Consistently weighted measures for complex network topologies Jobst Heitzig, J. J. F. Donges, Y. Zou, N. Marwan, J. Kurths Jobst Heitzig, 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 Jobst Heitzig Consistently weighted network measures EGU 2010
Motivation: Climate Networks Nodes represent grid cells, Nodes represent grid cells, cell size varies ≈ ≈ cos(latitude) cos(latitude) cell size varies Network measures Network measures are based on counting are based on counting (nodes, links, paths...) (nodes, links, paths...) (fictitious example) Jobst Heitzig Consistently weighted network measures EGU 2010
Motivation: Climate Networks Nodes represent grid cells, Nodes represent grid cells, cell size varies ≈ ≈ cos(latitude) cos(latitude) cell size varies Network measures Network measures are based on counting are based on counting 3 links to here 3 links to here (nodes, links, paths...) (nodes, links, paths...) 12 links to here 12 links to here some some node Polar regions are node Polar regions are over-represented over-represented Results can get biased Results can get biased or show artificial features or show artificial features (fictitious example) Jobst Heitzig Consistently weighted network measures EGU 2010
Motivation: Climate Networks Nodes represent grid cells, Nodes represent grid cells, cell size varies ≈ ≈ cos(latitude) cos(latitude) cell size varies Network measures Network measures are based on counting are based on counting (nodes, links, paths...) (nodes, links, paths...) artificially high artificially high Polar regions are Polar regions are clustering coefficient clustering coefficient over-represented over-represented around the North Pole around the North Pole Results can get biased Results can get biased or show artificial features or show artificial features Jobst Heitzig Consistently weighted network measures EGU 2010
Natural idea: Use weights Cell size Node ode weight weight Cell size N (fictitious example) Jobst Heitzig Consistently weighted network measures EGU 2010
Natural idea: Use weights Cell size Node ode weight weight Cell size N Almost no network measures Almost no network measures use node node weights already weights already use Existing measures using Existing measures using link weights don't help weights don't help link (fictitious example) Jobst Heitzig Consistently weighted network measures EGU 2010
Natural idea: Use weights Cell size Node ode weight weight Cell size N Almost no network measures Almost no network measures use node node weights already weights already use linked area of linked area of Existing measures using Existing measures using total weight A total weight A link weights don't help weights don't help link linked area of linked area of total weight B some total weight B some ≈ A A ≈ node node Find node-weighted Find node-weighted versions of measures versions of measures (degree, clustering coeff., (degree, clustering coeff., betweenness, spectrum, ...) betweenness, spectrum, ...) (fictitious example) Jobst Heitzig Consistently weighted network measures EGU 2010
Simple example: The “degree” measure Nodes v, i, ... v, i, ... Nodes node weights weights w w v , w i , ... node v , w i , ... Degree: Degree: k v = no. nodes linked to v v k v = no. nodes linked to Area-weighted connectivity: Area-weighted connectivity: k' v k' v k' v = sum of sum of w w i k' v = i v v for all i i linked to linked to v v for all w v w v (Tsonis et al. 2006) (Tsonis et al. 2006) i i w w i i (serving suggestion) Jobst Heitzig Consistently weighted network measures EGU 2010
Simple example: The “degree” measure Nodes v, i, ... v, i, ... Nodes node weights weights w w v , w i , ... node v , w i , ... Degree: Degree: k v = no. nodes linked to v v k v = no. nodes linked to Area-weighted connectivity: Area-weighted connectivity: k* v k* k' v = sum of sum of w w i v k' v = i for all i i linked to linked to v v for all (Tsonis et al. 2006) (Tsonis et al. 2006) Better version of Better version of weighted degree: weighted degree: k* v = k' v + w v k* v = k' v + w v (serving suggestion) 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 w w i Goal: Find the right way of using the node weights i in some given measure f f in some given measure (degree, clustering coeff., betweenness, spectrum, ...) (degree, clustering coeff., betweenness, spectrum, ...) Idea: Consider what happens to f f Idea: Consider what happens to when the grid is refined! when the grid is refined! (fictitious example) 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 w w i Goal: Find the right way of using the node weights i in some given measure f f in some given measure (degree, clustering coeff., betweenness, spectrum, ...) (degree, clustering coeff., betweenness, spectrum, ...) Idea: Consider what happens to f f Idea: Consider what happens to when the grid is refined! when the grid is refined! Example: Example: Under typical refinements, Under typical refinements, f should get more realistic should get more realistic f (fictitious example) Jobst Heitzig Consistently weighted network measures EGU 2010
Redundant refinements / General guideline Under “redundant” redundant” refinements refinements Under “ f should should not not change change f (fictitious example) Jobst Heitzig Consistently weighted network measures EGU 2010
Redundant refinements / Guiding notion Under “redundant” redundant” refinements refinements Under “ f should should not not change change f This vague requirement helps This vague requirement helps to find the weighted formula f* f* to find the weighted formula for a given measure f f ! ! for a given measure (fictitious example) Jobst Heitzig Consistently weighted network measures EGU 2010
Redundant refinements / Guiding notion Under redundant redundant refinements, refinements, Under f should should not not change change f This vague requirement helps This vague requirement helps to find the weighted formula f* f* to find the weighted formula for a given measure f f ! ! for a given measure Guiding notion: Call f* f* Guiding notion: Call “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: Jobst Heitzig Consistently weighted network measures EGU 2010
d Example: Clustering coefficient 2 n Measures how closely linked the neighbours of v v are. are. Measures how closely linked the neighbours of Usual formula: Usual formula: C v = rate of links between neighbours of of links between neighbours of v v C v = rate = = Σ Σ i Σ j a v a i a j / k v ( k k v – 1) 1) i Σ j a i a j a v / k v ( v – v i j i j v ? ? Node splitting invariant formula: Node splitting invariant formula: C* * v = Σ i Σ j a' ' v w i a' ' i w j a' ' j / k* * v k* * v C v = Σ i Σ j a i w a j w a v / k v k v i i j j v i j v = link density in the region linked to v v = link density in the region linked to In this, a a ij = 1 means i i and and j j are linked, are linked, In this, ij = 1 means and a' a' ij = 1 means i i and and j j are linked or equal are linked or equal and ij = 1 means 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 i with a' i ) (e.g. replace a j with a' ) i j i j j Replace edge counts by Replace edge counts by sums of weight products C v = = Σ Σ i Σ j a v a i a j / / k k v ( k k v – 1 1) ) sums of weight products C v i Σ a i a j a v v ( v – j v i j i j v Replace node counts by Replace node counts by sums of weights sums of weights C* v = Σ i Σ j a' ' v w i a' i w j a' ' j / / k k* * v k* * v C* v = Σ i Σ j a i w a' w a v v k v i i j j j v i j v Plug in weighted instead of Plug in weighted instead of unweighted measures unweighted measures (k* v instead of k v in this case) (k* v instead of k v in this case) Verify the result is indeed Verify the result is indeed node splitting invariant! node splitting invariant! Jobst Heitzig Consistently weighted network measures EGU 2010
Effect in climate networks Clustering coefficient averaged by latitude Clustering coefficient averaged by latitude C v C v climate climate (dark is high) (dark is high) C* v C* v network network C v C v spatially homogeneous spatially homogeneous random network random network C* v C* v south latitude north south latitude north Jobst Heitzig Consistently weighted network measures EGU 2010
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