CLIM Program: Remote Sensing Workshop, Multilayer Modeling and Analysis of Complex (Systems) Data - Manlio De Domenico, Feb 12, 2018
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The document discusses multilayer modeling and analysis of complex systems, emphasizing the interconnectedness and interdependence of real-world systems. It outlines various mathematical frameworks and challenges in representing these multilayer networks, including redundancy, dynamics, and the merging of different layers. Key contributions include predictive tools and approaches for improving data analysis and understanding the structure and function of complex networks.
![Basis for network information theory / machine learning
D1(⇢|| ) = Tr [⇢(log2 ⇢ log2 )]
Kullback-Leibler divergence Gain
J1(⇢|| ) = S1
✓
⇢ +
2
◆
1
2
[S1(⇢) + S1( )]
Jensen-Shannon divergence
p
J1(⇢|| )
Jensen-Shannon distance Distance
MDD & Biamonte, Phys. Rev. X 6, 041062 (2016)](https://image.slidesharecdn.com/dedomenico-180228193817/85/CLIM-Program-Remote-Sensing-Workshop-Multilayer-Modeling-and-Analysis-of-Complex-Systems-Data-Manlio-De-Domenico-Feb-12-2018-36-320.jpg)
![Properties:
• Bounded in [0,1]
• Symmetric
• is a metric distance
p
DJS
• Hierarchical clustering
• Suggest a greedy (and fast) way to merge
Jensen-Shannon: DJS(⇢|| ) = h(µ)
1
2
[h(⇢) + h( )]
Mixture: µ =
1
2
(⇢ + )
Similarity between layers
(Convex linear combination)
MDD, Nicosia, Arenas & Latora, Nature Comm. 6, 6864 (2015)](https://image.slidesharecdn.com/dedomenico-180228193817/85/CLIM-Program-Remote-Sensing-Workshop-Multilayer-Modeling-and-Analysis-of-Complex-Systems-Data-Manlio-De-Domenico-Feb-12-2018-37-320.jpg)
![Eigenvector
PageRank
HITS
W↵
v↵ = 1v
v = ( ↵
aW↵
) 1
u↵
(W†
W)i
j⌥i = 1⌥j
(WW†
)i
j i = 1 j
Katz j = [( aM) 1
]i↵
j Ui↵
Ri
j⇡i = ⇡j
Classical Multilayer
MDD, Sole, Omodei, Gomez & Arenas, Nature Comms. 6, 6868 (2015)
Mi↵
j ⇥i↵ = 1⇥j
Ri↵
j ⇧i↵ = 1⇧j
(MM†
)i↵
j i↵ = 1 j
(M†
M)i↵
j ⌥i↵ = 1⌥j](https://image.slidesharecdn.com/dedomenico-180228193817/85/CLIM-Program-Remote-Sensing-Workshop-Multilayer-Modeling-and-Analysis-of-Complex-Systems-Data-Manlio-De-Domenico-Feb-12-2018-63-320.jpg)
CLIM Program: Remote Sensing Workshop, Multilayer Modeling and Analysis of Complex (Systems) Data - Manlio De Domenico, Feb 12, 2018
- 1. Multilayer Modeling & Analysis of Complex (Systems) Data Manlio De Domenico Complex Multilayer Networks Lab Center for Information Technology, Bruno Kessler Foundation Pasadena 12 February 2018
- 2. We are part of an increasingly complex world Units are interconnected, interdependent, integrated
- 3. Real world systems: INTERCONNECTED
- 5. Beyond lattices Models: INTERCONNECTED Erdos & Renyi (1959) Barabasi & Albert (1999) Watts & Strogatz (1998) Homogeneous Random Small-world Scale-free
- 6. WWW, Airport networks, … Models: INTERCONNECTED Barabasi & Albert (1999) Scale-free
- 7. Real world systems: INTERDEPENDENT
- 8. Real world systems Acting on complex system? • Triggering cascade of effects on the system itself • Triggering cascade of effects on interdependent systems Classical complex networks failure Not able to reproduce observed: • Diffusion properties • Clustering statistics • Resilience • … NATURE CONSISTS OF SYSTEMS OF SYSTEMS
- 9. Real world systems: FRAGILITY Systems’ integration makes them more vulnerable to perturbations Eyjafjallajökull eruption (2010) Acting on complex system? • Triggering cascade of effects on the system itself • Triggering cascade of effects on interdependent systems Worldwide air traffic delay
- 10. Our contributions Mathematical formulation for multilayer systems Redundancy of multilayer systems and dimensionality reduction Spreading processes on multilayer systems Predictive tools MDD et al, Phys. Rev. X 3, 041022 (2013) MDD, A. Sole-Ribalta, S. Gomez & A. Arenas, PNAS 11, 8351 (2014) MDD, Sole, Omodei, Gomez & Arenas, Nature Comms. 6, 6868 (2015) MDD, Nicosia, Arenas & Latora, Nature Comms. 6, 6864 (2015) MDD, Lancichinetti, Arenas & Rosvall, Phys. Rev. X 5, 011027 (2015) MDD, Granell, Porter & Arenas, Nature Phys., 12, 901 (2016) Baggio, Burnsilver, Arenas, Magdanz, Kofinas & MDD, PNAS, 113, 13708 (2016) MDD & Biamonte, Phys. Rev. X 6, 041062 (2016) MDD, Phys. Rev. Lett. 118, 168301 (2017)
- 11. Mathematical formulation of multilayer networks Challenge: Representation MDD et al, Phys. Rev. X 3, 041022 (2013)
- 12. • Intra-layer links from data • Inter-layer links absent Edge-colored Interconnected • Intra-layer links from data • Inter-layer links from data Basic classification (a) (b) (c)
- 13. • Intra-layer links from data • Inter-layer links absent Edge-colored Interconnected • Intra-layer links from data • Inter-layer links from data Basic classification (a) (b) (c)
- 14. Basic classification Replicas / State nodes
- 17. Given the canonical basis in: W = NX i,j=1 wijEij = NX i,j=1 wijei ⌦ e† j , W 2 RN ⌦ RN RN 0 B B B B @ 1 0 0 0 0 1 C C C C A 0 B B B B @ 0 1 0 0 0 1 C C C C A1 23 4 5 0 B B B B @ 0 0 0 1 0 1 C C C C A 0 B B B B @ 0 0 0 0 1 1 C C C C A 0 B B B B @ 0 0 1 0 0 1 C C C C A Classical Networks
- 18. Given the canonical basis in: W = NX i,j=1 wijEij = NX i,j=1 wijei ⌦ e† j , W 2 RN ⌦ RN RN Weighted Adjacency Matrix Equivalent to rank-2 tensor: W⇥ = NX i,j=1 wijE⇥ (ij) = NX i,j=1 wije (i)e⇥(j) 0 B B B B @ 1 0 0 0 0 1 C C C C A 0 B B B B @ 0 1 0 0 0 1 C C C C A1 23 4 5 0 B B B B @ 0 0 0 1 0 1 C C C C A 0 B B B B @ 0 0 0 0 1 1 C C C C A 0 B B B B @ 0 0 1 0 0 1 C C C C A W = 0 B B B B @ 0 w1,2 0 w1,4 0 w1,2 0 w2,3 0 0 0 w2,3 0 w3,4 w3,5 w1,4 0 w3,4 0 0 0 0 w3,5 0 0 1 C C C C A Classical Networks
- 19. Intra-layer adjacency tensors: C↵ (˜k˜k) = W↵ (˜k) Layers are just rank-2 tensors in RN⇥N Multilayer Networks
- 20. Inter-layer adjacency tensors: Inter-connections are rank-2 tensors in RN⇥N C↵ (˜h˜k) Multilayer Networks
- 21. One multilayer adjacency tensor in M ˜⇤ ⇥˜⌅ = LX ˜h,˜k=1 C⇥ (˜h˜k)E˜⇤ ˜⌅ (˜h˜k) = LX ˜h,˜k=1 2 4 NX i,j=1 wij(˜h˜k)E⇥ (ij) 3 5 E˜⇤ ˜⌅ (˜h˜k) = LX ˜h,˜k=1 NX i,j=1 wij(˜h˜k)E ˜⇤ ⇥˜⌅ (ij˜h˜k) RN⇥N⇥L⇥L MDD et al, Phys. Rev. X 3, 041022 (2013) Multilayer Networks
- 22. How to work with tensors?
- 23. Kolda & Bader, SIAM review 51, 455 (2009) 4 x 4 1 x 16 Matricization: change dimension while preserving information
- 24. Gomez et al, Phys. Rev. Lett. 110, 028701 (2013) MDD et al, Phys. Rev. X 3, 041022 (2013) ↵ Mi↵ j : j i From multilayer adjacency tensor to supra-adjacency matrix
- 25. Gomez et al, Phys. Rev. Lett. 110, 028701 (2013) MDD et al, Phys. Rev. X 3, 041022 (2013) From multilayer adjacency tensor to supra-adjacency matrix
- 26. Gomez et al, Phys. Rev. Lett. 110, 028701 (2013) MDD et al, Phys. Rev. X 3, 041022 (2013) R(NL)⇥(NL) Supra-adjacency matrix in RN⇥N⇥L⇥L Multilayer adjacency tensor in Multilayer network model“Raw” data From data to matrix
- 27. Trade-off between importance of intra- and inter-layer links Take home message
- 28. Weak coupling Poor multilayer effects Approx: analyze as single layers Take home message
- 29. Strong coupling Multilayer effects Take home message
- 30. Is your network a multilayer? Structural reducibility Challenge: Redundancy MDD, Nicosia, Arenas & Latora, Nature Comm. 6, 6864 (2015) MDD & Biamonte, Phys. Rev. X 6, 041062 (2016)
- 31. Node’s inter-layer out-strength: Flow on multilayer networks Data analysis better/faster in a space smaller than Understanding structure and function easier when there are no redundant components R(NL)⇥(NL) Is the full multilayer the best representation in all cases? MDD, Nicosia, Arenas & Latora, Nature Comm. 6, 6864 (2015)
- 32. “Merge until no information is lost” MDD, Nicosia, Arenas & Latora, Nature Comm. 6, 6864 (2015)
- 33. 1. How to compare complex networks, non-trivially? 2. Problem not tractable for increasing system size Important issues
- 34. All possible combinations of M layers in groups of any size can be prohibitive (Bell number) B(M) = 1X n=0 Bn n! Mn = eeM 1 Bn = nX k=0 ⇢ n k Exhaustive search for optimal merging of layers is not feasible Super-exponential scaling wrt the # layers Which merging strategy?
- 35. Density matrix ⇢ von Neumann entropy S(⇢) = Tr(⇢ log2 ⇢) Diffusion propagator Normalizing factor ⇢ = e ⌧L Z Z = Tr e ⌧L ˙p(⌧) = Lp(⌧) Diffusion equation General solution p(⌧) = e ⌧L p(0) Our quantum-inspired approach MDD & Biamonte, Phys. Rev. X 6, 041062 (2016)
- 36. Basis for network information theory / machine learning D1(⇢|| ) = Tr [⇢(log2 ⇢ log2 )] Kullback-Leibler divergence Gain J1(⇢|| ) = S1 ✓ ⇢ + 2 ◆ 1 2 [S1(⇢) + S1( )] Jensen-Shannon divergence p J1(⇢|| ) Jensen-Shannon distance Distance MDD & Biamonte, Phys. Rev. X 6, 041062 (2016)
- 37. Properties: • Bounded in [0,1] • Symmetric • is a metric distance p DJS • Hierarchical clustering • Suggest a greedy (and fast) way to merge Jensen-Shannon: DJS(⇢|| ) = h(µ) 1 2 [h(⇢) + h( )] Mixture: µ = 1 2 (⇢ + ) Similarity between layers (Convex linear combination) MDD, Nicosia, Arenas & Latora, Nature Comm. 6, 6864 (2015)
- 38. • Merging “different” layers causes information loss and entropy increases • Merging “similar” layers causes no information loss and entropy ~ unchanged The main idea MDD, Nicosia, Arenas & Latora, Nature Comm. 6, 6864 (2015)
- 39. London Tube
- 40. Only Circle Line is “redundant” London Tube
- 41. Multilayer Gene-Protein Interaction Networks Layers are different types of Physical & Genetic interactions
- 42. Plasmodium Falciparum Mus MusculusGallus Gallus Drosophila Melanogaster Caenorhabditis Elegans Candida Albicans Bos TaurusArabidopsis Thaliana Rattus Norvegicus Saccharomyces Cerevisiae Schizosacch. Pombe Homo Sapiens Oryctolagus Cuniculus Xenopus Laevis Danio Rerio Epstein–Barr VirusHIV Virus Hepatitis C Virus
- 43. TransportHumanGene-Protein 50% MDD, Nicosia, Arenas & Latora, Nature Comm. 6, 6864 (2015)
- 44. Multilayer Functional Brain Networks • Resting state fMRI • Layers are functional networks per frequency band • 0.01 Hz to 0.25 Hz in step of 0.02 Hz MDD, Sasai & Arenas, Frontiers in Neuroscience 10, 326 (2016) MDD, Gigscience 6, 1 (2017)
- 45. A B
- 47. Diffusion on multilayer networks Challenge: From Structure to Dynamics MDD, C. Granell, M. Porter, A. Arenas, Nature Physics 12, 901 (2016) A. Lima, MDD, V. Pejovic, M. Musolesi, Nature Sci. Rep. 5, 10650 (2015)
- 48. Layer switching MDD, C. Granell, M. Porter, A. Arenas, Nature Physics 12, 901 (2016) Types of dynamics Single dynamics Coupled dynamics (a) (b)
- 49. Layer switching Nodejumping pj,⇥(t + 1) = T⇥⇥ jj pj,⇥(t) + LX =1 6=⇥ T ⇥ jj pj, (t) + NX i=1 i6=j T⇥⇥ ij pi,⇥(t) + LX =1 6=⇥ NX i=1 i6=j T ⇥ ij pi, (t) MDD, A. Sole-Ribalta, S. Gomez, A. Arenas, PNAS (2014) Single dynamics
- 50. Layer switching Nodejumping pj,⇥(t + 1) = T⇥⇥ jj pj,⇥(t) + LX =1 6=⇥ T ⇥ jj pj, (t) + NX i=1 i6=j T⇥⇥ ij pi,⇥(t) + LX =1 6=⇥ NX i=1 i6=j T ⇥ ij pi, (t) MDD, A. Sole-Ribalta, S. Gomez, A. Arenas, PNAS (2014) Transition dynamics
- 51. Layer switching Nodejumping pj,⇥(t + 1) = T⇥⇥ jj pj,⇥(t) + LX =1 6=⇥ T ⇥ jj pj, (t) + NX i=1 i6=j T⇥⇥ ij pi,⇥(t) + LX =1 6=⇥ NX i=1 i6=j T ⇥ ij pi, (t) MDD, A. Sole-Ribalta, S. Gomez, A. Arenas, PNAS (2014) Transition dynamics
- 52. Layer switching Nodejumping pj,⇥(t + 1) = T⇥⇥ jj pj,⇥(t) + LX =1 6=⇥ T ⇥ jj pj, (t) + NX i=1 i6=j T⇥⇥ ij pi,⇥(t) + LX =1 6=⇥ NX i=1 i6=j T ⇥ ij pi, (t) MDD, A. Sole-Ribalta, S. Gomez, A. Arenas, PNAS (2014) Transition dynamics
- 53. Layer switching Nodejumping pj,⇥(t + 1) = T⇥⇥ jj pj,⇥(t) + LX =1 6=⇥ T ⇥ jj pj, (t) + NX i=1 i6=j T⇥⇥ ij pi,⇥(t) + LX =1 6=⇥ NX i=1 i6=j T ⇥ ij pi, (t) MDD, A. Sole-Ribalta, S. Gomez, A. Arenas, PNAS (2014) Transition dynamics
- 54. MDD, A. Sole-Ribalta, S. Gomez, A. Arenas, PNAS (2014) p (t + 1) = T↵ p↵(t) Master equation Single-layer: ˙p (t) = L ↵ p↵(t) Transition matrix between nodes/layers Probability to find the walker in a node/layer Multi-layer: Normalized Laplacian matrix p ˜(t + 1) = T↵˜ ˜ p↵˜(t) p ˜(t) T↵˜ ˜ ˙p ˜(t) = L ↵˜ ˜ p↵˜(t) L ↵˜ ˜ = ↵˜ ˜ T↵˜ ˜ Probability rate evolution Transition dynamics
- 55. Gomez et al, Phys. Rev. Lett. 110, 028701 (2013) MDD, C. Granell, M. Porter, A. Arenas, Nature Physics 12, 901 (2016) Layer 1 Layer 2 Aggregated 1 10 100 1 100 Inter−layer coupling Smallestpositiveeigenvalue(Λ2) Layer 1 Layer 2 Aggregated 1 1 100 Inter−layer coupling Smallestpositiveeigenvalue(Λ2) 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 Connection probability in Layer 1 ConnectionprobabilityinLayer2 Regime 1 Regime 2 Regime 1 Regime 2 Unexpected phenomena: enhanced diffusion Diffusion in coupled networks might be faster than each layer separately
- 56. Using Structure & Dynamics to identify key units Challenge: Prediction MDD, Sole, Omodei, Gomez & Arenas, Nature Comms. 6, 6868 (2015)
- 57. Classical Multilayer In-degreek = W↵ u↵ Out-degreek↵ = W↵ u K↵ = M↵˜ ˜ U ˜ ˜ u K = M↵˜ ˜ U ˜ ˜ u↵ F⇢ = U⇢ ⇢U⇢ ⇢ “Ones” tensors Kroneker tensor F ˜ ✏˜⌘ = U ˜ ✏˜⌘ ˜ ✏˜⌘ u↵ Cotrace Tensor
- 58. In-degreek = W↵ u↵ Out-degreek↵ = W↵ u i-th node in- degree k(i) = k e (i) K↵ = M↵˜ ˜ U ˜ ˜ u K = M↵˜ ˜ U ˜ ˜ u↵ k(i) = K e (i) F⇢ = U⇢ ⇢U⇢ ⇢ “Ones” tensors Kroneker tensor F ˜ ✏˜⌘ = U ˜ ✏˜⌘ ˜ ✏˜⌘ u↵ Cotrace Tensor Classical Multilayer
- 59. In-degreek = W↵ u↵ Out-degreek↵ = W↵ u i-th node in- degree k(i) = k e (i) K↵ = M↵˜ ˜ U ˜ ˜ u K = M↵˜ ˜ U ˜ ˜ u↵ k(i) = K e (i) F⇢ = U⇢ ⇢U⇢ ⇢ “Ones” tensors Kroneker tensor F ˜ ✏˜⌘ = U ˜ ✏˜⌘ ˜ ✏˜⌘ u↵ Cotrace Tensor c(W↵ ) = W↵ ⇢ W⇢ W↵ W↵ ⇢ F⇢ W↵ Global clustering C(M↵˜ ˜ ) = N 1 M↵˜ ˜ M ˜ ✏˜⌘ M✏˜⌘ ↵˜ M↵˜ ˜ F ˜ ✏˜⌘ M✏˜⌘ ↵˜ Classical Multilayer
- 60. EigenvectorW↵ v↵ = 1v Classical Multilayer Mi↵ j ⇥i↵ = 1⇥j
- 61. Eigenvector PageRank “Google” tensor: W↵ v↵ = 1v Ri j⇡i = ⇡j Classical Multilayer Mi↵ j ⇥i↵ = 1⇥j Ri↵ j ⇧i↵ = 1⇧j
- 62. Eigenvector PageRank HITS W↵ v↵ = 1v (W† W)i j⌥i = 1⌥j (WW† )i j i = 1 j Ri j⇡i = ⇡j Classical Multilayer Mi↵ j ⇥i↵ = 1⇥j Ri↵ j ⇧i↵ = 1⇧j (MM† )i↵ j i↵ = 1 j (M† M)i↵ j ⌥i↵ = 1⌥j
- 63. Eigenvector PageRank HITS W↵ v↵ = 1v v = ( ↵ aW↵ ) 1 u↵ (W† W)i j⌥i = 1⌥j (WW† )i j i = 1 j Katz j = [( aM) 1 ]i↵ j Ui↵ Ri j⇡i = ⇡j Classical Multilayer MDD, Sole, Omodei, Gomez & Arenas, Nature Comms. 6, 6868 (2015) Mi↵ j ⇥i↵ = 1⇥j Ri↵ j ⇧i↵ = 1⇧j (MM† )i↵ j i↵ = 1 j (M† M)i↵ j ⌥i↵ = 1⌥j
- 64. Healthy Schizophrenia Mux Mux & Aggr. Discriminative Diagnostic accuracy Multilayer Aggregatevs 85% 70% MDD, Sasai & Arenas, Frontiers in Neuroscience 10, 326 (2016)
- 65. Take home messages Systems of systems: integration requires a new framework
- 66. Our finding: multilayer networks are more predictive Take home messages Aggregating complex data might lead to misunderstanding Systems of systems: integration requires a new framework
- 67. Our finding: multilayer networks are more predictive Take home messages Aggregating complex data might lead to misunderstanding New mathematical formulation & network information theory as natural frameworks for data analytics of complex systems Systems of systems: integration requires a new framework
- 68. Multilayer Theory of Data Systems? Network: the architecture of a data system Multilayer Network a natural framework for: • Structure: integrated data systems • Dynamics: information flow and integration Layers: software/methodology/…?
- 69. @manlius84 mdedomenico@fbk.eu THANK YOU! MDD et al, Phys. Rev. X 3, 041022 (2013) MDD, A. Sole-Ribalta, S. Gomez, A. Arenas, PNAS 11, 8351 (2014) MDD, Sole, Omodei, Gomez & Arenas, Nature Comms. 6, 6868 (2015) MDD, Nicosia, Arenas & Latora, Nature Comms. 6, 6864 (2015) MDD, Lancichinetti, Arenas & Rosvall, Phys. Rev. X 5, 011027 (2015) MDD, Granell, Porter & Arenas, Nature Phys., 12, 901 (2016) Baggio, Burnsilver, Arenas, Magdanz, Kofinas & MDD, PNAS, 113, 13708 (2016) MDD & Biamonte, Phys. Rev. X 6, 041062 (2016) MDD, Phys. Rev. Lett. 118, 168301 (2017) Methodology and theoretical background https://comunelab.fbk.eu/ MDD, Porter & Arenas, J. Complex Networks 3, 159 (2015) Computational tools http://muxviz.net