$v^*$ given the graph $G$ and ${\bf Y},$ where $t$ is an unknown parameter.

![An Example of Information Propagation[]{data-label="figure:ip_eg"}](figure/ip_eg.pdf "fig:"){width="1\columnwidth"}\

Figure \[figure:ip\_eg\] is an example of the infection process. The left figure shows the information propagation over time. The nodes on each dotted line are the nodes which are infected at that time slot, and the arrows indicate where the infection comes from (e.g., node $4$ is infected by node $2$).

The figure on the right is the network we observe, where the shaded nodes are infected nodes and others are susceptible or recovered nodes. The pair of numbers next to each node are the corresponding infection time and recovery time. For example, node $3$ was infected at time slot $2$ and recovered at time slot $3$. $-1$ indicates that the infection or recovery has yet occurred. Note that these two pieces of information are not available to us, and we include them in the figure to illustrate the infection and recovery processes. If we observe the network at the end of time slot $3,$ then the snapshot of the network is ${\bf Y}=\{0,1,0,1,0,1,1\},$ where the states are ordered according to the indices of the nodes.

Maximum Likelihood Detection
----------------------------

