Optimal Allocation of Interconnecting Links in Cyber ...oyagan/Talks/Interdependent.pdf · −...

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

[email protected]

Collaborators:

Douglas Cochran, Virgil Gligor, Armand Makowski,

Dajun Qian, Junshan Zhang, Jun Zhao


Research Overview

A. Wireless (Sensor) Networks

− Connectivity, security, performance evaluation, anddesign

B. Network Science

−Dynamical processes on coupled complex networks


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


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?


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


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


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.


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.


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


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?


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.


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


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


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.


Real-world examples

Goal: Mitigate catastrophic impacts

Plan of action: Model and quantify cascading failures &

Develop design strategies that improve robustness


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.


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.


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


Robustness of theBuldyrev et al. model

Network B

1

3

2

N

3

2

1

Network A

N


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


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


Fractio

noffu

nctio

nalnodes,SA

∞


Single ER network

≃ 0.66

Interdep.networks

∗ Interdependent networks are much more vulnerable to

attacks!


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

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0.8

0.9

1


Fraction

offu

nctionalnodes,SA

∞


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.


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


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)


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.


∗ 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}


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)


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


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


Fractio

noffu

nctio

nalnodes,SA

∞


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


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

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0.5

0.6

0.7

0.8

0.9

1


Fractio

noffu

nctio

nalnodes,SA

∞


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 %


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

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0.9

1


Fractio

noffu

nctio

nalnodes,SA

∞


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.


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

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0.8

0.9

1


Fractio

noffu

nctio

nalnodes,SA

∞


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.


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

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0.5

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0.8

0.9

1


Fraction

offu

nctionalnodes,SA

∞


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


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.


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)


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)


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


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


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


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


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).


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.


Thanks!

Visit www.andrew.cmu.edu/~oyagan for references..


2 3 4 5 6 7 8 9 100.1

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k

p c

a,b=3−−System 3a,b=3−−System 1a,b=6−−System 3a,b=6−−System 1

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0.25

0.3

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0.55

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k

p c

a,b=3−−System 2a,b=3−−System 1a,b=5−−System 2a,b=5−−System 1

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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-)


⋆ From J. Peerenboom


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


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.


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.

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