<p>The exploration of brain networks has reached an important milestone as relatively large and reliable information has been gathered for connectomes of different species^1,2. Analyses of connectome data sets reveal that the structural length follows the exponential rule³, the distributions of in- and out-node strengths^4,5 follow heavy-tailed lognormal statistics, while the functional network properties exhibit powerlaw tails⁶, suggesting that the brain operates close to a critical point where computational capabilities⁷ and sensitivity to stimulus is optimal⁸. Because these universal network features emerge from bottom-up (self-)organization, one can pose the question of whether they can be modeled via a common framework, particularly through the lens of criticality of statistical physical systems. Here, we simultaneously reproduce the powerlaw statistics of connectome edge weights and the lognormal distributions of node strengths from an avalanche-type model with learning that operates on baseline networks that mimic the neuronal circuitry. We observe that the avalanches created by a sandpile-like model⁹ on simulated neurons connected by a hierarchical modular network (HMN)^10,11 produce robust powerlaw avalanche size distributions with critical exponents of 3/2 characteristic of neuronal systems¹². Introducing Hebbian learning, wherein neurons that ‘fire together, wire together,’ recovers the powerlaw distribution of edge weights and the lognormal distributions of node degrees, comparable to those obtained from connectome data⁵. Our results strengthen the notion of a critical brain, one whose local interactions drive connectivity and learning without a need for external intervention and precise tuning¹³.</p>