to solve the maximum likelihood problem. To overcome this difficulty, we propose a sample path based approach which is discussed below.

Sample Path Based Detection
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Instead of using the MLE, we propose to identify the sample path ${\bf X}^*[0, t^*]$ that most likely leads to ${\bf Y},$ i.e., $$\begin{aligned}
{\bf X}^*[0,t^*]=\hbox{arg}\max_{t,{\bf X}[0,t]\in {\cal
X}(t)}\Pr\left({\bf X}[0,t]\right),\label{eqn:optsamplepath}\end{aligned}$$ where ${\cal X}(t)=\{{\bf X}[0,t]|{\bf F}({\bf X}(t))={\bf Y}\}.$ The source node associated with ${\bf X}^*[0,t^*]$ is then viewed as the information source.

Sample Path Based Detection On Tree Networks
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The optimal sample paths for general graphs are still difficult to obtain. In this section, we focus on tree networks and derive structure properties of the optimal sample paths.

First, we introduce the definition of eccentricity in graph theory [@Har_91]. The eccentricity $e(v)$ of a vertex $v$ is the maximum distance between $v$ and any other vertex in the graph. The Jordan centers of a graph are the nodes which have the minimum eccentricity. For example, in Figure \[figure:infectioneccentricity\], the eccentricity of node $v_1$ is $4$ and the Jordan center is $v_2,$ whose eccentricity is $3.$

![An Example Illustrating the Infection Eccentricity[]{data-label="figure:infectioneccentricity"}](figure/intro_eg.pdf){width="3in"}

Following a similar terminology, we define the infection eccentricity $\tilde{e}(v)$ given ${\bf Y}$ 