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\emph{Full-bond percolation} with parameter $p$ is the process in which,given a graph, for every edge independently, we delete the edge withprobability $1-p$. Bond percolation is motivated by problems in mathematicalphysics and it is studied in parallel computing and network science tounderstand the resilience of distributed systems to random link failure and thespread of information in networks through unreliable links.  Full-bond percolation is also equivalent to the \emph{Reed-Frost process}, anetwork version of \emph{SIR} epidemic spreading, in which the graph representscontacts among people and $p$ corresponds to the probability that a contactbetween an infected person and a susceptible one causes a transmission of theinfection.  We consider \emph{one-dimensional power-law small-world graphs} withparameter $\alpha$ obtained as the union of a cycle with additional long-rangerandom edges: each pair of nodes $(u,v)$ at distance $L$ on the cycle isconnected 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, thereare $\Omega(n)$ nodes that are reachable in $G_p$ from one another in $O(\logn)$ 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 connectedcomponents 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 epidemicspreading, had not been analyzed before.

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