In this talk we will discuss the influence of network topology on collective properties of a dynamical system defined upon it.For this purpose,we propose a network model in which links of a regular chain are rewired according to a probability p within a specific range r,much like a lace is done,the reason why we call these lace networks.Using this procedure we add long-range links to the initial backbone.We then focus on how the thermodynamical behavior of a dynamical system,the XY-rotors model,is affected by the controlled topological changes of(p,r)and we identify a quantity,the network dimension d(p,r)as a crucial parameter.Varying this dimension we are able to cross over from topologies with d < 2 exhibiting no phase transitions to ones with d > 2 displaying a second order phase transition,passing by topologies with dimensional = 2 which exhibit states characterized by infinite susceptibility and macroscopic chaotic/turbulent dynamical behavior.