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We prove that the radii of convergence of the solutions of a$p$-adic differential equation $${\fancyscript{F}}$$ over an affinoid domain$X$of the Berkovich affine line are continuous functions on$X$that factorize through the retraction of $${X\to\Gamma}$$ of$X$onto a finite graph $${\Gamma\subseteq X}$$ . We also prove their super-harmonicity properties. This finiteness result means that the behavior of the radii as functions on$X$is controlled by a$finite$family of data.

Berkovich spaces; Radius of convergence; Newton polygon; spectral radius; controlling graph; finiteness