Shockwavelike Behavior across Social Media

Online communities featuring “anti-X” hate and extremism, somehow thrive online despite moderator pressure. We present a first-principles theory of their dynamics, which accounts for the fact that the online population comprises diverse individuals and evolves in time. The resulting equation represents a novel generalization of nonlinear fluid physics and explains the observed behavior across scales. Its shockwavelike solutions explain how, why, and when such activity rises from “out-of-nowhere,” and show how it can be delayed, reshaped, and even prevented by adjusting the online collective chemistry. This theory and findings should also be applicable to anti-X activity in next-generation ecosystems featuring blockchain platforms and Metaverses.

Figure 1

Empirically observed (a) fusion and (b) total fission of in-built communities featuring anti-U.S. hate on VKontakte between day $t$ (yellow) and $t+1$ (blue). Red nodes are anti-U.S. communities that later got shut down (total fission); green nodes are those still not yet shut down; yellow links point to individuals (white dots) removed from the anti-U.S. community on day $t+1$; blue links point to individuals added to the anti-U.S. community on day $t+1$. Spatial layout results from (a) and (b) being close-ups of a fuller network plotted using ForceAtlas2, meaning that nodes appearing closer together are more interconnected. (b) Also shows that very few individuals are simultaneously also members of other communities (SM shows further proof). (c) Empirically observed clustering of antigovernment communities across platforms around U.S. Capitol riot. (d)–(f) The theory in this Letter incorporates (d) heterogeneous individuals aggregating (i.e., fusion) based on character similarity, (e) total fission with probability ${\nu }_f$, (f) time-varying population size $N(t)$.

Figure 2

Empirical data (symbols) and Eq. (2) theory predictions (lines) for in-built anti-X communities within and across platforms. (a) Size (i.e., number of members) of foreign anti-U.S. (jihadi) communities on VKontakte. (b) Size of domestic anti-U.S. government (pro-civil war) communities on Facebook. Insets: changing population size; time-averaged $F(t)$ which suggests that (b) reflects a heterophily fusion mechanism more than (a). (c) and (d) Complementary cumulative distribution (CCDF) of individual community sizes $s$ from (a) and (b). (e) Evolution of total size of all communities from (b). (f) CCDF at a higher scale, i.e., sizes of clusters of interlinked communities. Inset: empirically inferred interaction kernel $W(s_1,s_2)$ obtained from data across all platforms; axes $s_1$ and $s_2$ are sizes of interacting aggregates.

Figure 3

Mitigations suggested by Eq. (2) theory. (a) and (b) Changing newcomer flux $\dot{N}(t)=q{t}^{\beta }$. $\overline{F(t)}\equiv F$; $N(0)={10}^3$; $q=0.5$; uniform user distribution. (c) Changing user population distribution $\mathcal{P}(\{{\overrightarrow{y}}_i\})$. Starting with uniform user distribution, converge means $\mathcal{P}(\{{\overrightarrow{y}}_i\})$ evolves to single delta function (i.e., users all identical); diverge means $\mathcal{P}(\{{\overrightarrow{y}}_i\})$ evolves to two opposite peaks (i.e., users polarized). $t_1$ is the median time for the change. Analytic results (curves) and microscopic simulations (symbols).

Figure 4

(a) Approximate analytic solutions of Eq. (2) using a time-average $\overline{F(t)}\equiv F$ (see Eq. S64 in Sec. 2.2.3 in the Supplemental Material [15]) versus exact microscopic simulations (solid gray curves show average over 100 runs). $N(0)=500$; uniform heterogeneity distribution; $\dot{N}(t)=q=0.5$. (b) Empirical data (symbols) and approximate analytic predictions (solid curves) for coupled shock waves (Supplemental Material Sec. 2.7 [15]).