Parent-child book reading is important for fostering the development of various lifelong cognitive and social abilities in young children. Despite numerous reports describing the effects of familiarity on shared reading for children, the exact neural basis of the functional network architecture remains unclear. We conducted magnetoencephalographic (MEG) experiments using graph theory to elucidate the role of familiarity in shared reading in a child's brain network and to measure the connectivity dynamics of a child while listening to storybook reading (LSBR), which represents the daily activity of shared book reading between the child and caregiver. The LSBR task was performed with normally developing preschool- and school-age children (N = 15) under two conditions: reading by their own mother (familiar condition) vs. an experimenter (unfamiliar condition). We used the phase lag index (PLI), which captures synchronization of MEG signals, to estimate functional connectivity. For the whole brain network topology, an undirected weighted graph was produced using 68 brain regions as nodes and interregional PLI values as edges for five frequency bands. Behavioral data (i.e., the degree of attention and facial expressions) were evaluated from video images of the child's face during the two conditions. Our results showed enhanced widespread functional connectivity in the alpha band during the mother condition. In the mother condition, the whole brain network in the alpha band exhibited topographically high local segregation with high global integration, indicating an increased small-world property. Results of the behavioral analysis revealed that children were more attentive and showed more positive facial expressions in the mother condition than in the experimenter condition. Behavioral data were significantly correlated with graph metrics in the mother condition but not in the experimenter condition. In this study, we identified the neural correlates of a familiarity effect in children's brain connectivity dynamics during LSBR. Furthermore, these familiarity-related brain dynamics were closely linked to the child's behavior. Graph theory applied to MEG data may provide useful insight into the familiarity-related child brain response in a naturalistic setting and its relevance to child attitudes.