Why is it called network “topology”? Why not network “graph”?

Question

I want to understand why we use the term "network topology" as opposed to "network graph", or another term, to talk about the structure of networks. I'm working on a network design for a project, and want to make sure I don't use any terms that I don't truly understand.

Wikipedia defines network topology as "the arrangement of the various elements (links, nodes, etc.) of a computer network." This strikes me as interesting, because when I hear the words link and node, I immediately think of graph theory and the objects it is concerned with.

Topology, according to Wikipedia again, is "concerned with the properties of space that are preserved under continuous deformations..." And when you look at the basic examples of topological objects, you see coffee cups and Möbius strips, as opposed to the discrete vertices and edges you see with graph theory.

So why do we refer to networks as having a "topology"?

Answer 1

Topology is the study of fundamental properties that exist no matter the arrangement of elements (for example the ratio of points, edges and faces in a polygon that does not change as you add or reduce points).

This term fits networks well because what you are looking for with network topology is the different arrangements of the network to perform the same function, with the goal being to produce the simplest configuration that you can to perform the task (thus eliminating waste)

A graph has a much more limited definition in mathematics and is not really about discovering the fundamental properties of arrangements.

Answer 2

Each graph is a topological object. The main properties of a graph are the nodes as well as the links between them. Those are properties which are preserved under continuous deformations. Thus the term "network graph" can surely be used in exchange for "network topology". The latter is probably more widely used; it highlights the "topological" properties a network like "which node is connected to which other node(s)", whilst ignoring properties like line length, band-width or other "non-topological" / "non-graph" properties.

However, I am pretty sure that has simply historical reasons, and not because it really fits better in a mathematical sense. And maybe "network topology" just sounds "more cool" to a lot of people than "network graph", as @CormacMullhall wrote in his comment ;-)

Answer 3

Let me offer some perspective from a mathematical side. I'm certainly no PhD mathematician, but I did just graduate with a math degree and took two graduate courses in topology this past year (point-set and algebraic) so I'll offer the little bit of knowledge I can contribute :) This is my first time posting to this site so I hope I can help. Warning, this answer is a bit long and contains some pretty abstract math ideas but I'll try my best to make it digestible and worth your while.

Like a lot of things in math, a topology is basically, at its core, just a special type of set that satisfies a few special conditions (however, if you want to get fancy there is something called a category of topological spaces, it's a fundamental example of a category in the subject of Category Theory). This special set determines what kind of sets we call "open" sets. It's definitely doable but a bit weird if we define a topology without defining a topological space, so I'll do that first. Specifically, this is what the definition of a topological space is:

A topological space is an ordered pair of two sets $(X, \tau)$ where $X$ is any general, arbitrary set, and $\tau$ is the topology. A topology must satisfy the following conditions:

1. $X$ and the empty set, $\emptyset$, are elements (denoted by the $\in$ symbol), of $\tau$.
2. The arbitrary (read: finite or infinite, countable or not) union of elements of $\tau$ is in $\tau$, and
3. The finite intersection of any elements in $\tau$ is in $\tau$.

As long as this special set of subsets, $\tau$, satisfies these three properties listed above, $(X, \tau)$ is a topological space. There exists an analogous treatment of this definition that uses closed sets (a closed set is the set complement of an open set) and De'Morgan's laws but it's logically equivalent so it's not really important.

In math, continuity and continuous functions are everywhere and really important. The concept of continuity you see in calculus (explained more rigorously in real/complex analysis courses) can be generalized to the concept of mapping open sets to open sets. Well, now that we have this definition of topology and a topological space, we can see the utility of topology because now we can examine the topology that the continuous function is based on. More specifically, since there are many different kinds of topologies and topological spaces, and a continuous function just maps an open set from the domain to an open set in the range/codomain (these two words are not equal but in this case it doesn't matter), so now we can have some pretty unique kinds of continuous functions. Okay, so that's enough about the math, here is my idea about why it's called a network topology instead of a network graph.

I lied, before we delve into why I think it's called a network topology, there is just a bit more math that came to mind that I should probably discuss. In math, there are these things called simplicial complexes which are just sets containing points, lines, triangles, and their n-dimensional counterparts all connected in some way together. Basically, a simplicial complex is just a collection of these geometric objects listed above that are in some way connected together. For instance, if we take a triangular pyramid, have a random dot sitting outside of it, connect a line to the tippy top of the pyramid, then connect a triangle to the end of that line segment, that would be a 3-simplicial complex because the highest dimension of any of those geometric objects is 3.

Now, the abstraction of this is called (it has a boring name) an abstract simplicial complex. Nice name right? These objects are just combinatorial abstractions of standard simplicial complexes but they allow us to examine and study things called d-skeletons. A 1-skeleton of an abstract simplicial complex is just the underlying graph of the structure which is just an undirected graph. Now we can get into the network stuff, I swear.

As you know, networks are just graphs. If I had to guess, it's called a network topology because not only does the name sound cooler (believe me, this is definitely an important and valid reason for something to have a name. It makes people more interested and nobody can resist sounding smart by saying this word), but also because the structure of the network when viewed as a graph (like, literally draw the nodes and what not) is a 1-dimensional skeleton of some kind of abstract simplicial complex--an algebraic topological object.

Do I for sure know that the person or persons who coined this term knew all this? No, but I do doubt it anyways. A lot of algebraic topology is pretty recent math in terms of history so it's not impossible that a mathematician or computer scientist who coined the term network topology knew all this, but I wouldn't be surprised if they didn't. It's not a bad thing, it just is what it is. I conjecture that somewhere between 90%-100% of the reasoning that the term is "network topology" instead of "network graph" is because it sounds cooler and whoever coined the term had some kind of passing familiarity with the subject of topology and thought that what he was doing "was basically topology" so hence that is the term that won. This is just my opinion but if I was a betting man this is what I'd put my money on.