  The phi-invariant was introduced by Shou-Wu Zhang and Nariya Kawazumi. We
study the continuity property of the phi-invariants for degenerating graphs,
and show that this continuity property induces an adelic divisor on the moduli
space of Riemann surfaces, and then give an asymptotic expression of the
Zhang--Kawazumi's invariants for Riemann surfaces near the boundary of the
moduli space. We mainly use Yuan--Zhang's adelic divisors, and follow Yuan's
idea of globalization of phi-invariants.
