Two research groups

PHYSICS · Multi-part · 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 research groups, {1,2,3}\{1,2,3\} and {4,5,6}\{4,5,6\}, are each a fully connected trio (a triangle of collaborators), joined by a single collaboration edge 33-44. For this network the second-smallest Laplacian eigenvalue is λ2=(5−17)/2≈0.438\lambda_2=(5-\sqrt{17})/2\approx0.438, with eigenvector (the Fiedler vector) components v2=(1, 1, 0.562, −0.562, −1, −1)\mathbf v_2=(1,\ 1,\ 0.562,\ -0.562,\ -1,\ -1) for nodes 11 to 66 in order. ((a)) Apply the sign rule to v2\mathbf v_2 to obtain a bisection of the six collaborators. ((b)) Compute the cut size RR of that bisection. ((c)) Check RR against the bound R≥Nλ2/4R\ge N\lambda_2/4.

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