Each hour:
(can be done by a cron job)
Log in on each server and:
(can be done by ssh with pre-installed keys)
list all TCP connections
(can be done by netstat or cat /proc/net/tcp or similar)
count them
(can be done by wc)
E. g.:
for host in $(cat hostnames)
do
n=$(ssh "$host" 'cat /proc/net/tcp | wc -l')
[ "$n" -gt 100 ] && echo "$host: $n connections"
done >> logfile.txt