1 The definition of confluent
Confluent drawing is a graph drawing technique that eliminates crossings by allowing edges to overlap, as long as the join is smooth. This is analogous to junctions on a train track that allow carriages to switch directions without stopping. For convenience, we recall the original definition (Dickerson et al. ) of a confluent drawing for a graph :
There is a one-to-one mapping between the vertices in and , so that, for each vertex , there is a corresponding vertex , which has a unique point placement in the plane.
There is an edge in iff there is a locally-monotone curve connecting and in A.
A is planar. That is, while locally-monotone curves in A can share overlapping portions, no two can cross.
Bach et al.  claim to construct such a drawing by converting a power graph decomposition (PGD) into an auxiliary routing graph (ARG). Then, for each original edge, the shortest path through the ARG is used as the sequence of control points for a spline. Their concept is widely applicable and of great practical use—however, in the process of implementing it ourselves, we found that the method does not strictly satisfy the above definition, and have therefore written this comment to reframe its contribution within a more accurate context.
The first limitation in their methodology stems from the use of a force-directed method to lay out the ARG. It ensures that the first condition is met, but rules out the third because there is no guarantee of planarity within such methods. While the authors do recognize this, and make the distinction that their drawings are ‘non-planar confluent’, in the following two sections we will prove that the second condition is not met either.
1.2 Problems with B-splines
The second limitation regards the use of B-splines (presumably of degree
, although it is not specified) for interpolating along the ARG. These were likely chosen because they satisfy the convex hull property (which prevents crossings at shared control points; see Jia et al. for an example of poorly implemented splines that do not satisfy this property), and also offer local control (i.e. moving a control point only affects the surrounding segments), which guarantees that splines that share enough intermediate control points will overlap. Local control is what makes it possible for drawings to be confluent; with the right ARG, it is possible for edges to share enough control points (
or more, with a uniform knot vector) such that they produce the overlapping portions that denote a confluent drawing.
There are two problems with using B-splines in this context. The first is that splines that share fewer than control points will not overlap, but sharing even a single routing node in the ARG should indicate a bundle (as in [1, Fig. 2]). The authors identified this possibility, calling it the ‘crossing artifact’ [1, Fig. 4]
, and fix this by splitting routing nodes into two. This ad hoc fix is ambiguous on undirected graphs because it is not specified how to classify routing edges as ingoing or outgoing, and it also may not remove all crossings if a routing node has five or more edges attached. Such a fix also only guarantees two shared control points through the split, and so only strictly works for.
The second problem is also caused by local control, but has an opposite result: that splines will always overlap if they share or more control points. Given the right ARG, edges may overlap such to create the visual impression that extra edges, not in the original graph, exist (Figure 1, top). This violates the second condition in the definition of confluent. We find that such an ARG can result from the PGD to ARG conversion, as explained in the next section.
1.3 Ambiguous decomposition
The third limitation is that the conversion from a PGD to an ARG can cause ambiguity. A PGD is an extension to the node-link diagram, that compresses the number of edges by grouping similar vertices together into power groups, and merging edges shared among group members into a single power edge instead. This is then converted into an ARG by (1) connecting the members of each power group to a routing node corresponding to the group, and (2) connecting pairs of routing nodes whose corresponding power groups are connected by power edges (see [1, § 3.1] for more detail). The ambiguity arises because (1) and (2) both result in routing edges in the ARG. One problem with this is that multiple PGDs are converted into the same ARG (Figure 2, top), but what leads to breaking the definition of confluent is that this can produce a short-circuit effect (Figure 1, bottom).
The structure of groups within a power graph can be represented as a tree, where groups are represented by branches and vertices by leaves. Trees have the nice property of being geodetic (i.e. there exists a unique shortest path between any pair of vertices), but the short-circuits caused by power edges can invalidate this, to either produce ambiguity in which path to use (if the shortest paths are equal), or to route splines in the wrong direction and imply edges that should not exist (Figure 1, bottom). It may seem as if our counter-example is contrived and should not ever appear due to the redundant nested structure of power groups, but a similar structure appears in the optimal decomposition of a clique (Figure 2, middle).
2 Similar methods and future directions
Bundling using an ARG has been done various times in the past. Hierarchical information has been used before to produce a tree as the ARG, most notably by Holten  using metadata, and also Jia et al.  and recently Zheng et al. 
using hierarchical clustering algorithms.
Spatial proximity between edges has also been considered, with MINGLE by Gansner et al.  and Metro Bundling (MB) by Pupyrev et al.  being two techniques that do not route based on topology, but are instead formulated as an optimization problem on a predetermined layout. MINGLE tries to ‘reduce ink’ in its ARG through an agglomerative method, while MB is based on avoiding label boundaries whilst minimizing routing cost, a metric that includes multiple criteria including ink and path lengths.
We recommend classifying all such methods together within the wider context of ARG generation. This both highlights the contribution of Bach et al.  as unique in its construction of an ARG from a PGD, and will serve to better contextualize future work within the field. The concept of using the topology of the graph itself to generate an ARG is also an important contribution from Bach et al. , and future work may be directed towards finding more optimal solutions in this context. Allowing the user to manually construct their own ARG based on searching for certain motifs may also be useful within an interactive setting.
This does not mean that the eponymous algorithm is not useful, in fact we find that it produces excellent results on graphs that admit a good PGD. However the issues addressed in this article should not be overlooked, and are important for both practitioners who should be aware of its limitations, and researchers who may be misled by imprecise terminology. The general problem of finding confluent drawings of arbitrary graphs remains open.
-  B. Bach, N. H. Riche, C. Hurter, K. Marriott, and T. Dwyer, “Towards unambiguous edge bundling: Investigating confluent drawings for network visualization,” IEEE Transactions on Visualization and Computer Graphics, vol. 23, no. 1, pp. 541–550, 2017.
-  M. Dickerson, D. Eppstein, M. T. Goodrich, and J. Y. Meng, “Confluent drawings: visualizing non-planar diagrams in a planar way,” Journal of Graph Algorithms and Applications, vol. 9, no. 1, pp. 31–52, 2005.
-  Y. Jia, M. Garland, and J. C. Hart, “Social network clustering and visualization using hierarchical edge bundles,” Computer Graphics Forum, vol. 30, no. 8, pp. 2314–2327, 2011.
-  D. Holten, “Hierarchical edge bundles: Visualization of adjacency relations in hierarchical data,” IEEE Transactions on Visualization and Computer Graphics, vol. 12, no. 5, pp. 741–748, 2006.
-  J. X. Zheng, S. Pawar, and D. F. M. Goodman, “Confluent* drawings by hierarchical clustering,” in 26th International Symposium on Graph Drawing and Network Visualisation, Springer, 2018 (in press).
-  E. R. Gansner, Y. Hu, S. North, and C. Scheidegger, “Multilevel agglomerative edge bundling for visualizing large graphs,” in 2011 IEEE Pacific Visualization Symposium, pp. 187–194, IEEE, 2011.
-  S. Pupyrev, L. Nachmanson, S. Bereg, and A. E. Holroyd, “Edge routing with ordered bundles,” Computational Geometry, vol. 52, pp. 18–33, 2016.