NETWORK

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NETWORK

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:NETWORKA 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 NETWORKS 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 NETWORKS 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 NETWORKS 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 NETWORKS 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


for complexcalculated. small-world Thenodes procedure are 392,340 is then actors repeated linked if by theychoosing were cast together many seed in at least network composed of eight Hownodes. can weFor reconcile each value the exponential of the box size increase l B , in equation (1)nodes at random oneand film 15 the ; 1 Levich (3) average theInstitute biological ‘mass’ andnetworks Physics of theDepartment, resulting of protein–protein City clusters College interactionsthe number found inof Escherichia nodes incoli the(429 cluster) proteins) is calculated and Homoassapiens such that each box contains nodes separated by a distance l , l B .ofwe Newsearch York, New forYork,the number with self-similarity, of boxes needed or (in to tile other thewords) entirean network underlying length-scaleinvarianttopology? At the root of the self-similar properties that wec l, defined degree as terized by a(kM narrow New York 10031, USAn cluster) the a box-counting function of l (946 to obtain proteins)- Self similarity and fratal -2 Minerva thelinked following Center if there and scaling: isDepartment a physical of binding Physics, between Bar-IlanthemUniversity, ThisRamat procedure Gan is appliedunravel intothisseveralstudydifferentis a scale-invariantreal networks:renormalization(1) aprocedureuster-growing method of (database available 52900, Israel via the Database of Interacting Proteins 16,17 ,that we show to be valid for dissimilar complex networks.very node typically has the other PINs.............................................................................................................................................................................arekM discussed in the Supplementary Information), andc l


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


aps. We employ the community definition specifieduse none of the others in the literature satisfy all thesets simultaneously 21,24 .e community structure around a particular node in threecognitive sciences Genome (Fig. Database 2b) and(SGD) the molecular-biological network29 . For some proteins no function is yetof protein–proteinavailable. interactions Thus, the 27 (Fig. fact that 2c). they These showpictures up in our can approach serve asmembers of communities can be interpreted as a prediction oftests or validations of the efficiency of our algorithm. In particular,- Overlapping communities -their functions. One such example can be seen in the enlargedbe associated with his fields of interest. b, The communities of the wordFigure 3 | Network of the 82 communities in the DIP core list of the protein–protein interactions of S. cerevisiae for k 5 4. The areas of the circles anddivisinetwopiecebe seneceswith16: itrelatethosebecomso ondirecof refnity’a treeapprocommThways.archiof theFree Awordsurveour ponesing tProteSacchlink bconsi6,355Altlocalsomestatisworkthatphenincrecriticbetwecommway wthe nthe dcomm


2. Dynamics on complex networksApril 1312 PM

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