Mapper on Graphs for Network Visualization
Networks are an exceedingly popular type of data for representing relationships between individuals, businesses, proteins, brain regions, telecommunication endpoints, etc. Network or graph visualization provides an intuitive way to explore the node-link structures of network data for instant sense-making. However, naive node-link diagrams can fail to convey insights regarding network structures, even for moderately sized data of a few hundred nodes. We propose to apply the mapper construction—a popular tool in topological data analysis—to graph visualization, which provides a strong theoretical basis for summarizing network data while preserving their core structures. We develop a variation of the mapper construction targeting weighted, undirected graphs, called mapper on graphs, which generates property-preserving summaries of graphs. We provide a software tool that enables interactive explorations of such summaries and demonstrates the effectiveness of our method for synthetic and real-world data. The mapper on graphs approach we propose represents a new class of techniques that leverages tools from topological data analysis in addressing challenges in graph visualization.
Networks are often used to model social, biological, and technological systems. In recent years, our ability to collect and archive such data has far outpaced our ability to understand them. For instance, the Blue Brain Project—the worldâs largest-scale simulations of neural circuits—generates instances of the micro-connectome containing 10 million neurons and 88 billion synaptic connections for the rodent brain. The challenges for graph visualization (sometimes called network visualization) are two-fold: how to effectively extract features from such complex data; and how to design effective visualizations to communicate these features to the users.
We propose to address these challenges by leveraging the mapper construction , a tool in topological data analysis (TDA), to develop visualizations for large network data. Given a topological space equipped with a function on , the classic mapper construction from the seminal work of Singh et al.  provides a topological summary of the data for efficient computation, manipulation, and exploration. It has enjoyed tremendous success in data science, from cancer research  to sports analytics , among others [17, 64, 65, 91]; it is also a cornerstone of several data analytics companies, e.g., Ayasdi and Alpine Data Labs.
In this paper, we develop a variation of the mapper construction targeting weighted undirected graphs, called mapper on graphs. For the rest of the paper, we use networks to refer to the data and graphs as an abstraction to the data. The mapper construction connects naturally with visualization by providing a strong theoretical basis for simplifying large complex data while preserving their core structures. Specifically:
We propose a set of summarization techniques to transform large graphs into hierarchical representations and provide interactive visualizations for their exploration.
We demonstrate the effectiveness of our method on synthetic and real-world data using three different topological lenses that capture various properties of the graphs.
We provide open-sourced implementation together with our experimental datasets via GitHub (see the supplement material).
2 Related Work
Graph visualization. We limit our review to node-link diagrams, which are utilized by many visualization software tools, including Gephi , GraphViz , and NodeXL . For a comprehensive overview of graph visualization techniques, see .
One of the biggest challenges with node-link diagrams is visual clutter, which has been extensively studied in graph visualization . It is mainly addressed in three ways: improved node layouts, edges bundling, and alternative visual representations.
Tutte  provided the earliest graph layout method for node-link diagrams, followed by methods driven by linear programming , force-directed embeddings [37, 47], embeddings of the graph metric , and connectivity structures [13, 50, 52, 53]. TopoLayout  creates a hybrid layout by decomposing a graph into subgraphs based on their topological features, including trees, complete graphs, bi-connected components, and clusters, which are subsequently grouped and laid out as meta-nodes. One of many differences between TopoLayout and our work is that we use functions defined on the graph to automatically and interactively guide decomposition and feature extraction among subgraphs.
Edge bundling, which bundles adjacent edges together, is commonly used to reduce visual clutter on dense graphs . For massive graphs, hierarchical edge bundling scales to millions of edges , while divided edge bundling  tends to produce higher-quality visual results. Nevertheless, these approaches only deal with edge clutter, not node clutter, and they only support limited types of analytic tasks [4, 67].
Finally, alternative visual representations have been used to remove clutter, ranging from variations on node-link diagrams, such as replacing nodes with modules  and motifs , to abstract representations, such as matrix diagrams  and graph statistics .
Node clustering. The objective in node clustering (or graph clustering) is to group the nodes of the graph by taking into consideration its edge structure . Common techniques include spectral methods [26, 36, 54, 94], similarity-based aggregation , community detection [69, 70], random walks [48, 80], and hierarchical clustering [12, 16]. Edge clustering has also been studied [23, 35]. Broadly speaking, our approach is a type of graph clustering that simultaneously preserves relationships between clusters.
TDA in graph analysis and visualization. Persistent homology (the study of topological features across multi-scales) and mapper construction are two of the most widely used tools in TDA. A number of works use persistent homology to analyze graphs [29, 32, 46, 76, 77], and it has been applied to the study of collaboration networks [5, 20] and brain networks [21, 24, 56, 57, 58, 59, 79]. In terms of graph visualization, persistent homology has been used in capturing changes in time-varying graphs , as well as supporting interactive force-directed layouts .
The mapper construction  has been widely utilized in TDA for a number of applications [17, 64, 65, 73, 91]. Recently, it has witnessed major theoretical developments (e.g., [19, 25, 68]) that further adjudicate its use in data analysis. To the best of our knowledge, this is the first time the mapper construction is utilized explicitly in graph visualization.
3 Methods: Mapper on Graphs Construction
Suppose the data is a weighted graph equipped with a positive edge weight and a real-valued function defined on its nodes . Our mapper on graphs method—a variation of the classic mapper construction—provides a general framework to analyze, simplify, and visualize , as well as functions on .
An open cover of a topological space is a collection of open sets for some indexing set such that . A finite open cover is a good cover if every finite nonempty intersection of sets in is contractible.
The mapper on graphs construction starts with a finite good cover of the image of , such that . Let denote the cover of obtained by considering the connected components (i.e., maximal connected subgraphs) induced by nodes in for each .
Given a cover of , let denote the simplicial complex that corresponds to the nerve of the cover , that is, . We compute the nerve of , denoted as , and refer to its -dimensional skeleton as the mapper on a graph, denoted as ; see Figure 1 for an illustrative example.
Parameters. Mapper on graphs is inherently multi-scale; its construction relies on two sets of parameters: the first defines the function/lens , and the other specifies the cover . For simplicity, we normalize the range space to be within .
Topological Lens: The function plays the role of a topological lens through which we look at the properties of the data, and different lenses provide different insights [9, 88]. Mapper on graphs currently considers three graph-theoretic lenses, average geodesic distance (AGD), density estimation, and eigenfunctions of the graph Laplacian (see Figure 2), although our framework can be easily extended to include other lenses (see Section 6).
Cover: The range of , , is covered by , which consists of a finite number of open intervals as cover elements . A common strategy is to use uniformly sized overlapping intervals. Let be the number of intervals, and describes the amount of overlap between adjacent intervals (see Section 3.2 for details). Adjusting these parameters increases or decreases the amount of aggregation mapper on graphs provides.
3.1 Topological Lens
An interesting open problem for the classic mapper construction is how to formulate topological lenses beyond the best practice or a rule of thumb [9, 10]. In practice, height functions, distances from the barycenter of the space, surface curvature, integral geodesic distances, and geodesic distances from a source point in the space have all been proposed as reasonable choices . In the graph setting, we focus on graph-theoretic lenses defined on the nodes of a graph, as illustrated in Figure 2. Each lens is chosen to reflect a specific property of interest that is intrinsic to the structure of a graph. In particular, we use as lenses average geodesic distance (AGD)  that detects symmetries in the graph while being invariant to reflection, rotation, and scaling; density estimation  that differentiates dense regions from sparse regions and outliers; and eigenfunctions of the graph Laplacian  that capture geometric properties of the graph.
Average geodesic distance. Suppose a weighted graph is equipped with a geodesic distance metric . That is, measures the geodesic/graph distance between two nodes . can be computed by utilizing Dijkstra’s shortest path algorithm. The average geodesic distance, , is given by
This definition implies that the nodes near the center of the graph will likely have low function values, while points on the periphery will have high values. The function has been used extensively in shape analysis due to its desirable proprieties in detecting and reflecting symmetry  based on how the function values are distributed. Therefore, the as a topological lens captures the symmetric properties of a graph, which are described by all or parts of the graph that are invariant to transformations such as reflection, rotation, and scaling.
The mathematical notion of automorphism, in some sense, captures the symmetry of the space as it is a structural-preserving way of mapping a space to itself. More precisely, consider a graph as a metric space equipped with the geodesic distance, . A bijection is called an automorphism on if for every . Let denote the group of automorphisms on . A function is isometry invariant over if for every : . is, therefore, an isometry invariant scalar function. Indeed, let be an automorphism on , then for every , we can verify that , and . See Figure 2(a) and Figure 3(a) for examples of on graphs.
Density estimation. The density estimation function  is given by
where is the geodesic distance between two nodes in the graph and .
Since is completely defined in terms of the distance , it is not hard to see that is also isometry invariant. correlates negatively with as it tends to take larger values on nodes that are close to the center, see Figure 2(b) and Figure 3(b) for examples.
Eigenfunctions of the graph Laplacian. Let be the vector space of all functions . The unnormalized Laplacian of the graph is the linear operator defined by mapping to , where
The eigenvectors of form a rich family of scalar functions defined on with many geometric properties . First, the gradient of the eigenfunctions of tends to follow the overall shape of the data ; and these functions have been used in applications, such as graph understanding , segmentation , spectral clustering , and min-cut problems . Sorting the eigenvectors of by increasing eigenvalues, we use eigenvectors of the second and third smallest eigenvalues of as the lens, denoted as and .
These vectors usually contain low-frequency information about the graph, and they help to retain the shape of complex graphs. In particular, , commonly referred to as the Fiedler vector , has desirable geometric properties . For instance, the maximum and the minimum of the Fielder vector tend to occur at nodes with maximum geodesic distances , see Figure 2(c) and Figure 3(c) for examples.
Furthermore, there is a connection between mapper on graphs, spectral clustering, and graph min-cut. For instance, the Fielder’s vector can be used to bi-partition the graph (i.e., based on or ); such a partition could also be approximated by computing mapper on a graph with as the lens and setting and . not only provides a generalization of spectral clustering but also preserves the connections (which form the min-cut) between the clusters (for appropriately chosen ).
Given a graph equipped with a lens function , suppose the range of the function is normalized to be within , and for a finite good cover of the interval . We represent this cover visually by drawing long, colored rectangular boxes, as indicated in 4.
The mapper on graphs construction relies on the choice of a cover for the interval ; such a choice is rather flexible but also essential to achieve effective graph visualization. To obtain an initial cover, we start by splitting into (the resolution parameter) intervals with equal length, such that and . The overlap parameter is then used to obtain the initial cover consisting of cover elements for .
Choosing and has a significant impact on the mapper on graphs output, as illustrated in Figure 4. Broadly speaking, smaller leads to a smaller topological summary of the graph; and smaller captures fewer connections between clusters of nodes. In the examples shown in the paper, we find a smaller typically give a more effective visualization for large and highly connected graphs, e.g., , while appears to be sufficient for small graphs.
4 Visual Design and Interaction
We provide a linked-view interface to enable exploration of the structure of a graph. It connects the original graph to its (multi-scale) summary in the form of a mapper on a graph , through interactive cover manipulation that supports customization of .
4.1 Cover Visualization and Interaction
The mapper on graphs construction relies on the choice of two sets of parameters: a lens and a cover. Therefore, the cover visualization consists of two components: a histogram of the lens, showing the distribution of function values, and an interactive cover designer. Via interactive visualization, we treat the exploration and manipulation of these parameters as a vehicle to study and summarize the intrinsic structure of an input graph.
Histogram of a lens. Understanding the distribution of the values of a lens can be helpful in the mapper on graphs construction. Figure 5(left) shows an example of a histogram for the lens in gray. The histogram is split into a fixed number of bins within the range . We will illustrate later how the visual information encoded in the histogram can be utilized to optimize the choice of the cover. In addition, the histogram of a lens can be used to inform the choices for and —generally speaking, a uniformly distributed lens function requires smaller .
Cover visualization and interactive manipulation. The cover is visualized using a series of boxes, one per cover element, displayed next to the histogram of the lens. Each box is placed based upon the start and end values of its interval and colored based upon the midpoint. Figure 5 shows the cover as red and orange boxes.
While an initial, uniform cover is sufficient for most graph visualizations, we provide interactive manipulation of individual cover elements that can be used to obtain more desirable mapper on graphs output in some cases. Given an interval , the user can manipulate its endpoints dynamically via an interactive interface such that can be shrunk, expanded, or shifted, as illustrated in Figure 6. As a user manipulates the interval , the connected components within split, merge, appear, or disappear. The histogram of a lens can be used to inform the cover manipulation; for instance, the length of an interval could be inversely proportional to the density of the histogram.
4.2 Graph Drawing
For both and its summary , we apply a Fruchterman-Reingold force-directed layout  with the Barnes-Hut approximation for repulsive forces . Our approach is ultimately agnostic of the graph layout algorithm, and different layouts (e.g., layered approaches) may improve the presentation of certain graphs.
For a given lens, , a node in is colored by a saturated colormap (red for , green for , and purple for or ) based on its function value. A node in (associated with a connected component ) is colored similarly by taking the average of the function values of nodes in . The size of is proportional to .
For both graphs, edge thickness is drawn proportional to edge weight. For , an edge represents the nonempty intersection between and . Therefore, its edge weight, and thus thickness, is drawn proportional to the size of the intersection, .
For readability, only the largest component of the mapper on a graph is included in the visualization. Furthermore, mapper on a graph nodes are removed from the output if the size of their connected component is less than a user-selected value.
4.3 Interactive Structural Correspondences
We provide three mechanisms for exploring structural correspondences between and : cover element selection, node selection, and edge selection.
Cover element selection. When a cover element is selected, the action triggers the selection of nodes in , as well as nodes that represent connected components of in . As illustrated in Figure 7(a-c), after the selection of a cover element (top left in (a-c), highlighted in blue), our system selects the corresponding nodes in (top) and in (bottom). As previously noted, if the nodes of captured by a particular cover element need fine-tuning, the box may be dragged, expanded, or contracted. will update correspondingly.
Node Selection. Each node in corresponds to a connected component from the original graph . With the selection of a node , our interface recovers and highlights from , as well as the cover element that generates the node , see Figure 7(d-e).
Edge Selection. Each edge in is determined by two connected components and in . With the selection of an edge , our interface highlights the clusters and in associated with its endpoints. Specifically, the sets , , and are colored differently to highlight node memberships and the relationship between the clusters. Figure 8 illustrates this process. Nodes that are unique to each endpoint are in purple and sky blue, respectively; nodes that correspond to the intersection are in blue (see Figure 8(a)). For comparison, clusters of nodes attached to individual endpoints are also highlighted in blue in Figure 8(b).
To demonstrate our approach, we have implemented our approach using Java and Processing
Synthetic datasets. We apply mapper on graphs to synthetic datasets, all of which are generated using NetworkX . Figure 9 shows 16 synthetic graphs and their corresponding mapper on graphs outputs . For each example, certain structures are emphasized depending on the choice of the lens and cover, such as symmetry (e.g., (b), (c), (k)) and the overall shape of the data (e.g., (a), (f), (l)). The original graphs for (a) and (f) have a circular shape, so we choose the Fiedler’s vector as the lens as it has variability across the graph. for (f) is particularly interesting as the original graph comes from a torus mesh, and appears similar to its corresponding Reeb graph. also captures the dual structures of some graphs in the cases of Grid graph (j) and the Dorogovtsev-Goltsev-Mendes graph (c).
USAIR 97. The previous example illustrates a natural interpretation of the nodes in as clusters in . In Figure 10, we provide a natural interpretation of the edges in as connections between clusters. The USAIR 97 graph consists of nodes and edges . The nodes represent airports and the edges routes between airports. We use as the lens and explore the USAIR 97 dataset by utilizing interactive edge selection as described in Section 4.
First in Figure 10(a), selecting edge in allows us to inspect their corresponding clusters (light and dark blue nodes) and (dark blue and purple nodes) in . and have Bethel and Anchorage International as major airports, respectively. Edge corresponds to airports in (blue nodes), including the Aniak and the St Mary’s. clearly captures the fact that these airports are the hubs between Bethel and Anchorage international in .
Second, selecting edge in enables the exploration of and in , respectively. , as shown in Figure 10(b), has Aniak and St Mary’s; while contains the Juneau International Airport. Edge in is mainly represented by Anchorage International. captures the fact that Anchorage International serves as a hub—in order to go from any airport in to airports in one must travel through Anchorage.
Finally, via node selection, node in corresponds to a peripheral cluster on the outskirt of (see Figure 10(a)). is represented mainly by the Guam international airport. In order to pass from any airport in to airports in , one must pass from the Honolulu International, which is contained in edge in .
Map of science. The map of science graph  (see Figure 11(a)) consists of nodes and edges. Nodes represent and are colored by specialties within major scientific disciplines, and edges represent co-authorship of publications between those specialties. Since does not exhibit obvious symmetry, we choose the 3rd smallest eigenfunctions of the graph Laplacian, , as the lens to help to retain the shape of . As illustrated in Figure 11(b), both and are laid out by way of correspondence where is shown to preserve the overall structure of . The highlighted nodes in also capture certain clusters in .
We could utilize interactive cover manipulation to obtain an even better representation of the data in Figure 11(c) where nodes in are circled to highlight the majority scientific discipline from the underlying cluster. For instance, node in represents the humanities-labeled nodes in ; nodes and represent chemistry and biology, respectively. Node and merge at node in , which represents medical science and infectious diseases.
6 Scalable Computation
The running time of mapper on graphs algorithm relies heavily on the choice of a lens. While the lenses we discussed earlier (Section 3) are effective in capturing various structures of an underlying graph, they are expensive to compute for very large graphs. To address the scalability issue, we need a lens that can be computed efficiently for very large graphs while still carrying structural information. We, therefore, consider PageRank  as an additional, scalable lens. We consider a version of the PageRank algorithm applicable to undirected graphs . A PageRank vector is defined for every node ,
where is the set of neighbors of ; is the damping factor, which is typically set at . Using the formation of , PageRank yields an iterative algorithm that can be computed efficiently in practice [14, 74]. The existence of the PageRank vector is guaranteed by the PerronâFrobenius theorem . A high PageRank score at typically means that is connected to many nodes, which also have high PageRank scores. The PageRank has been shown to be a continuous function in . For example, Figure 12(a) bottom illustrates the continuous variation of on for a random geometric graph. For the lens, we utilized , as it provided a good distribution of function values.
We utilize the PageRank implementation in NetworkX  and ran mapper on graphs on two synthetic graphs generated using NetworkX  and five real-world large graphs obtained from Stanford Large Network Dataset Collection  with up to 3 million edges using a MacBook Pro with GHz Quad-Core Intel Core i5 with GB memory. We report the average computational time in terms of PageRank () and mapper on graphs () in Table 1.
Finally, we would like to demonstrate that not only our approach is scalable using the PageRank lens, it also produces meaningful visualization results. Figure 12 gives examples of applying mapper on graphs to seven large graphs using the PageRank lens. Figure 12(a-b) are both synthetic graphs: (a) is a random geometric graph  with a radius (, ), while (b) contains a balanced tree with branching factor and height (, ). Mapper on graphs for both graphs are shown to capture the global organizational principle of the original graph.
Figure 12(c) contains a co-purchasing graph based on the CWBTIAB feature (Customers Who Bought This Item Also Bought) on Amazon website: if a product is frequently purchased together with product , then the original graph contains an edge . While the node-link diagram of cannot be improved beyond a hairball, provides a very compact summary containing information regarding popular products serving as “hubs”. Similarly, Figure 12(d-g) show the mapper on graphs for (d) YouTube social network, (e) Condense Matter collaboration network, (f) Epinions social network, and (g) a 2nd Amazon product co-purchasing network on March 2nd, 2003. The original graphs for these figures are not shown because they, too, are essentially hairballs.
In this paper, we present a TDA approach for graph visualization using a variant of the mapper construction called mapper on graphs. Our approach is effective at detecting clusters in a graph and preserving the relations among these clusters. It is also flexible in capturing the structure of a graph across multiple scales based on different topological or geometric lenses. Mapper on graphs could potentially be applied to other visualization tasks, for instance, as a skeleton representation for the underlying graph in determining an initial layout in graph drawing. Traditionally, the PageRank vector is employed on the web graph, which could be considered as a temporal network—the PageRank vector at a previous instance of the web graph is used as an initial vector to obtain a fast PageRank solution for the current instance of the web graph. Therefore, Mapper on graphs in conjugation with the PageRank lens could be utilized for the study of temporal networks.
Our focus in this paper is applying mapper on graphs to graph visualization. We are also interested in the theoretical properties of mapper on graphs. For example, how do we measure the distance between a mapper on graphs and its underlying graph ? What is an appropriate metric under which converges to the Reeb graph of as the granularity of the cover goes to zero? What is the structural stability of mapper on graphs with respect to perturbations of the lens , graph , and cover ? Recent theoretical advances on the stability [15, 19] and convergence [3, 15, 18, 19, 25, 68] of mapper construction could address these questions, at least partially. However, questions remain that are unique to mapper on graphs: e.g., what is the relation between the shape of and the geometric and topological properties of the lens ?
This work was supported in part by the National Science Foundation NSF IIS-1513616 and NSF DBI-1661375.
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