Two regional distributors each link six warehouses

PHYSICS · Subjective · 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

Two regional distributors each link six warehouses, numbered

1

6

, by direct routes. Network

X

is two triangles joined by one route:

1

2

1

3

2

3

4

5

4

6

5

6

and

3

4

. Network

Y

is a ring with one cross route:

1

2

2

3

3

4

4

5

5

6

6

1

and

1

4

. Both are simple, undirected and unweighted. (a) Show that

X

and

Y

have the same degree sequence, and hence the same

p_k

\langle k\rangle

, heterogeneity ratio

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

and neighbour mean

\langle k\rangle_{\rm nbr}

. Compute these four numbers. (b) Compute the clustering coefficient

C

of each network from triangles and connected triples. (c) Find the mean distance

\ell

of each network over its

15

pairs of nodes. (d) In each network, find the leading eigenvalue

\kappa_1

and the ratio of a degree-

3

node's eigenvector centrality to a degree-

2

node's. Justify that the eigenvector you find is the eigenvector centrality. (e) Say which of these summaries can tell

X

from

Y

and which cannot, and why eigenvector centrality, though it is not a local count, still cannot.

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