We will be investigating the value of network effects, a term referring to phenomena where “the more people use a product, the more valuable the product becomes”. Now, this may seem intuitively true, but why should it? After all, while receiving some Lego set when I was 10, I am certain that I would not have enjoyed the product more knowing a child in Switzerland was too. Well, I was not influenced by my Swiss friend because the value of a network largely depends on the type of product. Industries like social media (or online article publishers) are valuable largely because they establish spaces where I can connect with other users (friends to communicate with, potential readers, etc.).
Definitions for components of a network (left) and an example of a small-scale network with 4 users and 3 connections (right)
So for particular products we see that their value increases as the number of users increase. But as scientists, it is not only desirable to know a relationship is generally positive, but also determine how that relationship changes. In other words, “how do we mathematically quantify a product’s value to its number of users?”
This article will provide an introductory treatment about how networks are valued through a couple of proposed mathematical formula, while emphasizing their assumptions, derivation, and applicability. For us, this will only require some basic combinatorics (counting)! We will then look at a more realistic model utilizing a matrix formulation which conveniently generalizes our observations about networks.
Let’s think of the simplest possible network with $n$ users: a network without any connections! This may seem a little silly, but it is often useful to think of these limit cases when analyzing models.
A simple network with 4 nodes and no connections
In a network with $n$ people and no connections, the only value we have comes from the people themselves. Where may this be applicable? Well, if we are tasked with selling something simple like lemonade (and our parents paid for the supplies), we would earn a fixed amount of money for every person served. So here, we could think of this customer base as a non-interacting network. It follows that since every customer returns a fixed value, the total value will just be proportional to the total number of customers/users:
$$ V_{1}(n) = k_1n $$
Here, $k_1$ is simply a constant of proportionality; think of this as changing units from “user amount” to the “dollar amount”.
This equation is called Sarnoff’s Law*. Though it may seem simple, this was actually proposed when looking at the value of TV networks, which are delivered to customers who do not “interact” while using the service. This could also apply to streaming services like Netflix, web portals, etc.
*A technical note: The network effect this article explores is determined by the user’s experience. In this example, though, we mention a company gaining value. However it’s plausible that as a company’s value increases, the cash available to improve the user’s experience also increases linearly.
Now let us think about the other limit case: our network has as many connections as possible! (Here, we assume that there is at most one connection between two users). Since we have both users and connections, how would we determine the value of our network? Well, let us think about a network which is built on many connections, like a community of buyers and sellers. For someone looking to join this selling network, they aren’t really as concerned about the amount of buyers and sellers but rather how many transactions occur between them. This is because we gain money in these transactions. Similarly, for a highly connected network, it seems reasonable to assume that the value is characterized by the connections and not the users.
So, that means to find our second value function $V_2$, we need to find the number of connections in this maximally integrated network. Well, by our definition of connections between users, every user is connected to every other user. We need to count how many “lines” we have resulting from these connections.
A network with 4 nodes which is maximally connected
Let’s go over a small case first. How many two-member teams (connections) do we have in the four-person group Alice, Bob, Cheryl, and Don? Well, we only need two people to form a connection. There are $4$ choices for the first teammate and $3$ choices to for the second teammate, giving a total of $12$ pairings. However, we actually double-counted! For example, we have including teams where Alice was first chosen, Bob was second chosen and vice-versa. So our final answer is $\frac{12}{2}=6$ total teams. Similar reasoning implies that the number of 2-people connections for a group of $n$ people is given by
$$ \text{connections}={n\choose2} = \frac{n(n-1)}{2}. $$
A sanity check: plugging $n=4$ into this equation yields $6$, exactly what we calculated for our small example. Since we assumed that the value of our network is primarily driven by our connection number, we finally have