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Visualizing Electrical Power Systems as Flow Fields


Abstract and Figures

We describe a method for visualizing data flows on large networks. We transform data flow on fixed networks into a vector field, which can be directly visualized using scientific flow visualization techniques. We evaluated the method on power flowing through two transmission power networks: a small, regional, IEEE test system (RTS-96) and a large national-scale system (the Eastern Interconnection). For the larger and more complex transmission system, the method illustrates features of the power flow that are not accessible when visualizing the power transmission with traditional network visualization techniques.
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Workshop on Visualisation in Environmental Sciences (EnvirVis) (2018), pp. 1–8
R. Bujack and K. Rink and D. Zeckzer (Editors)
Visualizing Electrical Power Systems as Flow Fields
Samantha Molnar1,2and Kenny Gruchalla1
1National Renewable Energy Lab, Golden, CO USA
2University of Colorado Boulder, Boulder, CO USA
We describe a method for visualizing data flows on large networks. We transform data flow on fixed networks into a vector
field, which can be directly visualized using scientific flow visualization techniques. We evaluated the method on power flowing
through two transmission power networks: a small, regional, IEEE test system (RTS-96) and a large national-scale system (the
Eastern Interconnection). For the larger and more complex transmission system, the method illustrates features of the power
flow that are not accessible when visualizing the power transmission with traditional network visualization techniques.
1. Introduction
We introduce a new technique to visualize flow on networks by
constructing vector fields based on the network’s flow of infor-
mation. Our method allows us to use standard flow visualization
techniques on large networks to highlight salient features without
removing important topological information. We explore applica-
tions of our method to electric power systems, but emphasize that
this technique is applicable to any complex network with informa-
tion exchange.
An electrical power system is a network of components con-
nected by lines that deliver electricity from generators to con-
sumers. Power systems around the world are undergoing signifi-
cant changes in recent years, with the addition of renewable gen-
eration sources such as wind and solar. It is crucial to understand
how renewable generation sources influence power flow on the ex-
isting system so we can determine critical junctions and interac-
tions in the system. Visualization of power flow can help maintain
and increase the reliability of power systems. There are generally
two categories of electric power grids: transmission systems which
span long distances carrying high-voltage power from generators to
electrical substations, and distribution systems which deliver power
from electrical substations to individual consumers. Distribution
systems typically have a radial topology forming acyclic connected
graphs, while transmission systems tend to have much more com-
plex interconnected networks forming cyclic non-planar graphs. At
a given instant in time, these networks can be considered directed
graphs with some magnitude of power flowing from one vertex
along an edge to another vertex. The magnitude and direction of
power flowing on these graphs is dynamic, varying in time. The
topology of these graphs can change with the operation of circuit
breakers or switches; however, these are relatively rare events. For
our investigation, we are only concerned with the dynamics of the
flow not the topology.
We evaluate our visualization approach on power flowing
through two transmission power networks: a small IEEE test sys-
tem (RTS-96) and a large system (the Eastern Interconnection). The
small test system is simple enough to use traditional visualization
techniques, allowing us to determine if the vector field represents
the power flow accurately. The Eastern Interconnection provides a
real-world test case that is too complex for existing visualization
techniques to convey the flow trends (see Figure 1).
There are documented needs to visualize general trends of flow
on complex network structures; for example, the flow of infor-
mation on biological networks [AKK10] and neural networks
[LSKK16], and the flow of power on electrical networks [CK15]
are compelling examples. Most techniques involve representing the
flow explicitly on line graphs with glyphs (e.g., arrows) and line
attributes (e.g., color and width) [OW00b]. These techniques are
often sufficient for small networks but are inefficient or unreadable
for larger networks.
To visualize large-scale networks some form of aggregation be-
comes neccessary. Two common approaches are chord diagrams
and edge bundling. A chord diagram represents directed inter-
relationships between groups of objects, where these objects are ar-
ranged along a circle and connected with Be´zier curves [KSB09].
Chord diagrams can be used to represent data flow on a directed
graph [GNB16] by grouping vertices into regions and representing
the net flow between regions in the chord diagram. The chord di-
agram provides a quantitative view of net flow, but abstracts that
flow away from the toplogy of the network.
Edge bundling is a technique to simplify a graph by spatailly
grouping similar graph edges, decluttering and revealing under-
lying structure in large networks [LHT17]. However, large trans-
mission networks, like the Eastern Inconnection, have dense multi-
scale mesh-like topologies that provide few similar graph edges for
edge bundling to a group. For example, we applied geometry based
submitted to Workshop on Visualisation in Environmental Sciences (EnvirVis) (2018)
2S. Molnar & K. Gruchalla / Visualizing Electrical Power Systems as Flow Fields
Figure 1: A model of the Eastern Interconnection transmission network presents a visualization challenge with flows across 70,000 trans-
mission lines. The labeled areas indicate independent system operators controlling power in those regions. We would like to understand how
and where these areas exchange power.
edge bundling (GBEB) [CZQ08] to the Eastern Interconnection.
GBEB bundles edges based on a kernel density estimation of the
primary edge directions. As seen in Figure 2, this method does well
with edges in the North East coming from Canada to the United
States, but it fails to declutter the majority of the Eastern Intercon-
nection. This is because the power system is mainly composed of
star-like motifs that come to a single node from various directions.
We also applied kernel density edge bundling (KDEB) [HET12],
which is the base algorithm for other edge bundling techniques that
use edge attributes as an indicator for bundling [PHT15]. KDEB
determines the density of edges, computes the gradient of this den-
sity, and iteratively moves edges in the direction of the gradient
to bundle edges. Figure 3 shows KDEB applied to the Eastern In-
terconnection, and while more edges are bundled than the GBEB
approach, it does little to alleviate the clutter of the network. Al-
though the network is dense, this is generally uniformly distributed
over space. Furthermore, many of the edges that are close together
in the topology do not have a uniform direction, which makes it un-
clear how and in what direction they should be bundled. The con-
clusion we can draw from applying these edge bundling techniques
is that due to the mesh-like structure of the Eastern Interconnec-
tion, there is little advantage to bundling the edges of the network
to try and reduce visual clutter and obtain useful information about
power flow across the network. An alternate approach is required
to understand power flow across the network.
Our technique does not visualize aggregated edges, but instead
visualizes aggregated flow across the network. We can find no de-
scription in the literature of transforming the flow across a com-
plex network into a vector field for the visualization of flow trends.
Lobov et al. [LSKK16] appear to use a similar technique to un-
derstand properties of neural networks in response to a stimulus;
however, their technique is not well documented, and it is unclear
how they form their vector field. Nevertheless, this provides an ex-
cellent example of the need to further analyze parameter choices for
this type of visualization in order to understand how these choices
affect analysis results.
Our method allows us to use standard flow visualization tech-
submitted to Workshop on Visualisation in Environmental Sciences (EnvirVis) (2018)
S. Molnar & K. Gruchalla / Visualizing Electrical Power Systems as Flow Fields 3
Figure 2: The result of geometry based edge clustering on the East-
ern Interconnection [CZQ08]. Long transmission lines coming
from Canada are bundled using this technique, but small star-like
motifs in the South Eastern United States are mostly untouched.
Figure 3: The result of kernel density edge bundling on the Eastern
Interconnection [HET12]. Longer transmission lines are moved to
odd positions which do not help lessen clutter in the network.
niques on large networks to highlight salient features without re-
moving important topological information. Furthermore, we will
show that insightful knowledge of power exchange on the system
can be obtained using our visualization technique.
2. Methods
We analyze general flow patterns across a network by creating a
vector field based on the network dynamics and topology. Let the
network in question be represented by Gwith Vvertices and E
edges. We take each directed edge ei,jEwhich connects nodes
i,jVand form a vector, ˆei,j, based on its location in space. Each
ˆei,jhas an associated flow, fi,j. To begin constructing the vector
field, we place the network in an n×nsquare mesh M. Each cell in
the mesh, ml,kM, where ldefines the row and kdefines the col-
umn, will contain a vector. The function that computes this vector
is shown in Equation 1.
Vtotal (l,k,p,σ) = Vf l ow(l,k) + Vgrad(l,k,σ) + Vkern(l,k,p)(1)
Where Vf low is the vector sum of all edge flows through a cell,Vgrad
is the gradient of a Gaussian convolution to attract the flow towards
the topology, andVkern is a Gaussian weighting of nearby cells.
2.1. Calculating Vf l ow
Vf low represents the flow of information on the network. For each
cell in the grid that contains ˆe, we compute the sum of the vector
flows, defined by Equation 2.
Vf low (l,k) =
δ(ˆes,ml,k) = ˆesml,k1
otherwise 0
A depiction of Vf low of the RTS-96 system is shown in Figure 4a.
Notice that very few of the cells in the mesh are filled with non-zero
vectors. This will cause problems for traditional flow techniques
that seed streamlines or path lines at random positions. There is
also little guarantee that flow lines seeded at node positions will
remain in the flow field without other forces pushing it back into
this area. For these reasons, we add a gradient force to the vector
2.2. Calculating Vgrad
The purpose of Vgrad is to fill cells that do not directly intersect the
network with vectors that point towards the topology of the net-
work. This would allow streamlines seeded away from the network
topology to converge toward the network topology. If we imagine
placing the network on a rubber sheet, we would see the sheet sink
down where the lines are, and balls rolling on the sheet would tend
to fall into the valleys, or the edges of the network (see Figure 4b).
To achieve this effect we first calculate the weighted sum of each
cell in the network, where the weight represents how much of each
edge in Ethe cell contains. We traverse each edge, ei,j, in unit
vector intervals, and for each step we compute which cell ml,kis
closest to the current position and add 1 to that cell’s weight wl,kin
the matrix W. We then compute Vgrad based on Equation 3.
Vgrad(l,k,σ) = 1
The only tunable parameter in Vgrad that we explore in this work is
σwhich, in our metaphor, represents how fast flow moves towards
the edges of G. As σincreases, the number of cells with a non-zero
vector decreases.
2.3. Calculating Vkern(l,k,p)
Finally, in order to make the vector field less abrupt at the bound-
ary of cells that contain network edges and cells that do not, we
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Figure 4: Vector field representation of the RTS-96 test system with
a25 ×25 grid. a) directly represents the flow on the network by
encoding Vf l ow.b) Vgrad directs flow back towards the topology of
the network.
add a small vector to each cell that is the average of its neighbors.
For example, if σ=1, then Vgrad points cells without flow directly
towards the topology, exhibited in Figure 5a. Vkern (l,k,p)smooths
this transition and slightly points cells without edges towards the
majority direction of its neighbors, shown in Figure 5b. In Equa-
tion 4, prepresents the size of the neighborhood we average over;
p=1 represents the surrounding ring of neighbors around a cell
(of which there are at most 8), then p=2 represents the next sur-
rounding ring and so on. The function g(p)is a gaussian function
that determines how much weight each neighbor cell contributes
to the average. In this paper, we focus on varying pto understand
how the size of a neighborhood changes the vector field rather than
changing the weight of the contribution from the neighborhood.
Vkern(l,k,p) =
g(p)(Vfl ow(s,r) + Vgrad(s,r)) (4)
The parameter space for Vtot al (l,k,p,σ)is large and various in-
Figure 5: A graph edge represented as a vector field. Black vectors
indicate cells that contain an edge with flow and red vectors indi-
cate cells that do not. a) is constructed without Vkern (l,k,p)and b)
is constructed with Vkern (l,k,p)and p =1.
puts can change the types of conclusions one could draw about the
network flow. In the following section we will explore how various
inputs of Vtotal change the observed flow dynamics on two power
system test cases.
3. Experimental Results - Electric Power Systems
There are few techniques for the visualization of power systems
data [MJC12]. The simple one-line diagram, a static diagram with
single lines and schematic symbols to indicate the path and com-
ponents on an electric circuit [vM06], is a standard representation
for the static structure of the system. Most visualization research
has focused on augmenting the one-line diagram with flow data.
Various researchers [MC93,LQ98] have proposed encoding elec-
trical flow as line width and color and including arrows for flow
direction. Two-dimensional contours have been used to visualize
network data [WO00,OW00b], market data [OW00a], and system
stability margins [OKW99,KW02]. However, these techniques do
not scale to large transmission systems with tens of thousands of
lines. The visualization of large transmission networks more typi-
cally aggregate the line flows to the net flow or interchange between
regions of the network. These aggregated flows have been repre-
sented as ribbons in a chord diagram [GNB16] (see Figure 6). Net
interchange visualizations provide critical information on power
transfer between large regions in aggregate but fail to show how
that interchange occurs. For example, in Figure 6 we see 3 GW
of power flowing from SERC to TVA, and 3 GW of power flow-
ing from TVA to PJM. But we do not know where along that in-
terconnection this exchange occurs. Furthermore, we can speculate
that some of the flow from SERC to TVA is transferred to PJM, but
submitted to Workshop on Visualisation in Environmental Sciences (EnvirVis) (2018)
S. Molnar & K. Gruchalla / Visualizing Electrical Power Systems as Flow Fields 5
we cannot decisively know this based on the chord diagram. Ad-
ditional locational information could be useful to determine where
additional transmission lines should go, which transmission lines
are under stress at various times, and potentially change market ex-
change calculations.
Figure 6: A chord diagram of a time step of the Eastern Renew-
able Generation Integration Study (ERGIS). Large transmission
networks have tens of thousands of lines, directly visualizing these
lines is impractical. Transmission studies such as ERGIS have re-
sorted to only visualizing aggregated flow between regions. There
is significant flow between SERC and TVA, and TVA to PJM, but it
is unclear if the flow from SERC to TVA is then going to PJM.
3.1. RTS-96 System – Validation Case
IEEE RTS-96 [GWA99] is a test system commonly used in trans-
mission power system reliability studies. The system is a cyclic
non-planar graph with 73 vertices connected by 120 edges com-
prised of three subnetworks. RTS-96 is small enough to directly
visualize flows on the network with arrow glyphs indicating direc-
tion and line properties (e.g., color, width) representing flow magni-
tude (see Figure 7). Clearly, a system this small is best represented
with these traditional techniques. We apply our method to RTS to
evaluate if our vector field construction represents the power flow
We depict these vector fields using streamlines, but other vec-
tor field visualization techniques (i.e., path lines and line interval
convolution) are directly applicable. To evaluate the correctness of
the constructed vector field, we systematically seeded stream lines
near sources (i.e., buses with electrical generation) on RTS-96 and
followed those stream lines to the sinks (i.e., nodes with electrical
loads). The streamlines do travel from sources to sinks on paths
that generally follow the network (see Figure 8a and b). However,
how closely they follow the network depends on the parameters of
the vector field construction. A significant flow difference can be
observed between Figure 8a and Figure 8b at the location of the
bottom orange seed area. The final destination of the streamlines
changes from one node to another. Since Figure 8b has a larger
neighborhood and based on the direction the streamlines originate
from, they go towards the bottom right nodes instead of the farthest
right node. We explore the vector field construction parameters n,
Figure 7: RTS-96 topology with line widths representing power
magnitude and arrows indicating direction. Generator buses (i.e.,
sources) are colored green, load buses (i.e. sinks) are colored or-
ange, and the swing bus is colored red.
p, and σon the RTS-96 system (a subset of that investigation is
shown in Figure 9). First, as the grid size nbecomes larger (and the
cells smaller), the network topology becomes more defined in the
vector field, as there is less aggregation per cell. For example, com-
pare n=100 in Figure 9a and n=1000 in Figure 9d. When there
are fewer cells, the larger flows dominate the vector space and show
more of the cumulative dynamics on the system. When there are
more cells, many of the smaller flows are captured. In areas where
a cell contains an intersection, the constructed field will push flow
in the aggregate direction of all edges in that cell weighted by their
magnitude. Second, as we increase σrelative to the grid size, the
number of cells with non-zero vectors increases and stream lines
can be seeded farther away from the network. Although, the stream
lines converge to the network, the visual features created as σin-
creases may be difficult to interpret. For example, in Figure 9c a
saddle has formed in the middle of the network that could be mis-
construed as a pathway in the network. This feature occurs because
σhas become large enough compared to the grid size to fill cells in
the very center of the network. Based on Vgrad cells to the right of
the saddle point towards the right part of the network and cells to
submitted to Workshop on Visualisation in Environmental Sciences (EnvirVis) (2018)
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Figure 8: a and b) Streamlines seeded near generators gener-
ally follow the network path to the system’s loads center. How the
stream lines follow the network path is impacted by the construc-
tion parameters. Colors are based on which area the streamlines
originate from, area 1 is purple, area 2 is orange and area 3 is
green. The images were both constructed with n =1000 and σ=1
and vary in kernel width a) p =10 b) p =20.
the left of the saddle point towards the left of the network. Under-
standing how domain experts might interpret these flow patterns
will be necessary and is a topic for future work. But with proper
parameter selection, many of these artifacts are avoidable. Finally,
the number of neighbors considered in Vkern changes how much in-
fluence larger flow magnitudes have on the direction of the overall
flow. We find that a smaller neighborhood stays closer to the true
topology of the system, while large neighborhoods provide clear
illustrations of the cumulative flow. The choice for this parameter
is dependent on how much aggregation one wishes to show in the
There is one notable artifact visible in RTS-96 for all of the
streamlines shown in Figure 9, a spiral sink occurs in the lower
right hand area. This feature forms in a non-planar area of the net-
work, where lines in the network cross but do not intersect in the
actual physical topology (i.e., they are not connected by a node).
While there is a sink located in the vicinity, the intersection of an
edge close to this sink causes a spiraling effect where there in fact
is none. Our treatment of edge crossings is beneficial if one wishes
to observe the general flow trend in the network, but it may not give
an accurate depiction of small-scale flow features. As the mesh grid
becomes finer, this effect is less obvious because the cell containing
the intersection plays a much smaller role in the overall vector field
which provides a more detailed view of the flow.
Figure 9: Streamlines of various inputs for RTS-96 vector field
where the left column has n =100 and the right column has
n=1000.a) p=1,σ=1b) p=1,σ=2c) p=4,σ=4d) p=10,
σ=1e) p=10,σ=2f) p=40,σ=4. We scale the neighbor-
hoods for the different grid sizes so that they represent the same
amount of space, and all of the vector fields use g(p) = .05ep1
200 .
Animated version here.
3.2. ERGIS System – Motivating Case
The Eastern Interconnection is the largest power system in North
America and one of the most complex power systems in the world.
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S. Molnar & K. Gruchalla / Visualizing Electrical Power Systems as Flow Fields 7
We applied our visualization technique to data from the Eastern
Renewable Generation Integration Study (ERGIS) [BTP16] that
models the Eastern Interconnection with over 70,000 transmission
lines. This network provides a real-world test case for our visualiza-
tion technique, where directly rendering the line flows is impracti-
cal. We compared the aggregated flow visualization from the study,
which shows us how much net power flows between the regions, to
our results (see Figure 10). In general, the path of the stream lines
are confirmed by the chord diagram. For example, there is signif-
icant flow between SERC and TVA and between SPP and MISO
(reference Figure 1 for region labeling). While the chord diagram
provides a quantitative measure of the net interchange, the stream-
lines provide more qualitative detail to where along the interface
that interchange is occurring. We can gain more insight into the
Figure 10: Streamlines, distributed proportionally to the vector
magnitude, showing detailed flow of power for the Eastern Inter-
connection (n =1000,σ=1, p =10). Large amounts of power is
flowing in to the mid-west region.
flow characteristics by isolating the streamline seeding in partic-
ular regions and analyzing their path. Figure 11 shows flow be-
tween different areas in the Eastern Interconnection. Based on the
aggregate flow in the chord diagram (Figure 6), we know that a
significant amount of power flows from SERC to TVA, and from
TVA to PJM. However, what we can now see from Figure 11 is
that a significant amount of power that flows from SERC to TVA
then flows to PJM. Most of that SERC-TVA flow originates in Al-
abama and not elsewhere in the region. Additionally, we can see
bi-directional flow between MISO and TVA; although, in Figure 6
we are only privy to relatively small net interchange coming in to
TVA from MISO. This is new directional information that we could
only speculate based on the original visualizations. Preliminary ex-
amination of these results with domain scientists indicate that this
novel vector field construction provides useful locational informa-
tion that can be used during development, testing, and analysis of
power flow models and simulations.
Figure 11: Streamlines colored based on which region they orig-
inated from. Flows originating from SERC traveling through TVA
into PJM are clearly visible. Likewise bi-directional flow can be
seen between TVA and MISO.
The vector field construction parameters have a similar effect on
the large-scale system as they did on the small test system. Denser
meshes better resolve individual transmission lines, while larger
kernel sizes provide more aggregated versions of the flow. A spi-
raling sink artifact, like the one apparent in RTS-96, can also be
found in our Eastern Interconnection construction in eastern Ne-
braska (see Figure 10). Like RTS-96, there appears to be a non-
planar cluster of transmission lines in this area of Nebraska. In all
likelihood, this is an artifact of the construction, but with alternate
parameters this artifact does disappear.
4. Conclusions and Future Work
We introduced a method for visualizing data flows on large net-
works by transforming those flows into a vector field. The method
clearly shows promise. On our small-scale test case we verified
that the technique represents the predominate flow features. On
our large-scale test case we were able to identify complex flow
exchanges between different areas of the Eastern Interconnection
transmission system. We have a better understanding of how power
flows between regions with our visualization than an aggregate
view can present and a traditional line glyph rendering cannot han-
However, the method has some limitations. The construction
captures the most dominate flows at the expense of averaging out
the details of the flows. More concerning, non-planar areas of the
graph can lead to spiraling sink artifacts. A more sophisticated vec-
tor field allowing for probabilistic and/or conditional cells could
improve these situations. Developing these multi-state cells and the
visualization algorithms to support them is left for future work.
Our parameter investigation indicates that larger grid sizes yields
more accurate flow information while larger neighborhoods causes
submitted to Workshop on Visualisation in Environmental Sciences (EnvirVis) (2018)
8S. Molnar & K. Gruchalla / Visualizing Electrical Power Systems as Flow Fields
larger flow magnitudes to dominate more of the vector space. Fu-
ture work will extend the investigation to time-varying vector field
visualization methods so we can understand how flow evolves on
a network over time. User studies will be necessary to understand
how domain experts interpret the flow visualizations in relation to
the flow on the underlying network, which may provide a mecha-
nism to better tune the parameters of the construction.
5. Acknowledgements
This work was supported by the U.S. Department of Energy under
Contract No. DE-AC36-08GO28308 with Alliance for Sustainable
Energy, LLC, the Manager and Operator of the National Renew-
able Energy Laboratory. Funding provided by U.S. Department of
Energy Office of Energy Efficiency and Renewable Energy. NREL
is a national laboratory of the U.S. Department of Energy, Office
of Energy Efficiency and Renewable Energy, operated by the Al-
liance for Sustainable Energy, LLC. The U.S. Government retains
and the publisher, by accepting the article for publication, acknowl-
edges that the U.S. Government retains a nonexclusive, paid-up, ir-
revocable, worldwide license to publish or reproduce the published
form of this work, or allow others to do so, for U.S. Government
purposes. The authors would like to thank Nicholas Brunhart-Lupo
and Kristi Potter for useful conversations and insight on this work.
We would also like to thank Aaron Bloom and Clayton Barrows for
preliminary examination of the results.
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submitted to Workshop on Visualisation in Environmental Sciences (EnvirVis) (2018)
... Flows in the urban areas include the movement of goods and materials, but can also be concerned with invisible phenomena such as electricity. For example, Molnar and Gruchalla (2018) visualize electrical power systems by producing a dense vector field from a sparse network and then applying classical flow visualization techniques. The visual analytics and information visualization communities have long investigated motorized and non-motorized vehicle flow as well as travel behavior of pedestrians (Chen et al. 2015), which is usually sparse trajectory data. ...
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Bundling techniques provide a visual simplification of a graph drawing or trail set, by spatially grouping similar graph edges or trails. This way, the structure of the visualization becomes simpler and thereby easier to comprehend in terms of assessing relations that are encoded by such paths, such as finding groups of strongly interrelated nodes in a graph, finding connections between spatial regions on a map linked by a number of vehicle trails, or discerning the motion structure of a set of objects by analyzing their paths. In this state of the art report, we aim to improve the understanding of graph and trail bundling via the following main contributions. First, we propose a data-based taxonomy that organizes bundling methods on the type of data they work on (graphs vs trails, which we refer to as paths). Based on a formal definition of path bundling, we propose a generic framework that describes the typical steps of all bundling algorithms in terms of high-level operations and show how existing method classes implement these steps. Next, we propose a description of tasks that bundling aims to address. Finally, we provide a wide set of example applications of bundling techniques and relate these to the above-mentioned taxonomies. Through these contributions, we aim to help both researchers and users to understand the bundling landscape as well as its technicalities.
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Edge bundling methods reduce visual clutter of dense and occluded graphs. However, existing bundling techniques either ignore edge properties such as direction and data attributes, or are otherwise computationally not scalable, which makes them unsuitable for tasks such as exploration of large trajectory datasets. We present a new framework to generate bundled graph layouts according to any numerical edge attributes such as directions, timestamps or weights. We propose a GPU-based implementation linear in number of edges, which makes our algorithm applicable to large datasets. We demonstrate our method with applications in the analysis of aircraft trajectory datasets and eye-movement traces.
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Synchronization of neural network response on spatially localized periodic stimulation was studied. The network consisted of synaptically coupled spiking neurons with spike-timing-dependent synaptic plasticity (STDP). Network connectivity was defined by time evolving matrix of synaptic weights. We found that the steady-state spatial pattern of the weights could be rearranged due to locally applied external periodic stimulation. A method for visualization of synaptic weights as vector field was introduced to monitor the evolving connectivity matrix. We demonstrated that changes in the vector field and associated weight rearrangements underlay an enhancement of synchronization range.
Recent work, using electrical distance metrics and concepts from graph theory, has revealed important results about the electrical connectivity of empiric power systems. Such structural features are not widely understood or portrayed. Power systems are often depicted using unenlightening single-line diagrams, and the results of loadflow calculations are often presented without insightful elucidation, lacking the necessary context for usable intuitions to be formed. For system operators, educators, and researchers alike, a more intuitive and accessible understanding of a power system's inner electrical structure is called for. Data visualization techniques offer several paths toward realizing such an ideal. This paper proposes various ways, in which electrical distance might be defined for empiric power systems, and records how well each candidate distance measure may be embedded in two dimensions. The resulting 2-D projections form the basis for new visualizations of empiric power systems and offer novel and useful insights into their electrical connectivity and structure.
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