Information cascades on degree-correlated random networks

We investigate by numerical simulation a threshold model of social contagion on degree-correlated random networks. We show that the class of networks for which global information cascades occur generally expands as degree-degree correlations become increasingly positive. However, under certain conditions, large-scale information cascades can paradoxically occur when degree-degree correlations are sufficiently positive or negative, but not when correlations are relatively small. We also show that the relationship between the degree of the initially infected vertex and its ability to trigger large cascades is strongly affected by degree-degree correlations.

Figure 1

Observed frequency of global cascades, as a function of threshold $\varphi$ and average degree $z$, for (a) disassortative $\left(r=-0.5\right)$, (b) uncorrelated $\left(r=0\right)$, and (c) assortative $\left(r=+0.5\right)$ random networks. The color bar corresponds to all panels. The asterisks denote the numerical solutions to Eq. (3). The dashed vertical lines at $\varphi =0.18$ correspond to the three curves in Fig. 2.

Figure 2

Observed statistics of global cascades as a function of average degree $z$ for $\varphi =0.18$, on assortative $\left(r=+0.5\right)$, uncorrelated $\left(r=0\right)$, and disassortative $\left(r=-0.5\right)$ random networks. Each data point is the frequency of global cascades observed on ten network instances. The lines correspond to the direct measurements of the extended vulnerable components $\left(S_e\right)$. The $x$ axis is scaled logarithmically.

Figure 3

Frequency ($F_{\text{gc}}$, left column) and size ($S_{\text{gc}}$, right column) of global cascades, as a function of average degree $z$ and assortativity $r$, for three threshold values: $\varphi =0.2$ (top row), $\varphi =0.25$ (middle row), and $\varphi =0.33$ (bottom row). The color bar corresponds to all panels. The dashed horizontal lines in panels (a)–(d) are at $r=0$, and the dashed vertical lines in panels (e) and (f) correspond to the data presented in Fig. 4. We were unable to obtain data in the hatched region (see Table ).

Figure 4

Frequency (asterisks) and size (circles) of global cascades, as a function of assortativity $r$, for $z=2.0$ and $\varphi =0.33$. The bars correspond to the sizes of the vulnerable ($S_v$, black) and peripheral ($S_p$, white) subcomponents of $S_e$. The dashed line is provided as a guide for the eyes.

Figure 5

Visualizations of random network properties with $z=2$ and disassortative ($r=-0.8$, top row), uncorrelated ($r=0$, middle row), and assortative ($r=+0.8$, bottom row) degree-degree correlations. In (a), (d), and (g), we visualize averages of the $E$ matrices for all ten network instances $\left(N=10000\right)$ used at each parameter combination. Shading (ranging from white to black) of matrix entries in $E$ correspond to the probabilities (ranging from 0 to 0.17) with which a randomly chosen edge connects vertices of degree $j$ and $k$; entries corresponding to vertices with $k>6$ occur with very low frequency and so are not shown. In (b), (e), and (h), we depict the extended vulnerable component $\left(S_e\right)$ for $\varphi =0.33$ for individual random networks with $N=200$ (for visual clarity). Peripheral vertices $\left(k>3\right)$ are denoted as white squares; vertices in the vulnerable component are denoted as circles and are colored according to degree: white $\left(k=1\right)$, gray $\left(k=2\right)$, and black $\left(k=3\right)$. The corresponding pie charts in (c), (f), and (i) denote the proportions of degrees found in these extended vulnerable components (b), (e), and (h).

Figure 6

Average cascade size with $\varphi =0.18$ as a function of average degree $z$ given that the initial seed was placed in high-degree vertices (in the top 10% of the degree distribution, squares), average degree vertices (circles), or low-degree vertices (in the bottom 10% of the degree distribution, diamonds), for random networks with (a) $r=-0.9$, (b) $r=-0.5$, (c) $r=0$, (d) $r=+0.5$, and (e) $r=+0.9$. Note that in (a), we were not able to obtain data for $z<3$ (Table ). Lines are provided as a guide for the eyes.