Actomyosin stress fibers (SFs) play key roles in driving polarized motility and generating traction forces, yet little is known about how tension borne by an individual SF is governed by SF geometry and its connectivity to other cytoskeletal elements. We now address this question by combining single-cell micropatterning with subcellular laser ablation to probe the mechanics of single, geometrically defined SFs. The retraction length of geometrically isolated SFs after cutting depends strongly on SF length, demonstrating that longer SFs dissipate more energy upon incision. Furthermore, when cell geometry and adhesive spacing are fixed, cell-to-cell heterogeneities in SF dissipated elastic energy can be predicted from varying degrees of physical integration with the surrounding network. We apply genetic, pharmacological, and computational approaches to demonstrate a causal and quantitative relationship between SF connectivity and mechanics for patterned cells and show that similar relationships hold for nonpatterned cells allowed to form cell-cell contacts in monolayer culture. Remarkably, dissipation of a single SF within a monolayer induces cytoskeletal rearrangements in cells long distances away. Finally, stimulation of cell migration leads to characteristic changes in network connectivity that promote SF bundling at the cell rear. Our findings demonstrate that SFs influence and are influenced by the networks in which they reside. Such higher order network interactions contribute in unexpected ways to cell mechanics and motility.cytoskeleton | active cable model | actin | myosin | cell migration