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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
2 / 46
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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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.
5 / 46
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
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
7 / 46
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
Morphology of clusters on a graph
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.
Morphology of clusters on a graph
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.
Results for percolation probability g and mean (finite) cluster size n for random and real world networks
[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.
Results for percolation probability g and mean (finite) cluster size n for random and real world networks
[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
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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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
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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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.
24 / 46
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
25 / 46
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 = 4 at p = 0.255, and their deconvolution according to degree for k = 0, 1, 2, . . . , 10
26 / 46
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 = 4 at p = 0.3, and their deconvolution according to degree for k = 0, 1, 2, . . . , 10
27 / 46
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.
31 / 46
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
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
33 / 46
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.
34 / 46
Repeated Percolation Experiments
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
35 / 46
Repeated Percolation Experiments
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 = 2
36 / 46
Repeated Percolation Experiments
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 = 4
37 / 46
Repeated Percolation Experiments
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 = 8
38 / 46
Repeated Percolation Experiments
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 = 16
39 / 46
Repeated Percolation Experiments
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 = 32
40 / 46
Repeated Percolation Experiments
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 = 64
41 / 46
Repeated Percolation Experiments
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, . . . , 19 and k ≥ 20, τ = 64
42 / 46
Repeated Percolation Experiments
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, . . . , 19 and k ≥ 20, τ = 128
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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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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).
45 / 46
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−%) − %).
46 / 46