Graph and Network Algorithms

Directed and undirected graphs, network analysis

Graphs model the connections in a network and are widely applicable to a variety of physical, biological, and information systems. You can use graphs to model the neurons in a brain, the flight patterns of an airline, and much more. The structure of a graph is comprised of “nodes” and “edges”. Each node represents an entity, and each edge represents a connection between two nodes. For more information, see Directed and Undirected Graphs.

Example plots of undirected and directed graphs

Functions

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Construction

`graph`	Graph with undirected edges
`digraph`	Graph with directed edges

Modify Nodes and Edges

`addnode`	Add new node to graph
`rmnode`	Remove node from graph
`addedge`	Add new edge to graph
`rmedge`	Remove edge from graph
`flipedge`	Reverse edge directions
`numnodes`	Number of nodes in graph
`numedges`	Number of edges in graph
`findnode`	Locate node in graph
`findedge`	Locate edge in graph
`edgecount`	Number of edges between two nodes
`reordernodes`	Reorder graph nodes
`subgraph`	Extract subgraph

Analyze Structure

`centrality`	Measure node importance
`conncomp`	Connected graph components
`biconncomp`	Biconnected graph components
`condensation`	Graph condensation
`bctree`	Block-cut tree graph
`toposort`	Topological order of directed acyclic graph
`isdag`	Determine if graph is acyclic
`transreduction`	Transitive reduction
`transclosure`	Transitive closure
`isisomorphic`	Determine whether two graphs are isomorphic
`isomorphism`	Compute isomorphism between two graphs
`ismultigraph`	Determine whether graph has multiple edges
`simplify`	Reduce multigraph to simple graph

Traversals, Shortest Paths, and Cycles

`bfsearch`	Breadth-first graph search
`dfsearch`	Depth-first graph search
`shortestpath`	Shortest path between two single nodes
`shortestpathtree`	Shortest path tree from node
`distances`	Shortest path distances of all node pairs
`allpaths`	Find all paths between two graph nodes (Since R2021a)
`maxflow`	Maximum flow in graph
`minspantree`	Minimum spanning tree of graph
`hascycles`	Determine whether graph contains cycles (Since R2021a)
`allcycles`	Find all cycles in graph (Since R2021a)
`cyclebasis`	Fundamental cycle basis of graph (Since R2021a)

Matrix Representation

`adjacency`	Graph adjacency matrix
`incidence`	Graph incidence matrix
`laplacian`	Graph Laplacian matrix

Node Information

`degree`	Degree of graph nodes
`neighbors`	Neighbors of graph node
`nearest`	Nearest neighbors within radius
`indegree`	In-degree of nodes
`outdegree`	Out-degree of nodes
`predecessors`	Node predecessors
`successors`	Node successors
`inedges`	Incoming edges to node
`outedges`	Outgoing edges from node

Visualization

`plot`	Plot graph nodes and edges
`labeledge`	Label graph edges
`labelnode`	Label graph nodes
`layout`	Change layout of graph plot
`layoutcoords`	Graph node and edge layout coordinates (Since R2024b)
`highlight`	Highlight nodes and edges in plotted graph

Objects

GraphPlot Graph plot for directed and undirected graphs

Properties

GraphPlot Properties

Graph plot appearance and behavior

Topics

Directed and Undirected Graphs
Introduction to directed and undirected graphs.
Graphs and Matrices
This example shows an application of sparse matrices and explains the relationship between graphs and matrices.
Modify Nodes and Edges of Existing Graph
This example shows how to access and modify the nodes and/or edges in a graph or digraph object using the addedge, rmedge, addnode, rmnode, findedge, findnode, and subgraph functions.
Add Graph Node Names, Edge Weights, and Other Attributes
This example shows how to add attributes to the nodes and edges in graphs created using graph and digraph.
Graph Plotting and Customization
This example shows how to plot graphs, and then customize the display to add labels or highlighting to the graph nodes and edges.
Label Graph Nodes and Edges
This example shows how to add and customize labels on graph nodes and edges.
Add Node Properties to Graph Plot Data Tips
This example shows how to customize GraphPlot data tips to display extra node properties of a graph.
Visualize Breadth-First and Depth-First Search
This example shows how to define a function that visualizes the results of bfsearch and dfsearch by highlighting the nodes and edges of a graph.

Related Information

Graph and Digraph Classes

Featured Examples

Build Watts-Strogatz Small World Graph Model

Construct and analyze a Watts-Strogatz small-world graph. The Watts-Strogatz model is a random graph that has small-world network properties, such as clustering and short average path length.

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Use PageRank Algorithm to Rank Websites

Use a PageRank algorithm to rank a collection of websites. Although the PageRank algorithm was originally designed to rank search engine results, it also can be more broadly applied to the nodes in many different types of graphs. The PageRank score gives an idea of the relative importance of each graph node based on how it is connected to the other nodes.

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Partition Graph with Laplacian Matrix

Use the Laplacian matrix of a graph to compute the Fiedler vector. The Fiedler vector can be used to partition the graph into two subgraphs.

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