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Real networks are often dynamic. In response to it, analyses of algorithms on{\em dynamic networks} attract more and more attentions in network science andengineering. Random walks on dynamic graphs also have been investigatedactively in more than a decade, where in most cases the edge set changes butthe vertex set is static. The vertex sets are also dynamic in many realnetworks. Motivated by a new technology of the analysis of random walks ondynamic graphs, this paper introduces a simple model of graphs with increasingthe number of vertices, and presents an analysis of random walks associatedwith the cover time on such graphs. In particular, we reveal that a random walkasymptotically covers the vertices all but a constant number if the vertex setgrows {\em moderately}.

How Many Vertices Does a Random Walk Miss in a Network with Moderately Increasing the Number of Vertices?