

To further evaluate STEP, we conducted a parameter study to compare it against three state-of-the-art techniques: massive sequence views (MSV),¹⁸ interleaved edge splatting (IES),¹⁷ and stacked edge splatting (SES).³³ In Section “Theoretical comparison,” we first compare the four techniques theoretically. Then, in Section “Generative data model,” we present the generative model we used to synthetically generate dynamic graphs. Finally, we qualitatively inspect these graphs in Section “Qualitative results inspection.”

Theoretical comparison

MSV, IES, SES, and STEP are all edge-based visualization techniques. In edge-based techniques, graph vertices are drawn only once for the entire sequence, while edges are repeated at each timepoint. In contrast, in small multiples approaches, graph vertices and edges are drawn repeatedly at each timepoint (see Figure 12). This distinction in modeling suggests that edge-based techniques could be more scalable with respect to the time dimension than small multiples approaches, allowing them to reveal the temporal events in the data. In contrast, in edge-based techniques, changes in graph vertices are not expressed by the vertices themselves but rather by the edges attached to them. Therefore, temporal changes related to the graph vertices are not easily recognized in these techniques (i.e., node addition/removal).

Figure 12.

A comparison between small multiples approaches (a) and edge-based techniques: (b) MSV, (c) IES, (d) SES, and (e) STEP. In small multiples approaches, graph vertices and edges are drawn repeatedly at each timepoint. In edge-based techniques, graph vertices are drawn only once for the entire sequence, while edges are repeated at each timepoint.

In the following, we compare the four techniques with respect to two aspects: the depiction of edges and the visualization of graphs’ structural and temporal patterns. Due to the space limitation, we refrain from describing the other techniques in detail. The interested reader, however, is encouraged to read the respective publications.

Edge depiction

As seen in Figure 12, in both IES and MSV, edges are depicted by drawing straight lines going from the source to the destination, resulting in a sequence of lines over time. These sequences of lines form block structures that cover certain areas of the graph, depending on the edge length and time span. In large dynamic graphs, these blocks are likely to be drawn on top of each other, resulting in a cluttered visual representation, where the structural and temporal patterns are hardly seen. In contrast, in STEP and SES, edges are partially drawn over time. This reduces the number of depicted lines by a significant amount, resulting in an edge-scalable visualization and revealing structural and temporal patterns.

In STEP, SES, and IES, the depiction of edges essentially relies on the edge slope or inclination. To amplify the recognition of slope in highly cluttered areas, color is used to encode this information. In contrast, in MSV, since all edges are drawn as vertical lines, the slope encoding is irrelevant. Hence, the technique makes use of color to encode the length and direction (top or down) of edges. However, relying on color alone to distinguish edges might be problematic since the color of an edge is the result of alpha compositing all edges underneath. Therefore, it is likely that the edge will have varying colors as the density of the graph changes over time.

Structural and temporal patterns

MSV and IES plot the complete graph structure repeatedly at each timepoint. Such an approach results in a substantial amount of overdrawing, impairing the detection of both the structural and temporal patterns. In STEP, the graph structure is drawn only once over a range of timepoints, minimizing overdrawing substantially. In STEP, each timepoint contributes to the graph structure with a unique strip of pixels depending on its order in the timeline. While SES works in a similar way to STEP, the distinction between the two lies in how individual timepoints contribute to the overall graph timeline. In STEP, the position of the strip of pixels taken from each timepoint depends on the order of the timepoint within the entire timeline. In contrast, in SES, all timepoints contribute the same position for the pixel strip (i.e., always the first x-pixels from each timepoint). This distinction allows SES to be effective at revealing the temporal patterns and STEP to be effective at providing an overview of the graph structure. However, it also comes with a drawback. In the case of SES, it hides the graphs’ structural information. In the case of STEP, it hinders the recognition of the temporal patterns.

To obtain the graphs’ structural information, SES relies on tracing the edges to the final timepoint (a.k.a., the representative timepoint), where an aggregated graph structure from all timepoints is plotted. However, such a step is mentally demanding since it requires reconstructing the graph structure of the individual timepoints from the aggregated one. This makes SES ideal for revealing temporal patterns rather than structural ones. STEP, on the other hand, offers an insightful overview of how the graph structure evolves over time. However, due to the same reason, the same graph structure might appear differently, depending on the point in time it occurs. Figure 13 shows how the fan-out pattern is depicted over time in each visualization approach. While the shape of the pattern remains unchanged in MSV, IES, and SES, it is gradually changing in STEP, affecting the recognition of the same pattern over time.

Figure 13.

The fan-out pattern depicted over time: (a) MSV, (b) IES, (c) STEP, and (d) SES. While the shape of the pattern remains unchanged in MSV, IES, and SES, it is gradually changing in STEP.

Generative data model

To generate synthetic dynamic graphs, we adopted the model proposed by Cooper et al.⁴⁶ This model extends the preferential attachment model introduced by Barabási and Albert⁴⁷ (abbreviated as Barabási) by considering four elementary processes that are present in many real networks: the addition and removal of edges and vertices over time. While the addition follows the preferential attachment model, the removal is done rather randomly. By suitable choice of the input parameters, the model produces scale-free graphs, where the degree distribution of vertices follows the power-law $p_{k} ~ k^{- γ}$ , a property that has been observed in many real-world networks.^48,49 The model has the following parameters:

1. $α, α_{0}, α_{1}$ the input parameters in the original model. $α$ has to satisfy the condition $α > 0.5$ . Choosing $α_{1} > α_{0}$ , assigns more weight to the probabilities of node events. In contrast, $α_{0} > α_{1}$ , assigns more weight to the probabilities of edge events.

2. m: defines how many edges are to be added/removed randomly at each timepoint. It also implicitly defines the number of neighbors for the new nodes

3. t: defines the number of timepoints (iterations).

We can maintain a relatively stable edge density over time by setting the $m$ parameter to the desired density accompanied by suitable α’s probabilities. The graph size $n$ , in contrast, is controlled by the number of timepoints $t$ and the value of the $α_{1}$ parameter. Setting $α_{1}$ close or equal to 1.0 increases the probability of adding a new node at each iteration, implying that the total size of the graph is close to the number of timepoints.

By manipulating the $m$ , $t$ , and $α$ parameters, we can generate dynamic graphs of varying complexities. We experimented with three density values $m = {1, 2, 4}$ , to generate dynamic graphs where the number of edges is {equal, double, or four times} the number of nodes. By choosing these values, we aim to capture the average density found in many real-world networks.^34,35 Similarly, we experimented with two sets of $α$ parameters ${α = 0.85, α_{0} = 0.01, α_{1} = 0.75}$ and ${α = 0.6, α_{0} = 0.35, α_{1} = 0.2}$ . By the first and second sets, we aim to evaluate the techniques against node-related and edge-related events, respectively. Both sets result in scale-free graphs with degree exponent $γ \approx 3$ , which lies in the range $2 \leq γ \leq 4$ . This range has been identified as the degree exponent range for most real-world networks.^48,50

We decided to use a fixed number of timepoints $t = 100$ . On the one hand, we wanted to narrow the scope of our evaluation to only against the edge density, which is the critical factor, considering the techniques under evaluation. On the other hand, we wanted to ensure that the resulting visualizations could fit in an internet browser. This results in a total of six dynamic graphs listed in Table 1. For simplicity, we assume that all edges have the same weight (i.e., unweighted graphs).

Table 1.

By experimenting with three density values and two different sets of $α$ parameters, we produce six different graphs of varying complexities (G1–G6).

Event type	Edge density
	1	2	4
Node-related events	G1	G2	G3
Edge-related events	G4	G5	G6

Qualitative results inspection

We visualized the graphs using the four techniques. The results are shown in Figures 14 and 15. We use the color map introduced earlier in Figure 4 to encode the lines’ slope or the direction, in the case of MSV. Please check the supplemental materials for the generated graphs data and the corresponding R script. In the following, we summarize the results of our qualitative comparison.

Figure 14.

Node-related-events graphs G1–G3 (columns, from left to right) are visualized using the four visualization techniques: MSV, IES, SES, and STEP (rows, from top to bottom).

Figure 15.

Edge-related-events graphs G4–G6 (columns, from left to right) are visualized using the four visualization techniques: MSV, IES, SES, and STEP (rows, from top to bottom).

Structural patterns

Both MSV and IES suffer from overdrawing problems. Nevertheless, one can still notice the overall connectivity within the graphs by observing the color hue. For instance, in graph G2, a more yellowish hue suggests increased connections originating from nodes at the bottom toward nodes at the top, while in graph G3, a darker hue implies the opposite. One can confirm this by looking at the resulting images from STEP and SES. The overdrawing problem exhibited by MSV and IES, while producing esthetically appealing results, leads to a misrepresentation of the graph’s true density not only within the graph but also when comparing different graphs. For instance, when comparing graphs G4–G6, it is hard to tell, without prior knowledge, which one is denser than the other.

In contrast, STEP and SES both provide a less cluttered and more accurate representation of the actual density of the graph. While STEP depicts the graph structure directly aligned with time, SES requires tracing the edges back to the representative graph to obtain such information, which might be a challenging task. The representative graph is an aggregated version of all timepoints. Finding out how the graph looked like at certain timepoints requires tracing the edges of these timepoints to the representative graph and mentally reconstructing the corresponding graph structure. This task can be particularly hard, especially when these edges are infrequent. Of the four techniques, STEP appears to provide a better overview of how the network structure evolves. Looking at graphs G1–G3, one can identify nodes exhibiting the preferential attachment property of the Barabási model (i.e., fan-in). While it might be possible to identify nodes with high in-degree in other techniques, the advantage of STEP is in providing an overview of the entire network structure at various points in time, presenting snapshots that are absent in other techniques.

Temporal patterns

When it comes to identifying the temporal patterns or events, SES provides a better overview of such patterns. Figure 16 shows the first 40 timepoints of graph G5 visualized using the four techniques. We annotated some of the temporal patterns that we could recognize in each technique. In SES, we identified a periodic behavior for four edges: ①, ②, ③, and ④. Additionally, we spotted one edge that lasts for exactly one timepoint: edge⑤; at timepoint 21. In STEP, while we can identify these patterns, they do not appear as intuitive as they do in SES. Identifying temporal patterns in STEP requires some training since the shape of the pattern may vary over time. For instance, consider edge ②. While the edge is consistently drawn in the same manner in SES at each timepoint, the segment of the edge drawn in STEP gradually changes over time. This may be hard to grasp for an untrained observer.

Figure 16.

A cut-off of the first 40 timepoints from graph G5 visualized using: (a) MSV, (b) IES, (c) SES, and (d) STEP. Links ①, ②, ③, and ④ exhibit a periodic behavior.

In MSV, it is challenging to spot edges with periodic behavior. This is because edges occurring at the same timepoint will be drawn over each other, and the final color of the top edge will be the accumulated color of all edges underneath. As a result, it is hard to recognize the same edge over time, due to the color change. In Figure 16(c), the first occurrence of edge ① took place between timepoints 1 and 13. However, looking at the result of MSV in Figure 16(a), we could recognize edge ① up until timepoint 4. After that, the color of the edge changes due to the appearance of other edges. The same thing can be noticed in edges ② and ③. This change in color misleads our perception of identifying periods of time where an edge persisted, making the detection of periodicity patterns hard.

Although IES suffers from the same overdrawing problem as MSV, the usage of the slope encoding allows us to recognize the edges despite the color change, for example, the periodicity of edges ①,②, and ④. However, we cannot see the patterns easily if the edges have similar slopes. For example, ③ is overlapping with ①. In IES, due to the interleaving, edges endpoints are not perfectly aligned in time. While the start point corresponds to the timepoint where the edge occurs, the endpoint is shifted in time, depending on the strip width. This could be misleading sometimes. As opposed to SES and STEP, IES and MSV emphasize the outliers (i.e., edge⑤) since edges are since edges are fully drawn, irrespective of how long they persist.

To conclude, MSV and IES might be useful for analyzing the evolving structural properties of sparse graphs. However, as graph density increases, both methods are prone to overdrawing issues. SES excels in revealing temporal patterns, but getting an overview of the structural properties might be a challenging task. On the other hand, STEP appears to be more effective in providing an overview of the evolving structural properties. However, since the shape of the patterns changes over time, it might be challenging to discern them, especially with dense graphs.

Case study: Coreless filament-wound structures

We apply STEP in the context of architecture, engineering, and construction (AEC)⁵¹ to visualize coreless filament-wound structures.⁵² We briefly introduce the domain problem and dataset characteristics in Sections “Coreless filament winding” and “Abstraction and dataset characteristics,” followed by an exploration of the respective dataset in Section “Dataset exploration.”

Coreless filament winding

Fiber composites have recently joined the spectrum of architectural materials.⁵³ The advancement of digital technologies has facilitated the realization of fibrous architectures full potential, allowing for the fabrication of high-performance, lightweight, and resource-efficient structures with highly differentiated directionality.

To fabricate a fiber composite component, a robotic fabrication technique called coreless filament winding⁵³ is used to deposit resin-impregnated fiber filaments with only minimal formwork. During this process, the robot arm follows a winding sequence that defines the order in which it visits the anchors on boundary frames. At each anchor point, the robot wraps a continuous fiber tow around it before progressing to the next anchor in the sequence. The fibers span freely in space, and the resulting fiber net forms through the sequential fiber-fiber interaction.⁵³

A fiber net is defined by vertices and edges (see Figure 17). The vertices are 3D points and, therefore, define the geometry. Vertices correspond to the positions of the anchor points (i.e., boundary frames) or intersection points between the fibers. The edges define how the vertices are connected and, therefore, define the topology. The topology of the fiber net and positions of the intersection points are determined through a simulation of the winding sequence and the resulting fiber-fiber interaction.⁵³

Figure 17.

Three-dimensional fiber net and corresponding graph abstraction. A continuous fiber tow spans freely between anchor points on two boundary frames (frame a and frame b ) and the sequential fiber-fiber interaction results in intersection points . The graph abstraction maps the anchor points to vertices and the fibers to edges.

The directionality of the fibers, given by the winding sequence, is crucial for the structural integrity and performance of the building component. Depending on boundary conditions and winding constraints, there could exist thousands or even millions of potential winding sequences, each presenting diverse topological and geometrical characteristics. Exploring such a huge space can be challenging, especially when designers are interested in esthetic criteria that cannot be easily quantified. To aid the designers in exploring such a space, we abstract the fiber nets into graphs and then utilize STEP to explore the results.

Abstraction and dataset characteristics

This case study explores a dataset containing 2000 fiber net specimens. The dataset was generated in coordination with one domain expert. Please check the supplemental materials for the dataset files. All specimens share the same boundary conditions. As seen in Figure 17, each of the specimens consists of two frames, designated as frame A and frame B , each with six anchors. Each fiber net specimen has exactly twelve fibers spanning between these anchors. To visualize the dataset using STEP, we transform each fiber net into a graph, wherein anchor points map to vertices and fibers to edges (see Figure 17). By this abstraction, we preserve the topology of the fiber net while sacrificing the geometrical information. Since the dataset does not have temporal information associated with the graphs (i.e., no time dimension), we opt to use quantitative properties of the fiber nets, such as the number of fiber intersections or the total fiber length, in addition to topological similarity, to establish an order on the x-axis.

Dataset exploration

To explore the dataset, we followed a pair analytics protocol⁵⁴ with one visualization expert and one architect. The exploration was done in two sessions, each lasting 1 h on average. The goal of the exploration was open-ended (i.e., no predefined analysis tasks).

With a screen resolution of 1920 × 1200, we are able to fit (i.e., without scrolling) three rows of STEP within the application window. The three rows are composed of 34 TEP multiples. Each is composed of 16 graphs, making a total of 528 (≈26% of the dataset). Figure 18 shows a cutoff of the first row when sorting the dataset using three criteria. The top row (by topological similarity), the middle row (by total fiber length in descending order), and the bottom row (by total fiber length in ascending order).

Figure 18.

A cutoff of the first row after sorting the dataset by topological similarity , by total fiber length in descending order , and by total fiber length in ascending order . Each row shows different topological features. The highlighted triangular areas at the top and bottom indicate where the self-frame connections would appear.

Sorting by topological similarity allows us to observe these components distinguished by purple diagonal lines. These components mimic winding sequences, where the A frame’s first, second, third, fourth, and fifth anchors are connected to the B frame’s first, second, third, fourth, and fifth anchors, respectively. Depending on the winding sequence, these fibers can be in the opposite direction, from frame B to frame A , and therefore appear as orange lines going from bottom left to top right. An example of one of these components is shown in Figure 19. Since, in reality, the anchors for both frames are aligned along the y-axis, these diagonal lines are actually horizontal fibers between the two frames (see Figure 19). These fibers could act as a structural bracing between the frames. However, in this example, these components are not ideal since they do not match the target angles desired by the expert.

Figure 19.

Three examples of fiber nets are at the bottom, and their graph representations are at the top. fiber net with many horizontal connections, a lot of intersections with no self-frame connections, few intersections with many self-frame connections.

Sorting by total fiber length (see Figure 18) reveals an interesting pattern in the data. Fiber nets with low total fiber length often exhibit numerous self-frame connections, that is, connections between two anchors on the same frame. This relationship is expected, as self-frame connections result in shorter edges. These self-frame connections appear in the top and bottom triangular areas in the middle part of the TEP small multiples (see Figure 18). Conversely, fiber nets with high total fiber length typically lack these self-connections. As depicted in Figure 18, these triangular areas appear remarkably clean. Sorting by the number of intersections results in the same pattern. Figures 19 and show the difference between both components.

While the abstraction revealed interesting patterns in the dataset, further work might be required to adapt STEP to this particular domain. The winding sequences depend not only on the topology but also on the order in which the fibers have been laid (i.e., start/end vertices and direction). The component shows a symmetric winding pattern, but the resulting fiber-fiber interaction is not symmetric, as it highly depends on the sequence. Similarly, the crossing diagonal fiber in the component would not have appeared if the sequence were executed in reverse order and the diagonal fiber laid first since it would have laid under all subsequent fibers. This illustrates the importance of the sequential information for the resulting geometrical outcome. For the same sequence topology, a different start vertex might produce a different geometric result. In the current implementation, we do not account for the sequence of fibers within each component. However, it could potentially be addressed either by using color to encode such information or by treating each fiber component as a sequence of graphs, following the order in which fibers have been laid.

Limitations and future work

Our evaluation mostly relied on qualitative results inspection (QRI).⁵⁵ While QRI is a valid method for evaluating visualization techniques,⁵⁶ a quantitative user evaluation would validate our findings and complement the qualitative insights with quantitative ones. In our parameter study, we attempted to maintain a degree of rigor to avoid slipping into the common pitfalls of qualitative evaluation.⁵⁵ We achieved this first by following a generative model to have better control over the graph parameters and second by evaluating the techniques based on their ability to depict the graphs’ structural properties and temporal patterns.

Developing a generative model that can simulate the graphs’ structural properties and temporal patterns remains an open challenge. To generate graphs that exhibit different structural properties, different network classes such as Barabási,⁴⁷ Erdős -Rényi,⁵⁷ or Watts-Strogatz⁵⁸ can be used for this purpose.⁷ However, the challenge is to fuse the temporal patterns such as periodicity, trends, or stability when generating multiple instances of the same class (i.e., to introduce the time dimension). To tackle this challenge, we used the model proposed by Cooper et al.⁴⁶ that extends the Barabási model by adding and removing nodes or edges in a random fashion over time. However, the model does not explicitly account for the temporal patterns, and therefore, it is not ideal for the purpose of the evaluation. Whether or not Barabási, or other network classes, exhibit these temporal patterns in the real world is a question worth asking. More so is the question of whether a network class can evolve to become another one. Answering these questions and coming up with a rigorous generative model⁵⁹ is an essential challenge to tackle, especially if a controlled user study is the desired method of evaluation.

We envisioned STEP as a balanced approach between depicting the networks’ structural properties and temporal patterns. The evaluation showed that STEP is more effective at showing the former than the latter, especially with dense networks. The partial drawing of edges makes it challenging to recognize edges in highly cluttered areas. To address this issue, we drew reference lines in the background and used color to encode the lines’ slopes. Additionally, we implemented various interaction strategies to filter edges based on their occurrence frequency, adjust the width of the lines, or zoom to individual timepoints. Nevertheless, using color to encode edge attributes remains limited and cannot scale, for example, to depict multivariate networks. This makes STEP better suited for obtaining an overview of the structural properties of univariate networks rather than detailed information associated with the edges of multivariate ones.

When it comes to revealing temporal patterns associated with edges, STEP comes second to SES. SES benefits from using a canvas space to depict network structural properties that is separate from the canvas space it uses to depict temporal patterns. However, due to such separation, it is required to move back and forth between the two spaces (i.e., through edge tracing) to construct a mental map of the network structure at a certain point in time, which is a mentally demanding task. While temporal clustering was proposed³³ as a way of mitigating this drawback, it is still worth investigating how the four techniques would compare with temporal clustering in place.

Conclusion

In this paper, we introduced sequence of time-aligned edge plots (STEP), a visualization for dynamic networks and graph ensembles. By partitioning the graph sequence into equal-sized subsequences, STEP maintains a good aspect ratio and scales up to long graph sequences. To evaluate STEP, we applied it to two real-world datasets. Additionally, we conducted a qualitative parameter study to compare it against the state-of-the-art. Furthermore, we challenged the technique by applying it to a graph ensembles dataset from the architecture domain. Our evaluation showed that STEP can provide a comprehensive overview of the graphs’ structural changes, which is hard to obtain by competing approaches. Nevertheless, as the density of the graph increases, it becomes challenging to recognize the temporal events associated with edges. Future work could involve conducting an empirical user evaluation, developing generative models for dynamic graphs, and exploring various temporal clustering and vertex ordering strategies.

Supplemental Material

sj-7z-1-ivi-10.1177_14738716241265111 – Supplemental material for STEP: Sequence of time-aligned edge plots

Supplemental material, sj-7z-1-ivi-10.1177_14738716241265111 for STEP: Sequence of time-aligned edge plots by Moataz Abdelaal, Fabian Kannenberg, Antoine Lhuillier, Marcel Hlawatsch, Achim Menges and Daniel Weiskopf in Information Visualization

Footnotes

Funding

The author(s) disclosed receipt of the following financial support for the research,authorship,and/or publication of this article: This work was supported by the Deutsche Forschungsgemeinschaft (DFG,German Research Foundation) under Germany’s Excellence Strategy - EXC 2120/1 - 390831618.

ORCID iDs

Moataz Abdelaal

Daniel Weiskopf

Supplemental material

Supplemental material for this article is available online

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Supplementary Material

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