Improved Optimal and Approximate Power Graph Compression for Clearer Visualisation of Dense Graphs

Tim Dwyer∗ Christopher Mears† Kerri Morgan‡ Todd Niven§ Kim Marriott¶ Mark Wallacek

Monash University, Australia

(a) Flat graph with 30 nodes and 65 (b) Power Graph rendering computed using the (c) Power Graph computed using Beam Search (see 5.1) finds 15 links. heuristic of Royer et al. [12]: 7 modules, 36 links. modules to reduce the link count to 25.

Figure 1: Three renderings of a network of dependencies between methods, properties and fields in a software system. In the Power Graph renderings an edge between a node and a module implies the node is connected to every member of the module. An edge between two modules implies a bipartite clique. In this way the Power Graph shows the precise connectivity of the directed graph but with much less clutter.

ABSTRACT visualise in a way that individual links can still be followed. Such Drawings of highly connected (dense) graphs can be very difficult graphs occur frequently in nature as power-law or small-world net- to read. Power Graph Analysis offers an alternate way to draw a works. In practice, very dense graphs are often visualised in a way graph in which sets of nodes with common neighbours are shown that focuses less on high-fidelity readability of edges and more on grouped into modules. An edge connected to the module then im- highlighting highly-connected nodes or clusters of nodes through plies a connection to each member of the module. Thus, the entire techniques such as force-directed layout or abstraction determined graph may be represented with much less clutter and without loss by community detection [8]. Dense edge clutter may be allevi- of detail. A recent experimental study has shown that such lossless ated to better show node labels by rendering the edges very faintly compression of dense graphs makes it easier to follow paths. How- or with aggregate techniques like bundling [6]. Though such ap- ever, computing optimal power graphs is difficult. In this paper, we proaches may give a rough indication of the general graph structure show that computing the optimal power-graph with only one mod- they make following precise edge paths difficult or impossible. ule is NP-hard and therefore likely NP-hard in the general case. We Such path following is even more difficult when graphs are di- give an ILP model for power graph computation and discuss why rected. Distinguishing direction on edges allows for up to twice as ILP and CP techniques are poorly suited to the problem. Instead, we many distinct edges in the graph. For people trying to understand are able to find optimal solutions much more quickly using a cus- the graphs to accurately follow directed paths, edge curves must be tom search method. We also show how to restrict this type of search drawn in enough isolation that any indications of direction (such as arXiv:1311.6996v1 [cs.CG] 14 Nov 2013 to allow only limited back-tracking to provide a heuristic that has tapering, arrowheads or gradients [5]) are clearly visible. better speed and better results than previously known heuristics. Recently, alternative approaches have been suggested that at- tempt to retain the fidelity of individual edge paths by introducing 1 INTRODUCTION drawing conventions that allow a large number of actual edges to be In real-world applications such as biology and software engineer- precisely implied by a small set of composite edges. In particular, ing it is common to find network structures that are too dense to so-called Power Graph Analysis constructs a hierarchy over nodes, such that nodes with similar neighbour sets are placed in the same ∗e-mail: [email protected] group or module. An edge connected to the module then implies a †e-mail:[email protected] connection to each member of the module. Power Graph Analysis ‡e-mail:[email protected] utilises lossless compression since—by contrast with bundling or §e-mail:[email protected] community-based clustering—no information is lost in the render- ¶e-mail:[email protected] ing. That is, the technique can be said to be information faithful [9] ke-mail:[email protected] such that the full graph can be reconstructed by careful inspection of the drawing. Figure 1 gives a small example of the application of Power Graph Analysis techniques to a small software dependency graph. Power Graph Analysis has been shown to have practical applica- tion to visualising biological networks [12], detecting communities in social and biological networks [14] and more recently in software dependency diagrams [3]. A recent user study has shown that—for path-finding tasks—power-graph style groupings were more read- able than flat graphs, even for people with very little training [3]. Relatively little attention has been devoted to algorithms for find- ing power graph decompositions, i.e. the best choice of modules in the power graph. Royer et al. [12] give a heuristic for finding the decomposition and Dwyer et al. [3] give a constraint programming formulation for finding the optimal decomposition with respect to various criteria, such as fewer edges or fewer edges crossing group boundaries. Unfortunately, an empirical evaluation of the two al- gorithms given in [3] reveals that the heuristic of [12] is not very effective at finding an optimal solution while the constraint pro- (a) A small graph with 23 edges. gramming approach is too slow for practical use, taking days to run for larger graphs. Thus, we need better methods to find power graph decompositions. This is the subject of this paper. In particular, we focus on finding power-graphs that are optimal or approximately optimal with respect to the number of edges in the decomposition. In Section 3 we prove that the problem of minimising the number of edges in a power graph decomposition with only one group is NP-hard. This is strong evidence that the general problem is NP- hard, though specific proofs for unrestricted modules and different goal criteria are needed. In Section 4 we look at declarative models of the Power Graph (b) A power graph decomposition of the same graph problem for input into general-purpose solvers. In particular, we can be drawn with only 7 edges. offer some refinements to the Constraint Programming model in- troduced in [3] (Section 4.1) and propose a new Integer Linear Pro- Figure 2: A small example of power-graph decomposition. gramming model (Section 4.2). In Section 5 we explore explicit search methods. In Section 5.1 we introduce a beam-search based approximate method that pro- fusion when reading the diagram, as considered by Dwyer et al. [3]. duces power graphs for a given flat graph, that are much closer to Royer et al. introduced a simple heuristic for obtaining the power optimal than previous greedy heuristics. In Section 5.2 we extend graph decomposition for a given graph in O(|V|3|E|log|E|) time. this method to a full-backtracking search strategy that is able to find Their procedure involved first computing a “candidate module hier- optimal power graphs relatively efficiently using a lower-bound cal- archy” over the nodes in the graph by repeated greedy matching of culation to cut branches of the search that are not useful. Our exper- nodes or modules with similar Jaccard index between their neigh- iments (Section 6) show that this approach is orders of magnitude bour sets. A second iterative phase involves greedily instantiating faster than solving the declarative models using generic solvers. the module that reduces the most edges. This is repeated until no more edge reduction is possible. 2 BACKGROUNDAND DEFINITIONS This algorithm is reasonably fast but—as we show in Section Royer et al. [12] coined the term Power Graph Analysis to describe 6—produces results that are far from optimal. The Beam Search a technique for reducing edge clutter by introducing a module hier- method described in Section 5.1 has faster run-time (depending on archy over an undirected graph, such that edges connected to mod- a beam size parameter) and finds solutions that are very close to ules imply a connection to every child of the module. More for- optimal. Furthermore, in Section 5.2, we show that it is easy to mally, for a directed graph G = (V,E) a power graph configuration introduce backtracking into this explicit search method, to produce is a set of modules M where each module m ∈ M is a subset of V. a search that returns an optimal solution. Such a complete search We assume that M includes the set of trivial modules {{v} | v ∈ V}. is orders of magnitude faster than the best we have been able to Our use of the term module is due to the similarity with modules achieve with generic solvers applied to declarative models, despite considered in Modular Decomposition. Like Modular Decomposi- extensive experimentation with redundant constraints to limit the tion, modules can be nested to form a hierarchy and if two mod- search space in those models, as described in Section 4. ules overlap one must be fully contained in the other. That is, for m,n ∈ M, if mTn 6= /0then n ⊂ m or m ⊂ n. As in Modular De- 3 COMPLEXITY ANALYSIS composition, a power graph decomposition is drawn with a set of In this section we show that computing the optimal power-graph representative edges R uniquely specified by E and M, such that a with a single module is NP-complete for undirected bipartite representative edge between two modules represents a complete bi- graphs. It then follows that the result holds for directed graphs partite graph in E over the children of the two modules. That is, for in general. As an optimal module for a single module case need two modules m and n, a representative edge (m,n) ∈ R represents not be a module in the optimal power-graph with multiple modules the set of edges rep(m,n) = {(u,v) | u ∈ m,v ∈ n}. In drawings of (see Figure 3), it seems likely that it is hard in the general case power graphs, the set of representative edges R must be minimal, too. Finding an optimal single module is equivalent to finding a i.e. @e1,e2 ∈ R s.t. rep(e1) ⊂ rep(e2). biclique subgraph (A,B) (here A and B denote the two independent Unlike Modular Decomposition, however, the power graph def- sets of nodes) that maximises the edge savings, that is, |A||B| − |A|, inition above allows a representative edge to span module bound- |A| ≤ |B|. The largest part, B, is taken to be the single module. aries. For example, in Figure 2(b), the edge from node 5 to module This problem is different to the Maximum Vertex Biclique Problem {0,2,4} implies that E contains edges (5,0),(5,2),(5,4) but says (MVBP) that finds an induced biclique that maximises the number nothing about any modules containing node 5. This permits greater of nodes, |A| + |B|, and to the Maximum Edge Biclique Problem compression than modular decomposition, but can also cause con- (MBP) that finds a biclique subgraph that maximises the number of Case 1 |A| > k: Suppose |A| = k + c, c ∈ (0,k]. Then  k(k − 1) (k − c)(k − c − 1)  |A||B| − |A| ≤ (k + c) + − (k + c) 2 2 k   c   = k3 − k2 − k − c2 + c + c2 + c − 2 Figure 3: The optimal solution for a single module (shaded) is not in 2 2 the optimal solution for two modules (dashed lines). < k3 − k2 − k. Case 2 |A| ≤ k: Suppose |A| = k − c, c ∈ [0,k]. Now as G has no clique of size ≥ k, the number of edges in the subgraph of 0 (k−2)(k+c)2 G induced by V − A (and hence |B |) is ≤ 2(k−1) by Turan’s´ theorem [13]. So  k(k − 1) (k − 2)(k + c)2  |A||B| − |A| ≤(k − c) + − (k − c) 2 2(k − 1) ≤k3 − k2 − k

1  2 2 2 2  Figure 4: The construction G0. − (c + 1)k − 2c k + c(c − 1)(k − 2) 2(k − 1) tree structure. Here we highlight several nodes in the very linear program that minimizes the number of edges. A more de- large search space of module configurations for the graph shown in tailed description of the model can be found in the Appendix. Figure 2. A solid arrow indicates a direct child relationship in the Input and Parameters search tree. That is, the child configuration is obtained from a single merge operation from the parent. A dotted arrow indicates the target n is the number of vertices of the input graph G. configuration is reachable from the source via multiple merges. V = {0,1,...,n − 1} represents the vertices of the input graph G. e(u,v) represents the edges of the input graph G as an incidence matrix. That is, e(u,v) = 1 if (u,v) is an edge of G and e(u,v) = 0 1. dis[m ,m ] + sub[m ,m ] + sub[m ,m ] = 1, m 6= m ∈ M. otherwise. 1 2 1 2 2 1 1 2 m is the number of modules with at least two elements (we consider 2. ∑{sVer[v1,v2,m1,m2] | m1,m2 ∈ M,m1 6= m2} ≤ 1, each singleton vertex to belong to its own module). v1,v2 ∈ V. M = {0,1,...,n + m − 1} represents the set of all modules. 3. sav[m1,m2] ≤ ∑{sVer[v1,v2,m1,m2] | v1,v2 ∈ V,v1 6= v2}, Integer decision variables m1 6= m2 ∈ M. sav[m1,m2] the number of edges that may be removed from G (to 4. sMod[m1,m2] ≤ sav[m1,m2], m1 6= m2 ∈ M. then be replaced by a single edge) if the modules m and m are 1 2 5. sav[m ,m ] ≤ n2sMod[m ,m ], m 6= m ∈ M. added. 1 2 1 2 1 2 Binary decision variables 6. ∑ mod[v,m1] = 1, m1 ∈ V. v∈V mod[v,m] takes the value 1 if and only if vertex v belongs to the 7. mod[m ,m ] = 1, m ∈ V. module m. 1 1 1 ind[v,m] takes the value 1 if and only if (v,u) is an edge, for all u in Constraint 1 states that for any two distinct modules m1 and m2, the module m. either m1 and m2 are disjoint, or one is a proper subset of the other. bic[m1,m2] takes the value 1 if and only if, for every vertex v ∈ m1 Constraint 2 says that no edge can be counted twice in the saving and every vertex u ∈ m2, the pair (u,v) is an edge in G. calculation. Constraint 3 defines the variables sav[m1,m2]. Con- dis[m1,m2] takes the value 1 if and only if the modules m1 and m2 straints 4 and 5 defines sMod[m1,m2]. Constraints 6 and 7 force are disjoint sets of vertices. each vertex to be a singleton module. sub[m1,m2] takes the value 1 if and only if the module m1 is a proper subset of the module m2. 5 EXPLICIT SEARCH METHODS mInd[v,m1,m2] takes value 1 if and only if v ∈ m1 and ind[v,m2] = As defined in Section 2, for a given flat graph, a configuration is a 1. set of modules, where each module is a set of vertices of the graph vMod[v,m1,m2] takes value 1 if and only if v ∈ m1 and v ∈ m2. and other modules such that the modules form a hierarchy. The sVer[v1,v2,m1,m2] takes value 1 if and only if (v1,v2) is an edge objective is to find a configuration that minimises the number of with v1 ∈ m1 and v2 ∈ m2 and the edge (v1,v2) can be removed if edges. m1 and m2 are added. A configuration can be constructed by adding one module at a sMod[m1,m2] takes value 1 if and only if sav[m1,m2] > 0. time. The order in which the modules are added has no effect, so Objective we impose a constraint that only top-level modules are added – that Maximise is, if the configuration is {m1,m2,...,mn}, then a new module must not be a subset of any mi. ∑{sav[m1,m2] − sMod[m1,m2] | m1,m2 ∈ M, m1 6= m2}. The set of possible module configurations is then a tree, with the configuration containing only trivial modules (the flat graph) at the Constraints root, as illustrated in Figure 5. Each node in the tree has one child The majority of the constraints in our ILP model are purely to cor- for each module that can be added to that configuration. rectly define binary decision variables and so we have omitted them To prevent multiple paths to the same configuration, we impose for brevity. an arbitrary ordering on the children of a node, and force that if m1 and m2 are two sibling child modules such that m1 appears to the Data: A starting configuration of modules C0 with left of m2, then the module m1 may not appear in any configuration representative edges R0. A beam size k ≥ 1. ? ? in the subtree of m2. Result: An improved configuration C with |R | ≤ |R0| Thus, this tree will contain every possible configuration of mod- beam ← priority queue of configurations ules, and a search for the best configuration may simply do a full such that first element in queue has maximum |R|; traversal of the tree; however, this is impractically slow. In the re- push(beam,C ); mainder of this section we describe how to improve the efficiency 0 S ← hashset of signatures of configurations in beam; of this search. repeat 5.1 Beam Search (improved,currentbeam) ← (false, copy beam); foreach C ∈ currentbeam do Using the above definitions, we can define greedy heuristics to ob- merges ← {(e,m,n) | tain a power-graph configuration with significantly fewer represen- e ← nedges(m,n),(m,n) ∈ C ×C}; tative edges than are required to draw the flat graph. kbest ← from merges take k triples (e,m,n) To limit the branching at each search-tree node t associated with with smallest e where sig(C,(m,n)) 6∈ S; a configuration C, we restrict children of t to be those obtainable foreach (e,m,n) ∈ kbest do through a merge of two top-level modules in C. That is, a merge of b ← first (i.e. worst) configuration in beam; 0 modules m and n creates a new configuration C = C ∪ {{v} | v ∈ if |beam| < k or |R(b)| > e then m ∪ n}. C0 ← merge(m,n); A simple best-first search is—for a given starting configuration push(beam,C0); C with nt top-level modules—to try all possible merges to give insert(S,sig(C0)); {C0 ,...,C0 } children and then simply take the one with the 1 nt (nt −1) improved ← true; fewest representative edges to be the next configuration. We repeat end until no further improvement is possible. if |beam| > k then Since the set of representative edges R used to draw the power pop(beam); graph must be minimal, such a merge operation may leave m or end n with no associated representative edges, i.e. rep(m, ) = /0 or end rep( ,n) = R /0.Since a module with no associated edge in serves end no purpose, we remove such modules from the merge result C0. until not improved; In practice, a full merge is not required just to calculate the re- return configuration in beam with smallest |R| duction in edges achieved by the merge. Consider a merge of two modules m and n in C with representative edges R(C) giving a con- Algorithm 1: Beam Search figuration C0 with edges R(C0). If the outgoing and incoming neigh- bour sets in R of m are given by N+(m) and N−(m), then the num- ber of edges in R(C0) will be precisely: edge between existing module m and new module n replaces all nedges(m,n) = |R(C)| − |N+(m) ∩ N+(n)| − |N−(m) ∩ N−(n)| previous edges between members of m and members of n. Since there must be at least one such previous edge, the number of edges Applied to a graph G = (V,E), O(|V|) such greedy merges are removed is at least the number of edges added. possible before no more improvement is possible. Since a given Every optimal configuration can be reached by a sequence of module may include all edges in its neighbour set, computing improving module additions. That is, every module added reduces nedges(m,n) can be O(|E|log|E|) and to find the best possible the number of edges. merge this must be computed for all pairs of modules. Therefore, An outline of the reasoning is as follows. Let the desired optimal 3 a na¨ıve implementation could take O(|V| |E|log|E|) time though configuration be C = {m1,m2,...,mn}. Starting from the empty in practice the number of modules that must be considered and the configuration, choose any module that contains no sub-modules. size of their neighbour sets diminishes quickly. Adding this module will reduce the number of edges, so add it to Beam search [10, p. 195] gives us a way to introduce some lim- the current configuration. This module must reduce the number of ited backtracking into this best-first strategy. Instead of only con- edges – if it didn’t, then C wouldn’t be optimal. Then after adding sidering the single best merge at each iteration, we maintain a beam this module, find another module that contains no sub-modules (or of the k best solutions found so far. The full process is shown in only sub-modules that have already been added) and add that mod- Algorithm 1. ule, and so on. One detail in this algorithm is the check before insert of a config- An optimal solution has at most n − 2 modules, where n is the uration into the beam that we have not already considered a config- number of vertices in the flat graph. A configuration can have at uration that is structurally identical. This is done by maintaining a most n − 1 modules. A configuration with n − 1 modules must in- hashset S with signatures of configurations previously inserted into clude a module that contains all vertices in the graph. However, the beam. For a configuration C the signature sig(C) is obtained by such a module is not an improving module and so cannot appear in a canonical (ordered) traversal of the module hierarchy. an optimal solution. Therefore, an optimal configuration can have at most n − 2 modules. 5.2 Optimal Search If adding module m to configuration C would remove e edges, We can show certain properties of the configuration tree that will then adding module m to configuration C0 ⊃ C would remove at help us to eliminate regions of the tree without missing any optimal most e edges. configurations. First, we restrict the definition of “optimal” to be a Adding a module m with members {a,b,...} to configu- configuration that has no redundant modules. That is, if two con- ration C can only remove edges by replacing existing edges figurations C and C ∪ {m} have the same objective cost, only C can {(a,v),(b,v),...} with a single edge (m,v) (and similar for be considered optimal. opposite-oriented edges). A configuration C0 ⊃ C has the same set Adding a module cannot increase the number of edges. Any new of edges as C, except that some edges are replaced by a smaller set edge added to the graph must replace some existing edges. A new of edges. Therefore, the module m can only replace the same set of edges {(a,v),(b,v),...} or a smaller set where some of those edges just that pair will reduce the number of edges. (Any modules that have been merged. become redundant during this construction can be removed.) These properties allow us to calculate a lower bound on the best The consequence of this property is that in the search algorithm, possible configuration in under a given configuration in the search the modules to be added at each step need only be these binary tree. Given a configuration C with m modules, we can compute the modules instead of all possible modules. This greatly reduces the minimum number of edges for any configuration that is a superset search space. of C. Let c ,c ,...,c be the modules permitted to be added to con- 1 2 n 6 EXPERIMENTS figuration C. Each of these can be given a score S(ci) that is the number of edges removed by adding that module: S(ci) = 6.1 Heuristic Methods E(C) − E(C ∪ {ci}). Since we can add at most n − 2 − m modules Following Dwyer et al. [3], we generate random scale-free directed to C, the most edges we can possibly remove from C is the sum of graphs with various numbers of nodes using the model of Bollobas´ the best n − 2 − m scores of the modules c1,c2,...,cn. et al. [1]. To generate a graph with |V| nodes we control for density Therefore, for a given configuration C we can calculate the score 3 3 2 of a hypothetical configuration C0 ⊃ C that has the minimum possi- such that the number of edges is roughly proportional to 2 |V| . For ble edges. (The configuration is only hypothetical because it likely example, a graph with 10 nodes will have around 50 edges, while a violates the hierarchy restriction on the modules.) If the objective graph with |V| = 100 will have around 1500 edges. cost of C0 is no better than the best-known solution, the search can All heuristics were implemented in C# and run on a modern 2 backtrack immediately from C. An outline of the exhaustive search tablet PC . The source-code is available under an open-source li- 3 algorithm is given in Algorithm 2. cense . Figure 7 compares the decompositions obtained by applying Data: Current configuration, incumbent configuration, flat our various heuristic powergraph decomposition methods to graphs graph. generated as described above. It is clear that the Greedy Jaccard Result: The optimal configuration. Clustering method [12] (JC) is easily beaten in terms of quality of the final solution, even by beam search with a beam size k of 1 (i.e. if current configuration is better than incumbent then best-first search with no backtracking at all). Increasing k to 10 record current configuration as incumbent does improve the results slightly even—in some cases—returning end the optimal (e.g. see Figure 6(d)). However, this is achieved at a bound ← compute bound of current configuration; ten-fold increase in running time (see Figure 8). Increasing k fur- if bound is not better than incumbent then ther to 100 only occasionally results in a slightly better solution. backtrack; Note that with k small compared to the total number of configura- end tions in the search space there is no guarantee that beam search will if current configuration below module limit then return the optimal. modules ← modules that can be added to current Figures 6(b), (c) and (d) show the renderings of the results ob- configuration; tained for a small instance with 10 nodes and 51 edges. The quality Sort modules with most edge-reducing modules first; of the renderings in terms of visual clutter, does seem to reflect the for each module m ∈ modules do numbers seen in Figure 7. There is a fairly marked visual improve- Recurse with m added to current configuration; ment between JC (Figure 6(b)) and Beam Search with k=1 (BS1) Upon backtracking, forbid m from modules; (Figure 6(c)), in particular BS1 can be rendered without edge-edge end intersections while for the JC decomposition it is impossible. How- end ever, there is a much less obvious improvement from BS1 to BS10 return incumbent configuration (Figure 6(d)): the latter has three fewer edges and four fewer edge- Algorithm 2: Outline of optimal search. module boundary crossings. The result found by BS10 is, in fact, the optimal as confirmed by running a full search. In terms of running time, Beam Search with k = 1 is easily the We can further reduce the number of modules considered at each fastest method, as seen in Figure 8. Generally, the running time step of the search. We only need to consider binary modules— (and also memory requirements) for beam search grow linearly in as produced by the merge operation for the beam search (Sec- k. For example, BS10 has very close to twice the run time of BS5. tion 5.1)— which have exactly two members (which themselves The exception is that the running time of BS1 is significantly less may be either modules or single vertices). The search is still guar- than 10% that of BS10 since we are able to apply all the merges anteed to find an optimal configuration. entirely in place and no copying of configurations is required nor Any optimal configuration has a counterpart that has only bi- checks for structural equivalence of configurations. nary modules, and each such binary module reduces the number of edges. Consider an optimal configuration C, constructed by the 6.2 Optimal Methods sequence of modules < m1,m2,...,mn >. Let mi be a non-binary module. We can convert mi into a binary module by grouping to- We have compared the efficiency of the exhaustive search methods gether two of its members arbitrarily. (This newly created sub- across a corpus of 10-vertex graphs. The running times on a single module will have no edges: if it was beneficial for this new sub- representative graph are given below. module to have edges, C would not have been optimal.) The new sub-module is inserted immediately before m in the construction ILP model > 160 CPU hours i CP model without redundant constraints > 72 hours sequence. If mi is still not binary, recurse. All new sub-modules created in this way reduce the number of CP model with redundant constraints 26.6 hours edges during construction. We know that adding the non-binary Optimal Search 8 minutes Optimal Search (edge-minimization only) 22 seconds module mi reduces the number of edges. From this we know that the members of mi share at least one common predecessor or com- 2 mon successor. Therefore, any pair of the members of mi share a Intel Ivy Bridge Core i7, up to 2.6GHz common predecessor or successor, and adding a module containing 3http://dgmlposterview.codeplex.com (a) Flat graph with |V| = 10,|E| = 51. An orthogonal layout style was (b) Greedy Jaccard Metric decomposition. Decomposition has 5 mod- used (Y-Files implementation of topology-shape metrics approach) in an ules, 23 edges and 21 crossings between edges and module boundaries. attempt obtain a layout and routing in which each individual edge path is traceable. Such traceability is necessary to support discerning reachabil- ity or path-finding tasks.

(c) Beamsearch decomposition k = 1. Decomposition has 6 modules, 19 edges (d) Optimal decomposition obtained with beam search k = 10 (optimality and 17 crossings between edges and module boundaries. proven with method described in Section 5.2). Decomposition has 6 modules, 16 edges and 13 crossings between edges and module boundaries.

Figure 6: Flat and power graph renderings of a graph with 10 nodes and 51 edges. It is interesting to note that the reduction in clutter from (b) to (c) is fairly obvious, however, the qualitative improvement from (c) to (d) is less dramatic. Layouts were obtained using the yEd software (http://yfiles.com) using a combination of automatic layout and manual refinement with automatic edge routing. emszsadGed acr lseigheuristic. various Clustering with Jaccard Search Greedy Beam and for sizes times beam running of Comparison 8: Figure Run Time (seconds) power various the for size heuristics. graph decomposition versus graph cost of Comparison 7: Figure Edges in Decomposition 1000 1200 0 1 2 3 4 5 6 7 200 400 600 800 0 20 10 30 20 40 30 Graph Size (numbernodes) of 40 50 Node Count Node 50 60 60 70 70 80 80 90 90 100 100 Clustering Greedy Jaccard Beamsearch k=100 Beamsearch k=10 Beamsearch k=1 Clustering Greedy Jaccard k=10 Search Beam k=5Beam Search k=1 Search Beam ehd htcnd oehn esnbewe lnrdrawing planar a when developing reasonable on something focus do of little can computability been that the has methods There on focus graphs. confluent to graph planar tended confluent have in- Current algorithms also drawing this components. decomposition bipartite power-graph identifying Like volves [2]. drawing graph the variants. for its required and is problem graph analysis power complexity general more above mentioned As as (such problem time. sepa- polynomial the require be even of may may functions) variations proof, or proofs goal that NP-hardness rate or likely general constraints different also a slightly case) is for module it ground-work one though (the strong problem is analysis which power-graph a for proof decompositions. optimal search Jaccard to the beam comparison bad by the how was just of method discovering clustering by development motivated as example, part large such For in methods was heuristic search. proven better beam has develop instances the us small able be helping for being least in not but at invaluable in may them visualisation, optimality compute solutions graph efficiently prove optimal power to and Truly practical find for to days. necessary able to is compared ex- that an minutes provided method have search we contrast, plicit simple, By defies im- models. still declarative many problem math- efficient been this such have for techniques technologies there programming solver improve- While ematical and significant compiler model. the and to CP provements model previous a ILP to known ments first the contributed 100 the for again memory. (e.g. and results time optimal tech- search more graph for beam still node the 1078 obtain of to applicability to the compared nique shown above edges also graph have 624 the We with for JC. e.g. decomposition optimal, a to closer achieves much are that tions graph dynamic data. live-streaming of decomposition continuous scenar- like new ios enable to improvement performance enough significant es rp ih10ndsad10 de a edcmoe in decomposed be can edges with a 1500 seconds and example, 0.35 nodes For 100 (JC). with [12] graph Clustering dense Jaccard Greedy the heuristic, rtcl ntepatclsd,tebs-rtsac ehd(rbeam (or method search best-first the search the- side, and practical practical the both On results, of oretical. number a presented has paper This optimal. found be methods to three it other proved The and solution solution. optimal optimal solverthe the generation find lazy-clause to the failed with run was CPX model Gurobi commercial CP the the 20 with for run and running was cores model 8 ILP with The parallel, in hours. run was solver ILP The CPU. crossings. more boundary do is compute must variant to and tie-breaking step space search The search each at larger model. work a CP explores the it as because breaking way slower other same the the and in edges, ties minimising the only of one variants two search: implemented For optimal we [3]. model, in CP as the modules number with of the comparison number minimising and by crossings ties boundary break module to of try model CP the of variants F 8 C 7 eae ehiu o ipiyn es rpsi confluent is graphs dense simplifying for technique related A NP-hardness known first the give we side theoretical the On have we problem, decomposition graph power optimal the For decomposi- computes also it that shown have experiments Our l xasiemtoswr u na ne oei .7GHz 2.67 i7 Core Intel an on run were methods exhaustive All both while edges, minimize to tries only model ILP the that Note 5 4 http://www.opturion.com/cpx.html http://www.gurobi.com/ 5 RHRWORK URTHER h L oe n Pmdlwtotrdnatconstraints redundant without model CP and model ILP The . ONCLUSION k = or 1 BS 10 BS nsa62eg eopsto)i reasonable in decomposition) edge 612 a finds 1 BS smc atrta h rvosbs available best previous the than faster much is ) 1 oprdt .4scnsfrJ.Ti sa is This JC. for seconds 2.74 to compared 4 solver, BS 1 is not possible. We think an optimisation-based approach related to the techniques described in this paper may have some success in this regard. Finally, we need better layout methods for power-graph decom- positions and clustered graphs generally. The examples in this pa- per were initially arranged with the Y-Files organic layout method but required significant manipulation to achieve pleasing alignment and to minimise crossings between edges.

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AILP MODEL 12. ∑ vMod[v,m1,m2] ≥ 1 − dis[m1,m2], m1 6= m2 ∈ M. v∈V The following is a complete description of the ILP model dis[m ,m ] = dis[m ,m ] m 6= m ∈ M used to minimise the number of power edges (or equivalently 13. 1 2 2 1 , 1 2 . to maximise the number of edges saved by adding modules). 14. ∑ (vMod[v,m1,m2] − mod[v,m1]) ≥ n(sub[m1,m2] − 1), Input and Parameters v∈V m1 6= m2 ∈ M. n is the number of vertices of the input graph G. V = {0,1,...,n − 1} represents the vertices of the input graph G. 15. ∑ (vMod[v,m1,m2] − mod[v,m1]) ≤ sub[m1,m2] − 1, m1 6= v∈V e(u,v) G represents the edges of the input graph as an incidence m2 ∈ M. matrix. That is, e(u,v) = 1 if (u,v) is an edge of G and e(u,v) = 0 16. ∑ vMod[v,m1,m2] ≤ ( ∑ mod[v,m2]) − sub[m1,m2], m1 6= otherwise. v∈V v∈V m is the number of modules with at least two elements (we consider m2 ∈ M. each singleton vertex to belong to its own module). M = {0,1,...,n + m − 1} represents the set of all modules. 17. dis[m1,m2] + sub[m1,m2] + sub[m2,m1] = 1, m1 6= m2 ∈ M. Integer decision variables 18. sVer[v1,v2,m1,m2] ≤ e(v1,v2), v1,v2 ∈ V, m1,m2 ∈ M. sav[m1,m2] the number of edges that may be removed from G (to 19. sVer[v1,v2,m1,m2] ≤ mod[v1,m1], v1,v2 ∈ V, m1,m2 ∈ M. then be replaced by a single edge) if the modules m and m are ! 2 20. sVer[v1,v2,m1,m2] ≤ mod[v2,m2], v1,v2 ∈ V, m1,m2 ∈ M. added. Binary decision variables 21. sVer[v1,v2,m1,m2] ≤ bic[m1,m2], v1,v2 ∈ V, m1,m2 ∈ M. mod[v,m] takes the value 1 if and only if vertex v belongs to the 22. ∑{sVer[v1,v2,m1,m2] | m1,m2 ∈ M,m1 6= m2} ≤ 1, v1,v2 ∈ module m. V. ind[v,m] takes the value 1 if and only if (v,u) is an edge, for all u in 23. sav[m1,m2] ≤ ∑{sVer[v1,v2,m1,m2] | v1,v2 ∈ V,v1 6= v2}, the module m. m1 6= m2 ∈ M. bic[m1,m2] takes the value 1 if and only if, for every vertex v ∈ m1 24. sMod[m1,m2] ≤ sav[m1,m2], m1 6= m2 ∈ M. and every vertex u ∈ m2, the pair (u,v) is an edge in G. 2 dis[m1,m2] takes the value 1 if and only if the modules m1 and m2 25. sav[m1,m2] ≤ n sMod[m1,m2], m1 6= m2 ∈ M. are disjoint sets of vertices. 26. ∑ mod[v,m1] = 1, m1 ∈ V. sub[m1,m2] takes the value 1 if and only if the module m1 is a v∈V proper subset of the module m2. 27. mod[m1,m1] = 1, m1 ∈ V. mInd[v,m1,m2] takes value 1 if and only if v ∈ m1 and ind[v,m2] = 1. Constraints 1 and 2 define the variables ind[v,m]. Constraints 3 vMod[v,m1,m2] takes value 1 if and only if v ∈ m1 and v ∈ m2. – 5 define mInd[v,m1,m2]. Constraints 6 and 7 define bic[m1,m2]. sVer[v1,v2,m1,m2] takes value 1 if and only if (v1,v2) is an edge Constraints 8 – 10 define vMod[v,m1,m2]. Constraints 11 – 13 de- with v1 ∈ m1 and v2 ∈ m2 and the edge (v1,v2) can be removed if fine dis[m1,m2]. Constraints 14 –16 define sub[m1,m2]. Constraint m1 and m2 are added. 17 says that for any two distinct modules m1 and m2, either m1 and sMod[m1,m2] takes value 1 if and only if sav[m1,m2] > 0. m2 are disjoint, or one is a proper subset of the other. Constraint 18 Objective – 21 defines the variables sVer[v1,v2,m1,m2]. Constraint 22 says Maximise that no edge can be counted twice in the saving calculation. Con- straint 23 defines the variables sav[m1,m2]. Constraints 24 and 25 ∑{sav[m1,m2] − sMod[m1,m2] | m1,m2 ∈ M, m1 6= m2}. defines sMod[m1,m2]. Constraints 26 and 27 force each vertex to be a singleton module. Constraints

1. ∑ (e(v,u)−1)mod[u,m1] ≥ n(ind[v,m]−1), v ∈ V, m1 ∈ M. u∈V

2. ∑ (e(v,u) − 1)mod[u,m1] ≤ ind[v,m1] − 1, v ∈ V, m1 ∈ M. u∈V

3. mInd[v,m1,m2] ≤ mod[v,m1], v ∈ V, m1 6= m2 ∈ M.

4. mInd[v,m1,m2] ≤ ind[v,m1], v ∈ V, m1 6= m2 ∈ M.

5. mInd[v,m1,m2] ≥ mod[v,m1] + ind[v,m1] − 1, v ∈ V, m1 6= m2 ∈ M.

6. ∑ (mInd[v,m1,m2]−mod[v,m1]) ≥ n(bic[m1,m2]−1), m1 6= v∈V m2 ∈ M.

7. ∑ (mInd[v,m1,m2] − mod[v,m1]) ≤ bic[m1,m2] − 1, m1 6= v∈V m2 ∈ M.

8. vMod[v,m1,m2] ≤ mod[v,m1], v ∈ V, m1 6= m2 ∈ M.

9. vMod[v,m1,m2] ≤ mod[v,m2], v ∈ V, m1 6= m2 ∈ M.