We need to talk about K = viN…

Every tech entrepreneur dreams about making a product that goes viral, but true virality is as rare as a unicorn and poorly understood. A proper understanding of the viral coefficient, K, can give insights that enable a more systematic approach to maximising it. Because as far as virality is concerned, K is hiding something pretty important…

I’m going to distinguish different types of virality and go into detail about the various ways to model it. I’m also going to introduce the idea of “pre-virality” and why it shouldn’t be discounted. The stakes are high, viral growth means companies literally don’t need to spend a penny on marketing, and yet achieve explosive popularity and success. However, unless you have a maths background, virality can be quite counter-intuitive. (Beware, this article contains equations!)

My experience of viral growth

I’m no stranger to virality. I’ve been watching and playing with viral experiments since 1999. I’ve witnessed abject failures as well surprising successes. My interest began with setting up an experimental R&D team within moonfruit.com (a dotcom started by my two besties from Imperial, Wendy and Eirik). My team built intelligent agents to help people discover the web by learning how and what they searched for. The agent project never came to light (funding was pulled), but, on the way, we released two interactive desktop toys to test out some of the cool tech that we were building, and thought they would be good viral branding experiments. Neither went viral, instead they both quickly died out upon release. That was the end of that. A few years later, I noticed that suddenly everyone, it seemed, was inviting me by email to sign up to a web service called Birthday Alarm. It was clearly going massively viral. I didn’t know it at the time, but it turned out this was the brainchild of one of my physics classmates from Imperial. Michael had been experimenting both systematically and persistently, and finally hit a jackpot. He went on to create and sell Ringo and then Bebo, so it was well-worth the effort. Then in 2009, Wendy masterminded a simple but ingenious Twitter competition which went so viral that at one point more than 2% of all tweets globally contained the #moonfruit hashtag. Twitter actually pulled the campaign it was so disruptive. Interestingly, copycat competitions failed to go viral. More recently, I had a surprising, albeit smaller, viral success in 2015 when an article about how to learn a language quickly that I wrote on Quora suddenly started sending so much traffic to my latest startup, Kwiziq, that we upgraded our servers to cope. It was something I’d written months before and completely forgotten about. Unbeknownst to me it had been growing slowly and then suddenly, boom!

Virality is no easy thing to achieve – and certainly comes with no guarantees – but all my experience has shown me that it is possible to approach it systematically and tip the odds in one’s favour. I decided to write this article to pull together all my own thoughts and learnings over the last sixteen years because at Kwiziq we are trying to grow as virally as possible.

The classic viral loop

This is how the classic “viral loop” is generally viewed:


Intuitively, we see that, if the average number of people invited by each user, Nmultiplied by the percentage that accept the invitation, i, is greater than 1, then a positive feedback loop is created since each user effectively generates at least one more.

For this loop then, we define the viral coefficient as:

 K = i N 

This is a great starting point for understanding virality, but as we will see, the problem with it is that it obscures more than it reveals. As a result, it gives almost no insight into what to do increase K. We also mustn’t simply accept this is a good model for the realities of word-of-mouth sharing, without question, because a bad model will misguide us. We are need to understand the viral process in more detail.

Viral Models

All models are wrong,” as we physicists like to say, “but some are useful.

How we model virality for a product depends very much upon when and how people can and do share it or invite their network. Models are idealisations and therefore always wrong in some way, but they have two very important roles to play in a startup:

  1. They can help predict growth, but more importantly :-
  2. They give us insights that can guide product and process design

So the choice of model is rather important. There are two popular models which display very different behaviour and yet both mathematically model the same loop.

The Steady Sharing Model

The simplest model (often the starting point for more sophisticated models used by viral blogger Andrew Chen) bases growth on the idea that user base will always bring in a proportion, k, of new users (via sharing) every viral cycle.

Note that we don’t actually need to know what the viral cycle length is, but we’ll see how important it is later.

If we start with a number of users,  U_{0} then the total number of users we have after one viral cycle, or  \Delta t is:

 U_{\Delta t} = U_{0} K

where  K = 1 + k

e.g. if k =15\% growth then K = 1 + 0.15 = 1.15

The number of users after an arbitrary time,t, is just this formula iteratively applied to itself  t / \Delta t times:

 U_{t} = U_{0} K^{(\frac{t}{\Delta t})}

This formula acts exactly the same as compound interest on a principle amount. So long as  k is more than zero, you’ll get some growth. But, the model is only valid if users do actually continuously share.

If we ignore network saturation for now – i.e. if we assume that for the first twenty cycles we still have plenty of room to grow – then Steady Sharing looks like this for varying values of K:


At very low values, there’s minimal contribution to growth.  Effects like this would easily be masked by any inbound or paid marketing channels. As K grows though, since the growth is exponential, the viral channel would soon make itself very clear indeed.

The Big Bang Theory

David Skok proposes a replacement model for Steady Sharing which I call the “Big Bang” viral model. In this model new customers invite / share to all the people in their circle of influence all in one go in the first loop of the cycle but then stop. The formula for growth in this model is slightly more complex, and  K here has a slightly different meaning.

If we start with a number of users,  U_{0} then if every new user subsequently successfully brings  K new users within one viral cycle, or  \Delta t is:

 U_{\Delta t} = U_{0} K

bringing the total users to  U_{\Delta t} + U_{0} .

(Note that K doesn’t have to be an integer – if a thousand users brings 33 more, say, then K would be 0.033).

Since only new users bring new growth, the equation for the number of users after time t is a little more complex:

U_t = U_{0} \cdot {K^{(\frac{t}{\Delta t}+1)} -1 \over K - 1}

(if you’re curious why it’s that, there’s a nice derivation here).

At large values for K, the two models behave in an increasingly similar manner (Steady Sharing lags by one cycle), however for low values (compare the graphs on the left for both models), the two mechanisms are starkly different:


A K value of 1.0 produces linear growth, above 1.0 produces exponential growth but below values of 1.0 this type of sharing dies out; the number of users reached approaches a limit.

Models versus reality

So, which of these models is true?

Neither. And both.

Skok claims the Steady Sharing Model is wrong, but it’s no more wrong than any other model. Which model is more accurate and applicable depends on the type of product, the user behaviour, and how sharing is enabled. Indeed, it’s possible for one, both or neither to apply.

Some products naturally support steady sharing – indeed multiple opportunities for users to share to the same people: content-based products like YouTube, for example. Others encourage Big Bang invites of entire contacts list, all in one go.

The Big Bang model is a very good fit for one-offs: articles, books, songs, videos etc. – things that people experience once and where there’s an immediate desire and opportunity to share but little incentive to return after. It’s also a good fit for products that haven’t achieved strong stickiness yet or where only one opportunity to invite people is given.

How do these models match reality though?

Learning from real examples

I don’t have enough data or insight for the moonfruit competition or Birthday Alarm, so I’ll pick two of my own examples.

Example #1 Desktop Toys

These were simple .exe files that we sent to a seed base of users – by email – and tracked the spread via code in them that pinged our servers when they were activated. They pretty much followed exactly the Big Bang model with a K value somewhere less than 0.5 (I can’t recall the seed user number exactly but we didn’t get a large multiple of the seed users). We were’t expecting miracles – we knew that sending a .exe install file was a pretty hefty ‘payload’ for a viral to carry and that a proportion of people would be suspicious (even though it was a harmless install).

We were disappointed naturally, but if I knew then what I knew now, I’d have had a better idea of how to measure and increase K.

Example #2 Quora article

The Quora also went viral in very much a Big Bang way, at first anyway, only slowing when it reached saturation in the primary network – roughly a million views (reaching what Chen calls the network’s “carrying capacity”).

However, the growth didn’t slow to zero as predicted by Big Bang.

In fact, it went pretty linear and still garners about 5-6k views a month which I’m more than happy with as it generally keeps me top of this list and continues to drive traffic to Kwiziq. In reality, it didn’t follow either model closely.


All models are wrong. In the real world, things are fantastically more complicated than our simple models. In reality, it’s quite likely that some users will be Steady Sharing (I know several teachers who share it ongoingly, for example), and others will Big Bang share when they encounter it; and others still who do both. It’s possible to a be a hybrid of both models.

Furthermore, and this is vital to understand, the source of new users (or readers in this case) is not limited to those generated by the sharing of the article. New readers are being brought in by Quora (internally via emails they send out highlighting popular articles, and via external search). They also get traffic from links from syndicated versions of the article such as these Huffington Post and Business Insider versions.

Neither of the models account for external sources to the viral cycle. In both though, this would add an extra boost to the number of users in any given cycle.

Finally, neither accounts for multiple sharing mechanisms.

The article gets shared by email, Facebook and Twitter as well, to name a few (Twitter being the most obvious to me as I often get an @gruffdavies mention).

So in fact, instead of one value for K, we really have one for each of channel and they will be different.

The important learning from this is that, even if the sharing mechanism is Big Bang and has a K lower than 1.0, if there are other mechanisms for introducing fresh users each cycle, then K becomes a magnifier.


If we have external sources of users (paid or inbound marketing for example), then even lower values of K than 1.0 are potentially still hugely valuable. Values between 0.5 and 1.0 are what I called pre-viral and still well worth attaining, because each user then multiplies into more, even though the effect is self-limiting:

K value Multiplying Effect
            0.5 200%
            0.7 333%
            0.8 495%
            0.9 891%

Using models as a guide to design

Models aren’t just good for telling us how a product or service might spread, but can guide our tactics and strategies underpinning growth. They can and should guide product and process decisions. 

It’s clear that non-viral sources of acquisition should not be overlooked. Any web-based service is going to have SEO and paid marketing channels – and these channels can help to continuously feed viral growth mechanisms even when the K values are low. It’s also clear that, if possible, steady sharing mechanisms are going to have a huge benefit in the long-term.

There are obvious major learnings to be had immediately from these models, and a third which may be less obvious:

  1. Big Bang is better than Steady Sharing for high values of K, but Steady Sharing produces continual growth at low values, therefore the ideal scenario should support and encourage a hybrid of both.
  2. In Steady Sharing for all K, and Big Bang for K > 1.0 reducing the sharing cycle time has an extraordinary impact on growth because it’s exponential. Optimise for the shortest possible cycle times.
  3. K is actually the combined result of multiple steps of a funnel. Each step reduces K by a factor. So decompose K into its components and then optimise each.

All this brings me to the title of this blog post.

We really need to talk about K = viN

Kwiziq already supports multiple sharing mechanisms and one invitation mechanism (teachers, tutors or studygroup coordinators can invites students to join). We haven’t supported users-inviting-users yet but this is high on our agenda. However, it seemed obvious to me that, with respect to virality, because we only support one language right now, this dramatically reduces K.

The classic viral loop is missing a vital detail: relevance to the user. A user won’t even think about accepting an invitation unless the offer is relevant to their needs.  In our case, if they aren’t learning French then they won’t consider accepting. So the viral loop really looks like this:


Only a percentage of users invited, v will find the product relevant, of which a further percentage, i will accept, which gives us:

 K = v i N 

Very high relevance is a key reason that Michael had such success with Birthday Alarm – everyone has a birthday and everyone has people whose birthdays they want to remember. It’s the reason moonfruit’s twitter campaign was so successful – almost everyone wanted a MacBook and the “cost” of retweeting a hashtag to have the chance win one was negligible to the user.

We estimate that a third of the world speaks another language badly (essentially our market) which puts an absolute upper limit on v. In reality, though it’s much smaller currently: it’s whatever proportion of people are interested specifically in learning French. So clearly, offering as many popular languages as possible has the potential to multiply the current value of v (and therefore K) by significant multiples.  By comparison, any efforts to optimise the acceptance rate, i, are going to be limited and far harder to achieve per percentage point than expanding into new languages.

We already know it’s possible to achieve virality with more languages since at least three of the major players in the language learning space grew virally to 30M – 150M users over the space of a just a few years after they started offering the most popular languages.

Analyse Feasibility: the Kevin test

The K = viN equation lets us do a decent feasibility study on whether we even stand a chance at making a product grow virally. Let’s plug some values in. We first need to know the maximum potential values for each based on our knowledge of the product and the market. Note, these will be way higher than anything we could achieve, we’re just looking for an absolute ceiling value because if this figure isn’t viral then we can’t ever hope to achieve it. We’re a looking to see if our ceiling value is greater than at least ten here because we need to be realistic about achieving only 10% of the ceiling = 1.0 for virality), or there’s just no point. Next we’ll use what we think our current starting values could be and then, target values that we hope are attainable.  For N, I’m going to use the median number of Facebook friends people have as this is a published figure: about 200.  The mean figure is higher at 338, but we should always use conservative estimates.

For Kwiziq, then, the K = viN feasibility analysis looks like this:

Relevance Invite
v i N = K
ceiling :  33% 50% 200 33 !!
est. starting : 5% 5% 200 0.5 pre-viral
target : 15% 10% 200 3 hugely viral

So, Kwiziq passes the “Kevin feasibility test”.

Detailed viral analysis

This is just the beginning of what we must do in analysing the viral loop though. It behooves us to understand every single step in the funnel, whether it’s a real step involving a physical action, or a step in the user’s decision process in their head. What we need is something reminiscent of the Drake Equation.

Here’s a an example as a starting point for a specific product viral loop, but in real life, this should be an ongoing process of identifying and optimising every possible step of attrition that can reduce  K, including the UX itself that could inhibit sharing at the top of this cycle.


There are fairly obviously a great deal of potential UX improvement steps between a user joining, wanting to and being easily able to invite others that will be worth careful attention.


“If you cannot measure it, you cannot improve it” – Lord Kelvin

One final piece of advice, measure everything. The minute you have a measure to track, you’ll start thinking of ways to improve it.

You may not always have direct measures (how do you measure real word-of-mouth, for example?) If, like Kwiziq, your product supports many different acquisition types and channels, and is a hybrid viral model, then you need to look for indirect measures that will indicate when they are working.

I would suggest you track something like the ratio of New Paying Users to New Registrations per week/month. This is especially revealing because paying users will contain most of your “sneezers” (they like your product enough to pay for it, right?). New users coming though viral channels will start to make this metric rise non-linearly at the beginning (it will eventually become asymptotic towards 100% if viral takes over). Your efforts to improve UX and address anything that’s reducing K ought to be visible here early. If your paying users are growing faster (at higher order) than new users, you’re doing something right!

For a “freemium Saas” product like Kwiziq, measuring the ratio of new paying users to all new users will reveal any viral contributions to growth before they become apparent in your overall acquisition measures

NB: you will need to understand and analyse any other mechanisms that can cause non-linearity here to be sure, but over time, a metric like this ought to detect viral growth well before it becomes visible in your overall acquisition metrics.


Good luck!

P.S. The flow diagrams in this blog post were created in seconds using Pidgin.

Published by Gruff Davies

Gruff is a British scientist, tech entrepreneur, and author. He was named one of the World's Top 50 Innovators by Codex in 2019 at the Royal Society, London.

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