network model, we show that such motor-driven networks with high concentrations of Arp2/3 complexes show inhibited dynamics because of the saturation of nucleation sites on actin filaments by the Arp2/3 complexes, while low Arp2/ 3 concentrations aggravate contractility of the networks with hallmarks of short contraction time and small actin clusters. At intermediate Arp2/3 concentrations, sudden collapses of actin clusters in the networks, or ''avalanches'', occur. We have implemented graph theory to quantify the higher-order organization among actomyosin networks, powerful to visualize the hierarchy of the complex networks as well as to extract unprecedented insights on the dynamics of actomyosin networks that can be validated experimentally.