A museum floor plan has four galleries joined in a loop (corridors ) and an entrance…

PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex Systems · Complex and collective systems · Complex and collective systems

Many Parts, One Behaviour, Ch Networks as graphs: degree, distance and clustering

Problem

A museum floor plan has four galleries

1,2,3,4

joined in a loop (corridors

12,23,34,41

) and an entrance lobby

5

that opens directly onto both gallery

1

and gallery

2

(corridors

15,25

), so that

1

2

and

5

form a small triangular hall. There are no other corridors. Treat the corridor plan as a simple, undirected, unweighted network. ((a)) Write the

5\times5

adjacency matrix (order

1,2,3,4,5

), give the degrees and the mean degree. ((b)) Give the degree distribution

p_k

and the heterogeneity ratio

\langle k^2\rangle/\langle k\rangle^2

. ((c)) Compute the clustering coefficient

C

. ((d)) Starting from

\mathbf x^{(0)}=(1,1,1,1,1)

, apply the adjacency matrix twice and report the ratio of gallery

1

's score to gallery

3

's score.

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