Actomyosin networks give cells the ability to move and divide. These networks contract and expand while being driven by active energy-consuming processes such as motor protein walking and actin polymerization. Actin dynamics is also regulated by actin-binding proteins, such as the actin-related protein 2/3 (Arp2/3) complex and calcium/calmodulin-dependent protein kinase II (CaMKII) complex. Arp2/3 generates branched filaments while CaMKII binds and bundles actin filaments, thereby changing the overall organization and dynamics of the network. While the structures of Arp2/3 and CaMKII have been explored extensively, how these complexes change the architectural dynamics of the actomyosin network raises many questions. In this work, the spatiotemporal patterns of dynamical actin assembly accompanying reorganization caused by Arp2/3 and CaMKII were studied using both a mass action kinetic model and a computational model (mechanochemical dynamics of active networks or MEDYAN). This computational model simulates actomyosin network dynamics as a result of chemical reactions whose rates are modulated by rapid mechanical equilibration. We show that branched actomyosin networks relax significantly more slowly than do unbranched networks. Also, branched networks undergo rare convulsive movements, termed “avalanches”, that release strain in the network. These avalanches are associated with the more heterogeneous distribution of mechanically-linked filaments displayed by branched networks. These far-from equilibrium events arising from the marginal stability of growing actomyosin networks provide a possible mechanism of the “cytoquakes” recently seen in experiments. We show that the multivalent interactions between CaMKII and actin enable rich structural arrangements of actin filaments. Unlike branched networks, networks with CaMKII resemble crystalline networks due to abundant connections between CaMKII and actin filaments.