PHYSICS · Subjective · Fermi University Physics · Thermal, Statistical & Complex Systems · Complex and collective systems · Complex and collective systems
Many Parts, One Behaviour, Ch Communities: finding the groups in a network
Problem
A ring is built from n identical modules, each a complete "friendship clique" Kq of q mutually acquainted people, with successive cliques joined by a single extra edge so the ring closes on itself (as for the ring of five-cliques, but here with q=6). (a) For two groups joined by a single edge, with total degrees d1, d2 and the whole network having M edges, derive ΔQmerge=M1−2M2d1d2 for the change in modularity on merging them. (b) Specialise to a ring of n cliques Kq, where every group has the same total degree dg=q(q−1)+2 and the whole ring has M=ndg/2 edges, and show that merging two neighbouring cliques raises Q whenever n>q(q−1)+2. (c) Evaluate the threshold for q=6, and state whether a ring of n=30 such cliques lies above or below it.