We will start with definitions and overview of the main mathematical objects that are involved in the inverse problems of our interest. These include the domains of definitions of the functions and operators, the boundary and spectral data and interpolation/extrapolation and restriction techniques.

A harmonic function is a twice continuously differentiable function f : U → R (where U is an open subset of Rn) which satisfies Laplace's equation.

The value of a harmonic function is a weighted average of its values at the neighbor vertices.

Corollary: the maximum (and the minimum) of a harmonic functions occurs on the boundary of the graph or the manifold.

Analytic continuation is an extension of the domain of a given analytic function.

We consider the following random walk of a particle on G with discrete time.

• At moment t = 0 the particle occupies a boundary vertex v of G.
• At moment t = n+1 the particle moves to a neighbor of its position at moment t = n.

Definition: the electrical network is a directed weighted graph with a boundary ${\displaystyle G=(V,\partial V,E,\gamma )}$ such that ${\displaystyle (a,b)\in E\rightarrow (b,a)\in E}$ and ${\displaystyle \gamma (a,b)=\gamma (b,a)}$. The weight function defined on the edges of the graph is called conductivity.