The complexity ofcellular networksWarning: Statistical physics.It only works on average.http://regan.med.harvard.edu/CVBR-course.php

1. Meet complex networks- Complexity and networks -Many differentcomponents• atoms, particles• spins, oscillators• cells, DNA, proteinsComplexity in:System as a whole:**NETWORK**A variety ofinteractions• gravity• electromagnetism• genetic regulation• signaling• topology of interactions• time evolution of the structure• dynamics on the structure

- Meet these networks -SocietyCommunication• Friendships, sexual contacts• Co-authorship, citations• Movie actors, business• Co-authorship, citations• Movie actors, businessBiologyDays of ThunderFar and AwayEyes Wide Shut• Internet• World Wide Web• Phone call networksYou mean to sayyou are going to talkabout ALL these? InAnd more…GENERAL?• Genetic regulation• Protein-protein interactions• Metabolic pathways• Food webs• Neuron networks• Airline routes• Word webs• Power grid

- How it all begun -Königsberg, capital of East Prussia1730’s, Euler’s timePregel River

- How it all begun -Königsberg, capital of East Prussia1730’s, Euler’s timeProblem: find a walk to crossall bridges once, but only once.Euler:• land: nodes• bridges: graphPregel RiverGraph theory+ TopologySolution: a general theorem.• Eulerian paths only exist ongraphs with no odd degreenodes, or exactly 2 odddegree nodes.• => none for old Königsberg

- Social science knows we network -Stanley Milgram’s experiment, Harvard, 1967START• 64 of the 296 letters made it• average path length : 5.5 or 6END• random individuals instarting towns• information packet:target name, occupationand city (about study,instructions)• if you know target,mail package to target• if not: mail it to apersonal aqquaintanceyou think might knowhim• send back tranckingpostcardSixdegrees ofseparation

- Paul Erdős rethinks graph theory -1950’s: the Erdős-Rényi Random Networkp = 1/6• statistical approach• the null model of large realnetworks• threshold probability value:➡ percolation transition =1➡ one giant connectedcomponent➡ Poisson degree distribution➡ SMALL WORLD< d > i,j ∼ log(N)

THE NEW SCIENCE OF **NETWORK**S 245- Birth of modern complex networks -OK: birth of our modern interest in themAR AR219-SO30-12.tex AR219-SO30-12.sgm LaTeX2e(2002/01/18) P1: IBC1998 Watts and Strogatz:Small World NetworksTHE NEW SCIENCE OF **NETWORK**S 2451999 Barabasi and Albert:Scale-Free Networks• high clustering• short average paths• US power grid• Movie actors• neural network ofC. ElegansFigure 1 (a) Schematic of the Watts-Strogatz model. (b) Normalized average shortestpath length L and clustering coefficient C as a function of the random rewiring parameterp for the Watts-Strogatz model with N = 1000, and 〈k〉 = 10.• WWW, U Notre Dame

THE NEW SCIENCE OF **NETWORK**S 245- Birth of modern complex networks -OK: birth of our modern interest in themAR AR219-SO30-12.tex AR219-SO30-12.sgm LaTeX2e(2002/01/18) P1: IBC1998 Watts and Strogatz:Small World NetworksTHE NEW SCIENCE OF **NETWORK**S 2451999 Barabasi and Albert:Scale-Free Networks• high clustering• short average paths• power law degreedistribution!• US power grid• Movie actors• neural network ofC. ElegansFigure 1 (a) Schematic of the Watts-Strogatz model. (b) Normalized average shortestpath length L and clustering coefficient C as a function of the random rewiring parameterp for the Watts-Strogatz model with N = 1000, and 〈k〉 = 10.• BA model➡ growth➡ preferentialattachment

- Explosion of data -• Internet

Business ties in US biotech-industry

Business ties in US biotech-industry

Survival signaling in largegranular lymphocye leukemiaL survival signaling network. Node and edge color represents the current knowledge of the signaling abnormalities inonstitutively active nodes are in red, down-regulated or inhibited nodes are in green, nodes that have been suggested to be deown-regulation) are in blue, and the states of white nodes are unknown or unchanged compared with normal. Blue edge in

Disease Network

Protein foldingpathways(a)(c)(b)10 010 -1P MD(k)10 -210 -310 -410 -5e k/k 0k -2400450500600700800900100010 0 1100 120010 1 10 2k

Food webs

You mean to sayyou are going to talkabout ALL these? InGENERAL?STRUCTURAL UNIVERSALITIESEVOLUTION OF STRUCTUREWHAT DO THEY DO? (and how?)

- Let’s use an example -• Simple graph (metabolites - nodes;reactions - links)=> degree properties=> paths and their statistics=> clustering, motifs=> degree mixing patterns=> communities, hierarchy, fractalsA + BAB> C + DCD• Directed graph (one-way reactions)• Weighted graph (rates, concentrations)• Bipartite graph (both metabolitesand reactions are network nodes)

- Degree distribution and its claim to fame -HomogeneousgraphMeet the hubs

- Rich gets richer -1999, Barabasi & Albert Scale-Free Model• Networks grow• New nodes pick popular old nodes:preferential attachment1965, Price, scietific citation network➡ measures powerlaw➡ builds model: cited papers getmore citations

- Statistical physics gets its handson scale-free models -• linear preferential attachment: critical for powerlaws• fitness-based attachment models: math similar to BEC• other growth models that result in preff. attachment• configuration model: ensemble of all networks with agiven degree distribution➡ scale-free networks are ultra-small: ~ log (log N)➡ vanishing number of triangles• physical constraints on link length➡ large cost of length forbids hubs➡ complex models for spatially embeddedscale-free networks

- Paths in a small world -The Kevin BacongameJohnTurturroThe ErdősNumberClintHowardGenePattersonErdős has a Bacon number of 4

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aps. We employ the community definition specifieduse none of the others in the literature satisfy all thesets simultaneously 21,24 .cognitive sciences (Fig. 2b) and the molecular-biological networkof protein–protein interactions 27 (Fig. 2c). These pictures can serve astests or validations of the efficiency of our algorithm. In particular,- Overlapping communities -e community structure around a particular node in threebe associated with his fields of interest. b, The communities of the word

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2. Dynamics on complex networksApril 1312 PM