Connectivity of Inhomogeneous Random K-Out Graphs

We propose inhomogeneous random K-out graphs <inline-formula> <tex-math notation="LaTeX">$\mathbb {H}(n; {\mu }, {K}_{n})$ </tex-math></inline-formula>, where each of the <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> nodes is assigned to one of <inline-formula> <tex-math notation="LaTeX">$r$ </tex-math></inline-formula> classes independently with a probability distribution <inline-formula> <tex-math notation="LaTeX">${\mu } = \{\mu _{1}, \ldots, \mu _{r}\}$ </tex-math></inline-formula>. In particular, each node is classified as class <inline-formula> <tex-math notation="LaTeX">$i$ </tex-math></inline-formula> with probability <inline-formula> <tex-math notation="LaTeX">$\mu _{i}>0$ </tex-math></inline-formula>, independently. With <inline-formula> <tex-math notation="LaTeX">${K}_{n} = \left ({K_{1,n}, \ldots, K_{r,n} }\right)$ </tex-math></inline-formula>, each class <inline-formula> <tex-math notation="LaTeX">$i$ </tex-math></inline-formula> node <italic>selects</italic> <inline-formula> <tex-math notation="LaTeX">$K_{i,n}$ </tex-math></inline-formula> distinct nodes uniformly at random from among all other nodes. A pair of nodes are adjacent in <inline-formula> <tex-math notation="LaTeX">$\mathbb {H}(n; {\mu }, {K}_{n})$ </tex-math></inline-formula> if at least one selects the other. Without loss of generality, we assume that <inline-formula> <tex-math notation="LaTeX">$K_{1,n} \leq K_{2,n} \leq \ldots \leq K_{r,n}$ </tex-math></inline-formula>. Earlier results on <italic>homogeneous</italic> random K-out graphs <inline-formula> <tex-math notation="LaTeX">$\mathbb {H}(n; K_{n})$ </tex-math></inline-formula>, where all nodes select the same number <inline-formula> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula> of other nodes, reveal that <inline-formula> <tex-math notation="LaTeX">$\mathbb {H}(n; K_{n})$ </tex-math></inline-formula> is connected with high probability (whp) if <inline-formula> <tex-math notation="LaTeX">$K_{n} \geq 2$ </tex-math></inline-formula> which implies that <inline-formula> <tex-math notation="LaTeX">$\mathbb {H}(n; {\mu }, {K}_{n})$ </tex-math></inline-formula> is connected whp if <inline-formula> <tex-math notation="LaTeX">$K_{1,n} \geq 2$ </tex-math></inline-formula>. In this paper, we investigate the connectivity of inhomogeneous random K-out graphs <inline-formula> <tex-math notation="LaTeX">$\mathbb {H}(n; {\mu }, {K}_{n})$ </tex-math></inline-formula> for the special case when <inline-formula> <tex-math notation="LaTeX">$K_{1,n}=1$ </tex-math></inline-formula>, i.e., when each class 1 node selects only one other node. We show that <inline-formula> <tex-math notation="LaTeX">$\mathbb {H}\left ({n;{\mu },{K}_{n}}\right)$ </tex-math></inline-formula> is connected whp if <inline-formula> <tex-math notation="LaTeX">$K_{r,n}$ </tex-math></inline-formula> is chosen such that <inline-formula> <tex-math notation="LaTeX">$\lim _{n \to \infty } K_{r,n} = \infty $ </tex-math></inline-formula>. However, any bounded choice of the sequence <inline-formula> <tex-math notation="LaTeX">$K_{r,n}$ </tex-math></inline-formula> gives a positive probability of <inline-formula> <tex-math notation="LaTeX">$\mathbb {H}\left ({n;{\mu },{K}_{n}}\right)$ </tex-math></inline-formula> being asymptotically <italic>not</italic> connected in the limit of large <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>. Simulation results are provided to validate our results in the finite node regime.

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