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. [Cooper et al 2010] [Delvenne et al., 2010]

Dual Perspective – Edge centered 'Classical' View Dual View Node centered 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 failing edge influenced edge influenced edge

How to quantify edge relations? Flow redistribution! failing edge failing edge influenced edge influenced edge Assuming a linear flow on the edges G=diag(g) incidence matrix incidence vector edge weight pseudoinverse of Laplacian independent of current injections

The flow redistribution matrix ◮ Flow redistribution matrix K E × E ≡ [ k 1 · · · k E ] ◮ 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 M E × E ≡ GB T L † B and the edge-embeddedness ε e ≡ 1 − g e b T e L † b e = 1 − g e R e

The edge to edge transfer function M E × E ≡ GB T L † 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 of the graph ◮ spectral properties of M

The edge-embeddedness ε e ≡ 1 − g e b T e L † b e = 1 − g e R e R e – 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 ◮ � ε e = #cycles in network ◮ Related to graph sparsification

Applications: a toy example (a) (b) 0.7 Embededness 0.6 0.5 0.4 0.3 0.2 0.1 (c) 0 50 100 150 200 250 Edge Variation of Information Principal component 0.8 1 No. Communities 3 0.1 0.6 2 |PCA| 10 5 6 0.05 0.4 0 7 0 10 0.2 −2 −1 0 1 10 10 10 10 9 Markov Time 0 50 100 150 200 250 Edge

Applications 1 – Iberian Power Grid (a) (c) 0.4 1 0.5 c1 |LODF| 100 0.2 0.4 200 1 c2 |LODF| 0.3 Edge 300 0.5 0.2 400 1 c3 |LODF| 500 0.1 0.5 600 0 0 0 100 200 300 400 500 600 100 200 300 400 500 600 Edge Edge (b) (d) 0.7 0.6 Embeddedness c3 0.5 c1 c2 0.4 0.3 0.2 (e) 0.1

Applications 1 – Iberian Power Grid (a) (c) 0.4 1 0.5 c1 |LODF| 100 0.2 0.4 200 1 c2 |LODF| 0.3 Edge 300 0.5 0.2 400 1 c3 |LODF| 500 0.1 0.5 600 0 0 0 100 200 300 400 500 600 100 200 300 400 500 600 Edge Edge (b) (d) 0.7 0.6 Embeddedness c3 0.5 c1 c2 0.4 0.3 0.2 (e) 0.1

Applications 1 – Iberian Power Grid (a) (c) 0.4 1 0.5 c1 |LODF| 100 0.2 0.4 200 1 c2 |LODF| 0.3 Edge 300 0.5 0.2 400 1 c3 |LODF| 500 0.1 0.5 600 0 0 0 100 200 300 400 500 600 100 200 300 400 500 600 Edge Edge (b) (d) 0.7 0.6 Embeddedness c3 0.5 c1 c2 0.4 0.3 0.2 (e) 0.1

Applications 2 – Street networks (a) London Boston New York (b) Embeddedness 0.2 0.4 0.6 0.8

Applications 2 – Street networks (a) London Boston New York (b) Embeddedness 0.2 0.4 0.6 0.8

Applications 3 – C. elegans (a) (c) Head Mid-Body Tail normalized Fiedler vector ASJ L sensory neuron (H) (B) (T) PHAR AIMR interneuron processing depth ASJ R ASIR AINL ASIL ALMR motor neuron AIML PVM PHAL AWAR AWAL HSNL AINR ASGL PHCR ADLR AWBR AVFL ASEL AWBL FLPR IL2L ADFL PVQL IL2R ASGR ALML VC05 PLNL ADLL LUAR Neurons AVM SDQL CEPDR ASER PDEL CEPVR AWCR IL2VL ADEL ASHL PLMR IL2VR BAGL BDUL PHBR ASHR PVPR PVDL RIH CEPDL ADFR AVHR BDUR AWCL ASKL RIR PQR URBR URXL PVQR AVHL PVNL ADER PHBL VC04 CEPVL FLPL OLLL PDER ALNL ASKR RIFR AFDR AUAL BAGR PLML SDQR ADAL LUAL PHCL AUAR AVG AFDL RIFL HSNR URYVR URYVL ALNR RIS URXR URADR PVT PVNR URAVR ADAR PVR PVWR URAVL AIYL PVPL IL1R OLLR URBL RMGR AIAR AVFR AIYR RICL AVJ R URADL ALA PVWL PVCR IL1VR SAAVL OLQVL RMGL AQR IL1L AIZL AIAL AIBL AVDR DVA URYDL RMFL VD1 1 Neurons DVC VA12 IL1VL URYDR AIZR RICR OLQVR RIAR AVDL SMBVR PDA SAAVR (b) OLQDR RIGL PVCL RMFR AVJ L VC02 DVB RIVR RIVL RIGR VB01 VB08 RIAL SMBVL IL1DL AIBR AVL IL1DR AVKR AVKL AVAL DB01 OLQDL RMHR SAADL VB1 1 VC03 RMEL RMHL SMBDR AVBL VB09 Sensory Interneurons Motor RIPL RIBL SAADR AVEL AVAR VB10 VB02 RMEV SIBDR SIADR VA07 AVER RID RIPR DB02 RMER RIBR SMBDL RIMR AVBR SIADL VB07 VA02 VB06 DB07 VA09 RMED SIAVR RMDL SIBVL VC01 AS1 1 DB04 RIML SIBVR SABD VB03 RMDDR SIBDL VA01 DA09 SIAVL VA08 RMDDL PDB SMDVL DB03 AS06 VD10 VB05 VA1 1 RMDR AS01 AS02 VA04 VB04 SMDVR RMDVR SABVR DA03 VD13 AS09 SMDDR DA01 RMDVL SMDDL DB05 VA06 DA04 VA03 DA02 AS03 Embeddedness SABVL DB06 DA08 AS04 Neurons DA05 VA05 AS07 AS05 VD01 AS10 DA07 DA06 VD09 DD05 VA10 0.3 0.4 0.5 0.6 0.7 0.8 0.9 VD07 VD12 AS08 DD01 VD08 (d) VD05 DD02 DD04 VD02 VD04 VD03 DD03 VD06 Neurons

Applications 3 – C. elegans (a) (c) Head Mid-Body Tail normalized Fiedler vector ASJL sensory neuron (H) (B) (T) PHAR AIMR interneuron processing depth ASJR ASIR AINL ASIL ALMR motor neuron AIML PVM PHAL AWAR AWAL HSNL AINR ASGL PHCR ADLR AWBR AVFL ASEL AWBL FLPR IL2L ADFL PVQL IL2R ASGR ALML VC05 PLNL ADLL LUAR Neurons AVM SDQL CEPDR ASER PDEL AWCR IL2VL CEPVR ADEL ASHL PLMR IL2VR BAGL BDUL PHBR ASHR PVPR PVDL RIH CEPDL ADFR AVHR BDUR AWCL ASKL RIR PQR URBR PVQR AVHL PVNL URXL ADER PHBL VC04 CEPVL FLPL OLLL PDER ALNL ASKR RIFR AFDR AUAL BAGR PLML SDQR ADAL LUAL PHCL AUAR AVG AFDL RIFL HSNR URYVR URYVL ALNR RIS URXR URADR PVT PVNR ADAR PVR PVWR URAVR URAVL AIYL PVPL IL1R OLLR URBL RMGR AIAR AVFR AIYR RICL AVJR URADL ALA PVWL PVCR IL1VR SAAVL OLQVL RMGL AQR AIZL AIAL IL1L AIBL AVDR DVA URYDL RMFL VD11 Neurons DVC VA12 IL1VL AIZR RICR OLQVR RIAR URYDR AVDL SMBVR PDA SAAVR (b) OLQDR RIGL PVCL RMFR AVJL VC02 DVB RIVR RIVL RIGR VB01 VB08 SMBVL IL1DL RIAL AIBR AVL IL1DR AVKR AVKL AVAL DB01 OLQDL RMHR SAADL VB11 VC03 RMEL RMHL SMBDR AVBL VB09 Sensory Interneurons Motor RIPL RIBL SAADR AVEL VB10 RMEV SIBDR SIADR AVAR VB02 AVER RID VA07 RIPR DB02 RMER RIBR RIMR SMBDL AVBR SIADL VB07 VB06 DB07 VA02 VA09 RMED SIAVR RMDL SIBVL VC01 AS11 DB04 RIML SIBVR SABD VB03 RMDDR SIBDL VA01 DA09 SIAVL VA08 RMDDL PDB SMDVL DB03 AS06 VD10 VB05 VA11 RMDR AS01 AS02 VA04 VB04 RMDVR SMDVR SABVR DA03 VD13 AS09 SMDDR DA01 RMDVL SMDDL DB05 VA06 DA04 VA03 DA02 AS03 Embeddedness SABVL DB06 DA08 AS04 Neurons DA05 VA05 AS07 AS05 VD01 AS10 DA07 DA06 VD09 DD05 VA10 0.3 0.4 0.5 0.6 0.7 0.8 0.9 VD07 VD12 AS08 DD01 VD08 (d) VD05 DD02 DD04 VD02 VD04 VD03 DD03 VD06 Neurons

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 money... The people.. ◮ ONR ◮ J. Lehmann (ABB) ◮ EPSRC ◮ S. N. Yaliraki ◮ Studienstiftung des ◮ M. Barahona 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 flow betweeness of connecting links changes Embeddedness of connecting links invariant