View source for Graph drawing

{{Short description|Visualization of node-link graphs}}
{{redirects|Network diagram|network diagrams in specific contexts|#Application-specific graph drawings}}
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'''Graph drawing''' is an area of [[mathematics]] and [[computer science]] combining methods from [[geometric graph theory]] and [[information visualization]] to derive two-dimensional (or, sometimes, three-dimensional) depictions of [[graph (discrete mathematics)|graph]]s arising from applications such as [[social network analysis]], [[cartography]], [[linguistics]], and [[bioinformatics]].<ref>{{harvtxt|Di Battista|Eades|Tamassia|Tollis|1998}}, pp. vii–viii; {{harvtxt|Herman|Melançon|Marshall|2000}}, Section 1.1, "Typical Application Areas".</ref>

A drawing of a graph or '''network diagram''' is a pictorial representation of the [[vertex (graph theory)|vertices]] and [[edge (graph theory)|edges]] of a graph. This drawing should not be confused with the graph itself: very different layouts can correspond to the same graph.<ref name="dett6">{{harvtxt|Di Battista|Eades|Tamassia|Tollis|1998}}, p. 6.</ref>  In the abstract, all that matters is which pairs of vertices are connected by edges.  In the concrete, however, the arrangement of these vertices and edges within a drawing affects its understandability, usability, fabrication cost, and [[aesthetics]].<ref name="dett-viii"/> The problem gets worse if the graph changes over time by adding and deleting edges (dynamic graph drawing) and the goal is to preserve the user's mental map.{{sfnp|Misue|Eades|Lai|Sugiyama|1995}}

==Graphical conventions==
[[File:4node-digraph-natural.svg|thumb|upright=0.5|[[Directed graph]] with arrowheads showing edge directions]]
Graphs are frequently drawn as node–link diagrams in which the vertices are represented as disks, boxes, or textual labels and the edges are represented as [[line segment]]s, [[Polygonal chain|polylines]], or curves in the [[Euclidean plane]].<ref name="dett-viii">{{harvtxt|Di Battista|Eades|Tamassia|Tollis|1998}}, p. viii.</ref> Node–link diagrams can be traced back to the 14th-16th century works of Pseudo-Lull which were published under the name of [[Ramon Llull]], a 13th century polymath. Pseudo-Lull drew diagrams of this type for [[complete graph]]s in order to analyze all pairwise combinations among sets of metaphysical concepts.{{sfnp|Knuth|2013}}

In the case of [[directed graph]]s, [[Arrow (symbol)|arrowheads]] form a commonly used graphical convention to show their [[Orientability|orientation]];<ref name="dett6"/> however, user studies have shown that other conventions such as tapering provide this information more effectively.<ref>{{harvtxt|Holten|van Wijk|2009}}; {{harvtxt|Holten|Isenberg|van Wijk|Fekete|2011}}.</ref> [[Upward planar drawing]] uses the convention that every edge is oriented from a lower vertex to a higher vertex, making arrowheads unnecessary.{{sfnp|Garg|Tamassia|1995}}

Alternative conventions to node–link diagrams include adjacency representations such as [[circle packing theorem|circle packings]], in which vertices are represented by disjoint regions in the plane and edges are represented by adjacencies between regions; [[intersection graph|intersection representations]] in which vertices are represented by non-disjoint geometric objects and edges are represented by their intersections; visibility representations in which vertices are represented by regions in the plane and edges are represented by regions that have an unobstructed line of sight to each other; confluent drawings, in which edges are represented as smooth curves within mathematical [[train track (mathematics)|train tracks]]; fabrics, in which nodes are represented as horizontal lines and edges as vertical lines;<ref name="Longabaugh 2012">{{harvtxt|Longabaugh|2012}}.</ref> and visualizations of the [[adjacency matrix]] of the graph.

==Quality measures==
Many different quality measures have been defined for graph drawings, in an attempt to find objective means of evaluating their aesthetics and usability.<ref>{{harvtxt|Di Battista|Eades|Tamassia|Tollis|1998}}, Section 2.1.2, Aesthetics, pp. 14–16; {{harvtxt|Purchase|Cohen|James|1997}}.</ref> In addition to guiding the choice between different layout methods for the same graph, some layout methods attempt to directly optimize these measures.
[[File:4node-digraph-embed.svg|upright=0.5|thumb|[[Planar graph]] drawn without overlapping edges]]
*The [[Crossing number (graph theory)|crossing number]] of a drawing is the number of pairs of edges that cross each other.  If the graph is [[planar graph|planar]], then it is often convenient to draw it without any edge intersections; that is, in this case, a graph drawing represents a [[graph embedding]]. However, nonplanar graphs frequently arise in applications, so graph drawing algorithms must generally allow for edge crossings.<ref>{{harvtxt|Di Battista|Eades|Tamassia|Tollis|1998}}, p 14.</ref>
*The [[Area (graph drawing)|area]] of a drawing is the size of its smallest [[bounding box]], relative to the closest distance between any two vertices. Drawings with smaller area are generally preferable to those with larger area, because they allow the features of the drawing to be shown at greater size and therefore more legibly. The [[aspect ratio]] of the bounding box may also be important.
*Symmetry display is the problem of finding [[Graph automorphism|symmetry group]]s within a given graph, and finding a drawing that displays as much of the symmetry as possible. Some layout methods automatically lead to symmetric drawings; alternatively, some drawing methods start by finding symmetries in the input graph and using them to construct a drawing.<ref>{{harvtxt|Di Battista|Eades|Tamassia|Tollis|1998}}, p. 16.</ref>
*It is important that edges have shapes that are as simple as possible, to make it easier for the eye to follow them. In polyline drawings, the complexity of an edge may be measured by its [[Bend minimization|number of bends]], and many methods aim to provide drawings with few total bends or few bends per edge. Similarly for spline curves the complexity of an edge may be measured by the number of control points on the edge.
*Several commonly used quality measures concern lengths of edges: it is generally desirable to minimize the total length of the edges as well as the maximum length of any edge. Additionally, it may be preferable for the lengths of edges to be uniform rather than highly varied.
*[[Angular resolution (graph drawing)|Angular resolution]] is a measure of the sharpest angles in a graph drawing. If a graph has vertices with high [[degree (graph theory)|degree]] then it necessarily will have small angular resolution, but the angular resolution can be bounded below by a function of the degree.{{sfnp|Pach|Sharir|2009}}
*The [[slope number]] of a graph is the minimum number of distinct edge slopes needed in a drawing with straight line segment edges (allowing crossings). [[Cubic graph]]s have slope number at most four, but graphs of degree five may have unbounded slope number; it remains open whether the slope number of degree-4 graphs is bounded.{{sfnp|Pach|Sharir|2009}}

==Layout methods==
[[File:Social Network Analysis Visualization.png|thumb|right|A force-based network visualization.{{sfnp|Grandjean|2014}}]]
[[File:Spectral graph drawing of small world graph.svg|thumb|Spectral graph layout visualization.]]
There are many different graph layout strategies:
*In [[force-based layout]] systems, the graph drawing software modifies an initial vertex placement by continuously moving the vertices according to a system of forces based on physical metaphors related to systems of [[spring (device)|springs]] or [[molecular mechanics]]. Typically, these systems combine attractive forces between adjacent vertices with repulsive forces between all pairs of vertices, in order to seek a layout in which edge lengths are small while vertices are well-separated. These systems may perform [[gradient descent]] based minimization of an [[energy function]], or they may translate the forces directly into velocities or accelerations for the moving vertices.<ref>{{harvtxt|Di Battista|Eades|Tamassia|Tollis|1998}}, Section 2.7, "The Force-Directed Approach", pp. 29–30, and Chapter 10, "Force-Directed Methods", pp. 303–326.</ref>
*[[Spectral layout]] methods use as coordinates the [[eigenvector]]s of a [[Matrix (mathematics)|matrix]] such as the [[Discrete Laplace operator|Laplacian]] derived from the [[adjacency matrix]] of the graph.<ref>{{harvtxt|Beckman|1994}}; {{harvtxt|Koren|2005}}.</ref>
*Orthogonal layout methods, which allow the edges of the graph to run horizontally or vertically, parallel to the coordinate axes of the layout. These methods were originally designed for [[VLSI]] and [[Printed circuit board|PCB]] layout problems but they have also been adapted for graph drawing. They typically involve a multiphase approach in which an input graph is planarized by replacing crossing points by vertices, a topological embedding of the planarized graph is found, edge orientations are chosen to minimize bends, vertices are placed consistently with these orientations, and finally a layout compaction stage reduces the area of the drawing.<ref>{{harvtxt|Di Battista|Eades|Tamassia|Tollis|1998}}, Chapter 5, "Flow and Orthogonal Drawings", pp. 137–170; {{harvtxt|Eiglsperger|Fekete|Klau|2001}}.</ref>
*Tree layout algorithms these show a rooted [[tree structure|tree]]-like formation, suitable for [[tree (graph theory)|trees]]. Often, in a technique called "balloon layout", the children of each node in the tree are drawn on a circle surrounding the node, with the radii of these circles diminishing at lower levels in the tree so that these circles do not overlap.<ref>{{harvtxt|Herman|Melançon|Marshall|2000}}, Section 2.2, "Traditional Layout – An Overview".</ref>
*[[Layered graph drawing]] methods (often called Sugiyama-style drawing) are best suited for [[directed acyclic graph]]s or graphs that are nearly acyclic, such as the graphs of dependencies between modules or functions in a software system. In these methods, the nodes of the graph are arranged into horizontal layers using methods such as the [[Coffman–Graham algorithm]], in such a way that most edges go downwards from one layer to the next; after this step, the nodes within each layer are arranged in order to minimize crossings.<ref>{{harvtxt|Sugiyama|Tagawa|Toda|1981}}; {{harvtxt|Bastert|Matuszewski|2001}}; {{harvtxt|Di Battista|Eades|Tamassia|Tollis|1998}}, Chapter 9, "Layered Drawings of Digraphs", pp. 265–302.</ref>
[[File:Goldner-Harary-linear.svg|thumb|Arc diagram]]
*[[Arc diagram]]s, a layout style dating back to the 1960s,<ref>{{harvtxt|Saaty|1964}}.</ref> place vertices on a line; edges may be drawn as semicircles above or below the line, or as smooth curves linked together from multiple semicircles.
*[[Circular layout]] methods place the vertices of the graph on a circle, choosing carefully the ordering of the vertices around the circle to reduce crossings and place adjacent vertices close to each other. Edges may be drawn either as chords of the circle or as arcs inside or outside of the circle. In some cases, multiple circles may be used.<ref>{{harvtxt|Doğrusöz|Madden|Madden|1997}}.</ref>
*[[Dominance drawing]] places vertices in such a way that one vertex is upwards, rightwards, or both of another if and only if it is [[reachability|reachable]] from the other vertex. In this way, the layout style makes the reachability relation of the graph visually apparent.<ref>{{harvtxt|Di Battista|Eades|Tamassia|Tollis|1998}}, Section 4.7, "Dominance Drawings", pp. 112–127.</ref>

==Application-specific graph drawings==
Graphs and graph drawings arising in other areas of application include
* [[Sociogram]]s, drawings of a [[social network]], as often offered by [[social network analysis software]]<ref>{{Harvard citation text|Scott|2000}}; {{Harvard citation text|Brandes|Freeman|Wagner|2014}}.</ref>
* [[Hasse diagram]]s, a type of graph drawing specialized to [[partial order]]s<ref>{{Harvard citation text|Di Battista|Eades|Tamassia|Tollis|1998}}, pp. 15–16, and Chapter 6, "Flow and Upward Planarity", pp. 171–214; {{Harvard citation text|Freese|2004}}.</ref>
* [[Dessin d'enfant]]s, a type of graph drawing used in [[algebraic geometry]]{{sfnp|Zapponi|2003}}
* [[State diagram]]s, graphical representations of [[finite-state machine]]s{{sfnp|Anderson|Head|2006}}
* [[Computer network diagram]]s, depictions of the nodes and connections in a [[computer network]]{{sfnp|Di Battista|Rimondini|2014}}
* [[Flowchart]]s and [[DRAKON|drakon-charts]], drawings in which the nodes represent the steps of an [[algorithm]] and the edges represent [[control flow]] between steps.
* [[Project network]], graphical depiction of the chronological order in which activities of a [[Project management|project]] are to be completed.
* [[Data-flow diagram]]s, drawings in which the nodes represent the components of an [[information system]] and the edges represent the movement of information from one component to another.
* [[Bioinformatics]] including [[phylogenetic tree]]s, [[protein–protein interaction]] networks, and [[metabolic pathway]]s.{{sfnp|Bachmaier|Brandes|Schreiber|2014}}

In addition, the [[placement (electronic design automation)|placement]] and [[Routing (electronic design automation)|routing]] steps of [[electronic design automation]] (EDA) are similar in many ways to graph drawing, as is the problem of [[greedy embedding]] in [[distributed computing]], and the graph drawing literature includes several results borrowed from the EDA literature. However, these problems also differ in several important ways: for instance, in EDA, area minimization and signal length are more important than aesthetics, and the routing problem in EDA may have more than two terminals per net while the analogous problem in graph drawing generally only involves pairs of vertices for each edge.

==Graph drawing algorithms==
There are many algorithms for graph drawing. Among them are:
* The Reingold-Tilford algorithm for tree drawing.{{sfnp|Reingold|Tilford|1981}}
* Kant's algorithm,{{sfnp|Kant|1992}} which constructs a polyline drawing of a 3-connected planar graph such that the size of the minimal angle among arcs is at least <math>\frac{1}{d}\pi</math>, where ''d'' is the maximal node degree; and its generaliztion, that also works well for other planar graphs, by Gutwenger and Mutzel.{{sfnp|Gutwenger|Mutzel|1998}}
* Tamassia's algorithm for minimizing the number of bends in an orthogonal representation of a planar graph.{{sfnp|Tamassia|1987}}
* The Magnetic Spring Model by Sugiyama and Misue.{{sfnp|Sugiyama|Misue|1995}}

==Software==
[[File:Gephi 0.9.1 Network Analysis and Visualization Software.png|thumb|A graph drawing interface ([[Gephi]] 0.9.1)]]Software, systems, and providers of systems for drawing graphs include:
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* [[BioFabric]] open-source software for visualizing large networks by drawing nodes as horizontal lines.
* [[Cytoscape]], open-source software for visualizing molecular interaction networks
* [[Gephi]], open-source network analysis and visualization software
* [[graph-tool]], a [[Free Software|free/libre]] [[Python (programming language)|Python]] library for analysis of graphs
* [[Graphviz]], an open-source graph drawing system from [[AT&T Corporation]]<ref>"Graphviz and Dynagraph – Static and Dynamic Graph Drawing Tools", by John Ellson, Emden R. Gansner, Eleftherios Koutsofios, Stephen C. North, and Gordon Woodhull, in {{harvtxt| Jünger|Mutzel|2004}}.</ref>
* [[Linkurious]], a commercial network analysis and visualization software for [[graph databases]]
* [[Mathematica]], a general-purpose computation tool that includes 2D and 3D graph visualization and graph analysis tools.<ref>{{citation |url=http://reference.wolfram.com/mathematica/tutorial/GraphDrawingIntroduction.html |title=Introduction to graph drawing|work= Wolfram Language & System Documentation Center|access-date=2024-03-21 }}</ref>
* [[Microsoft Automatic Graph Layout]], open-source .NET library (formerly called GLEE) for laying out graphs<ref>{{harvtxt|Nachmanson|Robertson|Lee|2008}}.</ref>
* [[NetworkX]] is a Python library for studying graphs and networks.
* [[Tulip (software)|Tulip]],<ref>"Tulip – A Huge Graph Visualization Framework", by David Auber, in {{harvtxt| Jünger|Mutzel|2004}}.</ref> an open-source data visualization tool
* [[yEd]], a graph editor with graph layout functionality<ref>"yFiles – Visualization and Automatic Layout of Graphs", by Roland Wiese, Markus Eiglsperger, and Michael Kaufmann, in {{harvtxt| Jünger|Mutzel|2004}}.</ref>
* [[PGF/TikZ]] 3.0 with the <code>graphdrawing</code> package (requires [[LuaTeX]]).<ref>{{harvtxt|Tantau|2013}}; see also the older [http://www.tcs.uni-luebeck.de/downloads/mitarbeiter/tantau/2012-gd-presentation.pdf GD 2012 presentation] {{Webarchive|url=https://web.archive.org/web/20160527034523/http://www.tcs.uni-luebeck.de/downloads/mitarbeiter/tantau/2012-gd-presentation.pdf |date=2016-05-27 }}</ref>
* [[LaNet-vi]], an open-source large network visualization software
* [[Open Graph Drawing Framework|OGDF]], an open-source library of C++ data structures and algorithms, mostly for graph drawing

== See also ==
* [[International Symposium on Graph Drawing]]
* [[List of Unified Modeling Language tools]]

==References==
===Footnotes===
{{reflist|colwidth=30em}}

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 | title = Graph Drawing: Algorithms for the Visualization of Graphs
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*{{citation
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 | isbn = 978-3-540-00881-1
 | publisher = Springer-Verlag
 | title = Graph Drawing Software
 | year = 2004}}.
{{refend}}

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{{refend}}

==Further reading==
{{refbegin|colwidth=30em}}
*{{citation
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 | journal = [[Computational Geometry (journal)|Computational Geometry: Theory and Applications]]
 | pages = 235–282
 | title = Algorithms for Drawing Graphs: an Annotated Bibliography
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 | year = 1994
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*{{citation|title=Drawing Graphs: Methods and Models|editor1-first=Michael|editor1-last=Kaufmann|editor2-first=Dorothea|editor2-last=Wagner|editor2-link=Dorothea Wagner|publisher=Springer-Verlag|series=[[Lecture Notes in Computer Science]]|volume=2025|year=2001|doi=10.1007/3-540-44969-8|isbn=978-3-540-42062-0|s2cid=1808286}}.
*{{citation|editor-first=Roberto|editor-last=Tamassia|editor-link=Roberto Tamassia|title=Handbook of Graph Drawing and Visualization|publisher=CRC Press|url=http://cs.brown.edu/~rt/gdhandbook/|year=2014|access-date=2013-08-28|archive-url=https://web.archive.org/web/20130815181243/http://cs.brown.edu/~rt/gdhandbook/|archive-date=2013-08-15|url-status=dead}}.
{{refend}}

==External links==
* [http://graphx.codeplex.com GraphX library for .NET] {{Webarchive|url=https://web.archive.org/web/20180126071742/http://graphx.codeplex.com/ |date=2018-01-26 }}: open-source WPF library for graph calculation and visualization. Supports many layout and edge routing algorithms.
* [http://gdea.informatik.uni-koeln.de/ Graph drawing e-print archive]: including information on papers from all [[International Symposium on Graph Drawing|Graph Drawing symposia]].

{{Graph representations}}

[[Category:Graph drawing| ]]