my %used;
for ( @{ $graph_data->{Edges} } ) {
    $edges{$_->{in}->[0]} //= {};
    $edges{$_->{in}->[0]}->{$_->{in}->[1]} //= [];
    push( @{ $edges{$_->{in}->[0]}->{$_->{in}->[1]} }, $_->{out} );
    $used{$_->{out}} //= 0;
    $used{$_->{out}}++;
    if ( grep { $used{$_} > 1 } keys( %used ) ) {
            die( 'Only one input per node!' );
    }
}
[download]

Perl has a thing called autovivification, which means that that could be simplified to:

my %used;
for ( @{ $graph_data->{Edges} } ) {
    push @{ $edges{ $_->{in}[0] }{ $_->{in}[1] } }, $_->{out};
    if ( ++$used{ $_->{out} } > 1 ) {
        die 'Only one input per node!';
    }
}
[download]

    eval {
        ...
    };
    if ( $! ) {
        say "Error starting ID $_->{id}: $!";
    }
[download]

The $! variable does nothing useful there.    You need to use the $@ variable instead.