Post on 21-Sep-2020
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
oyagan@andrew.cmu.edu
Collaborators:
Douglas Cochran, Virgil Gligor, Armand Makowski,
Dajun Qian, Junshan Zhang, Jun Zhao
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
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
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?
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
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
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.
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.
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
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?
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.
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
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
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.
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
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.
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.
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
CMU SV, April 16th, 2013 19
Robustness of theBuldyrev et al. model
Network B
1
3
2
N
3
2
1
Network A
N
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
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!
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.
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
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)
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.
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}
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)
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
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
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 %
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.
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.
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
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.
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)
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)
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
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
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
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
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).
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.
CMU SV, April 16th, 2013 43
Thanks!
Visit www.andrew.cmu.edu/~oyagan for references..
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-)
CMU SV, April 16th, 2013 45
⋆ From J. Peerenboom
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
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.
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.