Loops, not groups: Long cycles are responsible for discontinuous phase transitions in higher-order network contagions

TL;DR

This work presents a self-consistent solution that accounts for local group structure and global cycles where the process can feedback on itself and finds that only the latter mechanism can give rise to a discontinuous phase transition in the size of global cascades, which is a defining feature of complex contagions.

Abstract

We study a self-consistent approach to introduce higher-order effects in a branching process model of complex contagion on clustered networks. Branching processes operate over an infinite population such that they never circle back and interact with previously exposed parts of the system. This infinite, treelike, structure makes it tricky to account for complex contagion mechanisms such as group effects, peer pressure, or social reinforcement where multiple exposures interact in synergistic ways. Here we present a self-consistent solution that accounts for local group structure and global cycles where the process can feedback on itself. This allows us to distinguish multiple exposures that stem from a single transmission chain, from those occurring at the intersection of different transmission chains. We find that only the latter mechanism can give rise to a discontinuous phase transition in the size of global cascades, which is a defining feature of complex contagions. Group effects alone, without long cycles, produce standard continuous phase transitions.

Figures (6)

• Figure 1: Each node in a Newman-Miller network has $s$ single edges and $t$ triangles, where the probability that a uniformly-randomly selected node has $s$ single edges and $t$ triangles is $p_{s,t}$.
• Figure 2: The average size $S$ of the largest connected component (top left) and the probability $R$ ($R=S$ for simple contagion) that a node is in the largest connected component (right) for varying values of $T$, the base transmission probability. The curves are the value of $S$ from Eq. (\ref{['eq:simple_size']}) for simple contagion. These results are for a doubly-Poisson distributed network with $\nu = 1$, 2, and 3 and $\mu = 6.5 - 2\nu$. The points in the top panels represent the mean values of 100,000 simulations each on a network with at least 100,000 nodes. The bottom panels show the size and probability of a global cascade (from Eq. (\ref{['eq:simple_size']})) for varying $\nu$ and $T$, where $\mu = 6.5 - 2\nu$.
• Figure 3: The case where $\alpha_\Delta = \alpha_\curvearrowright = 10$; i.e., groups and loops. (Left) The stable (solid lines) and unstable equilibria (dashed lines) for the largest connected component size $S$ from Eq. (\ref{['eq:loops_groups_size']}) and (right) the probability $R$ of observing a supercritical cascade from Eq. (\ref{['eq:groups_loops_prob']}) for a doubly-Poisson distributed network with $\nu = 1$, 2 and 3 and $\mu = 6.5 - 2\nu$ for varying values of $T$, the base transmission probability. The points in the top panels represent the mean values for 100,000 simulations each on a network with at least 100,000 nodes. The unstable equilibria were calculated as in dodds2004universal. The bottom panels show the size and probability (from Eqs. (\ref{['eq:groups_loops_prob']}) and (\ref{['eq:loops_groups_size']})) for varying $\nu$ and $T$, where $\mu = 6.5 - 2\nu$.
• Figure 4: The mechanism of spread through a group. (Left) The first node to be activated in a group activates its neighbours, as in classic bond percolation, with transmission probability $T$. (Right) In the situation where two nodes out of three in a triangle are active, the activation probability of the third node depends on the context of activations around the active nodes. From left to right in the group effects panel: (i) If there is an active link exists between the two active nodes in the triangle, the third node is activated by the second node with probability $T_\Delta$, (ii) The same mechanism applies with probability $T_\Delta$ even if one of the active nodes was also reached by a neighbour outside the group, and (iii) if there is no transmission link between the two active nodes in the group, the second node activates the third node with probability $T_\curvearrowright$.
• Figure 5: The case where $\alpha_{\Delta}=10$ and $\alpha_{\curvearrowright}=1$; i.e., groups, no loops. (Left) The largest connected component size $S$ from Eq. (\ref{['eq:loops_groups_size']}) and (right) the probability $R$ of observing a supercritical cascade from Eq. (\ref{['eq:groups_loops_prob']}) for a doubly-Poisson distributed network with $\nu = 1$, 2 and 3 and $\mu = 6.5 - 2\nu$ for varying values of $T$, the base transmission probability. The points in the top panels represent the mean values for 100,000 simulations each on a network with at least 100,000 nodes.