Dimensional Analysis and Power Laws

Dimensional Analysis •  Often used in physics •  For reasons given thus far, many processes

should scale as a power law. •  Given some quantity, f, that we want to

determine, we need to intuit what other variables on which it must depend, {x1,x2,…,xn}.

•  Assume f depends on each of these variables as a power law.

•  Use consistency of units to obtain set of equations that uniquely determine exponents.

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f (x1,x2,...,xn ) = x1p1 x2

p2 ...xnpn

Example 1: Pythagorean Theorem

•  Hypotenuse, c, and smallest angle, θ, uniquely determine right triangles.

•  Area=f(c, θ), DA implies Area=c2g(θ).

θ

φ

φ

c a

b

Area of whole triangle=sum of area of smaller triangles

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a2g(θ) + b2g(θ) = c 2g(θ)⇒ a2 + b2 = c 2

θ

Example 2: Nuclear Blast

•  US government wanted to keep energy yield of nuclear blasts a secret.

•  Pictures of nuclear blast were released in Life magazine with time stamp

•  Using DA, G. I. Taylor determined energy of blast and government was upset because they thought there had been a leak of information

•  Radius, R, of blast depends on time since explosion, t, energy of explosion, E, and density of medium, ρ, that explosion expands into

•  [R]=m, [t]=s, [E]=kg*m2/s2, ρ=kg/m3

•  R=tpEqρk

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1= 2q − 3k0 = p − 2q0 = q + k

q=1/5, k=-1/5, p=2/5 R∝ (E / ρ)1/5 t2/5 ⇒ E∝ R5ρt2

m s kg

unknown constant coefficient can be determined from y-intercept of regression of log-log plot of time series

Pitfalls of Dimensional Analysis

•  Miss constant factors

•  Miss dimensionless ratios

•  But, can get far with a good bit of ignorance!!!

Summary •  Self-similarity and fractalsèPower Laws •  Behavior near critical point èPower Laws •  But, Power Lawsènear critical points •  Dimensional Analysis assumes power law

form and this is partially justified by necessity of matching units

If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is. -John von Neumann

Theories are approximations that hope to impart deeper understanding

We all know that art [theory] is not truth. Art [theory] is a lie that makes us realize truth, at least the truth that is given us to understand. The artist [theorist] must know the manner whereby to convince others of the truthfulness of his lies.

--Pablo Picasso

A little philosophy of science

•  Many general patterns are power laws

•  Can often explain these without knowledge of all the details of the system

•  Art of science is knowing system well enough to have intuition about which details are important

Single prediction models are not enough

•  Much better to predict value of exponent and not just that it is a power law

•  To really believe a theory we need multiple pieces of evidence (possibly multiple power laws) and need to be able to predict many of these

•  Understanding dynamics and some further details allows one to predict deviations from power law, and that is a very strong test and leads to very precise results

n−1/3

Barabasi et al.

Power laws in statistical distributions and scaling of growth of networks

Erdos-Renyi random graphs

Examples of distributions of real data

ActorsInternetPowergrid

Growth models of networks

1.  Start with m0 unconnected nodes 2.  Network grows one vertex at a time (gene duplication,

species invasion, etc)

3.  Add 1 node at a time and form m new connections between this node and existing nodes. (Why?)

4.  Connections are formed with probability where ki is the connectivity of node i.

(Preferential attachment/rich get richer/proportional model) This is a type of self similarity! Power laws are

to be expected. €

ki / k j∑

Scaling of connectivity

Total number of nodes at time t is m0+t Total number of edges is mt=

Connectivity at next time step is on average (edges can’t be lost):

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k j∑ /2

€

ki(t +1) = ki(t) + m[ki(t) / k j ]∑

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ki(t)∝ t

Numberofnewedgesatnext7mestep

Probabilityofthatconnec7ongoingtonodei

Scaling of connectivity

Scaling of probability density

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P(ki > k) =mk

"

# $

%

& '

mtm0 + t"

# $

%

& '

k

(

)

* * * *

+

,

- - - -

Connec7vityat7methatnodeicameintoexistence

Averageconnec7vityofanynodeat7met

Connec7vityforcomparisonandprobability

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P(ki < k) =1− P(ki > k) =1− m2tk 2(m0 + t)

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dPdk

∝ k−3Probabilitydensity

Scaling of probability density

Scaling of probability density

Scaling of probability density

Addingedgeswithoutaddingnewnodesdoesnotreachequilibrium

Metabolic networks

E coli metabolic substrate network

Archaea Ecoli

Celegans Averageacross43organisms

Scaling of probability density in metabolic networks

Distributions of pathway lengths

Scaling of connectivity

Scaling of lengths with node removal

Return to motifs

Similar frequencies of subgraphs in real networks

Universal scaling exponents in metabolic networks

γisdegreeexponentandαisclusteringexponent.Exponentscharacterizelocalandglobalnetworkorganiza7on

Statistically significant subgraphs (i.e., motifs) for power law networks

NODES

EDGES

Scaling of probability density of triangles

Tisnumberofselectedsubgraphspassingbyanode

Dis7nctcomponents

Scaling of probability density of penta-graphs

Giantcomponent

Tisnumberofselectedsubgraphspassingbyanode

Phase transitions in hex-graphs: Before transition are motifs

Aggrega7on/clusteringdecreaseswithnumberofedgesinsubgraph.Normalizedbysizeoflargestconnectedcomponent

Free Network software

1.  Networkworkbench(NWB)

2.  Fanmod

3.  Gephi

4.  Prism.m

Canusetheseinyourresearch!

Allhumansarecaughtinaninescapablenetworkofmutuality.-Mar7nLutherKing,Jr.

Candisplaysamenetworkindifferentways

Canlookatdifferentnetworkorlevelsforgivensystem

Canlookatdifferent7mes

Canlookatdifferent7mes

What have we learned? 1.  Evolu7onarytheory—selec7on,dri[,muta7ons,epistasis,neutrality,coalescence2.  Networktheory—mo7fs,subgraphstructure,branchinghierarchical,op7mal,clustering,growth,preferen7ala\achment(genes,proteins,foodwebs,disease)3.Kolmogorov/Diffusion/FokkerPlanck—howtocombinedirec7onalandnon-direc7onalprocesses,cellmigra7on,cancer,geneexpression,speciesabundance4.Scaling—powerlaws,selfsimilarity,asympto7cexpansions5.Howtoreadamodelingpaperandworkthroughthemathandlogicwithoutjustreadingfiguresandpunchlines6.Howtorelateassump7onstofiguresandfundamentalequa7onsandpredic7ons7.Fixedpoints,ODEs,sums,deltafunc7on,gammafunc7on

THANKS to all of you!

Good luck with your projects!