**Top Left.** *Success rate and homonymy as a function of time.* Success rate and homonymy are reported as a function of time (number of games), averaged over sliding time windows. Homonymy is here defined as the number of pairs of objects that have at least an associated word in common in an individual inventory, divided by the number of pairs

and averaged over all the agents. We define here a normalized link probability for the ER random graph as

, where

is the threshold in order to have, with probability one in the infinite size limit (

), a giant connected component equal to the whole graph. Similarly, a normalized time scale parameter is defined as

. Results are reported for

and for different values of the learning parameter

.

**Top Right.** *Word length distribution.* In the main figure the distribution of word length

*L* for different

values is reported, again fixing

.

is the observed empirical function [41] (fitting parameters

,

,

). In the top inset we show the same distribution fixing

for different values of

. Note that here the curves overlap, indicating that the word length distribution does not depend on the objects graph connectivity. In the bottom inset the

*average word length*,

*L*, as a function of

is reported, for different values of

.

**Bottom Left.** *Frequency-rank distribution for elementary forms*. In the main figure the frequency-rank distribution for elementary forms is shown for different values of the parameter

keeping fixed

. In the top inset we show the same distribution fixing

and for different

, showing that the distribution of elementary forms again does not depend on the objects graph connectivity. In the bottom inset both the

*number of distinct elementary forms* composing the lexicon and the

*effective number of distinct elementary forms*, as defined in the text, are reported, as a function of

and for different values of

.

**Bottom Right.** *Combinatoriality*. Combinatoriality

*C* (see the text for definition) for different values of

as a function of

. In the inset the

*normalized entropy*, as defined in the text, is reported, again for different values of

as a function of

. In all the measures reported above the number of agents and the number of objects in the environment are fixed respectively to

and

and results are averaged over 100 realizations of the dynamics on the same objects graph (see main text for further details).

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