Editing Directed acyclic graph (section)

=== Causal structures ===
{{main|Bayesian network}}
Graphs in which vertices represent events occurring at a definite time, and where the edges always point from an earlier time vertex to a later time vertex, are necessarily directed and acyclic. The lack of a cycle follows because the time associated with a vertex always increases as you follow any directed [[Path (graph theory)|path]] in the graph, so you can never return to a vertex on a path.  This reflects our natural intuition that causality means events can only affect the future, they never affect the past, and thus we have no [[causal loop]]s. An example of this type of directed acyclic graph are those encountered in the [[Causal sets|causal set approach to quantum gravity]] though in this case the graphs considered are [[#Transitive closure and transitive reduction|transitively complete]]. In the version history example below, each version of the software is associated with a unique time, typically the time the version was saved, committed or released. In the citation graph examples below, the documents are published at one time and can only refer to older documents.

Sometimes events are not associated with a specific physical time. Provided that pairs of events have a purely causal relationship, that is edges represent [[causality|causal relations]] between the events, we will have a directed acyclic graph.<ref>{{citation|title=Causal Learning|first1=Alison|last1=Gopnik|author-link=Alison Gopnik |first2=Laura|last2=Schulz|author2-link=Laura Schulz |publisher=Oxford University Press|year=2007|isbn=978-0-19-803928-0|page=4|url=https://books.google.com/books?id=35MKXlKoXIUC&pg=PA4}}.</ref> For instance, a [[Bayesian network]] represents a system of probabilistic events as vertices in a directed acyclic graph, in which the likelihood of an event may be calculated from the likelihoods of its predecessors in the DAG.<ref>{{citation|title=Probabilistic Boolean Networks: The Modeling and Control of Gene Regulatory Networks|publisher=Society for Industrial and Applied Mathematics|first1=Ilya|last1=Shmulevich|first2=Edward R.|last2=Dougherty|year=2010|isbn=978-0-89871-692-4|page=58|url=https://books.google.com/books?id=RfshqEgO7KgC&pg=PA58}}.</ref> In this context, the [[moral graph]] of a DAG is the undirected graph created by adding an (undirected) edge between all parents of the same vertex (sometimes called ''marrying''), and then replacing all directed edges by undirected edges.<ref>{{citation |last1= Cowell |first1= Robert G. |author2-link=Philip Dawid|last2=Dawid|first2=A. Philip|author3-link=Steffen Lauritzen|last3=Lauritzen|first3=Steffen L.|author4-link=David Spiegelhalter|last4=Spiegelhalter|first4=David J.|title= Probabilistic Networks and Expert Systems |publisher= Springer |year= 1999 |isbn= 978-0-387-98767-5 |chapter= 3.2.1 Moralization|pages= 31–33 }}.</ref> Another type of graph with a similar causal structure is an [[influence diagram]], the vertices of which represent either decisions to be made or unknown information, and the edges of which represent causal influences from one vertex to another.<ref>{{citation|title=The Technology Management Handbook|first=Richard C.|last=Dorf|publisher=CRC Press|year=1998|isbn=978-0-8493-8577-3|page=9{{hyphen}}7<!-- Do not conver this hyphen into a dash! It is a section-page number, not a range of page numbers. -->|url=https://books.google.com/books?id=C2u8I0DFo4IC&pg=SA9-PA7}}.</ref> In [[epidemiology]], for instance, these diagrams are often used to estimate the expected value of different choices for intervention.<ref>{{citation|title=Encyclopedia of Epidemiology, Volume 1|first=Sarah|last=Boslaugh|publisher=SAGE|year=2008|isbn=978-1-4129-2816-8|page=255|url=https://books.google.com/books?id=wObgnN3x14kC&pg=PA255}}.</ref><ref name="pearl:95">{{citation | last = Pearl | first = Judea | doi = 10.1093/biomet/82.4.669 | issue = 4 | journal = Biometrika | pages = 669–709 | title = Causal diagrams for empirical research | volume = 82 | year = 1995| url = https://escholarship.org/uc/item/6gv9n38c }}.</ref>

The converse is also true. That is in any application represented by a directed acyclic graph there is a causal structure, either an explicit order or time in the example or an order which can be derived from graph structure. This follows because all directed acyclic graphs have a [[#Topological ordering|topological ordering]], i.e. there is at least one way to put the vertices in an order such that all edges point in the same direction along that order.