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

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 effect shows 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.

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

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

from MIPS database, mips.gsf.de 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

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.

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

We need the minimum number of boxes: NP-complete optimization problem! 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!

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 1 2

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

Box covering in yeast: protein interaction network

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)

Metabolic networks are fractals

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

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

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.

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

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.

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 ?

Renormalization of WWW network with

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

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

Evolution of complex networks opening boxes time evolution

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

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

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

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

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

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

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

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

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

Quantifying correlations P(k1,k2): Probability to find a node with k1 links connected with a node of k2 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

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

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

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

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

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

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

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

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

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:

Time evolution in yeast network

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

at the expense of the “poor”= A new principle of network dynamics 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 less vulnerable to intentional attacks

Positions available: jamlab.org 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

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

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)

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

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.

Classes of genes in the yeast proteome

Renormalization following the phylogenetic tree P. Uetz, et al. Nature 403 (2000).