On-line micro-blogging and social service websites enable users to read and send text-based messages to certain topics of interest. The popularity of these topics is commonly measured by the number of postings about these topics [15, 19]. For instance on Twitter, Digg and Youtube, users post their thoughts on topics of interest in the form of tweets and comments. One special characteristic of social media that has been ignored so far is that users can contribute to the popularity of a topic more than once. We take this into account by denoting first posts on a certain topic from a certain user by the variable First Time Post (*FTP*). If the same user posts on the topic more than once, we call it a Repeated Post (*RP*). In what follows, we first look at the growth dynamics of *FTP*.

When a topic first catches people’s attention, a few people may further pass it on to others in the community. If we denote the cumulative number of *FTP* mentioning the topic at time *t* by {N}_{t}, the growth of attention can be described by {N}_{t}=(1+{\chi}_{t}){N}_{t-1}, where the {\chi}_{t} are assumed to be small, positive, independent and identically distributed random variables with mean *μ* and variance {\sigma}^{2}. For small {\chi}_{s}, the equation can be approximated as:

{N}_{t}\simeq \prod _{s=1}^{t}{e}^{{\chi}_{s}}{N}_{0}={e}^{{\sum}_{s=1}^{t}{\chi}_{s}}{N}_{0}.

(1)

Taking logarithms on both sides, we obtain log\frac{{N}_{t}}{{N}_{0}}={\sum}_{s=1}^{t}{\chi}_{s}. Applying the central limit theorem to the sum, it follows that the cumulative count of *FTP* should obey a log-normal distribution.

We now consider the persistence of social trends. We use the variable vitality, {\varphi}_{t}=\frac{{N}_{t}}{{N}_{t-1}}, as a measurement of popularity, and assume that if the vitality of a topic falls below a certain threshold {\theta}_{1}, the topic stops trending. Thus

log{\varphi}_{t}=log\frac{{N}_{t}}{{N}_{t-1}}=log\frac{{N}_{t}}{{N}_{0}}-log\frac{{N}_{t-1}}{{N}_{0}}\simeq {\chi}_{t}.

(2)

The probability of ceasing to trend at the time interval *s* is equal to the probability that {\varphi}_{s} is lower than a threshold value {\theta}_{1}, which can be written as:

\begin{array}{rl}p=& Pr({\varphi}_{s}<{\theta}_{1})=Pr(log{\varphi}_{s}<log({\theta}_{1}))\\ =& Pr({\chi}_{s}<log({\theta}_{1}))=F(log({\theta}_{1})),\end{array}

(3)

where F(x) is the cumulative distribution function of the random variable *χ*. We are thus able to determine the threshold value from {\theta}_{1}={e}^{{F}^{-1}(p)} if we know the distribution of the random variable *χ*. Notice that if *χ* is independent and identically distributed, it follows that the distribution of trending durations is given by a geometric distribution with Pr(L=k)={(1-p)}^{k}p. The expected trending duration of a topic, E(L), is therefore given by

E(L)=\sum _{0}^{\mathrm{\infty}}{(1-p)}^{k}p\cdot k=\frac{1}{p}-1=\frac{1}{F(log({\theta}_{1}))}-1.

(4)

Thus far we have only considered the impact of *FTP* on social trends by treating all topics as identical to each other. To account for the resonance between users and specific topics we now include the *RP* into the dynamics. We define the instantaneous number of *FTP* posted in the time interval *t* as {\mathit{FTP}}_{t}, and the repeated posts, *RP*, in the time interval *t* as {\mathit{RP}}_{t}. Similarly we denote the cumulative number of all posts-including both *FTP* and *RP*-as {S}_{t}. The resonance level of fans with a given topic is measured by {\mu}_{t}=\frac{{\mathit{FTP}}_{t}+{\mathit{RP}}_{t}}{{\mathit{FTP}}_{t}}, and we define the expected value of {\mu}_{t}, E({\mu}_{t}) as the active-ratio {a}_{q}.

We can simplify the dynamics by assuming that {\mu}_{t} is independent and uniformly distributed on the interval [1,2{a}_{q}-1]. It then follows that the increment of {S}_{t} is given by the sum of {\mathit{FTP}}_{t} and {\mathit{RP}}_{t}. We thus have

{S}_{t}-{S}_{t-1}={\mathit{FTP}}_{t}+{\mathit{RP}}_{t}={\mu}_{t}FT{P}_{t}={\mu}_{t}({N}_{t}-{N}_{t-1})={\mu}_{t}{\chi}_{t}{N}_{t-1}.

(5)

And also

\begin{array}{rl}{E}_{\mu}({S}_{t})& ={E}_{\mu}({S}_{t-1})+{a}_{q}({N}_{t}-{N}_{t-1})\\ ={E}_{\mu}({S}_{t-2})+{a}_{q}({N}_{t}-{N}_{t-2})=\cdots \\ ={E}_{\mu}({S}_{0})+{a}_{q}({N}_{t}-{N}_{0})={a}_{q}{N}_{t}.\end{array}

(6)

We approximate {S}_{t-1} by {\mu}_{t}{N}_{t-1}. Going back to Equation 5, we have

{S}_{t}\simeq {\mu}_{t}({\chi}_{t}+1){N}_{t-1}\simeq {\mu}_{t}{e}^{{\chi}_{t}}{N}_{t-1}.

(7)

From this, it follows that the dynamics of the full attention process is determined by the two independent random variables, *μ* and *χ*. Similarly to the derivation of Equation 3, the topic is assumed to stop trending if the value of either one of the random variables governing the process falls below the thresholds {\theta}_{1} and {\theta}_{2}, respectively. One point worth mentioning here is that, {\theta}_{1} and {\theta}_{2} are system parameters, i.e. not dependent on the topic, but only on the studied medium. The probability of ceasing to trend, defined as {p}^{\star}, is now given by

{p}^{\star}=Pr({\chi}_{t}<log({\theta}_{1}))Pr({\mu}_{t}<{\theta}_{2})=\frac{{\theta}_{2}-1}{2({a}_{q}-1)}p,

(8)

. The expected value of {L}_{q} for any topic *q* is given by

E({L}_{q})=\frac{2({a}_{q}-1)}{F(log{\theta}_{1})({\theta}_{2}-1)}-1.

(9)

Which states that the persistent duration of trends associated with given topics is expected to scale linearly with the topic users’ active-ratio. From this result it follows that one can predict the trend duration for any topic by measuring its user active-ratio after the values of {\theta}_{1} and {\theta}_{2} are determined from empirical observations.