An electrical network is a system composed of interconnected elements, such as resistors, capacitors, inductors, and voltage or current sources. Unlike a circuit, an electrical network does not necessarily form a closed path. In other words, while all circuits can be considered networks due to their interconnected nature, not every network qualifies as a circuit.

A circuit, on the other hand, is also an interconnected system of electrical elements but must contain one or more closed paths. These closed paths, or loops, allow current to flow continuously, making the operation of the circuit possible.

The structure of a circuit can be broken down further into several key components: branches, nodes, loops, and meshes. Understanding these components is crucial for analyzing and designing circuits.

A branch represents a single element of the circuit, like a resistor, a voltage source, or a current source, connected by two terminals. For instance, a circuit that includes a voltage source, a current source, and two resistors would consist of four branches.

Nodes are points in the circuit where two or more branches intersect. They are typically represented by a dot in circuit diagrams. If a short circuit connects multiple nodes together, they effectively become a single node.

A loop in a circuit is a closed path that passes through several nodes without traversing any node twice. Loops can be independent or dependent. An independent loop has at least one branch that is not shared with any other loop.

A mesh, on the other hand, is a special type of loop. It is a closed path that does not contain any other loops within it. Meshes are fundamental to certain types of circuit analysis, such as mesh analysis, which systematically solves for current values.

Lastly, all these elements are tied together by the fundamental theorem of network topology, which states that for a network containing 'b' branches, 'n' nodes, and 'l' independent loops, these quantities are related by a specific formula, given by

Understanding this relationship is crucial for the analysis and design of electrical networks and circuits.