Branching dynamics of viral information spreading

Despite its importance for rumors or innovations propagation, peer-to-peer collaboration, social networking, or marketing, the dynamics of information spreading is not well understood. Since the diffusion depends on the heterogeneous patterns of human behavior and is driven by the participants’ decisions, its propagation dynamics shows surprising properties not explained by traditional epidemic or contagion models. Here we present a detailed analysis of our study of real viral marketing campaigns where tracking the propagation of a controlled message allowed us to analyze the structure and dynamics of a diffusion graph involving over 31 000 individuals. We found that information spreading displays a non-Markovian branching dynamics that can be modeled by a two-step Bellman-Harris branching process that generalizes the static models known in the literature and incorporates the high variability of human behavior. It explains accurately all the features of information propagation under the “tipping point” and can be used for prediction and management of viral information spreading processes.

Figure 1

The viral message diffusion graph of our campaigns is a set of 7118 disconnected cascades like this one observed in Spain. Its 122 nodes (represented by dots) are grouped in eight generations (horizontal layers) that stem from the generation zero node at the top (seed node, black) and grow through a branching process driven by the active nodes (gray) in each generation. Its treelike structure is devoid of closed paths or triangles for a clustering coefficient $C=0$.

Figure 2

Active nodes' (seed + viral) cumulative probability distribution for campaigns in all markets (circles). Solid line is the fit to a power-law $P\left(r\right)=H_{\alpha \beta }/(\beta +r^{\alpha })$ whose pdf exponent is $\alpha =3.54\pm 0.02$ (see Table ). Dashed line is the prediction of a discrete Poisson distribution with the mean of the empirical data (${\overline{r}}_a=2.61$).

Figure 3

Cumulative distribution of viral cascade size in all markets (circles). The power-law line underneath the circles is not a fit of the data but the prediction of the Bellman-Harris branching model with a power-law pdf for the recommendations distribution. The line below it is the branching model prediction with a Poisson distribution (see Sec. 3).

Figure 4

Relationship between the time (days) a viral node remains active (infected) ${\tau }_{\mathrm{infected}}$ and the time elapsed until it resends its first message (${\tau }_{\mathrm{response}}$) for each viral node in our campaigns. The line is ${\tau }_{\mathrm{infected}}={\tau }_{\mathrm{response}}$. Nodes on the line sent the message to all contacts at once while those outside it remained spreaders for a longer period. Only early responders (${\tau }_{\mathrm{response}}<10$ days) have some likelihood of staying active for more than one forwarding session.

Figure 5

Flowchart example of cascades generated by the Bellman-Harris branching model used to explain diffusion of information in social networks: The cascade starts with a seed (labeled 0) which sends the information to $r=3$ of its social contacts after time ${\tau }_0$. Viral nodes 1 and 2 are “infected” and forward the message to $r=3$ and $r=2$ social contacts after times ${\tau }_1$ and ${\tau }_2$, respectively, while uninterested node 3 remains inactive. Values of $r$ and $\tau$ are independent and sampled from distributions $P\left(r\right)$ and $G\left(t\right)$. Propagation continues until there are no active nodes left. Time increases left to right.

Figure 6

Size of viral cascades as a function of ${\lambda }_1$ for markets in Table . Triangles represent the cascade size multiplier $1/(1-R_1)$ (left $Y$ axis). Dashed line is not a fit but the prediction of the branching model Eq. (27) which diverges at the “tipping point” [${\overline{r}}_1\simeq 5.27$, ${\left({\lambda }_1\right)}_c\simeq 0.19$] estimated by the correlation ${\overline{r}}_1=1+22.48{\lambda }_1$ existing between transmissibility and fanout coefficient (solid line, circles, right $Y$ axis).

Figure 7

Malthusian parameter ${\alpha }_1$ for the gamma distribution of response times given by Eq. (45) above (left, ${\mu }_1>1$) and below (right, ${\mu }_1<1$) the “tipping point.” The horizontal line is the Malthusian parameter for the Poissonian approximation with the same average response time (i.e., $\rho =1$).

Figure 8

Distribution of the average fraction of new participants as a function of the cascades start time in our campaigns (circles) compared with Eq. (47) (black line), the prediction of the Bellman-Harris model with $P\left(t\right)$ the log-normal distribution of Eq. (46) and with $P\left(t\right)$ exponential of the same mean (red). Dashed line is Eq. (48), the asymptotic approximation of the Bellman-Harris model with $P\left(t\right)$ log normal. (Inset) Time dynamics of $S\left(t\right)$, the cascade's average size (circles) accurately predicted by the model for $G\left(t\right)$ log normal. In red are predictions for $G\left(t\right)$ exponential.

Figure 9

Clustering coefficient $C_{\mathrm{cas}}$ for the cascades network obtained through simulations of the viral propagation model on a real email network (circles) with $C_{\mathrm{soc}}=0.22$ and $\langle {\overline{k}}_{nn}\rangle =18.9$ compared with the linear relationship in Eq. (3).