**A. Problem**

If you have a Twitter team who mutually retweet each other's posts, and each member posts the same number of original tweets per day, how large must the team be in order to send one tweet per second, one tweet per minute, or one tweet per hour in your chosen hashtag?

**B. Solution**

Suppose you have two groups of people that shall implement your Twitter campaign to promote a particular hashtag:

**Influencers--**Their task is to write original tweets using the hashtag.**Retweeters**--These are the Tweet amplifiers. Their job is to share the tweets of the Twitter influencers. They may be a pure retweeter, i.e. he doesn't write his own tweet, or he may also be an influencer, too.

Let $n_I$ be the number of Influencers and let $n_R$ to be the number of pure Retweeters. If $n_{T/I}$ is the average number of original tweets per influencer per day, then the total number of tweets that your influencers can write is $n_{T/I}n_I$ and the total number of retweets is $n_{T/I}n_I n_R$. Thus, the total number of tweets that your marketing team can produce per day is \begin{equation} N = n_{T/I}n_I n_R + n_{T/I}n_I = n_{T/I}n_I (n_R + 1). \end{equation} For example, if you have 5 influencers tweeting 6 times a day and you have 10 pure retweeters, then the number of tweets you can produce per day is $6(5)(10+1) = 330$.

If you include your influencers as part of your retweeter team, provided that they cannot retweet their own posts, then we need to modify our formula to \begin{equation} N = n_{T/I}n_I n_R + n_{T/I}n_I + n_{T/I}n_I(n_I - 1) = n_{T/I}n_I(n_R + n_I). \end{equation} For example, if you have 5 influencers tweeting 6 times a day and you have 10 pure retweeters, with your influencers allowed to retweet posts of other influencers, then the number of tweets that your marketing team can produce per day is $6(5)(10+5)= 450$.

One special case is when you have no pure retweeters, $n_R=0$, so that your influencers become a mutually retweeting network. Then the number of tweets your team can produce per day reduces to \begin{equation} N = n_{T/I}n_I^2. \end{equation} That is, the number $N$ of your tweets per day is proportional to the number $n_I$ of your influencers and the number $n_{T/I}$ of tweets per influencer per day. For example, if you have 5 mutually retweeting influencers with each influencer posting 6 original tweets per day, then the total number of tweets your team can produce per day is $6(5^2)=150$.

Now, let us assume that your team consists of $n_I$ mutually retweeting influencers who can produce a total of $N$ combined tweets and retweets. The time interval $\Delta t$ between tweets that your team produced for a particular hashtag for one day would then be \begin{align} \Delta t &= \frac{t_D}{N} = \frac{t_D}{n_{T/I}n_I^2},\\ \end{align} where $t_D = 24\ hr = 1,440\ min = 86,400\ s$. Solving for $n_I$, we get \begin{equation} n_I = \sqrt{\frac{t_D/\Delta t}{n_{T/I}}}. \end{equation} That is, the number $n_I$ of mutually retweeting influencers to produce one tweet every time interval $\Delta t$ is inversely proportional to the square root of the the number $n_{T/I}$ of tweets per influencer. For example, if you want to produce a tweet every hour at a rate of 6 tweets per influencer, then $t_D/\Delta t = 24$ and the number of mutually retweeting influencers needed is $n_I = \sqrt{24/6} = 2$. If you want to produce a tweet every hour, then $t_D/\Delta t = 1,400$ and $n_I = \sqrt{1,400/6} = 15.5$ influencers. And if you want to produce a tweet every second, then $t_D/\Delta t = 86,400$ and $n_I = \sqrt{86,400/6} = 120$ influencers. In other words, you can build a mutually retweeting army of 120 men, each tweeting 6 times per day, to produce one tweet per second for the hashtag of your choice.