Table 1 Notation table

q _uv

The probability an infected node u infects its neighbor node v

p _v

The probability an infected node v recovers

Y

The partial snapshot

X _v (t)

The state of node v at time t

X(t)

The states of all nodes at time t

X[ 0,t]

The sample path from 0 to t

$\mathcal{X}\left(t\right)$

The set of all valid sample paths from time slot 0 to t

${\mathcal{I}}_{\mathbf{\text{Y}}}$

The set of the observed infected nodes

${\mathcal{\mathscr{H}}}_{\mathbf{\text{Y}}}$

The set of the unobserved nodes

$\tilde{e}(v,{\mathcal{I}}_{\mathbf{\text{Y}}})$

The observed infection eccentricity of node v

v ^†

The estimator of the information source

v ^∗

The actual information source

$t_v^{\ast }$

The time duration associated to the optimal sample path in which node v is the

information source

$\mathcal{C}\left(v\right)$

The set of children of v

ϕ(v)

The parent of node v

${\mathcal{Y}}^k$

The set of infection topologies where the maximum distance from the source to

an infected node is k

T _v

The tree rooted in v

$T_v^{-u}$

The tree rooted in v without the branch from its neighbor u

$\mathbf{\text{X}}\left([0,t],T_v^{-u}\right)$

The sample path restricted to topology $T_v^{-u}$

$t_v^I,t_v^R$

The infection time and recovery time of node v

d(v,u)

The length of the shortest path between node v and node u