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We consider wireless sensor networks under a heterogeneous random keypredistribution scheme and an on-off channel model. The heterogeneous keypredistribution scheme has recently been introduced by Ya\u{g}an - as anextension to the Eschenauer and Gligor scheme - for the cases when the networkconsists of sensor nodes with varying level of resources and/or connectivityrequirements, e.g., regular nodes vs. cluster heads. The network is modeled bythe intersection of the inhomogeneous random key graph (induced by theheterogeneous scheme) with an Erd\H{o}s-R\'enyi graph (induced by the on/offchannel model). We present conditions (in the form of zero-one laws) on how toscale the parameters of the intersection model so that with high probabilityall of its nodes are connected to at least $k$ other nodes; i.e., the minimumnode degree of the graph is no less than $k$. We also present numerical resultsto support our results in the finite-node regime. The numerical results suggestthat the conditions that ensure $k$-connectivity coincide with those ensuringthe minimum node degree being no less than $k$.

Minimum node degree in inhomogeneous random key graphs with unreliable links