RBG Random Bipartite Graph

A random bipartite graph (RBG) is a type of graph that consists of two disjoint sets of vertices, where each edge connects a vertex from one set to a vertex in the other set. The term "bipartite" refers to the property that the vertices can be partitioned into two independent sets, and "random" indicates that the graph is generated according to a random process.

To understand RBG, let's break down the term:

1. Bipartite Graph: A bipartite graph is a graph whose vertex set can be divided into two non-empty sets such that no edge connects vertices within the same set. In other words, if we have a bipartite graph with vertex sets V1 and V2, every edge in the graph connects a vertex from V1 to a vertex in V2.
2. Random: The random aspect of RBG means that the graph is generated using a random process or algorithm. Randomness can be introduced in various ways, such as by randomly choosing the number of vertices, randomly assigning edges between the two sets of vertices, or randomly selecting properties of the vertices or edges.

Generating an RBG typically involves the following steps:

Determine the sizes of the two vertex sets: First, you need to decide the number of vertices in each of the two sets. Let's denote the sizes of the sets as |V1| and |V2|.

Create the vertex sets: Create two disjoint sets V1 and V2, each containing |V1| and |V2| vertices, respectively. These sets will represent the two parts of the bipartite graph.

Assign edges: Randomly assign edges between the vertices of V1 and V2. The number of edges can vary depending on the desired density of the graph. One common approach is to specify the average degree, which is the average number of edges connected to each vertex.

There are different ways to assign edges randomly. One method is to consider each possible pair of vertices from V1 and V2, and for each pair, decide whether to add an edge between them with a certain probability. This probability can be uniform (each pair has the same chance of being connected) or can be assigned based on some distribution.

Another approach is to fix the number of edges, randomly select pairs of vertices, and add an edge between them without replacement until the desired number of edges is reached.

Output the RBG: Once the edges have been assigned, the RBG can be represented as a set of vertices and edges or in an adjacency matrix form, where each row corresponds to a vertex in V1 and each column corresponds to a vertex in V2.

RBGs are used in various applications, such as modeling interactions between two distinct groups of entities or representing relationships between two types of objects. The random nature of RBGs allows for the generation of different graph structures, enabling the study of graph properties and the analysis of algorithms on randomly generated graphs.