Two independent research collaborations each consist of three scientists who all work…
PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Thermal, Statistical & Complex Systems · Linear algebra and vector spaces
Many Parts, One Behaviour, Ch Communities: finding the groups in a network
Problem
Two independent research collaborations each consist of three scientists who all work with one another (a triangle K3), with no collaboration links between the two collaborations. Label the six scientists 1,…,6, with {1,2,3} the first triangle and {4,5,6} the second, so the whole network has two connected components. (a) Starting from xTLx=21∑ijAij(xi−xj)2 for an arbitrary real vector x, prove that the graph Laplacian L=D−A of any network is positive semi-definite. (b) Prove that the number of zero eigenvalues of L equals the number of connected components of the network. (c) Find the Laplacian eigenvalues of a single triangle K3, and hence state the full spectrum of the six-scientist network. Confirm that the zero-eigenvalue count matches the number of components.