After Viral

Wherein I discuss, mathematically, what happens once your product stops going viral and starts getting real.

Is it viral?

This is a question so many start-up entrepreneurs and mature businesses launching new products ask. Lean Analytics describes it as the third stage a start-up goes through, preceded by Stickiness (getting people to use the product) and followed by Revenue (what businesses are supposed to generate from their product). It’s a good read, highly recommend. The Viral Coefficient is just a metric you track on a product, and it’s why every online retailer you visit has referral offers.

But the term is much older, obviously, growing out of basic concepts from epidemiology. And most of the writing in business about going viral focuses on the first part of the process, where number go up fast. But there’s more to it than that, and I’m going to write about it, because it’s kind of important to know what lies on the other side of viral, i.e. once your product is mature and the market is saturated.

So today we’re AcmeCo Inc, and we sell a Widget in our Widget Shop on Main Street. The Widget is unbreakable, and we have an infinite supply of them. And we’re going to talk about how we go viral, and what happens after that.

Going Viral

There’s two modes of growth to look at, one is just organic traffic, and the other is driven by word-of-mouth aka virality aka exponential growth. Both arise under a scenario of people walking into your shop.

Normal-y Growth

Let’s say there are $n$ people who walk past your storefront every day, with a probability $p_b$ of entering the store to buy one Widget. Then the rate of number of Widgets sold in a day would be $r = p \times n$. This rate is a derivative with respect to time, $n_W’$, and the rate-of-widgets-sold equation is simply

$$n_W’ = p_b \times n$$

and the solution to this is easy enough, it’s just $n_W(t) = p_b \times n \times t$, linear growth in the number of widgets sold.

This describes the baseline situation. You just opened and have no customers, but you do have people walking by your Widget store who could become customers. Some of them will walk into the store and buy a Widget! Your first sale! There’s a steady stream of customers but nothing big happening, at least not yet.

That’s not viral growth, but it’s the kickstarter: organic traffic.

Exponential Growth

Now, let’s say something social happens — people that bought your Widget are so excited that they tell their friends and with some probability their friends now go directly to your store to buy a Widget, so that the number of people who show up to buy a Widget is a mix of the organic traffic and people who heard about your Widgets from a happy Widget-owner. We’ll call the number of new purchasers that result from each happy customer $R$, and so now the rate of Widgets sold is proportional to the total number of Widgets sold plus the organic traffic:

$$n_W ‘ = p_b \times n + R \times n_W$$

I can solve this differential equation too, and it’s just

$$n_W = \frac{p_b \times n}{R} \left (e^{R t} - 1 \right )$$

And there it is, the exponential growth. This will look roughly linear in growth for $t < R$, then take off. R is the so-called viral coefficient we want to measure, make as big as possible, etc., and it tells us how fast the product will take off.

This is why every e-commerce site you visit has some offer for referring a friend. Having customers produce more customers is vastly more powerful than organic traffic, and quickly dominates the organic traffic part of your business. Because every customer could spawn two or three customers, who then spawn their own two or three customers, you’ve officially Gone Viral.

This picture works pretty well at first, because every new Widget-buyer tells a bunch of people who don’t own Widgets. But what happens when you get successful and the conversations stop looking like

“Have you seen these amazing Widgets?”

“No, tell me more!”

and start looking more like

“Have you seen these amazing Widgets?”

“Yes! I don’t know how I lived without one!”

The End of Viral

At some point, the finite number of humans on the planet starts to become a problem for your business model.

Everyone has a Widget already! And nobody needs more than one Widget! Which means that the number of customers in the organic sale term, $p_b \times n$, and the number of the people hearing about your wonderful product and wanting one, $R \times n_W$, is now modulated by the probability that they already have a Widget. Business-speak, this means that the number of people walking by your store or hearing word-of-mouth from Widget owners who already have a Widget and don’t need to buy a new one is going up, and fewer of those people are available as new customers to buy your Widget.

Logistic Growth Equation

The probability that they don’t already have a Widget is given by $(P - n_W)/P$, where here $P$ is the total population you’re selling to. This changes the differential equation above a little bit, so that now the rate you’re selling Widgets every day is

$$\frac{d n_W}{d t} = p_b \times n \times \frac{P - n_W}{P} + K \times n_W \times \frac{P - n_W}{P}$$

We can already kinda see what’s going to happen now just by looking at the crucial $P - n_W$ term. Once we get to $P$ Widgets sold, we will sell no further Widgets! The market is now saturated, and we either go out of business or figure out a new revenue stream, maybe it’s NuWidget or whatever.

I’ll spare the details of solving this first order nonlinear ordinary differential equation. Suffice it to say it involved partial fraction decomposition, a technique I haven’t thought about since I TA’d DiffEqs *mumbles* years ago, but it leads to this absolute unit of an equation:

$$n_W(t) = P \times \left \{ \frac{\exp \left [ \left ( 1 + \frac{p_b n}{P K} \right ) K t \right ] - 1}{\exp \left [ \left ( 1 + \frac{p_b n}{P K} \right ) K t \right ] + \frac{P K}{p_b n}} \right \}$$

The equation is $0$ at the beginning, and asymptotes to $P$ as time goes on. Did I show you that equation to impress you? Yeah, little bit. But it is instructive, and hits on a pretty key aspect of finite populations: at a certain point, you run out of customers.

This assumes that once you have a Widget, you don’t need another Widget ever again. That’s not totally true or accurate, and at some point a Widget breaks, or you release Widget2.0, and there are ways to model that too. I won’t, because diving into systems of nonlinear integro-differential equations and compartmental models is not really the point here. But the basic picture is pretty straightforward and robust to any changes to the model — at a certain point, you’ve been so successful your growth can’t be viral anymore.

What about my business?

Here’s roughly what this equation means for you as a business owner selling Widgets. In the very beginning, you need people to find your business, and your growth is entirely from organic happenstance — people walking by your storefront and deciding to buy a Widget. At some point, word of mouth starts to take over, and your business is driven more by that than random happenstance — folks hear your Widget is amazing, and come to get some. But eventually, everyone who could buy a Widget from you has bought a Widget, and you’re stuck with nobody to sell to, and that’s bad.

Let’s say your revenue comes from selling a Widget once. Then your daily revenue is going to be the price you charge, $p$ (I’m running out of letters fast, here), times the number of units you move per day, which is $n_W’$. Well, that looks like this as a function of total Widgets sold:

Your peak revenue will be at the point that you’ve sold $P \times (1 - p_b n / P K)/2$ units — roughly half the population if the entire addressable market doesn’t walk by your store every day — and give a revenue of $p \times P \times (1 + p_b n / P K)^2/4$ — or roughly a quarter of the addressable market under the same assumption. After that, your daily revenue goes down.

Now, what if your Widget has a software subscription that comes with it? Well, that describes your initial revenue for selling the Widget hardware, but now you get to charge every sold Widget the price of the subscription as well, and that means you’ll be collecting that money from the entire addressable market in perpetuity. It’s not as lucrative as the initial hardware rush, but it does give a steady stream of revenue atop the one-time sale.

And that, friends, is why Thing as a Service is so popular these days.

But even with that, that’s essentially stable revenue, and it’s why businesses don’t see the sort of explosive growth they see when they first hit the market with their new Widget.

You can’t.

At least, not after you’ve gotten past the stage of widespread adoption, and start having to contend with basic demographics.

Virality in a Competitive Market

Of course, you’re probably not the only Widget-seller in town. We live in a market economy, and someone’s going to see the success of your Widget store, and open a competitor. Then we can model all this with a small modification of the Competitive Lotka-Volterra Equations, which look like our logistic growth curve, but now we saturate when everyone in town has bout A Widget, and not just Your Widget.

$$\frac{d n^{(b)}_W}{d t} = p_b \times n \times \frac{P - \sum_b n^{(b)}_W}{P} + K^{(b)} \times n^{(b)}_W \times \frac{P - \sum_b n^{(b)}_W}{P}$$

where we’re summing over all businesses $b$ in the competitive space. Each business has their own $K$-value — the rate at which a happy customer will recommend a new customer — but the same rate of organic traffic $p_b \times n$.

It’s an interesting differential equation if you’re the type of person who finds differential equations interesting. But the upshot is basically the same as before: at a certain point, most people have a Widget, and they don’t need to buy a Widget, so your exponential growth phase of Widget sales will end. Being first to market probably helps with market share, as does being more viral (“I know they’re the same Widgets as across the street, but Acme Widget Company really walked me through the process and made me feel at ease with my Widget purchase”), but at some point finite population effects catch up with us all.

Business After Virality

Once your business starts hitting the end of virality, some sort of pivot is going to be necessary. Maybe it’s time to release Widget2. Maybe it’s time to produce an add-on to improve the functionality for Widget at cleaning bird baths. Perhaps it’s time to pivot to Widget as a Service and charge subscriptions to operate the Widget everyone bought. That is going to depend on your business model, what you think your customers will tolerate, and what the competitive landscape looks like.

But there are warning signs. The first is that the virality coefficient you’re measuring starts going down, and there’s nothing you seem to be able to do about it. This means you’re out of the exponential range, virality is starting to end, and pretty soon you’ll saturate as a business.

Another warning sign is the fraction of estimated total addressable market you’ve gobbled up. If you know how many major competitors you have in a market, and what the total addressable market is, you can start to estimate the saturation point from the dynamics of the Competitive Lotka-Volterra Equations above. A simple assumption would be that each business has the same $K$-value, and assume they’re the same as yours. Then you can figure out the saturation distribution of Widgets sold, based on a few assumptions. When you start getting close to half your allocation it’s safe to bet that you’re about to start saturating. Again, what you can do about this depends on your specific business model, but it’s an alert that it’s time to start thinking of the Next Big Thing.