Bond Percolation in Small-World Graphs with Power-Law Distribution

\emph{Full-bond percolation} with parameter $p$ is the process in which, given a graph, for every edge independently, we delete the edge with probability $1-p$. Bond percolation is motivated by problems in mathematical physics and it is studied in parallel computing and network science to understand the resilience of distributed systems to random link failure and the spread of information in networks through unreliable links. Full-bond percolation is also equivalent to the \emph{Reed-Frost process}, a network version of \emph{SIR} epidemic spreading, in which the graph represents contacts among people and $p$ corresponds to the probability that a contact between an infected person and a susceptible one causes a transmission of the infection. We consider \emph{one-dimensional power-law small-world graphs} with parameter $\alpha$ obtained as the union of a cycle with additional long-range random edges: each pair of nodes $(u,v)$ at distance $L$ on the cycle is connected by a long-range edge $(u,v)$, with probability proportional to $1/L^\alpha$. Our analysis determines three phases for the percolation subgraph $G_p$ of the small-world graph, depending on the value of $\alpha$. 1) If $\alpha < 1$, there is a $p<1$ such that, with high probability, there are $\Omega(n)$ nodes that are reachable in $G_p$ from one another in $O(\log n)$ hops; 2) If $1 < \alpha < 2$, there is a $p<1$ such that, with high probability, there are $\Omega(n)$ nodes that are reachable in $G_p$ from one another in $\log^{O(1)}(n)$ hops; 3) If $\alpha > 2$, for every $p<1$, with high probability all connected components of $G_p$ have size $O(\log n)$. The setting of full-bond percolation in finite graphs studied in this paper, which is the one that corresponds to the network SIR model of epidemic spreading, had not been analyzed before.