Optimal Allocation of Interconnecting Links in Cyber-Physical...
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CMU SV, April 16th, 2013 1
Optimal Allocation of Interconnecting Links inCyber-Physical Systems: Interdependence,
Cascading Failures and Robustness
Osman Yagan
CyLab
Carnegie Mellon University
Collaborators:
Douglas Cochran, Virgil Gligor, Armand Makowski,
Dajun Qian, Junshan Zhang, Jun Zhao
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CMU SV, April 16th, 2013 2
Research Overview
A. Wireless (Sensor) Networks
− Connectivity, security, performance evaluation, anddesign
B. Network Science
−Dynamical processes on coupled complex networks
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CMU SV, April 16th, 2013 3
A. Random graphs for wireless (sensor) networkapplications
• Random Graphs = Graphs generated by a random process
• Can model many types of relations and processes in physical,
biological, social, and engineering systems.
• Studied several problems derived from
⋆ Random key predistribution schemes for wireless sensor
networks → Dissertation topic
− Connectivity and mobility in wireless networks
− Modeling and analysis of social networks
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CMU SV, April 16th, 2013 4
Wireless sensor networks (WSNs) and security
• Distributed collection of small sensors that gather security-
sensitive data and control security-critical operations.
• Random key predistribution schemes are widely regarded as
the appropriate solutions for securing WSNs.
Evaluating random key predistribution schemes:
• How to select the parameters of a given scheme so that
certain desired properties hold with high probability?
• How do various schemes compare with each other w.r.t.
connectivity, security, memory load, and scalability?
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CMU SV, April 16th, 2013 5
My dissertation
• The Eschenauer-Gligor (EG) scheme
⋄ Connectivity under full visibility
† ISIT 2008, ISIT 2009, CISS 2010, IT 2012
⋄ Connectivity under an on-off channel model (unreliable
links)
† IT 2012
⋄ Diameter, clustering coefficient, and small-world properties
† Allerton 2009, GraphHoc 2009, IT 2013
• Published • In Review
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CMU SV, April 16th, 2013 6
My dissertation cont’d.
• The pairwise scheme of Chan, Perrig and Song
⋄ Connectivity under full visibility
† ISIT 2012, IT 2013
⋄ Connectivity under an on-off channel model
† ICC 2011, IT 2013
⋄ Scalability, gradual deployment
† WiOpt 2011, Perf Eval 2012
⋄ Security
† PIMRC 2011, TISSEC 2013
• Published • In Review
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CMU SV, April 16th, 2013 7
Postdoctoral work & Future directions
• Connectivity in Random Threshold Networks
⋄ IEEE JSAC: Social Networks, joint with A. M.
Makowski.
• k−connectivity of the EG scheme under an on-off channel
⋄ IT 2013, joint with J. Zhao and V. Gligor.
Future Directions:
• Connectivity, coverage, outage probability, and capacity
of tiered cellular networks.
• Analysis and performance evaluation of mobile data offloading
technologies; e.g., femtocell, Wi-Fi.
• Social network modeling, mobility in WSNs.
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CMU SV, April 16th, 2013 8
B. Network science
• An inter-disciplinary field bringing together researchers from
diverse backgrounds
⋄ engineering, mathematics, physics, biology, computer
science, sociology, epidemiology, etc.
• Tremendous activity over the past decade: special issues,
conferences, journals on network science.
⋄ DoD research initiatives, NSF grant programs
Main aim: Developing a deep understanding of the dynamics
and behaviors of social, biological and physical networks.
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CMU SV, April 16th, 2013 9
Dynamical processes on complex networks
∗ Spreading of an initially localized effect throughout the whole (or,
a very large part of the) network.
• Diffusion of information, ideas, rumors, fads, etc.
• Disease contagion in human and animal populations.
• Cascade of failures, avalanches, sand piles.
• Spread of computer viruses or worms on the Web.
† Searching on networks (WWW, P2P)
† Flows of data, materials, biochemicals.
† Network traffic, congestion.
∗ Barrat et al. Dynamical Processes on Complex Networks, 2008
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CMU SV, April 16th, 2013 10
Main Motivation
∗ Most research on complex networks focus on the limited case of a
single, non-interacting network.
∗ Yet, many real-world systems do interact with each other.
⋄ Major infrastructures depend on each other:
telecommunications, energy, banking and finance,
transportation, water supply, public health.
⋄ Social networks are coupled together:
Facebook, Twitter, Google+, YouTube, etc.
Q: Dynamical processes on interacting networks?
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CMU SV, April 16th, 2013 11
Contributions thus far
1. Cascading failures on interdependent cyber-physical systems
⋄ O. Yagan, D. Qian, J. Zhang and D. Cochran, IEEE Trans.
Parallel and Distrib. Syst. 23(9): 1708–1720, Sept. 2012
2. Influence propagation in social networks with multiple link
types
⋄ O. Yagan and V. Gligor, Phys. Rev. E 86, 036103, Sept. 2012
3. Information propagation in coupled social-physical networks
⋄ O. Yagan, D. Qian, J. Zhang and D. Cochran, IEEE JSAC:
Network Science, to appear.
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CMU SV, April 16th, 2013 12
Today
⋆ Cascading failures on interdependent cyber-physical systems
⋄ O. Yagan, D. Qian, J. Zhang and D. Cochran, “Optimal Allocation
of Interconnecting Links in Cyber-Physical Systems:
Interdependence, Cascading Failures and Robustness,” IEEE Trans.
Parallel and Distrib. Syst. 23(9): 1708–1720, Sept. 2012
Outline:
• Interdependent networks: definition, relevance, issues
• How to evaluate the robustness of interdependent networks
• Finding design strategies that improve robustness
• Optimum resource allocation strategy to maximize robustness
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CMU SV, April 16th, 2013 13
Interdependent networks?
• A collection of networks that depend on one another to provide
proper functionality.
• Interdependence is omnipresent in many modern systems.
⋄ National infrastructures: telecommunications, energy,
banking & finance, water supply, emergency services.
• Interdependence exists even at smaller scales: e.g., smart-grid
⋄ Power stations depend on communication nodes for control
while communication nodes depend on power stations for
their electricity supply.
Large, smart and more capable systems
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CMU SV, April 16th, 2013 14
But . . ., interdependent networks are fragile
Adversarial attacks, system failures, and natural hazards ⇒
• Node failures in one network may lead to failure of the
dependent nodes in other networks, and vice versa.
• Continuing recursively, this may lead to a cascade of failures.
• The failure of a very small fraction of nodes from a network
may lead to the collapse of the entire system.
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CMU SV, April 16th, 2013 15
Real-world examples
Goal: Mitigate catastrophic impacts
Plan of action: Model and quantify cascading failures &
Develop design strategies that improve robustness
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CMU SV, April 16th, 2013 16
A starting point: Buldyrev et al. (Nature, 2010)
Network B
1
3
2
N
3
2
1
Network A
N
Figure 1: Intra-topologies are not shown. Inter-links determine
support-dependence relationships.
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CMU SV, April 16th, 2013 17
Cascade dynamics
• Initially, a fraction 1− p of nodes are randomly removed from
Network A ⇒ Models random attacks or failures.
• A node is said to be functional at Stage i if
1) it has at least one inter-edge with a node that was
functional at Stage i− 1, and
2) it belongs to the largest connected component of the of its
own network
• Cascade of failures propagates alternately between A and B,
eventually leading to a steady state.
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CMU SV, April 16th, 2013 18
Robustness metrics
• SA∞: Fraction of functional nodes of network A at steady
state.
• SB∞: Fraction of functional nodes of network B at steady
state.
• 1− pc : Critical attack size = Largest attack that can be
sustained.
⋄ If more than 1− pc fraction is attacked ⇒ SA∞= SB∞
= 0
⋄ If less than 1− pc fraction is attacked ⇒ SA∞, SB∞
> 0
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CMU SV, April 16th, 2013 19
Robustness of theBuldyrev et al. model
Network B
1
3
2
N
3
2
1
Network A
N
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CMU SV, April 16th, 2013 20
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fraction of nodes attacked, (1 − p)
Fractio
noffu
nctio
nalnodes,SA
∞
critical fraction1 − pc ≃ 0 .18
Single ER network
≃ 0.66
Interdep.networks
Figure 2: Networks A and B are Erdos-Renyi (ER) with mean degree
d = 3
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CMU SV, April 16th, 2013 21
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fraction of nodes attacked, (1 − p)
Fractio
noffu
nctio
nalnodes,SA
∞
critical fraction1 − pc ≃ 0 .18
Single ER network
≃ 0.66
Interdep.networks
∗ Interdependent networks are much more vulnerable to
attacks!
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CMU SV, April 16th, 2013 22
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fraction of nodes attacked, (1 − p)
Fraction
offu
nctionalnodes,SA
∞
critical fraction1 − pc ≃ 0 .18
Single ER network
≃ 0.66
Interdep.networks
Interdep. Nets Single Nets
1 − p c ≃ 1 −
2.45d
1 − p c ≃ 1 −
1d
∗ Single network case provides a fundamental limit on the
robustness of interdependent networks.
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CMU SV, April 16th, 2013 23
Our goals
• Quantify robustness under more realistic interdependent
network models
∗ Multiple inter-links per node, rather than the one-to-one
correspondence model
• Develop design strategies
∗ Reveal trade-offs between the # of inter-links and
robustness
∗ Characterize optimum inter-link allocation strategies
Yagan et al., IEEE TPDS, Sept. 2012
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CMU SV, April 16th, 2013 24
A new interdependent network model
Network B
1
2 2
1
Network A
N
k
k−1
N
∗ Each node has exactly
k inter-edges
∗ Any one of its k
inter-connections can
provide the needed support
to a node
Quantities of interest:
1) SA∞, SB∞
⇒ fraction of functioning nodes at steady-state
2) 1− pc as a function of k (and intra-degree distributions)
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CMU SV, April 16th, 2013 25
General solution
∗ Let Ai, Bi denote the functioning giant components in Net A and
Net B at stage i with corresponding fractional sizes SAiand SBi
.
With p′A1= p and SA1 = pFA(p), we have the recursive relations
p′Bi= 1−
(
1− pFA(p′Ai−1
))k
; SBi= p′Bi
FB(p′Bi), i = 2, 4, 6, . . . .
p′Ai= p
(
1−(
1− FB(p′Bi−1
))k
)
; SAi= p′Ai
FA(p′Ai), i = 3, 5, . . .
pFA(p) : Fractional size of the giant component in A′, where A′ is
the subgraph of A induced by the pN functl. nodes (after failures).
A −→failure of (1 − p)-fraction A′ −→largest component A′′
|A′′|/N = pFA(p) ⇒ Depends on intra-degree distributions.
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CMU SV, April 16th, 2013 26
∗ This recursive process stops at an “equilibrium point” where we
have p′B2m−2= p′B2m
and p′A2m−1= p′A2m+1
so that neither network
A nor network B fragments further. Setting x = p′A2m+1, y = p′B2m
x = p(
1− (1− FB(y))k)
y = 1− (1− pFA(x))k
(1)
Obtaining the quantities of interest: Assume FA, FB are known
1. Obtain the stable solution of Eqn (1) for a given p and k.
2. Compute SA∞:= limi→∞ SAi
= xFA(x) and SB∞= yFB(y).
3. Finding pc : repeat steps 1 and 2 for various p to find the
smallest p that gives SA∞, SB∞
> 0.
pc = inf {0 ≤ p ≤ 1 : SA∞, SB∞
> 0}
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CMU SV, April 16th, 2013 27
Special case: ER networks
∗ Assume both networks are ER with mean intra-degrees a and b.
∗ It is known that: FA(x) = 1− fA where fA is the unique solution
of fA = exp{ax(fA − 1)}. This leads to
SA∞= p(1− fk
B)(1− fA),
SB∞=
(
1− (1− p(1− fA))k)
(1− fB).(2)
where fA and fB are given by the pointwise smallest solution of
fB = k
√
1− log fA(fA−1)ap if 0 ≤ fA < 1; ∀fB if fA = 1
fA = 1−1− k
√
1−log fB
(fB−1)b
pif 0 ≤ fB < 1; ∀fA if fB = 1.
(3)
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CMU SV, April 16th, 2013 28
0 1
1
a) 1 − p = 0.60 1
1
b) 1 − p = 0.55
fA
fB
0 1
1
c) 1 − p = 0.5
0 1
1
d) 1 − p = 0.440 1
1
e) 1 − p = 0.40 1
1
f ) 1 − p = 0.3
Figure 3: Possible solutions of the system (3) when a = b = 3 and
k = 2. The critical 1 − pc corresponds to the case when the two curves
are tangential to each other.
∗ 1− pc = 0.44 ⇒ With k = 2 system is robust against failures of
up to 44 % of the nodes. With k = 1, only against 18 %
∗ Phase transition is discontinuous, i.e., first order
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CMU SV, April 16th, 2013 29
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fraction of nodes attacked, (1 − p)
Fractio
noffu
nctio
nalnodes,SA
∞
critical fraction1 − pc ≃ 0 .44
k = 2
k = 1
Single ER network
≃ 0.18 ≃ 0.66
Figure 4: Net A and Net B are ER with mean degrees d = 3
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CMU SV, April 16th, 2013 30
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fraction of nodes attacked, (1 − p)
Fractio
noffu
nctio
nalnodes,SA
∞
critical fraction1 − pc ≃ 0 .44
k = 2
k = 1
Single ER network
≃ 0.18 ≃ 0.66
∗ With k = 2 system is robust against failures of up to 44 % of the
nodes. With k = 1, only against 18 %
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CMU SV, April 16th, 2013 31
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fraction of nodes attacked, (1 − p)
Fractio
noffu
nctio
nalnodes,SA
∞
critical fraction1 − pc ≃ 0 .44
k = 2
k = 1
Single ER network
≃ 0.18 ≃ 0.66
∗ For attacks of up to 30 % of the nodes, interdependent networks
with k = 2 are almost as robust as single networks.
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CMU SV, April 16th, 2013 32
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fraction of nodes attacked, (1 − p)
Fractio
noffu
nctio
nalnodes,SA
∞
critical fraction1 − pc ≃ 0 .44
k = 2
k = 1
Single ER network
≃ 0.18 ≃ 0.66k = 3 4 5
∗ As k gets larger, the robustness curve approaches to the
fundamental limit.
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CMU SV, April 16th, 2013 33
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Fraction of nodes attacked, (1 − p)
Fraction
offu
nctionalnodes,SA
∞
critical fraction1 − pc ≃ 0 .44
k = 2
k = 1
Single ER network
≃ 0.18 ≃ 0.66k = 3
4 5
1 − pc ≃ 1 −
1+1 .45 ·k−1.2
d
pc vs. k
Trade-off between # of inter-links per node vs. robustness
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CMU SV, April 16th, 2013 34
A design question
• In our model, each node has exactly k undirected inter-edges;
i.e., k bi-directional inter-links per node.
• Suppose that we are given a fixed number of uni-directional
inter-network edges, say 2kN .
• How should these edges be allocated in order to maximize the
robustness, i.e., in order to achieve the largest SA∞, SB∞
, 1− pc
• Regular vs Random, Bi-directional vs Uni-directional
∗ Yagan, Qian, Zhang, Cochran, NetSciCom, April 2011.
∗ Shao, Buldyrev, Havlin, and Stanley, Phys. Rev. E, March 2011.
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CMU SV, April 16th, 2013 35
Random allocation vs. regular allocation
Implementing random allocation strategy:
∗ Specify α = (α0, α1, α2, . . .) with∑∞
j=0 αj = 1
∗ αj : fraction of nodes with j inter-links
∗ Randomly partition both networks into subgraphs with sizes
α0N,α1N, . . ., and assign j bi-directional inter-edges to each
node in the jth partition. ⇒ Intra topologies are unknown
∗ We want to compare
1− pc(α), SA∞(α), SB∞
(α) vs. 1− pc(k), SA∞(k), SB∞
(k)
∗ Matching condition: k =∑∞
j=0 αjj (with integer k)
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CMU SV, April 16th, 2013 36
Theorem 1 Consider α = (α0, α1, α2, . . .) such that
k =∞∑
j=0
αjj.
Then, for all p, we have
SA∞(k) ≥ SA∞
(α),
SB∞(k) ≥ SB∞
(α).
Furthermore
1− pc(k) ≥ 1− pc(α).
Notation Regular Random
Frac. of func. nodes, Net A SA∞(k) SA∞
(α)
Frac. of func. nodes, Net B SB∞(k) SB∞
(α)
Critical attack size 1− pc(k) 1− pc(α)
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CMU SV, April 16th, 2013 37
Theorem 1 Let α = (α0, α1, . . .) s.t. k =∞∑
j=0
αjj. For all p,
SA∞(k) ≥ SA∞
(α),
SB∞(k) ≥ SB∞
(α).(4)
Furthermore
1− pc(k) ≥ 1− pc(α). (5)
Remarks:
∗ Random allocation yields highest robustness if αk = 1, αj 6=k = 0
∗ Regular allocation is better than ‘any’ random allocation
∗ Theorem 1 is valid for arbitrary intra-degree dist of Net A and B
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CMU SV, April 16th, 2013 38
Bi-directional vs. uni-directional inter-edges
∗ Consider an arbitrary probability distribution α = (α0, α1, . . .).
∗ Uni-directional strategy: Assign αj-fraction of nodes j inward
inter-edges; the supporting node is picked arbitrarily. We compare
pc,uni(α), SA∞,uni(α), SB∞,uni(α) vs. pc(α), SA∞(α), SB∞
(α)
Theorem 2 For any p, we have that
SA∞(α) ≥ SA∞,uni(α),
SB∞(α) ≥ SB∞,uni(α),
(6)
and that
1− pc(α) ≥ 1− pc,uni(α). (7)
∗ Bi-directional is better than uni-directional for any ~α
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CMU SV, April 16th, 2013 39
Lessons learned
∗ Assume that intra-topologies of the networks are not known. For
a given average number of inter-edges per node (the number of
nodes it supports plus the number of nodes it depends upon),
i) it is better (in terms of robustness) to use bi-directional
inter-links rather than unidirectional links, and
ii) it is best to deterministically allot each node exactly the
same number of bi-directional inter-edges.
Broader inter-degree distribution ⇒ Lower robustness
Optimal inter-link allocation strategy:
Regular allocation of bi-directional links
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CMU SV, April 16th, 2013 40
Intuition
∗ Without knowing which nodes play a key role in preserving the
connectivity, it is best to treat all nodes “identically.”
∗ Regular allocation of bi-directional links ensures that each node
supports (and is supported by) the same number of nodes.
⇒ Uniform support-dependence relationship
∗ Random allocation strategy disrupts this uniformity and leads
to a reduction in the system robustness.
∗ Uni-directional links is even worse because of the domino-effect.
BUT, for single networks against random attacks
Broader degree distribution ⇒ Higher robustness
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CMU SV, April 16th, 2013 41
Summarizing . . .
• We proposed a new interdependent network model, where
nodes are allowed to have multiple inter-links.
• We analyzed the robustness of this new model against
cascading failures via the critical attack size and the
functional network sizes at steady-state.
• We characterized the trade-off between the number of
inter-links allocated and the robustness achieved.
• We showed that the optimal inter-link allocation strategy is to
give all nodes exactly the same number of bi-directional
inter-links (when intra-topologies are unknown).
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CMU SV, April 16th, 2013 42
Some ideas for future work
• Optimal inter-link allocation with topology information
⋄ Assign more inter-edges to high intra-degree nodes?
⋄ Assign more inter-edges to nodes with high betweenness?
• More realistic rules for node failures
⋄ Based on fraction of failed neighbors rather than giant comp
• Multiple sources of failures
⋄ Net A is more vulnerable to one type of failures, while Net
B is more vulnerable to another type.
• Correlations between inter- and intra-edges due to nodes’
spatial locations.
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CMU SV, April 16th, 2013 43
Thanks!
Visit www.andrew.cmu.edu/~oyagan for references..
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CMU SV, April 16th, 2013 44
2 3 4 5 6 7 8 9 100.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
k
p c
a,b=3−−System 3a,b=3−−System 1a,b=6−−System 3a,b=6−−System 1
3 4 5 6 7 8 9 100.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
a=b
p c
k=2−−System 3k=3−−System 3k=4−−System 3k=2−−System 1k=3−−System 1k=4−−System 1
2 3 4 5 6 7 8 9 100.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
k
p c
a,b=3−−System 2a,b=3−−System 1a,b=5−−System 2a,b=5−−System 1
3 4 5 6 7 8 9 100.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
a=b
p c
k=2−−System 3k=2−−System 2k=2−−System 1k=5−−System 3k=5−−System 2k=5−−System 1
Figure 5: Sys 1(regular), Sys 2 (poisson, bi-direc.), Sys 3 (poisson, uni-)
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CMU SV, April 16th, 2013 45
⋆ From J. Peerenboom
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CMU SV, April 16th, 2013 46
An illustration of cascading failures
Initial set-up
3v
1v
2v
4v
5v
6v
3'v
1'v
2'v
4'v
5'v
6'v
3'v
1'v
2'v
4'v
5'v
6'v
3v
4v
5v
6v
Stage 1 Stage 3Stage 2 Steady state
1'v
4v
5v
6v
4'v
5'v
6'v
4v
5v
6v
5v
4'v
5'v
4v 4'v
5'v
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CMU SV, April 16th, 2013 47
Influence Propagation in Multiplex Networks
• We proposed a new social contagion model that allows
⋄ capturing the effect of content on the influence
propagation process
⋄ distinguishing between different link types in the social
network
• Under this new model, we obtained the condition,
probability and expected size of global spreading events.
• We showed how different content may have completely
different spreading characteristics over the same network.
• We showed that link classification and content-dependence
of links’ roles are essential for an accurate marketing analysis.
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CMU SV, April 16th, 2013 48
Information Propagation in CoupledSocial-Physical Networks
• Considered a coupled social-physical network, where a number
of online social networks overlay a physical information
network (that represents face-to-face interactions).
• Obtained critical conditions under which an information goes
viral, i.e., reaches out to a significant fraction of the network.
• Computed the probability of an information going viral along
with the resulting fraction of individuals that are informed.
• First analytical work that shows how the coupling among social
networks can lower the critical threshold, and extend the scale
of information propagation.