Editing Graph theory (section)

== Applications ==
[[File:Wikipedia multilingual network graph July 2013.svg|thumb|The network graph formed by Wikipedia editors (edges) contributing to different Wikipedia language versions (vertices) during one month in summer 2013.<ref>{{Cite book|arxiv=1312.0976|last1=Hale|first1=Scott A.|title=Proceedings of the 2014 ACM conference on Web science |chapter=Multilinguals and Wikipedia editing |date=2014 |doi=10.1145/2615569.2615684|pages=99–108|isbn=9781450326223|bibcode=2013arXiv1312.0976H|s2cid=14027025}}</ref>]]

Graphs can be used to model many types of relations and processes in physical, biological,<ref>{{cite journal |last=Mashaghi |first=A. |title=Investigation of a protein complex network |journal=European Physical Journal B |volume=41 |issue=1 |pages=113–121 |year=2004 |doi=10.1140/epjb/e2004-00301-0 |display-authors=etal|arxiv=cond-mat/0304207 |bibcode=2004EPJB...41..113M |s2cid=9233932 }}</ref><ref>{{Cite journal|last1=Shah|first1=Preya|last2=Ashourvan|first2=Arian|last3=Mikhail|first3=Fadi|last4=Pines|first4=Adam|last5=Kini|first5=Lohith|last6=Oechsel|first6=Kelly|last7=Das|first7=Sandhitsu R|last8=Stein|first8=Joel M|last9=Shinohara|first9=Russell T|date=2019-07-01|title=Characterizing the role of the structural connectome in seizure dynamics|journal=Brain|language=en|volume=142|issue=7|pages=1955–1972|doi=10.1093/brain/awz125|pmid=31099821|issn=0006-8950|pmc=6598625}}</ref> social and information systems.<ref>{{Cite journal|last1=Adali|first1=Tulay|last2=Ortega|first2=Antonio|date=May 2018|title=Applications of Graph Theory [Scanning the Issue]|url=https://ieeexplore.ieee.org/document/8349656|journal=Proceedings of the IEEE|volume=106|issue=5|pages=784–786|doi=10.1109/JPROC.2018.2820300|issn=0018-9219|url-access=subscription}}</ref> Many practical problems can be represented by graphs. Emphasizing their application to real-world systems, the term ''network'' is sometimes defined to mean a graph in which attributes (e.g. names) are associated with the vertices and edges, and the subject that expresses and understands real-world systems as a network is called [[network science]].

=== Computer science ===
Within [[computer science]], '[[cybernetics|causal]]' and 'non-causal' linked structures are graphs that are used to represent networks of communication, data organization, computational devices, the flow of computation, etc. For instance, the link structure of a [[website]] can be represented by a directed graph, in which the vertices represent web pages and directed edges represent [[Hyperlink|links]] from one page to another. A similar approach can be taken to problems in social media,<ref>{{Cite journal | volume = 3| issue = 1| last = Grandjean| first = Martin| title = A social network analysis of Twitter: Mapping the digital humanities community| journal =Cogent Arts & Humanities| date = 2016| pages = 1171458|doi=10.1080/23311983.2016.1171458| s2cid = 114999767| url = https://hal.archives-ouvertes.fr/hal-01517493/file/A%20social%20network%20analysis%20of%20Twitter%20Mapping%20the%20digital%20humanities%20community.pdf| doi-access = free}}</ref> travel, biology, computer chip design, mapping the progression of neuro-degenerative diseases,<ref>{{Cite journal | volume = 11| issue = 2| last = Vecchio | first = F| title = "Small World" architecture in brain connectivity and hippocampal volume in Alzheimer's disease: a study via graph theory from EEG data| journal =Brain Imaging and Behavior| date = 2017| pages = 473–485| pmid =26960946 | doi = 10.1007/s11682-016-9528-3| s2cid = 3987492}}</ref><ref>{{Cite journal | volume = 81| issue = 2| last = Vecchio | first = F| title = Brain network connectivity assessed using graph theory in frontotemporal dementia| journal = Neurology| date = 2013| pages = 134–143| doi = 10.1212/WNL.0b013e31829a33f8| pmid = 23719145| s2cid = 28334693}}</ref> and many other fields. The development of [[algorithm]]s to [[List of algorithms#Graph algorithms|handle graphs]] is therefore of major interest in computer science. The [[Graph transformation|transformation of graph]]s is often formalized and represented by [[graph rewriting|graph rewrite system]]s. Complementary to [[graph transformation]] systems focusing on rule-based in-memory manipulation of graphs are [[graph database]]s geared towards [[Database transaction|transaction]]-safe, [[Persistence (computer science)|persistent]] storing and querying of [[Graph (data structure)|graph-structured data]].

=== Linguistics ===
Graph-theoretic methods, in various forms, have proven particularly useful in [[linguistics]], since natural language often lends itself well to discrete structure. Traditionally, [[syntax]] and compositional semantics follow tree-based structures, whose expressive power lies in the [[principle of compositionality]], modeled in a hierarchical graph. More contemporary approaches such as [[head-driven phrase structure grammar]] model the syntax of natural language using [[feature structure|typed feature structure]]s, which are [[directed acyclic graph]]s. 
Within [[lexical semantics]], especially as applied to computers, modeling word meaning is easier when a given word is understood in terms of related words; [[semantic network]]s are therefore important in [[computational linguistics]]. Still, other methods in phonology (e.g. [[optimality theory]], which uses [[lattice graph]]s) and morphology (e.g. finite-state morphology, using [[finite-state transducer]]s) are common in the analysis of language as a graph. Indeed, the usefulness of this area of mathematics to linguistics has borne organizations such as [http://www.textgraphs.org/ TextGraphs], as well as various 'Net' projects, such as [[WordNet]], [[VerbNet]], and others.

=== Physics and chemistry ===
Graph theory is also used to study molecules in [[chemistry]] and [[physics]]. In [[condensed matter physics]], the three-dimensional structure of complicated simulated atomic structures can be studied quantitatively by gathering statistics on graph-theoretic properties related to the topology of the atoms. Also, "the [[Feynman diagram|Feynman graphs and rules of calculation]] summarize [[quantum field theory]] in a form in close contact with the experimental numbers one wants to understand."<ref>{{cite book|first1=J. D.|last1=Bjorken |first2=S. D. |last2=Drell |title=Relativistic Quantum Fields |url=https://archive.org/details/relativisticquan0000bjor_c5q0|url-access=registration|publisher=McGraw-Hill |location=New York |year=1965 |page=viii }}</ref> In chemistry a graph makes a natural model for a molecule, where vertices represent [[atom]]s and edges [[Chemical bond|bond]]s. This approach is especially used in computer processing of molecular structures, ranging from [[Molecule editor|chemical editor]]s to database searching. In [[statistical physics]], graphs can represent local connections between interacting parts of a system, as well as the dynamics of a physical process on such
systems. Similarly, in [[computational neuroscience]] graphs can be used to represent functional connections between brain areas that interact to give rise to various cognitive processes, where the vertices represent different areas of the brain and the edges represent the connections between those areas. Graph theory plays an important role in electrical modeling of electrical networks, here, weights are associated with resistance of the wire segments to obtain electrical properties of network structures.<ref>{{Cite journal|last1=Kumar|first1=Ankush|last2=Kulkarni|first2=G. U.|date=2016-01-04|title=Evaluating conducting network based transparent electrodes from geometrical considerations|journal=Journal of Applied Physics|volume=119|issue=1|pages=015102|doi=10.1063/1.4939280|issn=0021-8979|bibcode=2016JAP...119a5102K}}</ref> Graphs are also used to represent the micro-scale channels of [[Porous medium|porous media]], in which the vertices represent the pores and the edges represent the smaller channels connecting the pores. [[Chemical graph theory]] uses the [[molecular graph]] as a means to model molecules.
Graphs and networks are excellent models to study and understand phase transitions and critical phenomena.
Removal of nodes or edges leads to a critical transition where the network breaks into small clusters which is studied as a phase transition. This breakdown is studied via [[percolation theory]].<ref>{{Cite book| last = Newman| first = Mark| title = Networks: An Introduction| publisher = Oxford University Press| date = 2010| url = http://math.sjtu.edu.cn/faculty/xiaodong/course/Networks%20An%20introduction.pdf| access-date = 2019-10-30| archive-date = 2020-07-28| archive-url = https://web.archive.org/web/20200728132820/http://math.sjtu.edu.cn/faculty/xiaodong/course/Networks%20An%20introduction.pdf| url-status = dead}}</ref>

=== Social sciences ===
[[File:Moreno Sociogram 2nd Grade.png|thumb|Graph theory in sociology: [[Jacob L. Moreno|Moreno]] [[Sociogram]] (1953).<ref>Grandjean, Martin (2015). [http://www.martingrandjean.ch/social-network-analysis-visualization-morenos-sociograms-revisited/ "Social network analysis and visualization: Moreno’s Sociograms revisited"]. Redesigned network strictly based on Moreno (1934), ''Who Shall Survive''.</ref>]]
Graph theory is also widely used in [[sociology]] as a way, for example, to [[Six Degrees of Kevin Bacon|measure actors' prestige]] or to explore [[Rumor spread in social network|rumor spreading]], notably through the use of [[social network analysis]] software. Under the umbrella of social networks are many different types of graphs.<ref>{{cite book|last=Rosen|first=Kenneth H.|title=Discrete mathematics and its applications|publisher=McGraw-Hill|location=New York|isbn=978-0-07-338309-5|edition=7th|date=2011-06-14}}</ref> Acquaintanceship and friendship graphs describe whether people know each other. Influence graphs model whether certain people can influence the behavior of others. Finally, collaboration graphs model whether two people work together in a particular way, such as acting in a movie together.

=== Biology ===
Likewise, graph theory is useful in [[biology]] and conservation efforts where a vertex can represent regions where certain species exist (or inhabit) and the edges represent migration paths or movement between the regions. This information is important when looking at breeding patterns or tracking the spread of disease, parasites or how changes to the movement can affect other species.

Graphs are also commonly used in [[molecular biology]] and [[genomics]] to model and analyse datasets with complex relationships. For example, graph-based methods are often used to 'cluster' cells together into cell-types in [[Single-cell_analysis#Transcriptomics|single-cell transcriptome analysis]]. Another use is to model genes or proteins in a [[Biological pathway|pathway]] and study the relationships between them, such as metabolic pathways and gene regulatory networks.<ref>{{cite journal | last1=Kelly | first1=S. | last2=Black | first2=Michael | title=graphsim: An R package for simulating gene expression data from graph structures of biological pathways | journal=Journal of Open Source Software | publisher=The Open Journal | volume=5 | issue=51 | date=2020-07-09 | issn=2475-9066 | doi=10.21105/joss.02161 | page=2161| bibcode=2020JOSS....5.2161K |biorxiv=10.1101/2020.03.02.972471|s2cid=214722561| doi-access=free | url=https://www.biorxiv.org/content/biorxiv/early/2020/06/30/2020.03.02.972471.full.pdf }}</ref> Evolutionary trees, ecological networks, and hierarchical clustering of gene expression patterns are also represented as graph structures. 

Graph theory is also used in [[connectomics]];<ref>{{Cite journal|last1=Shah|first1=Preya|last2=Ashourvan|first2=Arian|last3=Mikhail|first3=Fadi|last4=Pines|first4=Adam|last5=Kini|first5=Lohith|last6=Oechsel|first6=Kelly|last7=Das|first7=Sandhitsu R|last8=Stein|first8=Joel M|last9=Shinohara|first9=Russell T|date=2019-07-01|title=Characterizing the role of the structural connectome in seizure dynamics|journal=Brain|language=en|volume=142|issue=7|pages=1955–1972|doi=10.1093/brain/awz125|pmid=31099821|issn=0006-8950|pmc=6598625}}</ref> nervous systems can be seen as a graph, where the nodes are neurons and the edges are the connections between them.

=== Mathematics ===
In mathematics, graphs are useful in geometry and certain parts of [[topology]] such as [[knot theory]]. [[Algebraic graph theory]] has close links with [[group theory]]. Algebraic graph theory has been applied to many areas including dynamic systems and complexity.

=== Other topics ===
A graph structure can be extended by assigning a weight to each edge of the graph. Graphs with weights, or [[weighted graph]]s, are used to represent structures in which pairwise connections have some numerical values. For example, if a graph represents a road network, the weights could represent the length of each road. There may be several weights associated with each edge, including distance (as in the previous example), travel time, or monetary cost. Such weighted graphs are commonly used to program GPS's, and travel-planning search engines that compare flight times and costs.