Scaling, renormalization and self-similarity in complex networks

1. Hernan A. Makse Levich Institute and Physics Dept. City College of New York Scaling, renormalization and self-similarity in complex networks Chaoming Song (CCNY) Lazaros Gallos (CCNY) Shlomo Havlin (Bar-Ilan, Israel) Protein interaction network

2. Are “scale-free” networks really ‘free-of-scale’? “If you had asked me yesterday, I would have said surely not” - said Barabasi. (Science News, February 2, 2005). Small World effectshows that distance between nodes grows logarithmically with N (the network size): OR Self-similar = fractal topology is defined by a power-law relation: Small world contradicts self-similarity!!! How the network behaves under a scale transformation.

3. R. Albert, et al., Nature (1999) nd.edu WWW 300,000 web-pages

4. P(k) k Internet Faloutsos et al., SIGCOMM ’99 Internet connectivity, with selected backbone ISPs (Internet Service Provider) colored separately.

5. Yeast Protein-Protein Interaction Map Individual proteins Physical interactions from the “filtered yeast interactome” database: 2493 high-confidence interactions observed by at least two methods (yeast two-hybrid). 1379 proteins, <k> = 3.6 J. Han et al., Nature (2004) Modular structure according to function! Colored according to protein function in the cell: Transcription, Translation, Transcription control,Protein-fate,Genome maintenance,Metabolism,Unknown, etc from MIPS database, mips.gsf.de

6. Metabolic network of biochemical reactions in E.coli Chemical substrates Biochemical interactions: enzyme-catalyzed reactions that transform one metabolite into another. J. Jeong, et al., Nature, 407 651 (2000) Modular structure according to the biochemical class of the metabolic products of the organism. Colored according to product class: Lipids, essential elements, protein, peptides and amino acids,coenzymes and prosthetic groups,carbohydrates,nucleotides and nucleic acids.

7. How long is the coastline of Norway? It depends on the length of your ruler. Fractals look the same on all scales = `scale-invariant’. Box length Fractal Dimension dB- Box Covering Method Total no. of boxes

8. Boxing in Biology Boxing in Biology How to “zoom out” of a complex network? • Generate boxes where all • nodes are within a distance • Calculate number of boxes, , • of size needed to cover the • network We need the minimum number of boxes: NP-complete optimization problem!

9. 1 0 2 0 1 Most efficient tiling of the network 8 node network: Easy to solve 4 boxes 5 boxes 300,000 node network: Mapping to graph colouring problem. Greedy algorithm to find minimum boxes

10. Larger distances need fewer boxes 1 2 -dB fractal log(NB) 3 non fractal log(lB)

11. Box covering in yeast: protein interaction network

12. Most complex networks are Fractal Biological networks Metabolic Protein interaction 43 organisms - all scale Three domains of life: archaea, bacteria,eukaria E. coli,H. sapiens,yeast Song, Havlin, Makse, Nature (2005)

13. Metabolic networks are fractals

14. Technological and Social Networks TOO WWW Hollywood film actors 212,000 actors Other bio networks: Khang and Bremen groups Internet is not fractal! nd.edu domain 300,000 web-pages

15. Two ways to calculate fractal dimensions Cluster growing method Box covering method In homogeneous systems (all nodes with similar k) both definitions agree: percolation

16. Box Covering= flat average Cluster Growing = biased exponential power law Different methods yield different results due to heterogeneous topology Box covering reveals the self similarity. Cluster growth reveals the small world. NO CONTRADICTION!SAME HUBS ARE USED MANY TIMES IN CG.

17. Is evolution of the yeast fractal? present day Archaea + Bacteria Animals + Plants Other Fungi Yeast ~ 300 million years ago Ancestral yeast Ancestral Fungus Ancestral Eukaryote 1 billion years ago Ancestral Prokaryote Cell 3.5 billion years ago Following the phylogenetic tree of life: COG database von Mering, et al Nature (2002) 1.5 billion years ago

18. Same fractal dimension and scale-free exponent over 3.5 billion years… Suggests that present-day networks could have been created following a self-similar, fractal dynamics.

19. Renormalization in Complex Networks NOW, REGARD EACH BOX AS A SINGLE NODE AND ASK WHAT IS THE DEGREE DISRIBUTION OF THE NETWORK OF BOXES AT DIFFERENT SCALES ?

20. Renormalization of WWW network with

21. The degree distribution is invariant under renormalization Internet is not fractal dB--> infinity But it is renormalizable

22. Repeatedly BOXING the network is the same as going back in time: from a single node to present day. Turning back the time THE RENORMALIZATION SCHEME renormalization present day network ancestral node 1 time evolution Can we “predict” the past…. ? if not the future.

23. opening boxes Evolution of complex networks time evolution

24. The boxes have a physical meaning = self-similar nested communities How does Modularity arise? How to identify communities in complex networks? renormalization present day network ancestral node 1 time evolution

25. ancestral cell Boxes are related to the biologically relevant functional modules in the yeast protein interactome Emergence of Modularity in PIN renormalization time evolution translation transcription protein-fate cellular-fate organization present day network

26. Emergence of modularity in metabolic networks Appearance of functional modules in E. coli metabolic network. Most robust network than non-fractals.

27. community degree factor<1 node degree How the communities/modules are linked? Theoretical approach renormalization k’=2 k=8 s=1/4 k: degree of the nodes k’: degree of the communities

28. WWW Theoretical approach to modular networks: Scaling theory to the rescue The larger the community the smaller their connectivity new exponent describing how families link

29. new exponent new scaling relation A theoretical prediction relating the different exponents Scaling relations distance boxes degree

30. The communities also follow a self-similar pattern Scaling relations WWW Metabolic prediction Scaling relation works scale-free fractals communities/modules

31. All the models fail to predict self-similarity The Barabasi-Albert model of preferential attachment does not generate fractal networks Other models fail too: Erdos-Renyi, hierarchical model, fitness model, JKK model, pseudo-fractals models, etc. Why fractality? Some real networks are not fractal INTERNET

32. What is the origin of self-similarity? Can you see the difference? FRACTAL NON FRACTAL Internet map E.coli metabolic map Yeastprotein map HINT: the key to understand fractals is in the degree correlations P(k1,k2) not in P(k)

33. Quantifying correlations P(k1,k2): Probability to find a node withk1 links connected with a node ofk2 links Internet map - non fractal Metabolic map - fractal high prob. low prob. log(k2) log(k2) P(k1,k2) low prob. high prob. log(k1) log(k1) Hubs connected with hubs Hubs connected with non-hubs

34. Quantify anticorrelation between hubsat all length scales Hub-Hub Correlation function: fraction of hub-hub connections hubs Renormalize hubs Hubs connected directly

35. Hub-hub connection organized in a self-similar way non-fractal The larger de implies more anticorrelation fractal (fractal) (non-fractal) Anticorrelations are essential for fractal structure

36. What is the origin of self-similarity? Non-fractal networks Fractal networks • very compact networks • hubs connected with other hubs • strong hub-hub “attraction” • assortativity • less compact networks • hubs connected with non-hubs • strong hub-hub “repulsion” • dissasortativity Internet All available models: BA model, hierarchical random scale free, JKK, etc WWW, PIN, metabolic, genetic, neural networks, some sociological networks

37. How to model it? renormalization reverses time evolution Song, Havlin, Makse, Nature Physics, 2006 Both mass and degree increase exponentially with time time offspring nodes attached to their parents renormalize (m=2) in this case Scale-free: Mode I Mode II

38. How does the length increase with time? Mode I: NONFRACTAL SMALL WORLD Mode II: FRACTAL

39. Combine two modes together Mode I with probability eMode II with probability 1-e time renormalize e=0.5

40. Predictions Model reproduces local small world, scale-free and fractality yeast h.sapiens • model with e=0.2 • repulsion between hubs leads to fractal topology • small world locally inside well defined communities • model with e=1 • attraction between hubs • non-fractal • small world globally

41. The model reproduces the main features of real networks Case 1: e = 0.8: FRACTALS Case 2: e = 1.0: NON-FRACTALS

42. Model predicts all exponents in terms of growth rates Each step the total mass scales with a constant n, all the degrees scale with a constant s. The length scales with a constant a, we obtain: We predict the fractal exponents:

43. Time evolution in yeast network

44. Multiplicative and exponential growth in yeast PIN Length-scales, number of conserved proteins and degree

45. 1930 solid-state physics big world 1960 Erdos-Renyi model small world democracy= socialism 1999 BA model “rich-get-richer”= capitalism 2005 fractal model “rich-get-richer” at the expense of the “poor”= globalization A new principle of network dynamics less vulnerable to intentional attacks

46. Summary • In contrast to common belief, many real world networks are self-similar. • FRACTALS: WWW, Protein interactions, metabolic networks, neural networks, collaboration networks. • NON-FRACTALS: Internet, all models. • Communities/modules are self-similar, as well. • Scaling theory describes the dynamical evolution. • Boxes are related to the functional modules in metabolic and protein networks. • Origin of self similarity: anticorrelation between hubs • Fractal networks are less vulnerable than non-fractal networks Positions available: jamlab.org

47. An finally, a model to put all this together A multiplicative growth process of the number of nodes and links m = 2 Analogous to duplication/divergence mechanism in proteins?? Probability e hubs always connected strong hub attraction should lead to non-fractal Probability 1-e hubs never connected strong hub repulsion should lead to fractal

48. Different growth modes lead to different topologies For the both models, each step the total number of nodes scale as n = 2m +1( N(t+1) = nN(t) ). Now we investigate the transformation of the lengths. They show quite different ways for this two models as following: Mode I: L(t+1) = L(t)+2 Then we lead to two different scaling law of N ~ L smaller smaller Mode II: L(t+1) = 2L(t)+1 Mode III: L(t+1) =3L(t)

49. Dynamical model Suppose we have e probability to have mode I, 1-e probability to have mode II and mode III. Then we have: or

50. Graph theoretical representation of a metabolic network (a) A pathway (catalyzed by Mg2+-dependant enzymes). (b) All interacting metabolites are considered equally. (c) For many biological applications it is useful to ignore co-factors, such as the high energy-phosphate donor ATP, which results in a second type of mapping that connects only the main source metabolites to the main products.