An online discussion forum contains two topic communities, "cooking" with members and…
PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Thermal, Statistical & Complex Systems · Complex and collective systems
Many Parts, One Behaviour, Ch Communities: finding the groups in a network
Problem
An online discussion forum contains two topic communities, "cooking" with n1=40 members and "cycling" with n2=60 members. In the stochastic block model, every possible pair of members is linked independently with a probability that depends only on the pair's block labels, prs. A snapshot of the forum's friendship network has m12=108 links between a cooking member and a cycling member. (a) Write the log-likelihood of the observed inter-block links as a function of p12, treating each of the n1n2 possible cooking-cycling pairs as an independent Bernoulli trial with success probability p12, and derive the maximum-likelihood estimate p^12=m12/(n1n2) by setting ∂lnL/∂p12=0. (b) Evaluate p^12 for the stated numbers, and convert it to an estimated mean number of cycling friends per cooking member, k^out=p^12n2.