Exemplar-based Layout Fine-tuning for Node-link Diagrams
Jiacheng Pan, Wei Chen, Xiaodong Zhao, Shuyue Zhou, Wei Zeng, Minfeng Zhu, Jian Chen, Siwei Fu, Yingcai Wu
EExemplar-based Layout Fine-tuning for Node-link Diagrams
Jiacheng Pan, Wei Chen, Xiaodong Zhao, Shuyue Zhou, Wei Zeng, Minfeng Zhu,Jian Chen, Siwei Fu, and Yingcai Wu
Figure 1. Case study with the Finan512 dataset [11] of 74,752 nodes and 261,120 edges: (a) in the node-link diagram generated withFM [27], we specify an exemplar (in red) and several similar substructures (in blue) are retrieved with k = , min = , max = , and ε = . ; (b) the exemplar; (c) five most similar (1-5) and five least similar (6-10) retrieved substructures; (d) we specify 14 substructuresaround the exemplar; (e) the modified exemplar; (f) modifications are transferred to 10 retrieved substructures; (g) modifications aretransferred to substructures around the exemplar. All modified substructures are merged into the entire graph. (h-k) Readabilitybefore (orange) and after (purple) modification transfer measured by four readability criteria (from top to bottom: crosslessness,minimum-angle metric, edge-length variant, and shape-based metric); error bars depict 95% confidence intervals. Abstract —We design and evaluate a novel layout fine-tuning technique for node-link diagrams that facilitates exemplar-basedadjustment of a group of substructures in batching mode. The key idea is to transfer user modifications on a local substructure to othersubstructures in the entire graph that are topologically similar to the exemplar. We first precompute a canonical representation foreach substructure with node embedding techniques and then use it for on-the-fly substructure retrieval. We design and develop alight-weight interactive system to enable intuitive adjustment, modification transfer, and visual graph exploration. We also report someresults of quantitative comparisons, three case studies, and a within-participant user study.
Index Terms —Node-link diagram, graph layout, graph visualization, user interactions
NTRODUCTION
Generating appropriate layouts of graph data has been a major re-search topic over the past decades, as witnessed by the extensive liter-ature [5, 12, 24, 30, 59]. Among many solutions, node-link diagramsare widely used, because they reveal topology and connectivities [23].When the number of nodes and edges increases, algorithms aimed atboth computational speed and readability are valuable. New force-directed layout algorithms have harnessed data features to layout alarge set of nodes and edges effectively [32]. However, additionallayout optimizations or manual modifications are typically required toimprove readability [61].The aesthetics of a graph layout is often subjective and may vary • Jiacheng Pan, Xiaodong Zhao, Shuyue Zhou, Minfeng Zhu, Yingcai Wu, andWei Chen are with the State Key Lab of CAD&CG, Zhejiang University,China. E-mail: { panjiacheng, zhaoxiaodong, zhoushuyue, minfeng zhu,ycwu } @zju.edu.cn, [email protected] Wu is also with Zhejiang Lab, China.Wei Chen and Yingcai Wu are the corresponding authors.• Wei Zeng is with Shenzhen Institutes of Advanced Technology, ChineseAcademy of Sciences, China. E-mail: [email protected].• Jian Chen is with Ohio State University, USA. E-mail: [email protected].• Siwei Fu is with Zhejiang Lab, China. E-mail: [email protected] received xx xxx. 201x; accepted xx xxx. 201x. Date of Publicationxx xxx. 201x; date of current version xx xxx. 201x. For information onobtaining reprints of this article, please send e-mail to: [email protected] Object Identifier: xx.xxxx/TVCG.201x.xxxxxxx with user preferences. Modern rule-based graph layout methods [15,31, 36, 61] have successfully integrated users’ preferences into layout.Nevertheless, such state-of-the-art solutions for interactive fine-tuningof node-link diagrams work only for either dragging individual nodesor the entire diagram (e.g., fisheye) [8, 14]. Interaction techniqueshave enabled graph exploration but not layout modification [4, 65]. Inparticular, fine-tuning of node positions in a layout has to be manuallyperformed, which is laborious and time-consuming.We design an interactive exemplar-based tuning algorithm for dis-playing node-link diagrams in which exemplar is a local substructureof the underlying graph specified by users, following the techniqueproposed in [4] (Figure 1b). The key to our solution is first to findtopologically similar structures to the user-chosen exemplar and then tomorph these structures automatically into a user-defined layout beforeembedding them in the original graph. Transferring the user’s inputhas two main challenges. First, substructures in a large node-link di-agram can have distinctive topologies. Identifying similar ones fromthe entire diagram and constructing the correspondences between thetwo substructures is a nontrivial task. Second, mapping the dynamicchange of one exemplar to another requires solving a two-dimensionalsubstructure transformation. Our solution to these challenges has threemain components: representation , retrieval , and morphing of substruc-tures , designed to efficiently fine-tune substructures containing a groupof user-specified nodes and edges. Compared with the baseline method(manual node dragging), our approach facilitates fast specifications ofsubstructures and local layout fine-tuning based on users’ preferences.This paper makes the following contributions:• A novel layout fine-tuning method that can simultaneously adjust a r X i v : . [ c s . G R ] S e p ayouts of multiple similar substructures to user preferences;• An efficient modification-transfer algorithm that can transfer fine-tuned results of an exemplar substructure to other topologicallysimilar substructures;• A set of quantitative and qualitative experiments that evaluate theefficiency of our approach. ELATED W ORK
We review two related areas: graph visualization techniques and inter-action techniques.
Two-dimensional (2D) graph drawing methods have been broadly re-ported in textbooks and surveys [5, 24, 59]. Force-directed and re-lated drawing methods are classified into three categories [24]: force-directed methods, dimension-reduction methods, and multi-level meth-ods. Force-directed methods simulate physical forces on nodes andedges to layout graphs; many extensions exist, e.g., spring-embeddedmethods [17, 19, 32], energy-based methods [21, 35], and probabilisticmethods [10, 38]. Dimensionality-reduction methods aim to embedhigh-dimensional information (e.g., the shortest path length betweentwo nodes) into a 2D space, using methods such as multidimensionalscaling [3], self-organizing maps [2], and t-SNE [37]. Multi-levelmethods focus on accelerating graph drawing using two main phases:coarsening (simplify a graph into several coarser graphs) and refinement(successively compute fine layouts from simple coarser graphs) [20,27].Besides these generic layout algorithms for node-link diagrams, othermethods aim to solve specific drawing problems. For example, or-thogonal layouts proposed in [8, 36] improve readability of node-linkdiagrams of power-grids, software, and financial markets.Unlike prior studies on layout algorithms, our work focuses oninteractive fine-tuning by capturing users’ layout preferences throughinteraction. Our algorithms can potentially support personalized andfine-tuned layout of these current state-of-the-art graph visualizations.
We categorize interaction techniques into three levels: data-level, view-level, and encoding-level.
Data-level interactions focus on selecting the data for display. Theuser can interact with the graphs to see similar structures. A system de-veloped in [60] uses user-defined subgraph or motifs to reveal selectedstructures but these motifs were predefined and could not be modifiedby the users. Several systems [22, 63, 69] use PathRings to definemotifs in biological pathways, but they do not find similar structures.Novel machine learning solutions utilized in [41] measure the similaritybetween two graphs, but it is not feasible because it does not locatesubstructures. A structure-based recommendation approach [4] detectssimilar substructures in a graph from user input and lets users subse-quently interact with the detected structures. We adopt this approach tomeasure similarity, thus reducing user input; we subsequently introducea new algorithm to further reduce users’ repetitive and effortful nodeediting through a substructure transformation algorithm.
View-level interactions mostly support graph navigation. Topol-ogy information can be exploited in browsing a large graph [49].Fisheyes enlarge the display space for items of user interest to im-prove readability. For example, SchemeLens [8] reveals orthogonallylaid-out diagrams. And the structure-aware fisheye proposed in [62]reduces spatial and temporal distortions. Compared to these solutions,our method supports the user’s defined input to customize layout.
Encoding-level interactions seek to manipulate the visual repre-sentation and layout of graph data. An appropriate layout can benefitanalysis tasks [34, 44]. However, generating visually pleasing anduseful layouts for large graphs is still challenging. NodeTrix [29]combines two schemes to show inter-community relationships usinga node-link diagram and intra-community relationships using the ma-trix representation. In many situations, analysts fine-tune the nodepositions. An authoring tool proposed in [16] introduces continuouslayout in response to user input. A method that could integrate multiple ( a ) S ec ti on . Graph data NodeEmedding .........
Node-LinkdiagramExemplar Query similar substructures TargetsubstructuresModifiedexemplar Modified targetsubstructuresModificationtransfer(c) Section 3.3 Global optimization(d) Section 3.4(b) Section 3.2
User specificationpre-layouted
Figure 2. Workflow of our end-to-end system: (a) pre-computing nodeembeddings and laying-out a node-link diagram; (b) detecting similartarget substructures with a specified exemplar; (c) transferring usermodifications to similar substructures; (d) merging modified substructuresinto the entire layout with global optimization. graph layouts preserved topological structures in graphs by control-ling the Euclidean distance between nodes of subgraphs [66]. Someconstraint-based layout editing methods [55, 56, 61] allow the user toedit and explore a layout with selected constraint rules. However, thesemethods aim to draw a constraint-based layout, and could not edit alayout freely on nodes to reach a fine-tuned layout and incorporateusers’ preferences.
AYOUT F INE -T UNING : W
ORKFLOW AND I NTERFACE
We have designed and implemented an end-to-end tool for exemplar-based layout fine-tuning to reduce the manual workload of refininglayout by suggesting fine-tuning candidates (similar substructures) andtransferring user modifications to those candidates (Figure 2).Our workflow has four steps:
Step
1. Our algorithm calculates the node embedding of the entiregraph to retrieve similar structures (Figure 2a). A node-linkdiagram is generated with an initial layout of the entire graph.
Step
2. The user specifies an exemplar from the entire graph. Our sim-ilar structure-query technique using the method in [4] retrievesseveral target substructures topologically similar to the exem-plar (Figure 2b). The user can also specify target substructuresfrom the node-link diagram.
Step
3. The user modifies the exemplar’s layout. Our modificationtransfer algorithm transfers the modifications to target sub-structures (Figure 2c).
Step
4. Our algorithm merges the modified substructures into the orig-inal graph through global optimization to smooth the bound-aries. The user can iterate from
Step
Step
Our approach uses a node-embedding technique to embed a node intoa low-dimensional vector subject to its local topology. For a givenexemplar, we employ the node-embedding-based representation to rep-resent and retrieve similar substructures from the entire graph. In thisway, we simplify the subgraph-retrieving problem to a similar multidi-mensional data-searching problem. Though various node-embeddingrepresentations [13, 26, 28, 45, 53] are compatible with our approach,we leverage GraphWave [13] following the study conducted in [4]. Wepre-compute node embeddings because this process is time-consuming.
The user can specify a substructure using the lasso interactions in thenode-link diagram (Figure 3a). We then use the similar structure-query igure 3. The user interface of our prototype system: (a) an exemplarview; (b) a control panel; (c) a suggestions gallery; (d) a node-link view;(e) a modification history view. technique in [4] to retrieve a set of substructures that are potentiallysimilar to the exemplar (Figure 3c). Four parameters are used in thesearching process. The parameter k is used in the k -nearest neighborsalgorithm for retrieving similar nodes. A large k may introduce manycandidate nodes in a huge connected substructure; it will be filteredout by parameter max . On the other hand, a small k limits the numberof candidate nodes, so that the probability of forming a connectedsubstructure is small. The parameter k should be tuned interactively.We eliminate similar substructures whose node number is less thanthe minimum count ( min ) or more than the maximum count ( max ). Wesuggest setting min and max to be close to the number of nodes in theexemplar (e.g., set min to be half nodes and max to be twice nodes ),so as to generate substructures of similar scale to the exemplar. We alsoremove substructures whose Weisfeiler-Lehman similarity is less thanthe minimum similar threshold ( ε ).Also, we let the user specify additional structures in the node-linkdiagram using lasso interactions. We regard both retrieved substructuresand user-specified substructures as target substructures. Our approach uses dragging interaction to interactively manipulate theexemplar’s layout. The modification transfer algorithm described inSection 4 can transfer the exemplar’s layout modifications to targetsubstructures’ layouts. We design an interaction mode called “formatpainter” (inspired by operations in Microsoft Word) to perform themodification transfer. After modifying the exemplar’s layout, the usercan transfer modifications to other target substructures using the “copy”and “paste” buttons. Our approach records modifications after theuser clicks the “copy” button and transfers modifications into targetsubstructures after the user clicks the “paste” button.
The exemplar and target substructures are parts of the entire graph.Directly merging the modified layout into the entire graph may lead toabrupt boundaries of the modified substructures (Figure 4b). Thus, weperform a global optimization to preserve the smooth boundaries of themodified substructures (Figure 4c). The optimization process is similarto the deforming step, like the stress-majorization layout [21] (seeSection 4.2). We preserve details of the entire graph by minimizing therelative position displacements of each node pair. However, optimizingthe entire graph is computationally expensive. We found that deformingthe layout of the surroundings of the exemplar and target is to someextent adequate to reach smoothness. The surroundings of a structureare the induced subgraph of the entire graph whose nodes’ distances areless than a given distance d to the structure, where d is the maximumedge length in the entire graph. This ensures that nodes adjacent inboth topology and Euclidean distance can be included. (a) (b) (c) Figure 4. Global optimization in the Finan512 dataset case study; (a)original layout; (b) merging the modified target substructure into entiregraph without any optimization; (c) merging modified target with ourtechnique.
We design and implement a visual interface, which consists of 4 parts:
The exemplar view (Figure 3a) supports exploring and modifying aspecified exemplar. When the user finishes modifications, the usercan use “format painter” to transfer modifications to the targets.
Thecontrol panel (Figure 3b) enables the user to adjust parameters of themodification transfer algorithm and the similar-substructure-retrievalalgorithm.
The suggestion gallery (Figure 3c) sequentially displayssimilar structures according to their Weisfeiler-Lehman similarities tothe exemplar in node-link diagrams. In the meantime, the user canspecify a structure in the node-link diagram as a suggestion; this isdisplayed at the top of the suggestion gallery.
The node-link view (Fig-ure 3d) provides visualizations with various graph-layout algorithms.The user can use the lasso to specify a substructure as an exemplar,which the exemplar view will then display.
The modification historyview (Figure 3e) records layout change history applied to the exemplar.Each record relates to a piece of modification on the exemplar. Thelayouts before and after modifications are shown side by side. The usercan reuse modifications in the history view for transferring. The mostrecent history is displayed at the top.
ODIFICATION T RANSFER IN G RAPH S TRUCTURES
Here we introduce a modification-transfer algorithm to transfer layoutadjustments from one graph structure to another. We define terms inTable 1. Here, given a source graph structure layout S = ( V s , E s ) , usermodifications change S into a new layout S (cid:48) . And given a target graphstructure layout T = ( V t , E t ) , we denote the modification transfer as aprocess of analogizing the modifications ( S → S (cid:48) ) to the target graph( T → ˜ T (cid:48) ) in three steps: Step Marker selection first aligns T and S with correspondences C generated by the graph-matching method and then selects somefinely matched correspondences as markers (Figure 5a). Step Layout simulation ( T S −→ ˜ T ) alters the layout of the targetfrom T to ˜ T to simulate S and expands M to ˜ M (Figure 5b). Step Layout simulation ( ˜ T S (cid:48) −→ ˜ T (cid:48) ) alters the layout of the targetfrom ˜ T to ˜ T (cid:48) to simulate S (cid:48) (Figure 5c).Here, we perform two rounds of layout simulate because S (cid:48) is usuallydifferent from T , directly deforming T into the shape of S (cid:48) can lead tounpleasing transfers. The modification transfer algorithm relies on the correspondences be-tween two structures, denoted as C = { ( c si , c ti ) } , ≤ i ≤ min ( | V s | , | V t | ) .Any graph-matching method that produces injective correspondencesis suitable for modification transfer. Six graph-matching meth-ods [7, 9, 25, 42, 67, 68] are examined (see Suppl. Material andSection 5.1 for comparison details). We employ Factorized GraphMatching (FGMU) [68] because it achieves the best efficiency.Because graph-matching methods may depend on the graph layout,we layout the exemplar and target substructures with the same algorithmbefore constructing correspondences. We employ FM [27] becauseit is one of the most efficient layout algorithms to our knowledge. https://zjuvag.org/publications/exemplar-based-fine-tuning/ able 1. Definition of symbols. Here G = ( V , E ) denotes a graph with itslayout, where V = { v , v ,..., v n } , v i ∈ R contains the positions of a setof n nodes and E = { e , e ,..., e m } is a set of m edges in G . Symbol Description S = ( V s , E s ) A source graph layout S (cid:48) A modified source graph layout T = ( V t , E t ) A target graph layout˜ T = ( ˜ V t , E t ) The target graph layout that simulates S ’s layout˜ T (cid:48) The target graph layout that simulates S (cid:48) ’s layout M The set of paired markers that matches V s to V t ( m si , m ti ) ∈ M A pair of markers where m si ∈ V s and m ti ∈ V t C Correspondences between S and T ( c si , c ti ) ∈ C A correspondence pair where c si ∈ V s and c ti ∈ V t ( v i [ x ] , v i [ y ]) The x and y positions of node iV ( k ) Positions of the node set V in the iteration k R A 3 × TCS
AligningFiltering M Layout Simulation
T~M ~ Layout Simulation
S’ T’ ~ (a) (b) (c) T → T~ S T → T’~ S’ ~ Figure 5. Modification transfer. (a) Marker selection: aligning layouts oftarget T and source S first, and then selecting a set of markers M fromgiven correspondences C between S and T ; (b) the first round of layoutsimulation: altering the layout T to simulate S , which produces a newtarget structure layout ˜ T and expands M to ˜ M ; (c) the second round oflayout simulation: altering the layout ˜ T to simulate S (cid:48) , which produces ˜ T (cid:48) . The graph-matching methods can generate unpleasant matching resultsbecause these methods build correspondences for all nodes even ifthey are not well matched. To examine their correspondences, we firstalign two graph structures S and T according to the correspondences(described in Section 4.2). If the graph-matching method generatescorrect correspondences, we align two corresponding nodes together inthe aligning step and almost all of their neighborhoods can possibly bematched. Thus, we implement the correspondences filtering algorithm(Algorithm 1) to select “fine” correspondences ( ( c si , c ti ) ) that satisfy:1) The distance between c si and c ti is less than the average length oftheir adjacent edges multiplied by a given ratio ( r d ); and2) c si ’s neighbors are mostly matched to c ti ’s neighbors (with a ratiogreater than r u ).We fix r d and r u to be 2 and 0 . r d and a larger r u lead to fewer, possibly more accurate correspondences.There is a trade-off between accuracy and number of correspondences.Here, we use these “fine” correspondences as a set of markers M for thelayout simulation. In addition, our approach also supports specifyingmarkers manually to match user preferences. The user can click on twonodes, one in each of the exemplar and target substructures, to specifya pair of markers. The goal of the layout simulation is to smoothly deform the markers ofthe target T to those of the source S , while preserving the original layoutof the target T as much as possible (Figure 6). We do this in threesteps: aligning , deforming , and matching . The aligning step scales,rotates, and translates T to minimize the dissimilarity to S (Figure 6d).The deforming step alters the node positions of T to simulate theshape of S (Figure 6e). The matching step constructs correspondencesbetween the nodes of T and S by searching their neighbors (Figure 6f).These steps iteratively deform T into the shape of S until no more newcorrespondences are constructed. Algorithm 1
Correspondences filtering
Input: S = ( V s , E s ) : a source graph; T = ( V t , E t ) : a target graph; C = (cid:8) ( c si , c ti ) (cid:9) : a set of correspondences; r u : a minimum commonneighbors ratio; r d : a maximum distance ratio; Output: M = (cid:8) ( m si , m ti ) (cid:9) : a set of markers; Init markers M = ∅ for each correspondence pair ( c si , c ti ) do ns ← c si ’s neighbors’ corresponding nodes nt ← c ti ’s neighbors nu ← ns (cid:84) nt if Count ( nu ) > Count ( ns ) × r u or Count ( nu ) > Count ( nt ) × r u then ds ← the mean length of adjacent edges of c si dt ← the mean length of adjacent edges of c ti d ← distance between c si and c ti if d < ds × r d and d < dt × r d then Add ( c si , c ti ) into M end if end if end for return M ; Aligning.
We assume that both the topology and the layout ofthe source structure are similar to the target. To minimize the layoutdifference, the node-link diagrams of S and T are aligned. This step onlytransforms the global location and orientation of the target structure,not the positions of individual nodes.We use a small set of predefined markers to achieve an optimalalignment. The markers are a set of paired nodes M = (cid:8) ( m si , m ti ) (cid:9) , m si ∈ V s , m ti ∈ V t (Figure 6c). The markers on the source and the target arealigned by an affine transformation R : R = scale × cos θ sin θ tx − sin θ cos θ ty ≈ s h tx − h s ty (1)where scale is the scale coefficient, θ is the rotation angle, and tx and ty are the translation components. For the sake of simplicity, we use alinear approximation of R (after the approximately equal sign). R iscalculated by solving the minimization problem:min R | M | ∑ i || R m ti − m si || , (2)where ( m ti , m si ) ∈ M denotes one pair of markers. The minimizationproblem is equivalent to the problem:min T ∑ M || A ( s , h , tx , ty ) T − b || , (3)where A contains the positions of the markers in the target and b contains the positions of the markers in the source: A = m ti [ x ] m ti [ y ] m ti [ y ] − m ti [ x ] , b = m si [ x ] m si [ y ] ... , i = ,..., | M | . (4) m ti [ x ] and m ti [ y ] are the positions of the target marker m ti and m si [ x ] and m si [ y ] are the positions of the source marker m si . The minimizationproblem can be solved by: ( s , h , tx , ty ) T = A † b , (5)where A † is the Moore-Penrose pseudoinverse [48] of A . Thus, thetransformation can be defined as a linear function of the markers in thesource. With the affine transformation matrix R , T is transformed toalign S by a linear transformation (Figure 6d). After modification trans-fer, the target layout is restored by an inverse process of the alignmentstep, so that it can be merged into the entire layout with the originalrotation and scale. FDBA HC E83 67 51 492
83 67 51 492 G FDBA HC E13 62 4 5789 (d)
Aligning
T → S
Iterate until no more new correspondences are built
G FDBA HC E13 62 4 5789 G FDBA HC E5<=>D2<=>B 5 63 4891 72
Deforming T → S (e) Matching
T → S (f)Source: S(a) Target: T(b)Markers: M(c)
G FDBA HC E5489 32 71 65→D2→B G FDBA HC E548 321 79 67<=>F9<=>H G FDBA HC E548 321 79 (g) 2nd round (h) 3rd round Deforming T → S Matching
T → S
Deforming T → S Matching
T → S
Figure 6. Layout simulation: altering the shape of a target structure T tosimulate the layout of a source structure S . (a) a source structure S ; (b) atarget structure T ; (c) a set of markers M ; (d) aligning T to S with markers M ; (e) deforming T into S with markers M ; (f) matching the nodes of T to the nodes of S ; two pairs of markers are constructed: { (2,B) and(5,D) } ; (g) the second round of deforming and matching; two markerpairs are constructed: { (7,F) and (9,H) } ; (h) the third round of deformingand matching, one pair of markers is constructed: { (8,G) } ; Iterations areperformed until no more new correspondences are built. Deforming.
The deforming step seeks to alter the shape of T tosimulate S . We design an energy function to represent the process: E = E S + γ E M , (6)where γ is a weight parameter. The deforming step is equivalent tominimizing E . It seeks to force positions of target markers m ti toapproach source markers m si ( E M ) while preserving the original layoutinformation to reach a smooth deformation ( E S ). Here, we denote E M as the sum of distances between pairs of markers: E M = | M | ∑ i || m ti − m si || . (7) E S represents the layout change between T and ˜ T , which is con-structed by two items: E S = α E O + β E D , (8)where α and β are two weights, E O is designed to preserve orientationsof vectors between node pairs after the aligning step, and E D is designedto preserve distances between node pairs. E O is defined as: E O = ∑ i < j w i j || norm ( v ti − v tj ) − norm ( ˜ v ti − ˜ v tj ) || . (9)Here, norm ( · ) denotes the normalization of a vector. E D is defined as: E D = ∑ i < j w i j ( || v ti − v tj || − || ˜ v ti − ˜ v tj || ) , (10)where w i j is the weight related to the node pair ( v ti , v tj ) , and ( ˜ v ti , ˜ v tj ) is anode pair of the target structure after deformation ( ˜ T ). w i j is defined v v v v v v v v v v v v v v v (a) (c) α = 1, β = 1, γ = 100 (b) α = 0, β = 1, γ = 100 Figure 7. Different weighting schemes. The node v is moved to a higherposition while nodes v and v are fixed in their original positions. (a)is the original layout. (b) is the layout that preserves the distances with α = , β = , and γ = . (c) is the layout that keeps both orientationsand distances with α = , β = , and γ = . as: w i j = (cid:40) w || v ti − v tj || − , if { i , j } ∈ E t || v ti − v tj || − , otherwise , (11)where w is a preservation degree on the edges. Setting w greater than1 makes the algorithm pay more attention to preserve orientations andlength of edges.Preferences on preservation of distances and orientations can beconfigured by balancing α and β . For example, when α is small, thedistances between node pairs can be mostly preserved (Figure 7b). Ifwe enlarge α , the orientations can be better preserved (Figure 7c). Bothweighting schemes are optional. The parameter γ is used to configurethe weight of moving marker positions in the target structure to theircounterparts. A large γ ensures that markers of S and T can be aligned.Following the optimization process in the stress majorization tech-nique [21], E O and E D can be minimized by iteratively solving: L V t ( k ) w V t ( k + ) = L V t w V t , (12)and L w V t ( k + ) = L V t ( k ) w V t ( k ) , (13)where V t ( k ) and V t ( k + ) are the target nodes in time k and k + L w and L V t w are two weighted Laplacian matrices defined as: ( L V t w ) i j = (cid:26) − w i j inv ( || V ti − V tj || ) , i (cid:54) = j ∑ l (cid:54) = i ( L V t w ) il , i = j ( L w ) i j = (cid:26) − w i j , i (cid:54) = j ∑ l (cid:54) = i w il , i = j , (14)and the definition of L V t ( k ) w is similar to L V t w except that v ti and v tj arereplaced by their counterparts in time k . The process is repeated untilthe target layout stabilizes. Matching.
The matching step constructs node correspondencesbetween S and T . Any node pair ( v si , ˜ v tj ) , i ≤ | V s | , j ≤ | ˜ V t | that satisfies || v si − ˜ v tj || < r j is identified as one candidate correspondence. Weconsider that r j should be adaptive to different ˜ v tj , and thus, we associate r j to the mean length of ˜ v tj ’s adjacent edges. By default, r j is set tobe twice the mean length of the adjacent edges to avoid filtering outtoo many candidate node pairs. To avoid overlapping, correspondencesshould be injective. This maximum assignment problem can be solvedby the Hungarian algorithm [39, 40]. Here, we use distances betweennode pairs as the cost in the Hungarian algorithm.Adequate correspondences can yield accurate modification transfer.Thus, the aligning , deforming , and matching steps are iteratively per-formed by using the already-built correspondences or markers. Forexample, Figures 6(e-f) show the first round of deformation. With threemarkers, the target can not faithfully mimic the shape of the source.Additional correspondences are constructed by searching neighbors(Figure 6f). Two more deforming and matching rounds improve theaccuracy (Figures 6(g-h)). The iteration stops until the number ofcorrespondences no longer increases. ˜ T is often similar to S after defor-mation (Figure 6h). After that, layout simulation is performed again toalter the deformed target ˜ T into the modified source S (cid:48) (Figure 5c). b) (a) (c) A cc u r ac y Spacing
GA PM SMAC RRWM FGMUSM OURS
Figure 8. Quantitative comparison of several conventional graph match-ing methods and our approach: (a) average accuracy of different framespacing in the CMU-house-image dataset; (b) average accuracy of dif-ferent numbers of outliers in the Motorbike-image dataset; (c) averageaccuracy of different numbers of outliers in the Car-image dataset.
ESULTS AND E VALUATION
We implement our system in a browser-based architecture. The front-end application is developed with JavaScript using React and D3. Theback-end server uses Python 3.7.5 with flask, networkx, numpy, andscipy. All experiments are performed on a Macbook Pro laptop with anIntel Core i7-7820HQ CPU (2.9 GHz) and 16 GiB RAM.
Our approach uses a set of markers generated by graph-matchingmethods for modification transfer. We compared conventional graph-matching methods to ours using the following benchmark datasets withmanually labeled ground truth:
1) The CMU-house-image dataset [68] contains 111 frames of ahouse with 30 landmarks. We randomly remove 5 landmarks andgenerate a graph with Delaunay triangulation that connects land-marks for each frame. Frames are paired spaced by 0, 25, 50, and75 frames, yielding 444 pairs.
2) The Car-and-Motorbike image dataset [43] has 30 pairs of carimages and 20 pairs of motorbike images. We used Delaunaytriangulation to generate graphs for each image, added 0, 4, 8, 12,16, and 20 outliers randomly, and removed unconnected nodes,yielding 222 pairs of graphs.Node-link diagrams of these datasets are generated by the well-studied force-directed layout algorithm. We compare the accuracy ofgraph matching results. Figure 8 shows the average matching accuracyon different datasets. Our approach works slightly better than FGMUand exceeds other methods, meaning that our improvements on FGMUcan generate more accurate results in most cases. Note that graphs inthese benchmark datasets are smaller than those in the case studies.
We show how our exemplar-based layout fine-tuning approach worksin three case studies.We used FM [27] to generate the layout of the Finan512 dataset from the University of Florida Sparse Matrix Collection [11], whichis generated from multistage stochastic financial modeling [57]. Thegraph consisted of 74,752 nodes and 261,120 edges rendered in WebGL(Figure 1a). We saw several “donut-like” substructures.Next, we specified a substructure (here called an exemplar ) forfine-tuning (Figure 1b). We retrieved similar substructures using k = min = max = ε = .
95 (Figure 1a). To verify thetopology of these substructures, we select target substructures as the fivemost similar and five most dissimilar substructures according to theirWeisfeiler-Lehman similarities to the exemplar (Figure 1c). In addition,to fine-tune the “donut” subgraph, we use substructures around it astarget substructures.We interactively modified the exemplar into a layout with a dis-tinguishable structure (Figure 1e). After modification transfer, thesesubstructures became clearer (Figures 1(f, g)).Our smooth merging scheme generated visually pleasing detailscompared to direct merging without any optimization. For example,the boundary of the substructure in Figure 4c is easier to distinguishthan the one without optimization in Figure 4b.
The Power-Network dataset is collected from the Network DataRepository [54], which abstracts a power system: the nodes encodebuses and edges are the transmission lines among the nodes. Thenetwork contains 662 nodes and 906 edges. A multilevel graph layoutimplemented by Tulip [1] and OGDF [6] is employed to layout thenetwork (Figure 9a).To reveal transmissions among a set of buses that may form a cycle,we specify a set of nodes as an exemplar (Figure 9a, in red). With min = max = k =
5, and ε = .
5, two overlapped structuresare retrieved (Figure 9a, in blue). The retrieved structures are nottopologically similar to the exemplar, because our technique detectsembedding-similar structures, which are potentially similar to the ex-emplar. Thus we explore the node-link diagram to specify targetsubstructures. Several sets of nodes that may form cycles are specifiedas target substructures (Figure 9b, in blue). Connections among thesenodes are obscured by the visual clutter. The exemplar is interactivelymodified into a circle. Each target is deformed into a circle-like shapeby transferring modifications (Figure 9c). Now the connections amongnodes are far more distinguishable (Figure 9d) than the original layout.We increase α to increase the degree of orientation preservation,which means that orientations of edges tend to remain unchanged.This makes the shape of the modified target substructure smoother(Figure 10c). Because the edge lengths before modification transferare not identical (Figure 10a), solely preserving distances can lead tounsatisfying deformations. For example, setting α in Equation 8 to bezero generates irregular polygons (Figure 10b). The Price 1000 dataset is a tree from tsNET [37] that consists of1000 nodes and 999 edges. We layout the graph with a simple radialtree layout algorithm [33] (Figure 11a), and find that sibling nodes areoverlapped due to the space constraint.We select one representative subtree as an exemplar. To reducevisual clutter,this is reconfigured into a radial tree layout (Figure 11b).To reconfigure other interested subtrees, we specify two nodes of theexemplar as markers, and the algorithm transfers modifications on theexemplar to other subtrees (Figure 11c).Although there are some unpleasing details, their layouts are similarto the exemplar’s. Rather than interactively reconfiguring these subtreesfrom the original layout, our approach requires only a few slight modi-fications according to the minimum angle and the symmetry aestheticmetrics [52] to tune the details (Figure 11d) because it generates aninitial layout for each subtree.
Readability.
To evaluate the readability of the results generatedby our approach, we use the measurements (crosslessness, minimum-angle metric, edge-length variation, and shape-based metric) in [41] totest readability improvement. All these measurements are normalized.Larger values of the measurements suggest higher readability exceptedge-length variation. Results of readability measurements for the Fi-nan512 dataset, the Power-Network dataset and the Price 1000 datasetare given in Figures 1(h,i,j,k), Figures 9(e,f,g,h), Figures 11(e,f,g,h),accordingly. Bars representing measurement values before modifica-tion transfer are in orange and bars after modification transfer are inpurple. Note that, in the case study with the Price 1000 dataset, wealso measure the readability after slight modifications (in light purple).Results show that our approach improves readability in most cases.
We conducted a within-participant experiment in which we asked par-ticipants to fine-tune structures layouts in three modes:
1) Baseline manual : mouse dragging without our approach;
2) Our semi-automatic method with markers specified by user;
3) Our fully automatic method with markers initialized by filter-ing the results of FGMU.
Task.
Participants performed a task involving modifying the struc-ture on the screen according to the expert’s modifications on the exem-plar. Twenty substructures from four real-world datasets are used.
Datasets.
A graph visualization expert helped us define the 20total substructures used in the study. He first chose five exemplarsubstructures from four real-world datasets and then specified threetarget structures for each of the five exemplars (Figure 12). Graphs ser modificationModification Transfer (a) (d)(c)(b) min = 10, k = 10max = 100, ε =0.5 (e)(h)(g)(f)
Figure 9. Case study with Power-Network dataset [54]: (a) an exemplar (in red) and two retrieved substructures (in blue, which are overlapped)overlaid on a network depicted using FM [27]. (b) Two retrieved substructures are discarded. And several target substructures are specifiedmanually.(c) The shape of the exemplar is changed to a circle. Modification transfer alters the node positions of targets to simulate the exemplar’sshape. (d) All modified substructures are merged into the graph by an automatic optimization. (e-h) Readability before (orange) and after (purple)modification transfer measured by four readability criteria (from top to bottom: crosslessness, minimum-angle metric, edge-length variant, andshape-based metric);; error bars depict 95% confidence intervals. (a) (b) (c) α = 0β = 1γ = 100 α = 20β = 1γ = 100 Figure 10. Different weighting schemes for the Power-Network dataset.(a) a target substructure; (b) a low preservation on orientations with α = , β = , and γ = ; (c) a large preservation on orientations with α = , β = , and γ = . generated from four real-world datasets have already been laid out withFM [27]. Substructures are extracted with node positions. The expertwas also asked to modify five exemplars’ layouts to support our task. The Email-Eu-core dataset [51] is a time-varying email contactnetwork in a large European research institution with 986 nodes and332,334 contacts. Email communications within every 24 hours form agraph, yielding a total of 803 snapshots with 855 connected subgraphs.We obtained the first exemplar and its three target structures fromEmail-Eu-core dataset (Figure 12a). The expert modified the exemplarinto a fan-like shape (Figure 12a-1). The Mouse-Brain dataset [18] consists of 986 nodes and 1,536edges. Nodes represent the mouse visual cortical neurons and edges arefiber tracts connecting one neuron to another. We obtained the secondexemplar and its three target structures from the Mouse-Brain dataset(Figure 12b). The expert modified the exemplar into a star-like shapein which the interior node stays in the center and the leaves are placedevenly around the interior (Figure 12b-1). The Euroroad dataset [58] is a road network mostly in Europe.Nodes represent cities and an edge between two nodes denote thatthey are connected. The network consists of 1,174 nodes and 1,417edges. We obtained the third exemplar and its three target structures(Figure 12c) are extracted from the Euroroad dataset [58]. The expertmodified the exemplar into a round circle (Figure 12c-1). The High-School-contact dataset collected from the SocioPatternsinitiative [47] consists of 180 nodes and 45,047 contacts. We createda temporal network following the procedure in [60]. The last twoexemplars and six target structures were obtained from the High-School-contact dataset (Figures 12(d, e)). The expert modified one exemplarinto a shape in which the inner circle is laid out as a regular polygon andthe surrounding nodes are placed orthogonally (Figure 12d-1). And hemodified the other exemplar into an orthogonal layout (Figure 12e-1).We ensured that within the same dataset, the Weisfeiler-Lehmansimilarities between three target substructures and the exemplar aregreater than 0.7. We recorded all modifications made by the expertalong with a list of instructions (see Suppl. Material).
Participants and apparatus.
Twelve volunteers were recruited toparticipate in the study (5 males, 7 females; aging from 23 to 27). All participants were students or researchers concentrating in computerscience. They are familiar with visualization and four of them majorin graph visualization. The study was conducted on a PC providedby us equipped with a mouse, keyboard, and 24-inch display. Theinterface was displayed within a window size of 1920 × α = , β = , γ = , and w = Study Conditions.
We tested the performance of different fine-tuning techniques ( baseline manual , semi-automatic , and fully au-tomatic ) on a small graph layout. Each participant was asked to pro-cess three target structures in all four cases (one from the Mouse-Brain dataset, one from the Euroroad dataset, and two from theHigh-School-contact dataset) with three techniques, yielding 432(12 participants × × × Procedure.
The study has two stages. We first trained participantson the three manipulation modes ( baseline manual , semi-automatic ,and fully automatic ). They viewed a demo video of an expert’s op-erations using data samples extracted from the Email-Eu-core dataset(Figure 12a), and then practiced till they felt comfortable with the tasks.In the formal study, they were then asked to manipulate three targets’shapes to simulate the exemplar for each case using all three techniques(4 cases × × Hypotheses.
We measure performance by participants’ completiontime and number of interactions. We anticipate that the quality ofthe modified exemplar and the targets’ layouts makes little differencebecause participants were asked to fine-tune the target layouts untilthey were satisfied. We formulated three hypotheses: H1 Our fully automatic method is more efficient than the baselinemanual method. H2 Our semi-automatic method is more efficient than the baselinemanual method. H3 There is no difference in performance between our semi-automaticmethod and our fully automatic method.
Results.
Participants spent about 45 minutes on average on the user a) (b) (c)(d)
Manually selected markersUser modificationModification transferUser modifications on our result (e)(h)(g)(f)
Figure 11. The Price 1000 dataset [37]. (a) A radial tree layout. (b) A specified exemplar in which we specify two nodes with the two largest degreesas markers. This is modified into a radial tree layout interactively. (c) Targets and their counterparts after modification transfer. (d) Modified targetsafter several slight user modifications (in red). (e-h) Readability before (orange) and after (purple) modification transfer measured by four readabilitycriteria (from top to bottom: crosslessness, minimum-angle metric, edge-length variant, and shape-based metric); error bars depict 95% confidenceintervals..
Target substructures(should be modified by participants)Exemplar Exemplar(modified by expert) (a)(b)(c)(d)(e) (a-1)(b-1)(c-1)(d-1)(e-1)
Emaile-Eu-coreMouse-BrainEuroroadHigh-School-contact-1High-School-contact-2
Figure 12. Data samples in the user study. Five exemplars and 15 targetsubstructures were extracted from the four datasets. (a) is extracted fromthe Email-Eu-core dataset [51]; (b) is from the Mouse-Brain dataset [18];(c) is from the Euroroad dataset [58]; (d) and (e) are from the High-School-contact dataset [47]. study and got a reward of around $5 on completion. We recorded thenumber of interactions (mouse clicking and dragging) that participantsperformed and completion times to reach a satisfying layout. The com-pletion time includes marker specification, algorithm computation, andlayout modification; and the number of interactions includes markerspecification and layout modification. Figures 13(a, b) summarizes theresults. We analyzed our results using significance tests with signifi-cance levels set to . χ ( ) = . , p < .
05) and thecompletion time ( χ ( ) = . , p < . baseline manual , semi-automatic , and fully au-tomatic ) on two measurements (the number of interactions and thecompletion time). The post-hoc analysis (Figures 13(a, b)) showed thatour semi-automatic method performed most efficiently in both two mea-surements, followed by the baseline method and last our semi-automaticmethod. Thus H1 held while H2 and H3 were rejected. Feedback.
We collected some representative participant feedback.Most of them made comments along the lines of, “
In fully automatic
B S FBSFBSFBSFBSFBSF BSFBSFBSFBSFBSF
Figure 13. User study results. Measurement components are repre-sented as stacked bars. (a) The distribution of the number of interactions;(b) the distribution of completion time; (c) the distribution of number ofinteractions on different cases; (d) the distribution of completion time ondifferent cases. Error bars depict 95% CIs. mode, most results are pretty close to exemplar’s results. I have tomake little effort to modify them, especially in complex cases. But Istill have to verify whether there is room for improvement ”. Many ofthem mentioned that they were encouraged to attempt higher quality bythe high-quality result generated by the fully automatic method. Someof them mentioned that “
It is boring to wait for the fully automaticmethod to calculate the result ”. Another complaint about our methodsis that markers are hard to determine. Most participants had littleexperience in graph visualization. Interestingly, several participantsmentioned that “
The user study is like a game, fine-tuning layoutsmakes me feel relaxed because I generate nice-looking results ”. Oneof them suggested expanding our user study into an online system tocollect more user data.
Discussion.
We split the number of interactions and the completiontime to look for deeper insights. The completion time consists of threeparts: marker specification, algorithm computation, and interactivelayout modification (Figure 13d). The computation time occupies asmall fraction (in green) in both the semi-automatic and fully automaticmethods. The marker specification (in orange) contributes a lot tothe completion time of our semi-automatic method. In most cases,participants spent most time on interactively modifying layouts. Welso calculated average completion time per interaction for the threemethods; participants spent an average of 2.4 second, 2.6 second, and3.3 second on each layout modifying interaction using the baselinemethod, our semi-automatic method, and our fully automatic method,respectively. Participants spent more time thinking about and verifyingresults generated by our fully automatic method. Each interaction formarker specification takes an average of 4.1 seconds. We observe thatalmost all participants tended to choose internal nodes in the star-likestructures (Figure 12b) as markers. However, for structures extractedfrom the High-School-contact dataset, markers were diverse amongparticipants. We report results of specified markers by an expert ongraph analysis in the Suppl. Material. A good pair of markers shouldbe able to assume the same role or status in the source and the target(e.g., cut nodes). This indicates that experience in and knowledge ofgraph analysis are necessary for marker specifications.
ISCUSSION AND L IMITATIONS
In terms of the performance of modification transfer, our algorithm out-performs the baseline method (manual node dragging), as demonstratedin Section 5.3. It reduces or eliminates the laborious interactions. Andin terms of layout editing, our modification transfer algorithm may bemore flexible than rule-based layout approaches [31, 36, 61]. Ratherthan pre-defining a set of rules or metrics, our algorithm supportsarbitrary modifications on the exemplar.
Usability.
Our visualization interface is implemented with a set offundamental interactions, such as lasso, drag, pan, and zoom. The usercan easily explore the entire graph and specify substructures. Comparedto box selection, lasso interaction enables the user to more freely specifya substructure with a closed path. However, for complex graphs, layoutalgorithms can lead to visual clutter. It is hard for the user to specifystructures in a virtual plane, so that selection interactions such as filterand query will be suitable for complex cases.
Scalability.
Our cases show that our approach can handle fine-tuningon large-scale networks. Our interface with a WebGL rendering en-gine supports visualizing large-scale graphs with rich user interactions.Three aspects influence the scalability:
1) The substructure retrieval algorithm has a computational com-plexity of O ( | V s | × N ) , where N denotes the node number of theunderlying graph [4]. However, heuristic user-adjustments of theparameter k (see Section 3.2) may reduce scalability.
2) Modification transfer consists of three parts: graph matching, cor-respondence filtering, and two rounds of layout simulation. Thetime complexity of FGMU [68] for matching S = ( V s , E s ) and T = ( V t , E t ) is O ( k × max ( | V t | , | V s | ) + | E t || E s | )) , where k isthe number of iteration for FGMU. The average time complexity ofcorrespondence filtering is O ( min ( | V t | , | V s | ) ×| E t || E s | / ( | V t || V s | )) .The first round of layout simulation involves several iterations. Thenumber of iterations depends on the number of markers. More mark-ers can lead to less iterations. For each iteration, the deforming stepemploys a procedure similar to the stress-majorization layout [21],whose time complexity is the same as the stress majorization. Thetime complexity of the matching step is dominated by the Hungarianalgorithm, whose complexity is O ( m ) , where m is the number ofnodes selected for matching. The second round of layout simulationruns one time because no more correspondences are built.
3) The global optimization runs as fast as the stress-majorizationlayout, which is sensitive to the number of nodes in the surroundingsto be optimized.
Robustness.
Case studies and user study indicate that our approachcan handle different kinds of datasets and layouts. Our approach is notsensitive to the original layout, because we layout the exemplar andtargets with the same force-directed algorithm before building corre-spondences. Although the user study suggests that our fully automaticmethod works efficiently, we found that participants still performed afew interactions based on results generated by our approach. The rea-son may be that our approach generates similar layouts as the exemplar,not the same layouts; participants must check whether generated resultscan be improved.
Limitations and future work.
This work has several limitations. First, the usability of the marker specification can be improved. Weplan to allow the user to interactively select markers from correspon-dences built by graph-matching algorithms. An algorithm that can ratethe correctness of correspondences can improve its usability. Second,we could also conduct a thorough user evaluation of readability. Wedesigned our method to transfer modifications among structures, andthus the readability of substructure layouts generated by our approachdepends largely on the exemplar’s modifications. Third, the substruc-ture retrieval algorithm detects potentially similar structures using nodeembeddings. Its accuracy depends on the embedding technique.In the future, we plan to perform both lab-based control studies aswell as insight-based studies in real-world settings on our prototypesystem to measure readability [46, 50, 64], to characterise the goals andeffects, user perception, and insights.
ONCLUSION
We designed and evaluated an exemplar-based graph layout fine-tuningapproach that reduces human labor by transferring modifications madeon an exemplar to other substructures. A user interface is developed toenable fine-tuning of graph layouts. A quantitative comparison of twodatasets with ground truth indicates that our approach can reach moreaccurate correspondences. Three case studies show that our approachworks well on different datasets and layouts. A user study showsthat our approach significantly reduces or even eliminates laboriousinteractions. A CKNOWLEDGMENTS
We wish to thank all the anonymous reviewers for their thoroughand constructive comments. We also thank the participants for theirtime and efforts. This work is partially supported by National Natu-ral Science Foundation of China (61772456, 61761136020), NSFC(61761136020), NSFC-Zhejiang Joint Fund for the Integration of In-dustrialization and Informatization (U1609217), and Zhejiang Provin-cial Natural Science Foundation (LR18F020001). J. Chen is partiallysupported by National Science Foundation NSF OAC-1945347, NSFDBI-1260795, NSF IIS-1302755, CNS-1531491, and NIST MSE-70NANB13H181. Any opinions, findings, and conclusions or rec-ommendations expressed in this material are those of the authors anddo not necessarily reflect the views of the National Science Founda-tion of China (NSFC), National Institute of Standards and Technology(NIST) or the National Science Foundation (NSF). R EFERENCES [1] D. Auber. Tulipa huge graph visualization framework. In M. J¨unger andP. Mutzel, editors,
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