How to find the maximum independent set in a network?

To find the maximum independent set in a network, we can use the greedy algorithm.

The maximum independent set is a set of vertices in a graph where no two vertices are adjacent. The greedy algorithm starts with an empty set and adds vertices to it one by one, making sure that each added vertex is not adjacent to any of the previously added vertices. This process continues until no more vertices can be added.

To implement the greedy algorithm, we can follow these steps:

1. Initialize an empty set S.
2. While there are still vertices in the graph:
a. Choose a vertex v that has the fewest number of neighbours.
b. Add v to S.
c. Remove v and its neighbours from the graph.
3. Return S.

Let's illustrate this algorithm with an example. Consider the following graph:

```
A---B---C
| | |
D---E---F
```

We start with an empty set S. Vertex A has the fewest number of neighbours (one), so we add it to S. We remove A and its neighbour B from the graph:

```
D---E---F
```

Now we choose vertex D, which has only one neighbour E. We add D to S and remove D and E from the graph:

```
F
```

Finally, we choose vertex F, which has no neighbours. We add F to S and return S:

```
S = {A, D, F}
```

Therefore, the maximum independent set in this graph is {A, D, F}.