in this paper, let us first introduce the crucial tool that we use to describe graphs determined by a finite number of points. We compute *persistent homology* of *sublevel-set filtrations* of one dimensional simplicial complexes. The vertices of the simplicial complex have associated values, determined by a graph. We define the *persistence map* as the a map assigning to each sublevel-set filtration its *persistence diagram* or (equivalently) *barcode*. Existing literature about persistent homology is very rich, see e.g. [@chazal; @comptop; @ZC] and references provided there. Here we briefly recall persistent homology theory to the extent it is required by our theory. We also prove some preliminary results about the preimage (fiber) of the persistence map.

Let $z\in{\mathbb{R}}^N$ be fixed. We define the persistence map as the map associating the filtered complex ${{\mathcal{S}_N^{{{\mathsf F}}}}}(z)$ its persistence diagram, or sometimes referred to as the *barcode* [@ghrist; @ZC].

Observe that ${{\mathcal{S}_N^{{{\mathsf F}}}}}(z)$ is an one dimensional filtered *cell complex*. The persistent homology of cell complexes is computed using by now standard *persistent homology* algorithms provided in [@comptop].

By *persistence diagram* we mean a set of points of the extended plane $$\mathbb{E} = {\mathbb{R}}\times {\mathbb{R}}\cup\{\infty\}.$$ We denote the space of persistence diagrams by $${\mathsf{Per}}\subset 