Threshold cascades with response heterogeneity in multiplex networks

Threshold cascade models have been used to describe the spread of behavior in social networks and cascades of default in financial networks. In some cases, these networks may have multiple kinds of interactions, such as distinct types of social ties or distinct types of financial liabilities; furthermore, nodes may respond in different ways to influence from their neighbors of multiple types. To start to capture such settings in a stylized way, we generalize a threshold cascade model to a multiplex network in which nodes follow one of two response rules: some nodes activate when, in at least one layer, a large enough fraction of neighbors is active, while the other nodes activate when, in all layers, a large enough fraction of neighbors is active. Varying the fractions of nodes following either rule facilitates or inhibits cascades. Near the inhibition regime, global cascades appear discontinuously as the network density increases; however, the cascade grows more slowly over time. This behavior suggests a way in which various collective phenomena in the real world could appear abruptly yet slowly.

Figure 1

Illustration of the threshold model with heterogeneous responses on a two-layer multiplex network. Edges of the two layers are drawn as green (solid) and orange (dashed) lines, respectively. Circle-shaped nodes use the OR rule, while square-shaped nodes use the AND rule. Three cases for different fractions of OR nodes $\mathcal{E}$ are shown: (a) all nodes follow the OR rule $(\mathcal{E}=1)$; (b) all nodes follow the AND rule $(\mathcal{E}=0)$; and (c) nodes following either rule are mixed together $(0<\mathcal{E}<1)$. The numeric label on a node denotes the step at which the node activates, starting from the initially active seeds (labeled “0”). Unlabeled nodes are those remaining inactive.

Figure 2

(a) The $(z,R)$-phase diagram of global cascades starting from ${\rho }_0={10}^{-3}$ on duplex ER networks for various fractions $\mathcal{E}$ of OR nodes (lines) and for a single-layer network (gray shaded region). Global cascades occur on the left of the boundaries, which are obtained by finding local maxima of the number of iterations (NOI) of the recursion (2) iterated until successive iterates differ by less than ${10}^{-10}$ (step sizes $\Delta z=0.1,\Delta R=0.01$). The dashed and solid lines indicate the continuous and discontinuous transitions, respectively. (b) Cascade size $\rho$ and (c) NOI versus the mean degree $z$ with fixed threshold $R=0.18$ [indicated by the vertical dotted line in (a)] and $\mathcal{E}=0.2,0.5,1.0$. The lines are from theoretical calculation (1); the dots are from numerical simulations with $N={10}^6$ nodes, averaged over ${10}^2$ realizations. The type of small-$z$ transition changes from continuous $\left(\mathcal{E}\in \{0.5,1\}\right)$ to discontinuous $\left(\mathcal{E}=0.2\right)$. (c, inset) NOI at the small-$z$ transitions versus $\mathcal{E}$, displaying a peak at ${\mathcal{E}}_c\approx 0.28$.

Figure 3

The $(z,R)$-phase diagram of global cascades on duplex ER networks for the multiplex model (green line) and the single-layer model that was designed to replicate the multiplex model (black line) with $\mathcal{E}=0.2$. The dashed and solid lines indicate the continuous and discontinuous transition, respectively. The black dot separates two different transition types. For the single-layer model, the mean degree is $z_{\mathrm{single}}=2z$, and the thresholds are given by $R/2$ for OR nodes and $R$ for AND nodes. For the duplex model, the mean degree is $z_{\mathrm{duplex}}=z$ in each layer, and the thresholds are given by $R$ for all nodes.

Figure 4

(Main) Graphical solutions for the fixed points of the recursion (2) for duplex ER networks with $z=1.5,1.8,2.13,2.3$ and $R=0.18$, for $\mathcal{E}=0.2$. (Inset) Bifurcation diagram of the roots of $\mathbf{g}\left(\mathbf{q}\right)-\mathbf{q}$, with solid and dotted curves denoting the stable and unstable solutions, respectively. The red curve denotes the physical solutions for cascades starting from the small seed ${\rho }_0={10}^{-3}$. Global cascades appear discontinuously at $z\approx 2.13$.

Figure 5

(Main) The $(z,\mathcal{E})$-phase diagram of global cascades of duplex ER networks with threshold $R=0.18$. The gray region indicates where global cascades occur. Continuous and discontinuous transitions occur along the dotted and solid curves, respectively, separated by the cusp point $\mathbf{C}=({\mathcal{E}}_c,z_c)=(0.28,1.36)$ marked by a dot. (Inset) The scaling relation along the continuous transition curve in the new coordinate system $\left({\mu }_{\mathcal{E}},{\mu }_z\right)$ centered at a point on the transition curve for various $\mathcal{E}$. The best fit to the straight line is observed for the coordinate system centered at C. The straight line has slope 2, drawn as a guideline.

Figure 6

(a–d) Example cascade snapshots in a small duplex ER network of $N={10}^3$ nodes with $\mathcal{E}=0.2$. The red and blue nodes are active nodes that follow the OR and AND rules, respectively. Starting from a random 1% of seed nodes (a), the OR nodes activate in greater numbers in the first few steps (b). As time proceeds, however, AND nodes activate in greater numbers (c) and eventually dominate (d). (e) Cumulative numbers of active OR and AND nodes in a numerically simulated cascade on a duplex ER network of $N={10}^6$ nodes with $z=2.3$, just above the small-$z$ transition point for $\mathcal{E}=0.2$. Initial exponential growth (I, yellow) is rapidly slowed due to the “stubborn” AND nodes (II, green), but once enough nodes are active the activation resumes the exponential growth, and AND nodes overtake OR nodes (III, blue), finally reaching saturation (IV, gray).