Using functional resonance imaging (fMRI), we studied the relationship between perceptual organization and network topology within the primary visual cortex (V1). Twenty-six humans (male and female) were recorded during active observation of two Global and Local Navon letters. Correlations between fMRI fluctuations from different V1 sites were measured (after removing stimulus-evoked signals) in windows specific to each condition. Intra-V1, like brain-wide networks, presented an overall decrease of correlations during stimulation compared to baseline but with increased statistical dimensionality. Massive edgewise testing and network based-statistics (both corrected by FDR) identified differences in connection strengths between stimuli that were mapped to the visual field. Global letters elicited long links mainly connecting V1 sites mapping the lower left/right visual quadrants. Shorter links were associated with local letters, primarily mapped within the lower-left visual quadrant. These connections were not observed in the time-locked (feedforward) responses shared across participants, and their lengths exceeded V1 population receptive field sizes. Thus, these networks reflect activity idiosyncratic to each participant, possibly generated by interactions within or feedback to V1. Perception would sculpt V1 connectivity, with specific increases in link strengths (in a background of decreases). These findings could help shed light on V1 as a "cognitive blackboard".