Table 1 Symbol table for our method
Notation | Description |
|---|---|
G(V, E) | Graph with vertex set V and edge set E |
\(N=\mid V\mid\), \(M=\mid E\mid\) | Size of vertex set or edge set |
n, m | \(n=\mid V\mid\), \(m=\mid E\mid\) |
k | Degree of a vertex (number of edges emanated from a vertex) |
\(N_s\) | Size of seed set |
l | Number of hops |
i, j | Index of vertex |
\(\partial i\backslash j\) | Set of the nearest neighbors of vertex i but not including j |
\(e^{-y}\) | Penalty factor for minimizing the size of VC, y is an inverse temperature parameter |
\(i\rightarrow j\) | Link from vertex i to j |
\({\hat{\pi }}_{i}^{(0)}\) | Probability variable of never covered state 0 |
\({\hat{\pi }}_{i}^{(1)}\) | Probability variable of covered state 1 |
\({\hat{\pi }}_{i}^{(*)}\) | Probability variable of sometimes covered and sometimes not joker state \(*\) |
set of \(\partial Ball(j,l-1)\) | Set of the \(l-1\) nearest neighbors of vertex j |
\(N_{s}\) | Number of seeds |
\(\langle k\rangle\) | Average degree: 2M/N |
\(\beta\) | Infection probability |
S(t), I(t), R(t) | Cumulative probability of each state of S, I, or R at time t |
\(P_{i}^{I}(t)\), \(P_{i}^{R}(t)\), and \(P_{i}^{s}(t)\) | Probability of state S, I, and R for a vertex i at time t |
\(\mid VC \mid\) | Size of set of vertexes as vertex cover |
\(d_{i,j}\) | Distance of i and j defined by the shortest path length between them |
\(t_c\) | Convergent time until all infected vertexes are recovered |
D | Diameter of network as the maximum distance of the shortest path between vertexes |