I would have done it something like this:

my %net = ( ... );
my $start = '1.2.3.4';
my $end = '1.2.3.6';
print find_pathr( \%net, $end, $start ) . "\n";

sub find_pathr {
    my ( $net, $end, @list ) = @_;
    my $curr;
    do {
        $curr = shift @list;
    } until ( ! @list or exists $$net{ $curr } );
    return "No path!" unless exists $$net{$curr};
    return $end if $curr eq $end;
    unshift @list, @{ delete $$net{ $curr } };
    return "$curr->" . find_pathr( $net, $end, @list );
}
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Now this quick implementation isn't perfect. It doesn't pretend to find the most efficient path, it finds the first path based on the order of the elements of %net. It is destructive to %net. I don't like the do ... until loop.

But I do think it is an elegant solution, IMNSHO ;)

Update: This is a depth-first search, not breadth-first as is limbic~region's.