Quantifying uncertainty in a predictive model for popularity dynamics

The Hawkes process has garnered attention in recent years for its suitability to describe the behavior of online information cascades. Here we present a fully tractable approach to analytically describe the distribution of the number of events in a Hawkes process, which, in contrast to purely empirical studies or simulation-based models, enables the effect of process parameters on cascade dynamics to be analyzed. We show that the presented theory also allows predictions regarding the future distribution of events after a given number of events have been observed during a time window. Our results are derived through a differential-equation approach to attain the governing equations of a general branching process. We confirm our theoretical findings through extensive simulations of such processes. This work provides the basis for more complete analyses of the self-exciting processes that govern the spreading of information through many communication platforms, including the potential to predict cascade dynamics within confidence limits.

Figure 1

Example of the Hawkes process. (a) Intensity function $\lambda \left(t\right)$ given by Eq. (1) along with the background intensity $\mu \left(t\right)$ without any self-excitation, shown by the dashed line. (b) Corresponding event sequence of five events at times ${\tau }_i$.

Figure 2

Schematic of the model. A discussion is started by the seed at time ${\tau }_0=0$, represented by the horizontal blue line. At time ${\tau }_1$ the seed receives a reply, shown by the blue circle, with probability $\mu \left({\tau }_1\right)d\tau$, which starts its own subtree shown by the top horizontal red line. At time ${\tau }_2$ the comment receives a reply highlighted by the red circle, with probability $\xi \varphi ({\tau }_2-{\tau }_1)d\tau$. The seed receives another reply at time ${\tau }_3$, with probability $\mu \left({\tau }_3\right)d\tau$, which again starts another subtree. Finally, the subtree started at time ${\tau }_1$ receives another reply at time ${\tau }_4$, which occurred with probability $\xi \varphi ({\tau }_4-{\tau }_1)d\tau$, itself starting a new branch. We now observe the entire tree size, i.e., the number of comments, to be 5 at time $\mathrm{\Omega }$.

Figure 3

Numerical simulation of the Hawkes process over ${10}^6$ realizations. (a) Mean number of events for a Hawkes process with exponential memory kernel ($\beta =3$) and constant background intensity ${\lambda }_0=0.1$ for a range of fitness values. The lines represent the theoretical values for the expected number of events given by Eq. (24), while the circles represent the corresponding results from simulation. (c) The CCDF of the number of events in a scenario where $\xi =0.5$ and $\beta =3$ for three different values of observation time. Again, the lines represent the theoretical CCDF from inversion of the PGF and circles represent the results from simulation. (b) and (d) Corresponding plots for a power-law memory kernel with $c=0.01$ and $\beta =0.3$.

Figure 4

Numerical simulation of the Hawkes process over ${10}^6$ realizations. (a) Mean number of events for a purely self-exciting Hawkes process with exponential memory kernel ($\beta =3$) and equivalent background intensity with a fitness value of $\xi =0.8$. The black solid line represents the theoretical values for the expected number of events, while the circles represent the corresponding results from simulation. The gray dash-dotted lines indicates the 95% prediction interval, i.e., the values between which 95% of the distribution is centered at each time point, while the colored dashed lines indicate the equivalent percentiles from simulation at three time points $t=2,5,8$. (b) The CCDF of the number of events at the same three time points.

Figure 5

Schematic of the model described in Sec. 3. For a given process, we observe its evolution over what we refer to as the observation window from the time ${\tau }_0$ of the seed comment until a future time $T$. During this window we observe $n=5$ events, including the seed event itself, at times $\{{\tau }_0,{\tau }_1,\cdots ,{\tau }_4\}$. Our model then considers how the age of each of these events, $a_i=T-{\tau }_i$, influences the distribution of number of events over a prediction window of length $r=\mathrm{\Omega }-T$, where $\mathrm{\Omega }$ is the time at which we observe what has occurred in this window. In the ensemble shown above, four further events took place by the observation time.

Figure 6

Numerical simulation and theoretical results for two Hawkes processes where one realization is obtained over the interval [0,10] before proceeding to simulate ${10}^6$ ensembles based upon these events over the interval (10,20]. We show the average number of events along with the 95% percentiles from the distribution of events at a number of time points. In addition, we show the theoretical expected number of events and 95% prediction intervals obtained from numerical inversion of Eq. (38). (a) Case of background intensity ${\lambda }_0=1.5$, fitness $\xi =0.8$, and exponential memory kernel with mean time 3 ($\beta =1/3$). (b) Case with shifted power-law memory kernel ($\beta =1$, $c=0.01$, $\xi =0.5$, and ${\lambda }_0=1$).

Figure 7

Comparison of numerical simulations of two half-synthetic Hawkes processes with analytical results. In both cases, the constant background activity is ${\lambda }_0=0.1$ and the excitation function is given by an exponential distribution with mean time 3 ($\beta =1/3$) and fitness $\xi =0.8$. Moreover, in both cases, we observe $n=10$ events before $T$, although in this case these events are generated synthetically so that (a) $\tau =\{0,1,2,3,4,5,6,7,8,9\}$ and (b) $\tau =\{0,1,2,3,5,8,9,9.5,9.6,9.8\}$. Then the rest of the process is numerically computed and the prediction of future events is done analytically up to time $\mathrm{\Omega }=20$. The mean number of events is shown in (a) and (b) for the different observation times. In (a) and (b) the circles represent the average over ${10}^6$ Monte Carlo simulations, while the blue line shows the theoretical mean calculated directly from the inversion of the PGF given by Eq. (53). Note that for this particular case of a Hawkes process we may also obtain an analytical expression for this average through Eq. (54). We also show the $95\%$ prediction intervals for the popularities at each time point with the gray lines, again via inversion of Eq. (53). (c) and (d) The corresponding CCDFs for the number of events at a number of different time points are shown, where the lines are the theoretical distributions and the circles are observed CCDFs from the numerical simulations.

Figure 8

Simulations equivalent to the one in Fig. 7, i.e., with a synthetic evenly spaced series of events before $T=10$, but with different fitness values for the rest of the evolution up to time $\mathrm{\Omega }$. (a) Observed series along with the mean number of events during the prediction window. (b) Probability of zero replies (posts) in a prediction window of length $r$, $q_n\left(r\right)$.