In a completely connected network, how many lines are required for connecting 6 nodes?
15
Understanding Completely Connected Networks and Calculating Connections
In a completely connected network, also known as a full mesh topology, every single node is directly connected to every other node in the network. This type of topology provides high redundancy but can be very complex and costly to implement, especially as the number of nodes increases.
To find the number of connections (lines) required in a completely connected network with a specific number of nodes, we can use a simple mathematical formula.
Formula for Connections in a Full Mesh Topology
The number of lines required to connect 'n' nodes in a completely connected network is given by the formula:
$$ \text{Number of connections} = \frac{n \times (n - 1)}{2} $$
Let's understand why this formula works:
- Each node needs to connect to every other node. If there are 'n' nodes, each node needs 'n-1' connections to the other nodes.
- If we multiply 'n' (number of nodes) by '(n-1)' (connections per node), we get $n \times (n-1)$. However, this counts each connection twice (e.g., the connection from Node A to Node B is counted, and then the connection from Node B to Node A is also counted).
- Since the connection between two nodes is just one line, we divide the result by 2 to get the actual number of unique lines.
Calculating Lines for 6 Nodes
In the given question, we have a completely connected network with 6 nodes. So, the number of nodes, 'n', is 6.
Using the formula:
$$ \text{Number of connections} = \frac{n \times (n - 1)}{2} $$
Substitute n = 6:
$$ \text{Number of connections} = \frac{6 \times (6 - 1)}{2} $$
$$ \text{Number of connections} = \frac{6 \times 5}{2} $$
$$ \text{Number of connections} = \frac{30}{2} $$
$$ \text{Number of connections} = 15 $$
Therefore, 15 lines are required to connect 6 nodes in a completely connected network.
Analysis of Options
Let's check the given options based on our calculation:
- Option 1: 12 - Incorrect. Our calculation resulted in 15.
- Option 2: 15 - Correct. This matches our calculated number of lines.
- Option 3: 30 - Incorrect. This would be the result of $n \times (n-1)$, which counts each connection twice.
- Option 4: 24 - Incorrect. This does not match the formula or calculation.
The correct number of lines needed for 6 nodes in a completely connected network is 15.
Revision Table: Network Connections
| Network Type | Description | Formula for Connections (n nodes) | Example (n=6) |
|---|---|---|---|
| Completely Connected (Full Mesh) | Every node connected directly to every other node. | $$ \frac{n(n-1)}{2} $$ | $$ \frac{6(5)}{2} = 15 $$ |
| Bus Topology | Nodes connected to a single main cable. | 1 (the bus itself) + connections to bus (n) | 1 + 6 = 7 (simplified view) |
| Star Topology | Nodes connected to a central hub/switch. | n (connections from each node to central point) | 6 |
| Ring Topology | Nodes connected in a closed loop. | n (each node connects to two neighbours) | 6 |
Additional Information: Network Topologies Explained
Network topology refers to the physical or logical arrangement of connections (nodes and links) in a network. Understanding different topologies is crucial in network design.
- Bus Topology: All devices share a single communication line or cable. Simple and inexpensive for small networks, but a break in the bus stops all transmission.
- Star Topology: Each device has a dedicated point-to-point link only to a central controller (hub or switch). Less costly than mesh and easier to install/manage. If one link fails, only that device is affected. If the central hub fails, the entire network goes down.
- Ring Topology: Each device has a dedicated point-to-point connection with only two devices on either side of it. A signal passes along the ring in one direction. Simple but a break in the ring can disable the network unless it's a dual ring.
- Mesh Topology: Every device is connected to every other device (full mesh) or to some other devices (partial mesh).
- Full Mesh: As discussed, offers high reliability and dedicated links, but requires a large number of connections and is complex.
- Partial Mesh: Some devices are connected to all others, while others are only connected to one or two. More practical than full mesh for large networks.
- Hybrid Topology: A combination of two or more different basic topologies. For example, a network might connect several star topology networks together using a bus backbone.
The completely connected network discussed in the question is an example of a full mesh topology, known for its complexity in terms of wiring due to the high number of required lines.
Important Questions from Networking
In Gmail, what do you use when you receive a mail and want to send the mail in response of the same mail and to the original sender only?
- _______ are rules that exist at several levels in a telecommunication connection.
_______ is a type of dedicated file storage device that provides local-area network (LAN) users with centralized, consolidated disk storage through a standard Ethernet connection.
_______ is also known as protocol convertor.
Which of the following Internet protocol specifies how data is exchanged over the Internet and how it should be broken into IP packets?
