  A dynamic model of a society is studied where each person is an uncorrelated
and non-interacting random walker. A dynamical random graph represents the
acquaintance network of the society whose nodes are the individuals and links
are the pairs of mutual friendships. This network exhibits a novel percolation
like phase transition in all dimensions. On introducing simultaneous death and
birth rates in the population we show that the friendship network shows the six
degrees of separation for ever after where the precise value of the network
diameter depends on the death/birth rate. A SIS type model of disease spreading
shows that this society remains always healthy if the population density is
less than certain threshold value.
