By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. Stack Overflow for Teams is a private, secure spot for you and your coworkers to find and share information. I'm trying to find an efficient algorithm to generate a simple connected graph with given sparseness. Something like:. Start with one node. In each iteration, create a new node and a new edge.

The edge is to connect the new node with a random node from the previous node set. After all nodes are created, create random edges until S is fulfilled. Make sure not to create double edges for this you can use an adjacency matrix.

The partition-based answer by ypnos is a good start, but bias is introduced by always selecting a visited node for one end of a new edge. By randomly selecting a visited node at each iteration, nodes that are visited towards the beginning have more iterations from which they have a chance to be chosen. Therefore, earlier nodes are more likely to have a high degree number of edges than those picked later.

Bias isn't always a bad thing. It turns out this kind of bias is good for generating spanning trees that are similar to real world computer networks. However, in order to create a truly random connected graph the initial spanning tree must be picked uniformly from the set of possible spanning trees see Wikipedia's Uniform Spanning Tree article.

One approach to generating a uniform spanning tree is through a random walk.

### Random graph

Start at any vertex and do a simple random walk on the graph. Each time a vertex is first encountered, mark the edge from which it was discovered. When all the vertices are discovered, the marked edges form a random spanning tree. This algorithm is easy to code up, has small running time constants, and has a nice proof that it generates trees with the right probabilities. This works well for a simple connected graph, however if you need an algorithm for a directed graph then read the paper further as it describes Wilson's Algorithm.

Here is another resource for random spanning trees and Wilson's Algorithm.GitHub is home to over 50 million developers working together to host and review code, manage projects, and build software together. Work fast with our official CLI. Learn more. If nothing happens, download GitHub Desktop and try again. If nothing happens, download Xcode and try again.

If nothing happens, download the GitHub extension for Visual Studio and try again. This is a Python program to randomly generate sets of connected graphs for research purposes.

The generated graphs can be exported as GraphML files. It requires Python 2. A simple command line interface is avaiable for easy generation of graphs, but Classes can be used directly for further customisations. Command line syntax given below:. GraphML files. We use optional third-party analytics cookies to understand how you use GitHub. You can always update your selection by clicking Cookie Preferences at the bottom of the page. For more information, see our Privacy Statement.

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**The evolution of the G(n,p) random graph**

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Failed to load latest commit information. View code. Connected Graph Generator This is a Python program to randomly generate sets of connected graphs for research purposes.

A set of graphs can be generated to have a common graph. Node and Graph attributes can also be generated randomly.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. Stack Overflow for Teams is a private, secure spot for you and your coworkers to find and share information. I want to be able to generate random, undirected, and connected graphs in Java. In addition, I want to be able to control the maximum number of vertices in the graph.

I am not sure what would be the best way to approach this problem, but here are a few I can think of:.

Then, somehow randomly link vertices together maybe generate a random number per vertex and let that be the number of edges coming out of said vertex. Traverse the graph starting from an arbitrary vertex say with Breadth-First-Search and let our random graph G be all the visited nodes this way, we make sure that G is connected.

This would be the adjacency matrix for our graph the diagonal of the matrix should then either be all 1 's or all 0 's. Make a data structure from the graph and traverse the graph from any node to get a connected list of nodes and call that the graph G. Any other way to generate a sufficiently random graph is welcomed. Note : I do not need a purely random graph, i. I simply need lots and lots of graphs for testing purposes of something else. Here is the Graph class I am using you can tell why I am only interested in connected graphs at the moment :.

Whatever you want to do with your graph, I guess its density is also an important parameter. Otherwise, you'd just generate a set of small cliques complete graphs using random sizes, and then connect them randomly.

With this model, if your p is large enough, then it's highly probable your graph is connected cf. In any case, if you have several components, you can also force its connectedness by creating links between nodes of distinct components. First, you have to identify each component by performing breadth-first searches one for each component. Then, you select pairs of nodes in two distinct components, create a link between them and consider both components as merged.

You repeat this process until you've got a single component remaining. The only tricky part is ensuring that the final graph is connected. To do that, you can use a disjoint set data structure. Keep track of the number of components, initially n. Repeatedly pick pairs of random vertices u and v, adding the edge u, v to the graph and to the disjoint set structure, and decrementing the component count when the that structure tells you u and v belonged to different components.

Stop when the component count reaches 1. Note that using an adjacency matrix simplifies managing the case where the edge u, v is already present in the graph: in this case, adj[u][v] will be set to 1 a second time, which as desired has no effect. Learn more.How can we create a random graph with random number of vertices and edges, so that the graph is connected? To start, you can generate a random, connected tree by doing a random walk, except each step of the walk actually creates a the edge.

This approach runs in O V. From there, you can add random edges to the graph. I have not found a great approach to this, so the current method I use is generate all possible edges that can be created ignoring the weights between any two nodes and then randomly choose from that list.

That generation of combinations takes O V 2 time. Not great but currently good enough for my purposes.

## How to generate a random connected graph by its adjacency matrix with a fixed number of edges?

Smart HR Software. Get the algorithms textbook. How to generate a random connected graph? Please log in or register to add a comment. Please log in or register to answer this question. If matrix can be used to represent the graph, one possible solution may be as follows "import java. Moreover, return how many edges the graph has now.

Is there a way to divide your method into smaller subroutines? Related questions. Give an example of a graph without an articulation point, a Hamiltonian cycle, or a 2-coloring. Graph with no 3-clique that needs at least 4 colors. Graph that has 10 vertices, is 3-colorable and number of edges is maximized.

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We want an algorithm to generate a n, m, k -bipartite selected uniformly at random in the set of all n, m, k -bipartite graphs. The idea is to first create a connected bipartite graph with the minimum number of edges by iteratively picking isolated nodes and connected them to the other independent set of the bipartite graph. For that we will create two sets N and M and create a first random edge connecting N to M. Then we will pick a random isolated node from either N or M and connected it to a random non-isolated node from the other set.

To add these last edges we simply pick random pairs of nodes from NxM. The bias comes from the correlation between the node degrees and the time at which they have been added to the connected component. Which means that in the worst case the bias is still slightly less important than in the first solution. Sign up to join this community. The best answers are voted up and rise to the top. Generating a random connected bipartite graph Ask Question.

Asked 1 year, 7 months ago. Active 1 year, 7 months ago. Viewed times. For now the only algorithm I have in mind doesn't achieve uniform sampling: The idea is to first create a connected bipartite graph with the minimum number of edges by iteratively picking isolated nodes and connected them to the other independent set of the bipartite graph. Here is the python code corresponding to that algorithm import networkx as nx import random random. Is there a know result for that?

Or any existing result that may help? Active Oldest Votes. Sign up or log in Sign up using Google.Sign in to comment. Sign in to answer this question.

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Open Mobile Search. Trial software. You are now following this question You will see updates in your activity feed. You may receive emails, depending on your notification preferences. How to generate a random connected graph by its adjacency matrix with a fixed number of edges? Schneebaer on 19 Dec Vote 0. Hello, I need to generate a random graph with n nodes an m edges, where n and m are fixed.

Moreover, the graph has to be connected. How can I do it? My ideas so far were to generate an adjacency matrix with random number of edges, to count the number of non-zeros afterwards and to correct the number then.

For the connectedness I counted the number of connected components and linked every component by an edge. But I'm sure that this can be done better. Answers 0. See Also. Tags adjacency matrix random graph connected graph. Start Hunting!In mathematicsrandom graph is the general term to refer to probability distributions over graphs.

Random graphs may be described simply by a probability distribution, or by a random process which generates them. From a mathematical perspective, random graphs are used to answer questions about the properties of typical graphs. In other contexts, any graph model may be referred to as a random graph. A random graph is obtained by starting with a set of n isolated vertices and adding successive edges between them at random.

The aim of the study in this field is to determine at what stage a particular property of the graph is likely to arise. Except in the trivial cases when p is 0 or 1, such a G almost surely has the following property:.

It turns out that if the vertex set is countable then there is, up to isomorphismonly a single graph with this property, namely the Rado graph. Thus any countably infinite random graph is almost surely the Rado graph, which for this reason is sometimes called simply the random graph.

However, the analogous result is not true for uncountable graphs, of which there are many nonisomorphic graphs satisfying the above property. Another model, which generalizes Gilbert's random graph model, is the random dot-product model.

A random dot-product graph associates with each vertex a real vector. This model is extensible to directed and undirected; weighted and unweighted; and static or dynamic graphs structure. Random regular graphs form a special case, with properties that may differ from random graphs in general.

Once we have a model of random graphs, every function on graphs, becomes a random variable. The study of this model is to determine if, or at least estimate the probability that, a property may occur. The term 'almost every' in the context of random graphs refers to a sequence of spaces and probabilities, such that the error probabilities tend to zero. The theory of random graphs studies typical properties of random graphs, those that hold with high probability for graphs drawn from a particular distribution.

Percolation theory characterizes the connectedness of random graphs, especially infinitely large ones. Percolation is related to the robustness of the graph called also network. Localized percolation refers to removing a node its neighbors, next nearest neighbors etc. Random graphs are widely used in the probabilistic methodwhere one tries to prove the existence of graphs with certain properties. In particular, the moment the last isolated vertex vanishes in almost every random graph, the graph becomes connected.

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