This paper describes a hybrid Genetic Algorithm (GA) that is used to improve the layout of a graph according to a number of aesthetic criteria. The GA incorporates spatial and topological information by operating directly with a graph based representation. Initial results show this to be a promising technique for positioning graph nodes on a surface and may form the basis of a more general approach for problems involving multi-criteria spatial optimisation

Hobbs, M.H.W.

Rodgers, Peter

English

Kent Academic Repository

1Representing Space: A Hybrid Genetic Algorithm for Aesthetic GraphLayoutM.H.W. Hobbs and P.J. RodgersUniversity of Kent at Canterbury, UK.email: M.H.W.Hobbs@ukc.ac.uk, P.J.Rodgers@ukc.ac.ukAbstractThis paper describes a hybrid Genetic Algorithm (GA) that is used to improve the layout of a graphaccording to a number of aesthetic criteria. The GA incorporates spatial and topological information byoperating directly with a graph based representation. Initial results show this to be a promising techniquefor positioning graph nodes on a surface and may form the basis of a more general approach for problemsinvolving multi-criteria spatial optimisation.1. Introduction and BackgroundMany spatial problems have a common need tolocate interacting objects on a surface.Applications as diverse as geographical dataanalysis and molecular modelling deal withlocation as a special feature within a vector ofattributes that describe a problem. Generic GAshave been used in applications such as electronicengineering design problems [6]; molecularconformational analysis [13,14,]; network andgraph optimisation [15,17]. These systems dealwith a variety of graph based representations buttypically use a one dimensional encoding whichmeans that either spatial information is lost, orextra, problem specific constraints have to beintroduced.Hybrid GAs which model the specific spatial needsof a problem have been developed [4,10], andother evolutionary techniques, using richerrepresentation structures such as GeneticProgramming [9] have been applied to spatialproblems. This paper proposes that spatialproblems need a specifically spatial representationwhich, combined with the more generic aspects ofGAs can be used as a general solution technique.The problem considered here is to locate the nodesof an arbitrarily connected graph so that itconforms to aesthetically pleasing principles oflayout [1]. It is inherently difficult to optimise evenone criteria such as minimising edge crossings[18]. Current research has focused on force models[7] and simulated annealing [5] but these are morelikely to be trapped by local optima than a GA.The GA also has considerable potential to tacklelarge problems as the time complexity of theoptimisation technique is not dependent on the sizeof the graph. Additionally, it is possible to reducethe time complexity for calculating the fitnessfunction by using approximate methods andcaching partial results.The GA represents the spatial nature of theproblem by using a graph based representation.The crossover operator is performed directly on thegraph by splitting parent graphs and joining theresultant subgraphs into a child graph. This has theadvantage of maintaining the geometric and graphtheoretic association of the nodes negating theneed for any separate constraint satisfaction.The crossover operator also attempts to retaingeometrically close subgraphs from each parent inan attempt to carry better subgraphs into the nextgeneration. This  contrasts with other GA graphdrawing work that takes nodes (or nodeneighbourhoods) randomly from the two parentgraphs [11] or which translates the graphs to a onedimensional representation and uses constraints ontraditional crossover [12].Section two of this paper briefly describes themain features of the genetic algorithm; sectionthree  presents some initial results and section fourconcludes by summarising our findings anddiscussing the current and future work for thisproject.2. The Graphical Genetic AlgorithmThe initial population is created by a randomlyproduced geometric node layout with the numberof nodes and edges controlled by the user. Twographs are selected for reproduction by choosingthe fittest out of a four randomly selected  from thepopulation. This is a simple tournament selectionmechanism with a low selection pressure.22.1. Fitness FunctionWe use a multiobjective fitness function based onfive well known measurable aesthetic criteria forgraphs:1. Total edge length. This is simply the sum ofthe lengths of each edge.2. Graph area. This is a calculation of the area ofthe bounding rectangle of the nodes in thegraph.3. Node overlap. A weighted penalty is given tonodes that are too close or overlap each other.4. Angular resolution. This is a calculation of theangle between edges connected to the samenode and penalises connections that formacute angles.5. Edge crossing. Applies a penalty functionbased on the number of edge crossings in thegraph.While these criteria are useful measures ofaesthetic properties of graphs, this is not anexhaustive list and there are other measures thatcan be used [3,16].The calculation of the fitness function follows thesum of weighted global ratios approach [1]. Foreach criteria i = 1..5, given above, the fitness ratiori is calculated from the criteria value, fi, asfollows:ri = 1 + (fi - Max Valuei)/(MinValuei - Max Valuei)Where Max Valuei is the highest fi found for thatcriteria in the operation of the GA, and Min Valueiis the lowest fi found. This formula converts thecriteria from an absolute range to a ratio that ranksit against the best found so far by the algorithm.This means that for any given generation we cantell which are the fitter individuals but thatcomparison across generations is dependent on theoverall progress made in each of the fitnesscriteria. The fitness value, f, for a graph is then theweighted sum of these values:f = w1r1 + w2r2 + w3r3 + w4r4 + w5r5This fitness calculation is particularly suitablewhere there are a large number of criteria that needto be combined and the complex interrelationsbetween them are not easily determined [8].Weights are determined experimentally on a testgraph to achieve the desired results and as theyapply to ratios rather than absolute values thesesettings are suitable for graphs of different sizesand topologies.2.2. Crossover and MutationWe use a geometrically based gradient crossoverfunction. This adds nodes to the child graph bytaking nodes along a random gradient, one parentgraph ascending the gradient and one descendingthe gradient. A node is taken from each parentgraph in turn, unless the corresponding node isalready in the child graph, in which case the nextunused node is taken. This splits each parent graphinto two subgraphs with equal number of nodespreserving  spatial relationships and the mainfeatures of the topology.Figure 1. shows two parent graphs, where thehighlighted nodes form the child graph. Figure 2.shows the resultant child. The nodes are sorted onthe indicated gradient (shown as a directed linenext to each graph), with nodes from the first graphtaken from top left to bottom right and nodes fromFigure 1. The two parent graphs undergoing crossoverFigure 2. The Child graph3the second graph taken from bottom right to topleft.We mutate a graph by picking nodes from thegraph randomly, and moving them a randomdistance along the X and Y axis from their currentlocation. The likelihood of a mutation and the sizeof the movement are controlled by the user.3. ResultsFigure 3. shows a randomly generated graph with159 edge crossings and a graph from the finalpopulation which has four edge crossings producedby  running the GA with a  population of 20 for1000 generations. Typically, most of theimprovements were seen by around 250generations with small improvements due to minorchanges caused by the mutation operator.A number of small test graphs were generatedwhich could be optimised by hand. The resultsfrom the GA showed that in 3 out of 10 cases theoptimum number was found and on average theGA produced graphs with just over 2 extra edgecrossings.4. Conclusions and Further WorkWe have introduced a new geometrically basedcrossover technique for graph drawing GAs. Thishas been used within a hybrid GA that implicitlyprocesses spatial information via the mainoperators of crossover and mutation. This hasdemonstrated that it is possible to hybridise the GAto implicitly process spatial information via themain operators of crossover and mutation withoutresort to special optimisation or constraintenforcement techniques. Much work remains evenin this limited application area but this is a starttowards a more general goal of providing a genericframework for a wide range of spatial problems.There are two types of application that areappropriate to graph based GAs. The first areapplications that use graphs where the graphtheoretic structure is maintained throughout theGA process, as with graph drawing described here.An interesting area of this type is integrated circuitdesign, where the connections between circuits areconstant, and the aim is to minimise the physicallayout of circuits to improve the performance ofthe final chip. The second type of application iswhere the graph theoretic structure of the graphmay alter during the process of the GA. Oneexample is network routing, where in order toimprove communication between pairs of nodesshort paths are sought through a graph representinga communication network. For this second type,new graph crossover strategies may be required inorder to produce graphs that are different, but stillappropriate to the application area.There are a number of areas that require work inthe specific domain of graph drawing. Theselection of aesthetic criteria needs to incorporateideas of symmetry and even spacing betweennodes before the results can compete withtraditional methods for small graphs. However,even now the GA provides an excellent pre-processor to these methods where larger graphsizes make an exhaustive, deterministic approachFigure 3. An initial graph and the aesthetically optimised result4slow or prone to trapping in local optima. The GAis adaptable because of the general way in whichthe fitness function is calculated and scalable dueto the inherent efficiencies of the technique.The overall speed of the fitness calculation can beimproved by combining the calculation of thevarious criteria into one pass across the graph.Additionally, results for particular sub-graphs canbe cached and used where these patterns re-appear.To speed up the GA, a less general layouttechnique could be employed, using a grid wherenodes are restricted to appearing on certainregularly spaced points. The resolution of this gridwill be sufficient so that location of a node oneither of two adjacent points to will have aminimal effect on any of the fitness criteria. Itwould also allow the automatic enforcement ofnode overlap and minimum inter-node distance.Before any conclusions can be drawn about thewider context of general spatial problem solvingthis technique will be tested on a number ofdifferent applications. However, it can be expectedthat a system that can dispense with ad-hocoptimisations and constraints will be easier togeneralise and will be more sympathetic to theunderlying processes of the GA technique.References1. Di Battista, P. Eades, R. Tamassa and I. Tollis,Algorithms for drawing graphs: an annotatedbibliography. Computing Geom. Theory Appl. 4: pp235-282. 1994.2. P.J. Bentley and J.P. Wakefield. FindingAcceptable Solutions in the Pareto-Optimal Rangeusing Multiobjective Genetic Algorithms. WSC2Second Online World Conference on SoftComputing. June 1997.3. F.J. Brandenburg, M. Himslot and C. Rohrer. AnExperimental Comparison of Force-Directed andRandomised Graph Drawing Algorithms.Symposium on Graph Drawing, GD’95. LNCS1027. Springer-Verlag. 1995.4. R. Cooley, M. Hobbs, D. Pack. Genetic Algorithmsand the Analysis of Spatially Referenced Data.Applied Artificial Intelligence 2:11 pp. 151-170.1997.5. R. Davidson and D. Harel, Drawing Graphs NicelyUsing Simulated Annealing. 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A Genetic Algorithm for Survivable Network Design.

A Heuristic for Graph Drawing.

A standard GA Approach to Native Protein Conformation Prediction.

Algorithms for Directed Graphs: A unique approach using genetic algorithms.

Algorithms for drawing graphs: an annotated bibliography.

An Experimental Comparison of Force-Directed and Randomised Graph Drawing Algorithms.

An Overview of Evoutionary Algorithms in Multiobjective Optimization,

Automating the Layout of Network Diagrams with Specified Visual Organization.

Crossing Numbers of Graphs, Lower Bound Techniques and Algorithms.

Discovering Patterns in Spatial Data using Evolutionary Programming Genetic Programming

Drawing Graphs Nicely Using Simulated Annealing.

Finding Acceptable Solutions in the Pareto-Optimal Range using Multiobjective Genetic Algorithms.

Flexible ligand docking using a genetic algorithm.

Genetic Algorithms and the Analysis of Spatially Referenced Data.

Network Optimization Using Evolutionary Strategies.

Spatial Clustering with a Genetic Algorithm. In

Suaraweera Encoding Graphs for Genetic Algorithms: An Investigation using the Minimum Spanning Tree Problem.

Validating Graph Drawing Aesthetics.

Representing Space: A Hybrid Genetic Algorithm for Aesthetic Graph Layout

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Representing Space: A Hybrid Genetic Algorithm for Aesthetic Graph Layout

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