Anomalous Transport in Metapopulation Networks
SERGEI FEDOTOV
School of Mathematics
The University of Manchester, UK
Collaboration with Helena Stage (Manchester)
Ginzburg Conference on Physics
Sergei Fedotov (University of Manchester)
P.N.Lebedev Physical Institute
May 29 - June 3, 2012
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Content
1
INTRODUCTION
Scale-Free Networks
Metapopulation Transport on Networks
2
FRACTIONAL TRANSPORT EQUATIONS ON NETWORKS
Axiom of Cumulative Inertia
Fractional Diffusion Equation and Subdiffusion
Anomalous Aggregation on Network
U-form Residence Time PDF and Empirical Evidence
Sergei Fedotov (University of Manchester)
P.N.Lebedev Physical Institute
May 29 - June 3, 2012
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Transport in Metapopulation Networks
Scale-Free Network: The power-law probability that a given node has k
links (order k) to other nodes: P(k) ∼ k −γ , γ ∈ [2, 3].
Figure :
Barabási-Albert network,
(∗)
Colizza and Vespignani, Phys. Rev.
Lett. 99, 148701 (2007).
Sergei Fedotov (University of Manchester)
P.N.Lebedev Physical Institute
May 29 - June 3, 2012
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Transport in Metapopulation Networks
Scale-Free Network: The power-law probability that a given node has k
links (order k) to other nodes: P(k) ∼ k −γ , γ ∈ [2, 3].
Mean field transport equation:
X
Ik ′ (t)
dNk (t)
= −Ik (t) + k
P(k ′ |k) ′
dt
k
′
k
Figure :
Barabási-Albert network,
(∗)
Colizza and Vespignani, Phys. Rev.
Lett. 99, 148701 (2007).
Sergei Fedotov (University of Manchester)
Nk : mean number of individuals in node of
order k;
Ik : mean flux out of node of order k;
P(k ′ |k) : the probability of a link between
nodes of order k → k ′ . For Ik (t) = λNk (t):
hNi
Nkst = k
hki
– well-connected nodes are more populous.(∗)
Human activity is not Poissonian!(†)
(†) A.-L.
Barabási,
P.N.Lebedev Physical Institute
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Axiom of Cumulative Inertia in Network Theory
Axiom of Cumulative Inertia:
An individual’s escape probability from a node decreases with the
(residence) time T spent in the node.
This is an empirical sociological law. The escape rate γk decreases with
residence time
µk
γk (τ ) =
, µk , τ0 > 0
τ + τ0
Probability density function (PDF) of a residence time is
µk
µk
τ0
ψk (τ ) =
∼ 1/τ 1+µk ,
τ + τ0 τ + τ0
Fedotov and Stage, Phys. Rev. Lett. 118, 9 (2017).
Sergei Fedotov (University of Manchester)
P.N.Lebedev Physical Institute
May 29 - June 3, 2012
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Anomalous subdiffusive transport:
Mean square displacement of Brownian particle:
< B 2 (t) >= 2Dt
Macroscopic transport equation:
∂ρ
∂2ρ
= D 2,
∂t
∂x
Sergei Fedotov (University of Manchester)
x ∈R
P.N.Lebedev Physical Institute
May 29 - June 3, 2012
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Anomalous subdiffusive transport:
Mean square displacement of Brownian particle:
< B 2 (t) >= 2Dt
Macroscopic transport equation:
∂ρ
∂2ρ
= D 2,
∂t
∂x
x ∈R
Mean square displacement for subdiffusion:
< X 2 (t) >∼ t µ
0<µ<1
What is the macroscopic equation for the concentration ρ?
Sergei Fedotov (University of Manchester)
P.N.Lebedev Physical Institute
May 29 - June 3, 2012
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Anomalous transport: fractional order PDE
Macroscopic equation for the concentration ρ:
∂2 ∂ρ
= Dµ 2 Dt1−µ ρ ,
∂t
∂x
(1)
where the Riemann-Liouville (fractional) derivative Dt1−µ is defined as
Z t
1 ∂
ρ (x, u) du
1−µ
Dt ρ =
(2)
Γ(µ) ∂t 0 (t − u)1−µ
Sergei Fedotov (University of Manchester)
P.N.Lebedev Physical Institute
May 29 - June 3, 2012
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Anomalous transport: fractional order PDE
Macroscopic equation for the concentration ρ:
∂2 ∂ρ
= Dµ 2 Dt1−µ ρ ,
∂t
∂x
(1)
where the Riemann-Liouville (fractional) derivative Dt1−µ is defined as
Z t
1 ∂
ρ (x, u) du
1−µ
Dt ρ =
(2)
Γ(µ) ∂t 0 (t − u)1−µ
Sergei Fedotov (University of Manchester)
P.N.Lebedev Physical Institute
May 29 - June 3, 2012
6 / 11
Anomalous subdiffusion: < X 2 (t) >∼ t µ
0<µ<1
• Subdiffusion is due to trapping inside dendritic spines
.
Non-Markovian behavior of particles performing random walk occurs when
particles are trapped during the random time with non-exponential
distribution.
Power law waiting time distribution
φ (t) ∼
1
t 1+µ
with 0 < µ < 1 as t → ∞.
The mean waiting time is infinite.
Sergei Fedotov (University of Manchester)
P.N.Lebedev Physical Institute
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How Does the Axiom of Cumulative Inertia Affect the
Flux?
Instead of the classical flux Ik = λNk (t), the Axiom of Cumulative Inertia
leads to a flux
Ik =
1
D 1−µk Nk (t),
Γ(1 − µk )τ0µk t
for
µk < 1,
Dt1−µk is the Riemann-Liouville fractional derivative.
Classical results are transient with no steady state.
Main result: ultimately all individuals are attracted to the node with
µk < 1.
The node’s mean residence time hT i → ∞ (anomalous trapping).
Sergei Fedotov (University of Manchester)
P.N.Lebedev Physical Institute
May 29 - June 3, 2012
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Anomalous Aggregation
What is happening inside the trapping (anomalous) node?
Consider the structural density of individuals at time t with residence time
τ , ntrap (t, τ ).
ntrap (t, τ ) →
N
Γ(1 − µk )Γ(µk )τ µk (t − τ )1−µk
,
N is the total number of individuals in the network.
→ Most individuals have been there for a long time, or are new arrivals.
Is this realistic?
Sergei Fedotov (University of Manchester)
P.N.Lebedev Physical Institute
May 29 - June 3, 2012
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Yes! Data: American MidWest
0.5
0.4
n1
N
0.95
N1
N
0.3
0.9
Eq. (13)
0.85
0.2
10
11
12
13
Time [Log(s)]
0.1
0.0
0.0
0.2
Figure :
Sergei Fedotov (University of Manchester)
0.4
0.6
0.8
1.0
τ/t
Fedotov and Stage, PRL 118, 9 (2017)
P.N.Lebedev Physical Institute
May 29 - June 3, 2012
10 / 11
Conclusions
• The mesoscopic description of non-Markovian reaction-transport
processes on the network is still an open problem.
Sergei Fedotov (University of Manchester)
P.N.Lebedev Physical Institute
May 29 - June 3, 2012
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