The Routing of Complex Contagion in Kleinberg's Small-World Networks

In Kleinberg's small-world network model, strong ties are modeled as deterministic edges in the underlying base grid and weak ties are modeled as random edges connecting remote nodes. The probability of connecting a node $u$ with node $v$ through a weak tie is proportional to $1/|uv|^\alpha$, where $|uv|$ is the grid distance between $u$ and $v$ and $\alpha\ge 0$ is the parameter of the model. Complex contagion refers to the propagation mechanism in a network where each node is activated only after $k \ge 2$ neighbors of the node are activated. In this paper, we propose the concept of routing of complex contagion (or complex routing), where we can activate one node at one time step with the goal of activating the targeted node in the end. We consider decentralized routing scheme where only the weak ties from the activated nodes are revealed. We study the routing time of complex contagion and compare the result with simple routing and complex diffusion (the diffusion of complex contagion, where all nodes that could be activated are activated immediately in the same step with the goal of activating all nodes in the end). We show that for decentralized complex routing, the routing time is lower bounded by a polynomial in $n$ (the number of nodes in the network) for all range of $\alpha$ both in expectation and with high probability (in particular, $\Omega(n^{\frac{1}{\alpha+2}})$ for $\alpha \le 2$ and $\Omega(n^{\frac{\alpha}{2(\alpha+2)}})$ for $\alpha > 2$ in expectation), while the routing time of simple contagion has polylogarithmic upper bound when $\alpha = 2$. Our results indicate that complex routing is harder than complex diffusion and the routing time of complex contagion differs exponentially compared to simple contagion at sweetspot.