  Consider an algebraic identity between elliptic modular graphs where several
vertices are at fixed locations (and hence unintegrated) while the others are
integrated over the toroidal worldsheet. At any unintegrated vertex, we can
glue an arbitrary expression involving elliptic modular graphs which has the
same unintegrated vertex. Integrating over that vertex, we obtain new algebraic
identities between elliptic modular graphs. Hence this elementary process of
convoluting the original "seed" identity with other graphs yields infinite
number of new identities. We consider various seed identities in which two of
the vertices are unintegrated. Convoluting them with families of elliptic
modular graphs, we obtain new identities. Each identity is parametrized by an
arbitrary number of links in the graphs as well as the positions of
unintegrated vertices. On identifying the unintegrated vertices, this leads to
an algebraic identity involving modular graphs where all the vertices are
integrated over the worldsheet.
