Prove a network is a feedforward network if and only if the numbering of its cells satisfy these conditions

Undergraduate math student here attempting to understand neural networks. Picked up a text on sale, "Neural Network Learning and Expert Systems" (Gallant), and I'm just starting on the exercises for chapter 1.

The full question is:

Prove that a network is a feedforward network if and only if its cells are numbered in a way such that:

• all $p$ input cells are numbered from $1$ to $p$.

• whenever cell $u_j$ is connected to cell $u_i$, then $j

The definition of a feedforward network, according to Gallant, is a directed network such that there are no cycles between any nodes.

I'm confused for a few reasons.

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This sounds like I'll need at least a basic understanding of graph theory, which I don't have. Any quick and dirty resources that should cover what I need for this kind of material? I don't even know how to represent a network like this mathematically.

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I'm used to the numbering of things being pretty arbitrary, for example, numbering basis vectors in a set just to differentiate between them. I don't really have an intuitive idea for why the numbering of input cells is a factor in deciding if a network has directed cycles or not.

Thanks in advance.

One direction (given such a numbering, there are no cycles) is elementary, and I encourage you to work it out on your own. The other direction is topological sorting.

StackExchange Computer Science Q#74964, answer score: 3