The organizational principles that distinguish the human brain from those of other species have been a long-standing enigma in neuroscience. Here, we leverage advances in algebraic topology to uncover the structural properties of the human brain at subcellular resolution. First, we reveal a much higher perisomatic branching density in pyramidal neurons when comparing homologous cortical regions in humans and mice. Traditional scaling methods consistently fail to interpret this difference, suggesting a distinctive feature of human pyramidal neurons. We next show that topological complexity leads to highly interconnected pyramidal-to-pyramidal and higher-order networks, which is unexpected in view of reduced neuronal density in humans compared to mouse neocortex. We thus present robust evidence that reduced neuronal density but increased topological complexity in human neurons ultimately leads to highly interconnected cortical networks. The dendritic complexity, which is a defining attribute of human brain networks, may serve as the foundation of enhanced computational capacity and cognitive flexibility.