Clustering and percolation on superpositions of Bernoulli random graphs

Let $W=\{w_1,\dots, w_m\}$ be a finite set. Given integers $x_1,\dots,x_n\in[0, m]$ and numbers $q_1,...,q_n \in [0,1]$, let $D_1,\dots,D_n$ be independent uniformly distributed random subsets of $W$ of sizes $|D_i|=x_i$, $1\le i\le n$. Let $G_n$ be the union of independent Bernoulli random graphs $G(x_i, q_i),$ $1\le i\le n$, with vertex sets $D_1,\dots, D_n$. For $m,n\to+\infty$ such that $m=\Theta(n)$ we show that $G_n$ admits a tunable (asymptotic) power law degree distribution and non-vanishing global clustering coefficient. Moreover, the clustering spectrum admits a tunable scaling $k^{-\delta}$, $0\le \delta\le 2$, where $k$ is the vertex degree. Furthermore, we show a phase transition in the size of the largest connected component and examine the relation between the clustering spectrum and bond percolation threshold.