Competition-Induced Criticality in a Model of Meme Popularity

Heavy-tailed distributions of meme popularity occur naturally in a model of meme diffusion on social networks. Competition between multiple memes for the limited resource of user attention is identified as the mechanism that poises the system at criticality. The popularity growth of each meme is described by a critical branching process, and asymptotic analysis predicts power-law distributions of popularity with very heavy tails (exponent $\alpha <2$, unlike preferential-attachment models), similar to those seen in empirical data.

Figure 1

Schematic of the model. Time runs horizontally and nodes of the network are listed vertically; the screen color of each node indicates the meme it currently holds. At time $t_1$, node 1 retweets (RT) the blue meme to its followers (nodes 2 and 3). At time $t_2$, node 1’s screen is overwritten by the red meme, which was tweeted by one of the nodes followed by node 1.

Figure 2

Popularities $n_r(t)$ and $n_b(t)$ of red and blue memes in a single realization, starting from ten screens each. Note the different time scales in each figure, and the fact that the vertical scale in (a) is logarithmic. The inset boxes in (b) and (c) show the area of the previous figures.

Figure 3

Complementary cumulative distribution functions (CCDFs)—the fraction of memes with popularity $\ge n$—for numerical simulations, compared with the theory of Eq. (2). Dashed lines correspond to CCDFs for power law popularity distributions $P_n\propto n^{-\alpha }$. (a) $p_k={\delta }_{k,10}$, $N={10}^5$, $\mu =0$. (b) $p_k\propto k^{-\gamma }$ for $k\ge 4$ with $\gamma =2.5$ (mean degree $z=10.6$), $N={10}^6$, $\mu =0.01$. (c) Twitter network of the Spanish 15M movement [40, 41], $N=87$, 559, $z=69$, $\mu =0.05$.