Investigate Invertibility of Sparse Symmetric Matrix

In this paper, we investigate the invertibility of sparse symmetric matrices. We show that for an $n\times n$ sparse symmetric random matrix $A$ with $A_{ij} = \delta_{ij} \xi_{ij}$isinvertiblewithhighprobability\mathrm{.}Here,$\delta_{ij}$s,$i\ge j$arei\mathrm{.}i\mathrm{.}d\mathrm{.}Bernoullirandomvariableswith$$\mathbb{P}$ \left(\xi_{ij}=1 \right) =p \ge n^{-c}$,$\xi_{ij}, i\ge j$ are i.i.d. random variables with mean 0, variance 1 and finite forth moment $M_4$, and $c$ is constant depending on $M_4$. More precisely, $$s_{\mathrm{m}\mathrm{i}\mathrm{n}}(A)>\epsilon \sqrt{\frac{p}{n}}\mathrm{.}$$ with high probability.