Dimensional Analysis and Power Laws
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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.
€
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
€
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
€
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):
€
k j∑ /2
€
ki(t +1) = ki(t) + m[ki(t) / k j ]∑
€
ki(t)∝ t
Numberofnewedgesatnext7mestep
Probabilityofthatconnec7ongoingtonodei
Scaling of connectivity
Scaling of probability density
€
P(ki > k) =mk
"
# $
%
& '
mtm0 + t"
# $
%
& '
k
(
)
* * * *
+
,
- - - -
Connec7vityat7methatnodeicameintoexistence
Averageconnec7vityofanynodeat7met
Connec7vityforcomparisonandprobability
€
P(ki < k) =1− P(ki > k) =1− m2tk 2(m0 + t)
€
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!
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