Counting connected components in random temporal networks

Verena Schamboeck Spring meeting WSC 2017, Antwerpen, 15.5.2017

Smoluchowski coagulation problem

•  Coagulation problem: Cr+CsàCr+s

•  Master equation:

•  Applications in

–  Colloidal physics

–  Crowd dynamics

–  Social sciences

–  Astrophysics

–  Polymer chemistry

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M.vonSmoluchowskiDreiVorträgeüberDiffusion,Brown‘scheMolekularbewegungundKoagula=onvonKolloidteichenPhysik.Z.,17(1916),pp.557–571

•  Master equation: Reaction constant: ar,s=rs

à inviscid Burgers equation: (M1(t) – 1. moment of cr)

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Smoluchowski coagulation problem

Transformation using generating function:

Substitution:

J.WaCsAnintroduc=ontomathema=calmodelsofcoagula=on-fragmenta=onprocesses:Adiscretedeterminis=cmean-fieldapproachPhysicaD,222(2006),pp.1–20

•  Already highly non-trivial behavior:

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Smoluchowski coagulation problem

Mass conservation violated after critical time tc

Change in the asymptotic decay of the size distribution at tc: exponential à algebraic

Tota

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Time

Size of aggregate

Pro

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J.WaCsAnintroduc=ontomathema=calmodelsofcoagula=on-fragmenta=onprocesses:Adiscretedeterminis=cmean-fieldapproachPhysicaD,222(2006),pp.1–20

Cr+CsàCr+s

•  Smoluchowski coagulation problem:

–  Aggregates treated as a ‘blob’ without inner structure

•  Challenge: Find a model for growing discrete structures, e.g. highly crosslinked networks evolving in time

•  From local to global:

–  Local topology determined by arbitrary process that grows the network

–  Modeling the network as a random graph to extract global properties

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What about the topology?

Network theory

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•  Graph: –  Set of nodes connected by edges

•  Nodes: Degree (number of edges), various attributes •  Edges: Directional (in & out), bidirectional

–  Topology of a structure –  No spatial information

Nodes Edges

Connected components 1

2 3

4

5

in

out

bidirectional

Network theory

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•  Random graph with arbitrary degree distribution: –  Degree distribution u(i,j,k):

•  Probability of a node for having –  i in-edges –  j out-edges –  k bidirectional edges

–  Random graph: •  Probability distribution over all possible graphs •  Maximizes the entropy for a given degree distribution

{0,1,0}

{i,j,k}={1,0,1} {1,1,1}

{0,1,0}

{1,0,0}

Network theory

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•  Degree distribution: –  Imagine global process governing the degree distribution

•  t0: No component with s>1

•  t0<t<tc: Growth of component

•  tc: Phase transition – Emergence of the ‘giant component’ Giant component: scales linear with size of system

•  t>tc: Weight fraction of giant component increases

•  t∞: No aggregate with s>1, only giant component

Contains no giant component

Contains giant component

Set of all degree distributions

t0 tc

t∞

•  State of a node defined by –  Properties that alter the probability of a node to connect: v, p –  # in-edges, i –  # out-edges, j –  # bidirectional edges, k

•  Concentration of states of nodes determined by polymerization reactions:

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The degree distribution

Information on history of edge formation

Master equation: Reaction network:

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•  Goal: Extract size distribution of connected components from local information

•  Random graph with arbitrary degree distribution u(i,j,k) determined by the reaction mechanism

•  Transformation to the domain of generating function:

–  Original degree distribution:

–  Probability distribution for a node reached by an edge to be connected to further nodes:

•  Reached by in-edge

•  Reached by out-edge

•  Reached by bidirectional edge

From local to global

Minimum connectivity: 1

Normalization more in-edges à higher probability to be

reached

•  Connected components:

–  Size distribution w(s) à W(x)

–  Biased size distribution: Size distribution of connected component reached by

•  In-edge win(s) à Win(x)

•  Out-edge wout(s) à Wout(x)

•  Bidirectional edge wbi(s) à Wbi(x)

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From local to global

Uin …

Uout …

Ubi …

U … …

…

•  Size distribution of connected component reached by in-edge win

•  Size distribution of i out-components: Convolution: wout(s)*wout(s)*…*wout(s) à Wout(x)i

•  Size distribution of i out-, j in- and k bidirectional components: wout*…*wout*win*…*win*wbi*…*wbi à Wout

i Winj Wbi

k

•  GF of size distribution for component reached by in-edge: Win=x Σ uin(i,j,k) Wout

i Winj Wbi

k

Win=x Uin(Wout,Win,Wbi) Wout=x Uout(Wout,Win,Wbi)

Wbi=x Ubi(Wout,Win,Wbi)

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From local to global

Uin

Uout Ubi

Uin

…

Win

……Wout Wbi

Win

i k

j

i-times

i-times j-times k-times

i,j,k

∞

Win=x Uin(Wout,Win,Wbi) Wout=x Uout(Wout,Win,Wbi)

Wbi=x Ubi(Wout,Win,Wbi)

•  3 coupled functional equations:

Win= x Uin(Wout,Win,Wbi) Wout= x Uout(Wout,Win,Wbi)

Wbi= x Ubi(Wout,Win,Wbi)

à solve numerically

•  GF of size distribution of connected

component W = x U(Wout,Win,Wbi)

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From local to global

Uout

Uin

Ubi

U

Uout Ubi

Uin

…

Win

……Wout Wbi

Wout … …

Wbi

W

Win

•  Back transform:

–  Derivatives

•  Problem: Numerically unstable (only feasible until s~30)

–  Inverse Fourier transform:

•  Advantage: stable, faster (s log(s)) •  Disadvantage: evaluation for all s necessary

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From local to global

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Component size distribution

t<tc: w(s)∝e-s t=tc: w(s)∝s-3/2 t>tc:w(s)∝e-s

w(s)∝s-3/2

Violation of the conservation of mass after tc

•  Weight average component size defined as with with the partial moments μlmn of u(i,j,k)

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Analytic criterion for existence of the giant component

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•  Growing particles have discrete structure

•  Modeling the structure as a network

–  Random graph with arbitrary trivariate degree distribution

–  Degree distribution determined by Master equation, reaction network, …

–  Extract size distribution of connected components

–  Ananlytic criterion for emergence of giant component

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Summary

•  Computational polymer chemistry group at the University of Amsterdam

Piet Iedema Ivan Kryven Yuliia Orlova

•  Océ Technologies B.V. Technology Foundation STW

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Acknowledgements