Why is a power law interesting? 2. it begs a question about mechanism: How do networks come to have power-law degree distributions in the first place?

Transcription

1 you ve got the power Why is a power law interesting? 1. it is scale-free 2. it begs a question about mechanism: How do networks come to have power-law degree distributions in the first place?

2 powerful mechanisms combination of exponentials random walks (e.g. probability of first return time) 1 loss of scale at critical points self-organized criticality (forest fire model, sandpile model) 2 multiplicative processes (lognormal distribution) 3 rich-get-richer mechanisms (Yule process, preferential attachment)

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4 1 more is different, suddenly n=100 phase transition p=0.001 p=0.005 p=0.009 p=0.01 p=0.012 p=0.02 giant component

5 2 start a multiplicative process positive random number multiplicative process: by the central limit theorem, the logs obey a normal distribution:

6 2 a tail of another world is lognormal if is normal

7 2 end log of lognormal looks like a power law if the range of data covers only a few orders of magnitude..., or if the variance is large

8 3 start smart people Herbert Simon ( ) Derek de Solla Price ( )

9 citation networks cumulative distribution power-tail citations to all papers catalogued by ISI between 1981 and 1997 Derek de Solla Price ( )

10 you name it the rich get richer -effect cumulative advantage preferential attachment Matthew-effect For every one that hath shall be given......but from him that hath not, that also which he seemeth to have shall be taken away Matthew 25:29

11 de Solla Price model of citation networks nodes in network, now fraction of nodes with in-degree add a new node with fixed out-degree average out-degree is fixed at...thus mean in-degree is fixed as well: probability that a new edge attaches to any node of degree:

12 ...+1 # of incident edges: P of node with incident edges P of attachment

13 de Solla Price model master-equation (rate-equation): at stationarity......the master-equation becomes:

14 3 end de Solla Price model beta-function for large k

15 Barabasi & Albert: preferential attachment model same as Price, but undirected edges and new nodes appear with initial degree m (average degree of the network is now 2m) each link attaches to another node with probability proportional to that node s degree new node has initial degree m, fixed solve like Price model, via a master equation

16 clustering coefficient: are the friends of my friends my friends? each triangle contributes to 3 triples a triple 5 a triangle probability that two vertices that are neighbors of the same vertex will themselves be neighbors triangle, 8 triples: C=3/8 2 5

17 clustering coefficient: are the friends of my friends my friends? a triple 5 a triangle /6 0 5 C=1/5 x 13/6 = 13/

18 small world 1 mean vertex-vertex distance clustering coefficient o r C/ C 0.5 l/l max max rewiring probability p

19 so far our concern was global, now for local structure

20 patterns are rules A pattern language is a set of patterns used by a process to generate artifacts. Each pattern is a kind of rule that states a problem to be solved and a solution to that problem in a particular context. Christopher Alexander

21 elements of style levels of scale strong centers boundaries alternating repetition background simple shape local symmetry interlock & ambiguity contrast gradients roughness echoes the void essential connectedness source: Wired, March 2004, and

22 software patterns patterns are everywhere Leaky Bucket of Credits Problem: How can one processor know whether another processor is capable of handling more work? Forces: For a peripheral to be able to reject new work when the system is overloaded, it must be able to recognize when the system is overloaded. But having the bottleneck processor take up valuable realtime cycles to inform the (potentially large number of) peripherals would further reduce its capacity. And what happens if it gets so bogged down that it can t send out the Stop sending me work! messages? Solution: The bottleneck processor tells the peripherals when it is capable of accepting more work. It does so by sending credits to the peripherals. Each peripheral tracks a leaky bucket of credits received from the bottleneck processor (BP). As requests are sent to the BP, or simply as time passes, the bucket leaks until it is empty. When the system is not at capacity, the bucket is continuously refilled by new credits sent from the BP; however, if the system is at capacity, the BP will not send credits and the peripherals will hold back new work... (Meszaros, 1996) source: James Coplien, Software Patterns, Bell Labs (2000)

23 patterns are everywhere protein 3D motifs (domains) binding DNA... basic helix-loop-helix spanning a membrane... receptor tyrosine kinase binding a phosphorylated site... sequence motifs DNA binding site... regulatory site of dnaj

24 transcription network motifs in E.coli patterns are everywhere S. Shen-Orr, R. Milo, S. Mangan, U. Alon, Nature Genetics, 31, 64 (2002) E. Yeger-Lotem et al, PNAS, 101, 5934 (2004) network N. Rosenfeld & U. Alon, JMB, 329, 645 (2003) N. Rosenfeld, M. Elowitz & U. Alon, JMB, 323, 785 (2002) S. Mangan, A. Zaslaver & U. Alon, JMB, 334, 197 (2003) S. Mangan & U. Alon, PNAS, 100, (2003) G. Conant & A. Wagner, Nature Genetics, 34, 264 (2003) Shai Shen-Orr says E.coli thinks ahead because it feeds forward.

25 network patterns transcription network: nodes: operons edge from i to j: operon i contains a transcription factor for operon j detect whether certain subgraphs occur more frequently than expected. those that do are presumably solutions to some problem(s) figure out the problem(s) technically, it is a comparison of an actual network vs a random network of identical statistical profile (number of nodes, total number of edges, number of incoming and outgoing edges per node)

26 randomizing a graph uniformly randomizing a network with the switching algorithm: D D B B A A C C exchange only if no self-edges or multiple edges are generated uniformly randomizing a network with the matching algorithm: discard and start from scratch (!) if a self-edge or multiple edge is generated in a move

27 randomizing a graph matching algorithm if the network is not discarded upon formation of a multiple edge, the procedure will undersample this configuration 1 configuration 90 configurations

28 ChIP chip protein-dna interaction studies Chromatin Immuno Precipitation followed by a microarray

29 network motifs feedforward motif (FL) single input motif (SIM) -/+ dense overlapping regulons (DOR)...

30 overall picture

31 the feed forward loop the motives of motifs a delay mechanism Y X + AND Z pulse-filter X Y Z

32 multi-site phosphorylation is a pulse-filter too another delay mechanism "signal" target with 5 phosphoepitopes "output" phosphatase concentration of components concentration of components time time WF & D. Krakauer, unpublished

33 error correction through kinetic proof-reading and yet another delay mechanism + on off 1 R + on off 2 > off 1 R minimum error fraction: f 0 = off 1 /off 2 = exp - G/RT + on Q R off 1 off 1 + on Q R off 2 > off 1 off 2 minimum error fraction: f 0 2

34 Sx Sy Y X coherent Z most abundant in E.coli and yeast incoherent fairly abundant in yeast accelerates response time in Z

35 single input module the motives of motifs

36 transcriptional regulation & protein-protein interaction