Structure of complex networks: Quantifying edge-to-edge relations by - - PowerPoint PPT Presentation

Structure of complex networks: Quantifying edge-to-edge relations by failure induced flow redistribution

Netsci 2014 Berkeley Higher Order Models in Network Science Satellite meeting Michael T. Schaub Department of Mathematics Imperial College London

• J. Lehmann, S. N. Yaliraki and M. Barahona

June 3, 2014

Outline

◮ Background: why edge relations? ◮ Edge relations through flow redistribution ◮ Using edge relations for network analysis

Background: why edge relations?

◮ Network Analysis so far mainly node centric ◮ Communities, node roles, centralities, etc. [Delvenne et al., 2010] [Cooper et al 2010]

Dual Perspective – Edge centered

'Classical' View Node centered Dual View Edge centered

◮ Circuit Theory: voltage vs currents ◮ Computational mechanics: displacement vs stress ◮ Optimization: Primal vs Dual variables ◮ Systems engineering, estimation theory, etc...

How to quantify edge relations? Flow redistribution!

failing edge influenced edge

How to quantify edge relations? Flow redistribution!

failing edge influenced edge failing edge influenced edge

How to quantify edge relations? Flow redistribution!

failing edge influenced edge failing edge influenced edge

Assuming a linear flow on the edges

edge weight pseudoinverse of Laplacian G=diag(g) incidence matrix incidence vector

independent of current injections

The flow redistribution matrix

◮ Flow redistribution matrix

KE×E ≡ [k1 · · · kE]

◮ Independent of current injections ◮ Describes topological feature of system in the edge space:

edge-to-edge coupling

Characterising the flow redistribution matrix

The flow redistribution matrix can be decomposed

K = M [diag(ε)]−1

into the edge-to-edge transfer function matrix

ME×E ≡ GBTL†B

and the edge-embeddedness

εe ≡ 1 − ge bT

e L†be = 1 − geRe

The edge to edge transfer function

ME×E ≡ GBTL†B

◮ Transfer function – describes how input on edge translates

into flow on other edges

◮ Physics interpretation – discrete Green’s function (edge

space)

◮ Projection matrix (idempotent) – into the weighted cut space

• f the graph

◮ spectral properties of M

The edge-embeddedness

εe ≡ 1 − ge bT

e L†be = 1 − geRe

Re – resistance distance between endpoints of edge e

◮ related to the projection into the cycle space ◮ High embeddedness – edge features in many cycles

(weighted)

◮ Zero embeddedness – edge defines a cut (disconnects the

network)

◮ Unweighted networks – probability of not finding the edge in

a randomly selected spanning tree

◮ εe = #cycles in network ◮ Related to graph sparsification

Applications: a toy example

Edge Embededness Principal component

(a) (b) (c)

1 3 5 9 7 0.8 0.6 0.4 0.2 50 100 150 200 250 0.2 0.3 0.4 0.5 0.6 0.7 0.1 50 100 150 200 250 Edge |PCA| Markov Time Variation of Information

• No. Communities

10

−2

10

−1

10 10

1

10 10

2

0.05 0.1

6

Applications 1 – Iberian Power Grid

Embeddedness

0.7 0.6 0.5 0.4 0.3 0.2 0.1

(a) (b)

c1c2 c3

c2 c3 c1

(c)

0.5 1 0.5 1 0.2 0.4 100 200 300 400 500 600

Edge

|LODF| |LODF| |LODF| 100 200 300 400 500 600 100 200 300 400 500 600 0.1 0.2 0.3 0.4 0.5 1

Edge Edge

(d) (e)

Applications 1 – Iberian Power Grid

Embeddedness

0.7 0.6 0.5 0.4 0.3 0.2 0.1

(a) (b)

c1c2 c3

c2 c3 c1

(c)

0.5 1 0.5 1 0.2 0.4 100 200 300 400 500 600

Edge

|LODF| |LODF| |LODF| 100 200 300 400 500 600 100 200 300 400 500 600 0.1 0.2 0.3 0.4 0.5 1

Edge Edge

(d) (e)

Applications 1 – Iberian Power Grid

Embeddedness

0.7 0.6 0.5 0.4 0.3 0.2 0.1

(a) (b)

c1 c2 c3

c2 c3 c1

(c)

0.5 1 0.5 1 0.2 0.4 100 200 300 400 500 600

Edge

|LODF| |LODF| |LODF| 100 200 300 400 500 600 100 200 300 400 500 600 0.1 0.2 0.3 0.4 0.5 1

Edge Edge

(d) (e)

Applications 2 – Street networks

(b) (a)

London Boston 0.2 0.4 0.6 0.8 Embeddedness New York

Applications 2 – Street networks

(b) (a)

London Boston 0.2 0.4 0.6 0.8 Embeddedness New York

Applications 3 – C. elegans

(a)

Head (H) Mid-Body (B) Tail (T)

(b) (c)

Neurons Neurons Sensory Interneurons Motor Neurons Neurons

(d)

IL2VL IL2L IL1VL URADL IL1DL OLLL IL1L URYDL OLQDL URYVL RIPL OLLR IL2R URBL IL1DR URYDR URADR IL1R URAVL OLQVL RMED URBR OLQDR RIPR IL2VR RMEL CEPVL BAGR BAGL OLQVR URAVR RMER IL1VR URYVR CEPVR RMEV CEPDL RMDVL SAAVL SMDVL URXL RID ALA RMDVR CEPDR AVAL RIAL SAAVR RMDL URXR SMDVR AVAR RIAR ASKR ASKL AVEL ADLL ADFL RMDR AFDL AFDR SIBDL RIH AWBL AVER RMDDL AWCL ADFR ASGL SAADL ADLR AWAL AWBR ASIL ASHL SIBDR ASGR AIBL ASHR AWCR AWAR SIBVL RIVL SMDDL SAADR RMHL RMDDR ASIR AVHL AVHR RIVR AIBR RIBL RMFL AVBL SIBVR ASEL AVJ R AUAL SIADL RMHR AVJ L ASER AVBR RIBR RMFR SMDDR AIAL RIR SMBDL RIML ASJ L RIMR AUAR AVDR SMBVR AVDL AINL SMBVL ASJ R AINR SIADR AVL RICL AIAR SMBDR AIZR SIAVL SIAVR RICR AIZL AIYR AIMR AIML RIS AIYL VB02 AVKR AVKL AVFR SABVL FLPL FLPR SABVR AVFL AQR RIFR ADEL VB01 ADER DB02 ADAR ADAL RIGR RMGL RMGR RIFL AVG VA01 SABD RIGL DD01 DB01 AS01 VD01 DA01 VD02 VA02 VB03 AS02 DB03 DA02 VD03 BDUR BDUL VA03 SDQR VB04 VC01 DD02 AS03 VD04 DA03 VA04 VB05 VC02 DB04 AS04 VD05 VA05 AVM VB06 DA04 DD03 ALML ALMR VC03 AS05 VD06 VA06 VB07 DB05 AS06 VD07 VC04 DA05 VC05 HSNR HSNL VA07 VB08 AS07 DD04 VD08 VA08 VB09 DB06 AS08 PDEL VD09 PDER SDQL DA06 PVDL PVM VA09 VB10 AS09 DD05 VD10 VA10 VB1 1 DB07 AS10 DA07 VD1 1 VA1 1 AS1 1 VD12 PVPR PVT VA12 PVPL DA08 DA09 PDB VD13 PDA DVB DVA DVC PVQR PHAL PHAR PVQL LUAL PVCL PHBR ALNL PHBL LUAR ALNR PVCR PQR PVR PVWL PVWR PLNL PHCR PHCL PVNR PLMR PVNL PLML

processing depth normalized Fiedler vector Embeddedness

0.3 0.4 0.5 0.6 0.7 0.8 0.9 sensory neuron interneuron motor neuron

Applications 3 – C. elegans

(a)

Head (H) Mid-Body (B) Tail (T)

(b) (c)

Neurons Neurons Sensory Interneurons Motor Neurons Neurons

(d)

IL2VL IL2L IL1VL URADL IL1DL OLLL IL1L URYDL OLQDL URYVL RIPL OLLR IL2R URBL IL1DR URYDR URADR IL1R URAVL OLQVL RMED URBR OLQDR RIPR IL2VR RMEL CEPVL BAGR BAGL OLQVR URAVR RMER IL1VR URYVR CEPVR RMEV CEPDL RMDVL SAAVL SMDVL URXL RID ALA RMDVR CEPDR AVAL RIAL SAAVR RMDL URXR SMDVR AVAR RIAR ASKR ASKL AVEL ADLL ADFL RMDR AFDL AFDR SIBDL RIH AWBL AVER RMDDL AWCL ADFR ASGL SAADL ADLR AWAL AWBR ASIL ASHL SIBDR ASGR AIBL ASHR AWCR AWAR SIBVL RIVL SMDDL SAADR RMHL RMDDR ASIR AVHL AVHR RIVR AIBR RIBL RMFL AVBL SIBVR ASEL AVJR AUAL SIADL RMHR AVJL ASER AVBR RIBR RMFR SMDDR AIAL RIR SMBDL RIML ASJL RIMR AUAR AVDR SMBVR AVDL AINL SMBVL ASJR AINR SIADR AVL RICL AIAR SMBDR AIZR SIAVL SIAVR RICR AIZL AIYR AIMR AIML RIS AIYL VB02 AVKR AVKL AVFR SABVL FLPL FLPR SABVR AVFL AQR RIFR ADEL VB01 ADER DB02 ADAR ADAL RIGR RMGL RMGR RIFL AVG VA01 SABD RIGL DD01 DB01 AS01 VD01 DA01 VD02 VA02 VB03 AS02 DB03 DA02 VD03 BDUR BDUL VA03 SDQR VB04 VC01 DD02 AS03 VD04 DA03 VA04 VB05 VC02 DB04 AS04 VD05 VA05 AVM VB06 DA04 DD03 ALML ALMR VC03 AS05 VD06 VA06 VB07 DB05 AS06 VD07 VC04 DA05 VC05 HSNR HSNL VA07 VB08 AS07 DD04 VD08 VA08 VB09 DB06 AS08 PDEL VD09 PDER SDQL DA06 PVDL PVM VA09 VB10 AS09 DD05 VD10 VA10 VB11 DB07 AS10 DA07 VD11 VA11 AS11 VD12 PVPR PVT VA12 PVPL DA08 DA09 PDB VD13 PDA DVB DVA DVC PVQR PHAL PHAR PVQL LUAL PVCL PHBR ALNL PHBL LUAR ALNR PVCR PQR PVR PVWL PVWR PLNL PHCR PHCL PVNR PLMR PVNL PLML

processing depth normalized Fiedler vector Embeddedness

0.3 0.4 0.5 0.6 0.7 0.8 0.9 sensory neuron interneuron motor neuron

Take home messages

◮ Flow redistribution can characterise edge-to-edge relations ◮ Flow redistribution matrix – describes topological property

in edge space

◮ Decomposable in measures with graph theoretic meaning:

◮ Edge transfer function matrix (discrete Greens function) ◮ Edge-embeddedness (projection into cycle space,

sparsification)

◮ Ability to detect non-local effects in the edge coupling

Thank you!

The people..

◮ J. Lehmann (ABB) ◮ S. N. Yaliraki ◮ M. Barahona

The money...

◮ ONR ◮ EPSRC ◮ Studienstiftung des

• dt. Volkes

Everybody else – Thanks for listening!

Things left in the dark...

Schaub, M.T.; Lehmann, J.; Yaliraki, S.N. & Barahona, M., Structure of complex networks: Quantifying edge-to-edge relations by failure-induced flow redistribution, Network Science, April 2014, Vol. 2(1), pp. 66-89

QUESTIONS?

Embeddedness vs betweenness centrality

betweenness centrality

Embeddedness vs flow betweenness centrality

Embeddedness of connecting links invariant flow betweeness of connecting links changes