Heterogeneous Micro-Structure of Percolation in Complex ... · Heterogeneous Micro-Structure of...

Heterogeneous Micro-Structure of Percolationin Complex Networks

Reimer Kuhn

work done in part with Tim Rogers (Bath)

Disordered Systems GroupDepartment of Mathematics

King’s College London

Probability Seminar, QMUL 22 Jan 2020

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Outline

1 Setting the ScenePercolation on Complex NetworksPercolation – PhenomenologyPercolation – Heterogeneity

2 Message PassingMessage Passing/Cavity ApproachProbabilistic CharacterizationPercolation ProbabilityAverage Cluster SizeHeterogeneous Micro-Structure

3 Random Graph Ensembles

4 Results

5 Repeated Percolation Eperiments

6 Summary

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Outline




4 Results


6 Summary

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Percolation

Consider a complex network (or a lattice) of N vertices.

PercolationRandomly & independently occupy sites or bondsof the network with probability p.Look at distribution of sizes of clustersof contiguously occupied sites or bonds.

For N � 1 find

there is a critical occupancy-probability pc, s.t.for p < pc all clusters of bonds/sites are finite,for p > pc there is a giant/percolating cluster of size Sg with

Sg ∼ gN , with g > 0

Used

as model for porous media,to assess resilience of networks under attackto investigate spread of epidemics/information. . .

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Percolation – Phenomenology

Results for percolation probability g and mean (finite) cluster size n for random and real world networks

[from Karrer, Newman, Zdeborova, PRL (2014)]. Representative of results obtained in the field.

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Percolation – Heterogeneity

Average percolation probability and average sizes of finite clustersmiss heterogeneity in the problem.

Consider percolation process on local configurations below.

Propability Pg to be connected to giant component (GC):Left: Pg ∼ pk Right: Pg ∼ 1− (1− p)k

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Outline




4 Results


6 Summary

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Percolation – Message-Passing/Cavity Method

Consider graph of N � 1 vertices, and a given realization of thepercolation process.

Let Si be the cluster containing vertex i, and let si = |Si| be its size.

Let S(i)j be the cluster that can be reached via j ∈ ∂i on the cavity

graph from which i and all other edges connecting to it are removed,

and let s(i)j = |S(i)

j | be its size.

Morphology of clusters on a graph

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Percolation – Message-Passing/Cavity Method


Then si = 1 +∣∣⋃

j∈∂i S(i)j

∣∣ and s(i)j = 1 +

∣∣⋃`∈∂j\i S

(j)`

∣∣.Note: (i) S

(i)j = ∅, if edge (ij) is removed in the given realization.

(ii) On a tree S(i)j ∩ S

(i)k = ∅ for k 6= j.

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Percolation – Message-Passing/Cavity MethodProbabilistic Characterization — [Karrer, Newman, Zdeborova, PRL (2014)]

Denote by πi(s) the probability thatvertex i belongs to a cluster of finitesize s, s = 1, 2, 3 . . . .

Note that πi is not a normalized pmf,as a fraction of sites may sit in thegiant cluster of the system.


One hasπi(s) =

∑s∂i

π(i)(s∂i) δs,1+∣∣⋃

j∈∂i S(i)j

∣∣where π(i)(s∂i) denotes the probability that i connects to a set ∂i ofvertices which themselves belong to clusters of sizes s∂i = (sj)j∈∂i onthe cavity graph.On a tree:

πi(s) =∑s∂i

[ ∏j∈∂i

π(i)j (sj)

]δs,1+

∑j∈∂i sj

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Percolation – Message-Passing/Cavity MethodProbabilistic Characterization

Have expressed πi(s) in terms of cavity-probabilities {π(i)j (sj)}.

πi(s) =∑s∂i

[ ∏j∈∂i

π(i)j (sj)

]δs,1+

∑j∈∂i sj

Same line of reasoning (on a tree) gives

π(i)j (sj) = (1− p)δsj ,0 + p

∑s∂j\i

[ ∏`∈∂j\i

π(j)` (s`)

]δsj ,1+

∑`∈∂j\i s`

in which p is the probability that the link (ij) is actually present.

To analyze, re-express in terms of generating functions/z-transforms

Gi(z) =∑s≥0

πi(s) zs , H

(i)j (z) =

∑s≥0

π(i)j (s) zs

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Percolation – Message-Passing/Cavity MethodProbabilistic Characterization

Get

Gi(z) = z∏j∈∂i

H(i)j (z)

H(i)j (z) = 1− p+ pz

∏`∈∂j\i

H(j)` (z)

Note

Equations decouple in z.

Solve {H(i)j }-system iteratively on a given large single instance.

Although exact only on trees, get very accurate results on locallytree-like systems.In random graphs with finite mean connectivity: loops have lengthO(log(N)), and results asymptotically exact as N →∞.Alternatively: average equations over random graph ensemble.[Callaway et al. PRL (2000)].

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Percolation Probability

Note: Gi(1) =∑

s≥0 πi(s) is the probability that site i belongs to acluster of any finite size s, hence

gi = 1−Gi(1) = 1−∏j∈∂i

H(i)j (1)

is the probability that site i belongs to the giant cluster, and

g =1

N

N∑i=1

gi

gives the average percolation probability .

Find H(i)j ≡ H

(i)j (1) by solving the system

H(i)j = 1− p+ p

∏`∈∂j\i

H(j)`

Note H(i)j ≡ 1 is always a solution ⇒ gi ≡ 0.

Percolation transition where this solution becomes unstable.13 / 46

Average Cluster Size

The expected size of finite cluster to which i belongs is

〈si〉 =∑

s sπi(s)∑s πi(s)

=G′i(1)

Gi(1)= 1 +

∑j∈∂i

H(i)j′(1)

H(i)j (1)

.

Requires the z-derivative H(i)j′(1) = H

(i)j′(z)∣∣z=1

, which is obtained

from the equation for H(i)j (z) as

H(i)j′(1) = p

[1 +

∑`∈∂j\i

H(j)`′(1)

H(j)` (1)

] ∏`∈∂j\i

H(j)` (1) ,

giving

s =1

N

∑i

〈si〉

as the average size of finite clusters in the system.

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Percolation Probability and Average Cluster Size

Framework used by Karrer et al. to obtain results below.


[from Karrer, Newman, Zdeborova, PRL (2014)]

But missed heterogeneity of the gi and the 〈si〉 in their data.

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Percolation Probability and Average Cluster Size

Framework used by Karrer et al. to obtain results below.


[from Karrer, Newman, Zdeborova, PRL (2014)]

But missed heterogeneity of the gi and the 〈si〉 in their data.

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Heterogeneous Micro-Structure

Quantify heterogeneous micro-structure in percolation by looking at

ϕN (g) =1

N

∑i

δ(g − gi)

ψN (σ) =1

N

∑i

δ(σ − 〈si〉)

g

p

Grey-scale heat-map of ϕN (g) and of the mean percolation probability g as functions of p for

the gnutella P2P file sharing platform, a network of N = 62, 586 nodes, [RK, T Rogers, EPL (2017)].

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Heterogeneous Micro-Structure

Quantify heterogeneous micro-structure in percolation by looking at

ϕN (g) =1

N

∑i

δ(g − gi)

ψN (σ) =1

N

∑i

δ(σ − 〈si〉)

g

p

Grey-scale heat-map of ϕN (g) and of the mean percolation probability g as functions of p for

the gnutella P2P file sharing platform, a network of N = 62, 586 nodes, [RK, T Rogers, EPL (2017)].

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Outline




4 Results


6 Summary

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Limiting Distributions for Random Graph Ensembles

Evaluate limiting distributions

ϕ(g) = limN→∞

ϕN (g) and ψ(σ) = limN→∞

ψN (σ)

for models in configuration model class[maximally random subject to a degree distribution {pk}k≥0].

Recall

H(i)j = 1− p+ p

∏`∈∂j\i

H(j)`

H(i)j′ = p

[1 +

∑`∈∂j\i

H(j)`′

H(j)`

] ∏`∈∂j\i

H(j)`

⇔ stochastic recursion for (H(i)j , H

(i)j′) in the thermodynamic limit.

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Self-Consistent Estimation of Probabilities

Estimate π(h, h′) defined by

π(h, h′)dhdh′ = Prob(H(i)j ∈ (h, h+ dh], H

(i)j′ ∈ (h′, h′ + dh′])

self-consistently.From

H(i)j = 1− p+ p

∏`∈∂j\i

H(j)`

H(i)j′ = p

[1 +

∑`∈∂j\i

H(j)`′

H(j)`

] ∏`∈∂j\i

H(j)`

get

π(h, h′) =∑k≥1

k

cpk

∫ [ k−1∏`=1

dπ(h`, h′`)]δ(h−

(1− p+ p

k−1∏`=1

h`

))

×δ(h′ − p

[1 +

k−1∑`=1

h′`h`

] k−1∏`=1

h`

)21 / 46

Population Dynamics

Solve

π(h, h′) =∑k≥1

k

cpk

∫ [ k−1∏`=1

dπ(h`, h′`)]δ(h−

(1− p+ p

k−1∏`=1

h`

))

×δ(h′ − p

[1 +

k−1∑`=1

h′`h`

] k−1∏`=1

h`

)by a stochastic algorithm (population dynamics) [Mezard, Parisi, EPJB (2001)].

From solution obtain

ϕ(g) =∑k≥0

pk

∫ [ k∏`=1

dπ(h`, h′`)]δ(g − 1 +

k∏`=1

h`

)

ψ(σ) =∑k≥0

pk

∫ [ k∏`=1

dπ(h`, h′`)]δ(σ − 1−

k∑`=1

h′`h`

).

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Outline




4 Results


6 Summary

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Erdos-Renyi Graphs

ϕ(g)

g

Distribution of percolation probabilities for an Erdos-Renyi graph of mean degree c = 2

at p = 0.75. Full solution and simulation of a system of N = 10,000 vertices, using

5000 realizations of the percolation process.

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Erdos-Renyi Graphs

ϕ(g)

g

ψ(σ)

σ

Distribution of percolation probabilities (top) and of mean cluster sizes (bottom) for an Erdos-Renyi graph of

mean degree c = 2 at p = 0.75, and their deconvolution according to degree for k = 0, 1, 2, . . . , 10

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Erdos-Renyi Graphs

ϕ(g)

g

ψ(σ)

σ



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Erdos-Renyi Graphs

ϕ(g)

g

ψ(σ)

σ



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Scale-Free Graphs

ϕ(g)

g

Distribution of percolation probabilities for a scale free graph with pk ∝ k−3, k ≥ 2 at p = 0.5,

and its deconvolution according to degree for k = 2, 3, . . . , 9, and k ≥ 10. [RK, T Rogers, EPL (2017)]

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Scale-Free Graphs

ψ(σ)

σ

Distribution of mean cluster sizes for a scale free graph with pk ∝ k−3, k ≥ 2,

at p = 0.5 and its deconvolution according to degree for k = 2, 3, . . . , 9 and k ≥ 10

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Weakly-Nonlinear Expansion Near pc

Asymptotic results in the vicinity of pc for single instances. For

p− pc = ε� 1, expand H = (H(i)j ) as

1−H = εa+ ε2b+ ε3c+ . . .

Matching powers of ε in FPEs for H allows to determine a, b etc. interms of bi-orthonormal system of eigenvectors of Hashimotonon-backtracking matrix B.

Get expansion of gi and 〈si〉 up to second order in p− pc, and bounds,

α(p− pc)vminki . gi . α(p− pc)vmaxki

αvmin

|p− pc|ki . 〈si〉 .

αvmax

|p− pc|ki .

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Results – Weakly-Nonlinear Expansion

gi

p

〈si〉

p

Top: Probability gi to appear in the percolating cluster . Bottom: Expected size 〈si〉 of finite clusters containing given nodes.

Solutions of the cavity equations (solid lines) are compared with results of a second order expansion (dashed lines) for the same

system. A selection of results are shown for vertices from a scale free graph with pk ∝ k−3, k ≥ 2.

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Further Results

Closed form approximations for ϕ(g) and ψ(σ) in the large c limit of Erdos-Renyi graphs, for which the percolationtransition occurs at pc = 1/c. With scaling p = %/c:

ϕ(g) 'exp

{− c

2

(1 + 1

ηlog(1− g)

)2− log(1− g)

}√

2π/c η,

ψ(σ) '√c

√2πγ

exp

{−

c

2γ2(σ − 1− γ)2

},

where η = % +W (−%e−%), with W the Lambert W-function, and γ = %/(e%+W (−%e−%) − %).

ϕ(g)

g

Distribution percolation probabilities for a c = 8 at p = 0.2 (% = 1.6). Large c approximation

compared with simulation results for an ER network of size N = 1000. [RK, T Rogers, EPL (2017)]

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Outline




4 Results


6 Summary

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Repeated Percolation Experiments

Study probability for a node i to be on giant component in all oft = 1, 2, . . . , τ independent percolation experiments:

gi(τ) = 〈ni(τ)〉 =∏t∈τ

(1−

∏j∈∂i

(1− pg(i)j (t)

)

where g(i)j (t) is the probability that i is linked to the GC via

bond j in experiment #t,

g(i)j (t) = 1−

∏`∈∂j\i

(1− pg(j)` (t)

)Relevant e.g. for critical components of distribution grids.

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ϕ(g)

g

Distribution of mean cluster sizes for a scale free graph with pk ∝ k−3, k ≥ 2, at p = 0.5

and its deconvolution according to degree for k = 2, 3, . . . , 11 and k ≥ 12, τ = 1

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ϕ(g)

g



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ϕ(g)

g



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ϕ(g)

g



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ϕ(g)

g


and its deconvolution according to degree for k = 2, 3, . . . , 11 and k ≥ 12,τ = 16

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ϕ(g)

g



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ϕ(g)

g



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ϕ(g)

g



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ϕ(g)

g



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Outline




4 Results


6 Summary

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Summary

Looked at bond-percolation in complex networks.

Argued that percolation probabilities cannot be uniform across sitesof a network.

Similarly average size of a cluster to which a site belong varies fromsite to site.

Analyzed using message-passing/cavity method.

Evaluated distribution ϕ(g) of percolation probabilities and ψ(σ) ofaverage cluster sizes across a network.

Results are relevant for analysis of epidemics on networks (SIR-modelcan be mapped on bond percolation).

⇒ relevant for design of optimal immunization strategies.

Generalization for collection of independent percolation experiments(critical components of distribution grids).

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Closed Form Approximation

Closed form approximations for ϕ(g) and ψ(σ) in the large c limit ofErdos-Renyi graphs, for which the percolation transition occurs atpc = 1/c.

Equations solved by π(h) = δ(h− h∗) with h∗ solving

h∗ = 1− p+ pe−c(1−h?)

Use scaling p = %/c, with % > 1 to find

ϕ(g) 'exp

{− c

2

(1 + 1

η log(1− g))2− log(1− g)

}√2π/c η

,

ψ(σ) '√c√

2πγexp

{− c

2γ2(σ − 1− γ)2

},

in which η = %+W (−%e−%), with W the Lambert W-function, andγ = %/(e%+W (−%e−%) − %).

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