# Examining Virality

Virality: The act of content on the web being spread by users sharing it, bringing new users to the original content and therefore adding additional utility.

Virality is the noun for the adverb viral and it has been thrown about these days when it comes to the online world. Virality refers to a sort of new metric system used to determine the “pace” in which information is transmitted over the internets.Borrowed from biology and perhaps misused from the silicon tech and digital marketing community. Ironically the words “infection” or “infected” have been left out of the social media jargon perhaps due to the negative connotation it might have. Coming to think of it users who abstain  from social media platforms are immune  to virality.  So abstainace helps your “immune” system and keeps you in the dark ages. But humour aside an attempt should be made to formulate this phenomenon from a digital point of view.

The success of a viral application relies on number of posible receptors in an online ecosystem and the percentage of users that where indeed infected or converted. The formula below depicts the virality coefficient.

$k=icdot conv%$

The virality coefficient $kappa$  , where $i$ equals the number of posible receptors(or invites) and $conv%$ the conversion percentage. The assumption here is that each user sends his invites once is a single batch.

The following formula measures users over time.

$U(t)=U(0)cdotfrac{K^{(frac{1}{p}+1)}-1}{K-1}$

The time need for a new user to send invites is given by p. The number of epochs the invite process has gone through is represented by $frac{t}{p}$. The significance of introducing $p$ in the formula is that it shows that it’s easier to increace $U(t)$ by reducing $p$ rather than increasing $K$.

Verbalizing the aforementioned, lowering the amount fo time necessary for a user to invite other users to a site may be more effective than increasing the numer of invitations users send or the rate at which invited non-users convert. Lowering $p$ increases the power while increasing $K$ only increases the base.

In most cases $p$ is ignored. It is more likely than $K$ to be amenable to change. So perhaps it would be a good idea to invest in minimizing $p$.

An example of how the formula works is shown on the table below indicating how invitations increase the size of a user base over time. Say $K=2$ and $U(0)=5$ and $N$ the number of completed Epochs.

Epochs 0 1 2 3 4 5
New Users added this Epoch 10 20 40 80 160
Total Users 5 15 35 75 155 315

It is apparant that the New Users  row doubles every round. The number of New Users for round $i$  , is given by

$U(0)K^i$

and the Total Users  is the running sum of New Users; hence the total number of users is given by the summation

$U(t)=sumlimits_{i=0}^N U(0)K^i = U(0)cdot sumlimits_{i=0}^N K^i$

There is a known identity for sums of powers.

$sumlimits_{i=0}^{M-1}r^i=frac{1-r^M}{1-r}$

We use it here with $N=M-1$

$U(0)cdotsumlimits_{i=0}^{N}K^i=U(0)cdotfrac{1-K^{(N+1)}}{1-K}$

Multiply the term on the right by $-1/-1$

$U(0)cdotfrac{1-K^{(N-1)}}{K-1}$

Replace $N$  with $frac{t}{p}$

$U(t)=U(0)cdotfrac{K^{(frac{1}{p}+1)}-1}{K-1}$

Bringing us back to the original formula; hence in oder to increace $U(t)$ maximize $K$ and minimize $p$.

In one sentence: Make the users send out more invites and most importantly faster, you knew it, now you got the proof  :-)

Now that you know the math you understand why online poker became so popular the past 10 years.

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