Influence function of a variational problem

The article considers an object that can serve a mathematical model for very diverse classes of real physical systems united by one common property. Each of such systems is defined by a finite set of onedimensional fragments interacting with each other in various ways. It can be an electrical or information network, or a system of acoustic pipes, gas or oil pipelines, elastic cables (mathematical strings), and much more. Such objects are included in mathematics not so long ago. The work proves the existence of the influence function and its properties. All the results proposed in this paper are new because they extend to the previously known results of problems in graphs in essentially general conditions in solutions. In addition, earlier for problems in networks, questions of the forms of stability loss studied in this paper are not posed and therefore are not discussed.