[PPT] - Consistently weighted measures for complex network topologies Jobst PowerPoint Presentation

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Jobst Heitzig Consistently weighted network measures EGU 2010

Consistently weighted measures for complex network topologies

Jobst Heitzig, Jobst Heitzig, J. J.

F. Donges, Y. Zou, N. Marwan, J. Kurths
F. Donges, Y. Zou, N. Marwan, J. Kurths

Potsdam Institute for Climate Impact Research Potsdam Institute for Climate Impact Research Transdisciplinary Concepts and Methods Transdisciplinary Concepts and Methods

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Jobst Heitzig Consistently weighted network measures EGU 2010

Nodes represent grid cells, Nodes represent grid cells, cell size varies cell size varies ≈ ≈ cos(latitude) cos(latitude) Network measures Network measures are based on counting are based on counting (nodes, links, paths...) (nodes, links, paths...)

Motivation: Climate Networks

(fictitious example)

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Jobst Heitzig Consistently weighted network measures EGU 2010

Motivation: Climate Networks

Nodes represent grid cells, Nodes represent grid cells, cell size varies cell size varies ≈ ≈ cos(latitude) cos(latitude) Network measures Network measures are based on counting are based on counting (nodes, links, paths...) (nodes, links, paths...) Polar regions are Polar regions are

ver-represented
ver-represented

Results can get biased Results can get biased

r show artificial features
r show artificial features

3 links to here 3 links to here 12 links to here 12 links to here some some node node

(fictitious example)

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Jobst Heitzig Consistently weighted network measures EGU 2010

Motivation: Climate Networks

Nodes represent grid cells, Nodes represent grid cells, cell size varies cell size varies ≈ ≈ cos(latitude) cos(latitude) Network measures Network measures are based on counting are based on counting (nodes, links, paths...) (nodes, links, paths...) Polar regions are Polar regions are

ver-represented
ver-represented

Results can get biased Results can get biased

r show artificial features
r show artificial features

artificially high artificially high clustering coefficient clustering coefficient around the North Pole around the North Pole

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Jobst Heitzig Consistently weighted network measures EGU 2010

Cell size Cell size   N Node

de weight

weight

Natural idea: Use weights

(fictitious example)

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Jobst Heitzig Consistently weighted network measures EGU 2010

Cell size Cell size   N Node

de weight

weight Almost no network measures Almost no network measures use use node node weights already weights already Existing measures using Existing measures using link link weights don't help weights don't help

Natural idea: Use weights

(fictitious example)

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Jobst Heitzig Consistently weighted network measures EGU 2010

Cell size Cell size   N Node

de weight

weight Almost no network measures Almost no network measures use use node node weights already weights already Existing measures using Existing measures using link link weights don't help weights don't help Find node-weighted Find node-weighted versions of measures versions of measures (degree, clustering coeff., (degree, clustering coeff., betweenness, spectrum, ...) betweenness, spectrum, ...)

Natural idea: Use weights

linked area of linked area of total weight A total weight A linked area of linked area of total weight B total weight B ≈ ≈ A A some some node node

(fictitious example)

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Jobst Heitzig Consistently weighted network measures EGU 2010

Simple example: The “degree” measure

Nodes Nodes v, i, ... v, i, ... node node weights weights w wv

v , w

, wi

i , ...

, ... Degree: Degree: k kv

v = no. nodes linked to

= no. nodes linked to v v Area-weighted connectivity: Area-weighted connectivity: k' k'v

v =

= sum of sum of w wi

i

for all for all i i linked to linked to v v

(Tsonis et al. 2006) (Tsonis et al. 2006)

(serving suggestion)

k' k'v

v

w wv

v

v v

w wi

i

i i

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Jobst Heitzig Consistently weighted network measures EGU 2010

Simple example: The “degree” measure

Nodes Nodes v, i, ... v, i, ... node node weights weights w wv

v , w

, wi

i , ...

, ... Degree: Degree: k kv

v = no. nodes linked to

= no. nodes linked to v v Area-weighted connectivity: Area-weighted connectivity: k' k'v

v =

= sum of sum of w wi

i

for all for all i i linked to linked to v v

(Tsonis et al. 2006) (Tsonis et al. 2006)

Better version of Better version of weighted degree: weighted degree: k* k*v

v = k'

= k'v

v + w

+ wv

v

(serving suggestion)

k* k*v

v

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Jobst Heitzig Consistently weighted network measures EGU 2010

Why k* and not k'? And what about more complex measures?

Goal: Find the right way of using the node weights Goal: Find the right way of using the node weights w wi

i

in some given measure in some given measure f f

(degree, clustering coeff., betweenness, spectrum, ...) (degree, clustering coeff., betweenness, spectrum, ...)

Idea: Consider what happens to Idea: Consider what happens to f f when the grid is refined! when the grid is refined!

(fictitious example)

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Jobst Heitzig Consistently weighted network measures EGU 2010

Goal: Find the right way of using the node weights Goal: Find the right way of using the node weights w wi

i

in some given measure in some given measure f f

(degree, clustering coeff., betweenness, spectrum, ...) (degree, clustering coeff., betweenness, spectrum, ...)

Idea: Consider what happens to Idea: Consider what happens to f f when the grid is refined! when the grid is refined! Example: Example: Under typical refinements, Under typical refinements, f f should get more realistic should get more realistic  

(fictitious example)

Why k* and not k'? And what about more complex measures?

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Jobst Heitzig Consistently weighted network measures EGU 2010

Under “ Under “redundant” redundant” refinements refinements   f f should should not not change change

Redundant refinements / General guideline

(fictitious example)

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Jobst Heitzig Consistently weighted network measures EGU 2010

Under “ Under “redundant” redundant” refinements refinements   f f should should not not change change This vague requirement helps This vague requirement helps to find the weighted formula to find the weighted formula f* f* for a given measure for a given measure f f ! !

Redundant refinements / Guiding notion

(fictitious example)

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Jobst Heitzig Consistently weighted network measures EGU 2010

Under Under redundant redundant refinements, refinements,   f f should should not not change change This vague requirement helps This vague requirement helps to find the weighted formula to find the weighted formula f* f* for a given measure for a given measure f f ! ! Guiding notion: Call Guiding notion: Call f* f* “node splitting invariant” “node splitting invariant” if it doesn't change under if it doesn't change under this kind of node splitting: this kind of node splitting:

Redundant refinements / Guiding notion

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Jobst Heitzig Consistently weighted network measures EGU 2010

2n

d Example: Clustering coefficient

Measures how closely linked the neighbours of Measures how closely linked the neighbours of v v are. are. Usual formula: Usual formula: C Cv

v = rate

= rate of links between neighbours of

f links between neighbours of v

v = = Σ Σi

i Σ

Σj

j a

av

v

i

i a

ai

i

j

j a

aj

j

v

v / k

/ kv

v (

(k kv

v –

– 1) 1) Node splitting invariant formula: Node splitting invariant formula: C C* *v

v = Σ

= Σi

i Σ

Σj

j a

a' 'v

v

i

i w

wi

i a

a' 'i

i

j

j w

wj

j a

a' 'j

j

v

v / k

/ k* *v

v k

k* *v

v = link density in the region linked to

= link density in the region linked to v v In this, In this, a aij

ij = 1 means

= 1 means i i and and j j are linked, are linked, and and a' a'ij

ij = 1 means

= 1 means i i and and j j are linked or equal are linked or equal ? ?

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Jobst Heitzig Consistently weighted network measures EGU 2010

Useful techniques for formula construction

Consider each node a Consider each node a neighbour of itself neighbour of itself (e.g. replace a (e.g. replace ai

i

j

j with a'

with a'i

i

j

)

) Replace edge counts by Replace edge counts by sums of weight products sums of weight products Replace node counts by Replace node counts by sums of weights sums of weights Plug in weighted instead of Plug in weighted instead of unweighted measures unweighted measures (k* (k*v

v instead of k

instead of kv

v in this case)

in this case) Verify the result is indeed Verify the result is indeed node splitting invariant! node splitting invariant!

C Cv

v

= = Σ Σi

i Σ

Σj

j a

av

v

i

i a

ai

i

j

j a

aj

j

v

/ / k kv

v (

(k kv

v –

– 1 1) ) C* C*v

v = Σ

= Σi

i Σ

Σj

j a

a' 'v

v

i

i w

wi

i a'

a'i

i

j

j w

wj

j a

a' 'j

j

v

/ / k k* *v

v k

k* *v

v

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Jobst Heitzig Consistently weighted network measures EGU 2010

Effect in climate networks

latitude latitude south south north north Clustering coefficient averaged by latitude Clustering coefficient averaged by latitude climate climate network network spatially homogeneous spatially homogeneous random network random network C Cv

v

C Cv

v

C* C*v

v

C* C*v

v

(dark is high) (dark is high)

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Jobst Heitzig Consistently weighted network measures EGU 2010

Final example: Newman's random walk betweenness

weighted weighted

Gulf Stream, Gulf Stream, Canary Current Canary Current Antarctic Antarctic Circumpolar Circumpolar Current Current El El Ni Niñ ño,

,

Equatorial Equatorial Currents Currents

Measures “importance” of nodes Measures “importance” of nodes based on Kirchhoff's equations based on Kirchhoff's equations Unweighted and weighted versions Unweighted and weighted versions highlight slightly different features highlight slightly different features

unweighted unweighted

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Jobst Heitzig Consistently weighted network measures EGU 2010

References

J. Heitzig, J.
J. Heitzig, J.
F. Donges, Y. Zou, N. Marwan, J. Kurths (2010),
F. Donges, Y. Zou, N. Marwan, J. Kurths (2010),

Consistently weighted measures for complex network topologies, Consistently weighted measures for complex network topologies, under review. under review. A. A.

A. Tsonis, K.
A. Tsonis, K.
L. Swanson, P. Roebber (2006),
L. Swanson, P. Roebber (2006),
Bull. Am. Meteorol. Soc. 87, 585.
Bull. Am. Meteorol. Soc. 87, 585.