dynamical processes on ‘coupled’ complex networks...

Bell Labs, 16 November 2012 1

Dynamical Processes on ‘Coupled’ ComplexNetworks: Cascading Failures, Information

Epidemics, and Complex Contagions

Osman Yagan

CyLab

Carnegie Mellon University

[email protected]

Joint work with

Dajun Qian, Junshan Zhang, Douglas Cochran, and

Virgil Gligor


Network science

• An inter-disciplinary field bringing together researchers from

many areas.

⋄ 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


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?


My contributions

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. Complex contagions (influence propagation) in social networks

with multiple link types

⋄ O. Yagan and V. Gligor, Phys. Rev. E 86, 036103, Sept. 2012

3. Information propagation (simple contagions) in coupled

social-physical networks

⋄ O. Yagan, D. Qian, J. Zhang and D. Cochran, IEEE J. Sel.

Areas Commun., Issue on Network Science, to appear.


Today

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. Complex contagions (influence propagation) in social networks

with multiple link types

⋄ O. Yagan and V. Gligor, Phys. Rev. E 86, 036103, Sept. 2012


Cascading failures oninterdependent cyber-physical networks


Interdependent networks?

• A collection of networks that depend on one another to provide

proper functionality.

• Interdependence is omnipresent in many modern systems.

• Major national infrastructures: telecommunications, energy,

banking and finance, water supply systems, 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 complex 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 number 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 giant (i.e., the largest) component of the

subnetwork formed by the nodes (of its own network) that

satisfy condition 1.

• Cascade of failures propagates alternately between A and B,

eventually (i.e., in steady state) leading to either

i) residual functioning giant components in both networks, or

ii) complete failure of the entire system


Robustness metrics

• SA∞: Fractional size of the functional nodes of network A at

steady state.

• SB∞: Fractional size of the 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:

⇒ no functional giant component at Stage ∞; SA∞= SB∞

= 0


Main findings by Buldyrev et al.

• Let Networks A and B be Erdos-Renyi (ER) with mean degree

d. They established

• 1− pc ≃ 1− 2.45d

⇒ e.g., if d = 3, the system is robust against

the removal of up to 18 % of the nodes.

• For a single ER network: 1− pc = 1− 1d

⇒ with d = 3, can

sustain the failure of up to 66 % of the nodes.

⇒ Interdependent networks are much more vulnerable!

• Broader degree dist.⇒More vulnerable to random failure

• Exact opposite of the case for single networks

• Intuition: High degree nodes may inter-connect low degree ones


Rise of a new field

• The paper by Buldyrev et al. received 250 citations thus far

• Follow-up works concentrate on two main directions

⋄ Obtaining analogous results under more realistic models

∗ autonomous nodes, multiple networks, correlation

between inter- and intra-links, different network models,

multiple inter-links per node

⋄ Developing design strategies that improve robustness

∗ selection of autonomous nodes (nodes with high degree,

or betweenness), new metrics for robustness, optimum

inter-link allocation strategies

-Yagan et al., IEEE TPDS


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

∗ It suffices to have one

inter-connection with a

functioning 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 2: 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

Fraction of failed nodes (1 − p)

Fra

ction

offu

nctionalnodes,

SA

∞

critical fraction1 − pc ≃ 0 .44

k = 2

k = 1

Single ER network

≃ 0.18 ≃ 0.66

Figure 3: Net A and Net B are ER with mean degrees a = b = 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

0.4

0.5

0.6

0.7

0.8

0.9

1


Fra

ction

offu

nctionalnodes,

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

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Fra

ction

offu

nctionalnodes,

SA

∞


k = 2

k = 1

Single ER network

≃ 0.18 ≃ 0.66

∗ Phase transition is discontinuous in interdependent networks;

but, continuous in the single network case.


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


Fra

ction

offu

nctionalnodes,

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

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Fra

ction

offu

nctionalnodes,

SA

∞


k = 2

k = 1

Single ER network

≃ 0.18 ≃ 0.66k = 3

4 5

Figure 4: a = b = 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

0.4

0.5

0.6

0.7

0.8

0.9

1


Fra

ction

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


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

• Bi-directional vs Uni-directional, Regular vs Random

∗ 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:

∗ Consider a discrete probability distribution P : N → [0, 1], s.t.

P (j) = αj , j = 0, 1, . . ., 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

∗ With α = (α0, α1, . . .), we want to compare

pc(α), SA∞(α), SB∞

(α) vs. pc(k), SA∞(k), SB∞

(k)

∗ Matching condition: k =∑

∞

j=0 αjj (with integer k)


Theorem 1 Under the condition k =∞∑

j=0

αjj, for all p, we have

SA∞(k) ≥ SA∞

(α),

SB∞(k) ≥ SB∞

(α).(4)

Furthermore

1− pc(k) ≥ 1− pc(α). (5)

Proof. (Outline) Obtain recursive relations concerning the

functional network sizes at each stage. Use induction and exploit

convexity in the obtained relations via the Jensen’s inequality.

∗ Random allocation yields highest robustness when αk = 1, αj = 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


Intuitions

∗ Without knowing which nodes play a key role in preserving the

connectivity, it is best to treat all nodes “identically” and give

them equal priority in inter-edge allocation.

∗ 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.


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.


Influence propagation onmultiplex networks


A dynamical process: Binary decisions withexternalities

• Each individual must decide between two actions, e.g.,

⋄ To buy or not to buy a smart phone

⋄ To vote for Democrats or Republicans

• There is an inherent incentive for individuals to coordinate

their decisions with those of their immediate acquaintances.

Linear Threshold Model (D. Watts, PNAS, 2002)

• Nodes can be in either one of the two states: active or inactive.

• Each node is initially given a threshold τ drawn from Pth(τ).

• An inactive node with m active neighbors and k −m inactive

neighbors will turn active if mk≥ τ .


Global cascades

• Start by activating a few nodes (give incentives, free samples)

• Assumption: Once active, a node can not be deactivated.

• Global Cascades: A linear fraction of nodes (in the

asymptotic limit) eventually becomes active

∗ Watts [PNAS 99, 5766]: Condition and Probability of global

cascades when an arbitrary node is made active

∗ Gleeson & Cahalane [Pyhs. Rev. E 77, 46117]: Expected size


Our motivation

• Existing work considers simplex networks that have only a

single type of link.

• But, individuals engage in different types of relationships;

e.g., family, friends, office-mates, college-mates, etc.

• Existing work provides limited insight regarding the role of

“content” in the spreading process.

• Does a birth control pill have the same spreading

characteristics with a smart phone?

• What about joining a riot vs. deciding on vacation spot?


Networks with multiple link types:Multiplex networks

• Multiplex networks may provide better insight into the

cascade process by allowing link classification.

• May also allow capturing the effect of content

• Each link type may play a different role in different processes.

⋄ A video game would be more likely to be promoted among

high school classmates rather than family members;

exactly opposite in the case of a cleaning product.

⋄ Belief propagation: Links between distant Facebook

friends vs. links between close office-mates


A Facebook snap shot – Classifying ‘friends’

Exploiting the role of ‘link types’ can pave the way for

developing better marketing strategies


A content-dependent threshold model formultiplex networks

• Consider a multiplex network where links can be of r types.

• For a given content (e.g., rumor, product, political view),

consider positive scalars c1, . . . , cr, such that ci quantifies the

relative bias a type-i link has in spreading this content.

• Nodes switch state if their perceived proportion of active

neighbors exceeds a threshold τ . Namely, an inactive node

connected to mi active neighbors and ki −mi inactive

neighbors via type-i links will turn active if

c1m1 + c2m2 + . . .+ crmr

c1k1 + c2k2 + . . .+ crkr≥ τ


Comparison of threshold rules

• Watts, PNAS 2002 :

m1 +m2 + . . .+mr

k1 + k2 + . . .+ kr=

m

k≥ τ

• Yagan & Gligor, PRE 2012 :

c1m1 + c2m2 + . . .+ crmr

c1k1 + c2k2 + . . .+ crkr≥ τ

Content-dependent coefficients c1, . . . , cr will play a key role in the

dynamics of influence propagation


Network model

∗ Let r = 2; i.e., assume that there are only two link types.

∗ F: Random network of type-1 links with degree distribution {pfk}.

∗ W: Random network of type-2 links with distribution {pwk }.

∗ F,W : Independently generated with the configuration model

∗ H: System model F ∪W with colored distribution {pk}

pk = pfk1· pwk2

, k = (k1, k2)

∗ With c := c1/c2 an inactive node will become active if

cm1 +m2

ck1 + k2≥ τ

Q: Condition, probability, expected size of global cascades?


Condition and probability of global cascades

Simplex networks

• If the network is locally tree-like (i.e., no cycles), initially

influence can only diffuse through vulnerable nodes.

⋄ A node is deemed vulnerable if its state can be changed by

a single active neighbor; i.e., if 1k≥ τ ⇔ k ≤ 1

τ

• Global cascade condition = Existence of a giant vulnerable

component (GVC); a component comprising a positive

fraction of nodes.

• Probability of global cascades = Fractional size of the

extended giant vulnerable component

⋄ Extended GVC = Nodes that have a link with at least one

node in GVC: activating any node in EGVC will activate GVC.


Multiplex Networks with content-dependent threshold rule

• We need to define two notions of vulnerability:

⋄ A node is F-vulnerable if it becomes active by a single

active neighbor in F; i.e., if cck1+k2

≥ τ .

⋄ A node is W-vulnerable if it becomes active by a single

active neighbor in W; i.e., if 1ck1+k2

≥ τ .

− We say that a node is simply vulnerable, if it is vulnerable

w.r.t. at least one type.

• If c 6= 1, a node can be F-vulnerable but not W-vulnerable, or

vice versa. ⇒ An active vulnerable node does not necessarily

activate all of its vulnerable neighbors.

Ordinary definition of a vulnerable component becomes

vague!


A directed graph on vulnerable nodes

∗ In our formulation, subgraph of vulnerable nodes forms a

directed graph: i → j means that if active, i will activate j.

⋄ A potentially bi-directional F-link between nodes i and j will

have the direction from i to j (resp. j to i) only if j (resp. i) is

F-vulnerable (similarly for W-links).


Components of a directed network

• Out-component of a vertex is the set of vertices that are

reachable from it.

• In-component of a vertex is the set of nodes that can reach it

• Giant out-component (GOUT): Set of nodes with infinite

in-component.

• Giant in-component (GIN): Set of nodes with infinite

out-component.

• Giant strongly-connected component (GSCC): Intersection of

GIN and GOUT. ∗ Any pair of nodes

in GSCC are con-

nected via a

directed path.


A subtle picture

• Condition for global cascades = Existence of GIN (Existence

of nodes with infinite out-component)

• Probability of global cascades = Fractional size of the

extended giant in-component (EGIN)

⋄ Extended GIN = Nodes in GIN plus nodes that, once can

activated, can activate a node in GIN

• Giant vulnerable component = GSCC

⋄ A set of vulnerable nodes s.t. activating any node (in this

set) leads to the activation of all nodes in the set.

Is Existence of GIN ⇔ Existence of GSCC?


No, GIN may exist even if GSCC does not!

For n → ∞

• A positive fraction of nodes have infinite out-component. ⇒

GIN exists!

• The largest strongly connected component consists of two

nodes. ⇒ No GSCC!

Global cascades can take place evenwithout a giant vulnerable component!

⋄ Contradicts all previous models!


Analytic results

Branching Process for Exploring Out-Components

• Start by activating an arbitrary node, and then recursively

reveal the largest number of vulnerable nodes reached and

activated by exploring its neighbors.

⋄ gi(x) : Generating function for the finite number of nodes

reached by following a type-i link. (i = 1, 2)

⋄ ρk,i : Probability that a node with colored degree k is

i-vulnerable. (i = 1, 2)

⋄ G(x) : Generating function for the finite number of nodes

reached and activated.

Generating function of a rv Z ⇒ G(x) =∑

∞

ℓ=0 xℓ · P [Z = ℓ]


Recursive relations

G(x) = x∑

k=(k1,k2)pk · g1(x)

k1g2(x)k2

g1(x) = x∑

k

k1pk< k1 >

ρk,1g1(x)k1−1g2(x)

k2 +∑

k

k1pk< k1 >

(1− ρk,1) (8)

g2(x) = x∑

k

k2pk< k2 >

ρk,2g1(x)k1g2(x)

k2−1 +∑

k

k2pk< k2 >

(1− ρk,2) (9)

∗ We want to find the stable solution of (8)-(9) for x = 1.

∗ Trivial fixed point: g1(1) = g2(1) = 1 ⇒ G(1) = 1.

∗ G(1) = 1 ⇒ Conservation of probability implies that all

out-components are finite ⇒ no GIN ⇒ no global cascades

∗ But, the trivial solution g1(1) = g2(1) = 1 may not be stable.

∗ Shall check the Jacobian matrix of (8)-(9).


The Jacobian

Jp =

<(k21−k1)ρk,1>

<k1>

<k1k2ρk,1>

<k1>

<k1k2ρk,2>

<k2>

<(k22−k2)ρk,2>

<k2>

∗ Let σ(Jp) = max{|λi| : λiis an eigenvalue of Jp} : spectral radius

∗ If σ(Jp) ≤ 1, then the solution g1(1) = g2(1) is stable; i.e., GIN

does not exist whp.

∗ If σ(Jp) > 1, another solution with g1(1), g2(1) < 1 exists and

becomes the stable solution of (8)-(9) ⇒ G(1) < 1

∗ 1−G(1) = prob. of activating infinite nodes = size of EGIN

Global Cascade Condition: σ(Jp) > 1

Probability of Global Cascades, Ptrig: 1−G(1)


Expected size of global cascades

• Analysis is based on the approach by Gleeson & Cahalane

[Pyhs. Rev. E 77, 46117]: via the tools developed for analyzing

zero-temperature random field Ising model on Bethe lattices.

• Construct a tree with a single node at the top level ℓ = ∞.

⋄ qi,ℓ : Probability that a node at level ℓ, connected to its

unique parent by a type-i link, is active given that its parent is

not. (i = 1, 2)

• Recursive relations for each ℓ = 0, 1, . . ..

qi,ℓ+1 =∑

k

kipk< ki >

k1−δi1∑

l=0

k2−δi2∑

j=0

F ((l, j),k)

(

k1 − δi1l

)

ql1,ℓ

× (1− q1,ℓ)k1−l−δi1

(

k2 − δi2j

)

qj2,ℓ(1− q2,ℓ)k2−j−δi2


• Under the assumption that nodes do not become inactive once

they turn active, the quantities q1,ℓ and q2,ℓ are non-decreasing

in ℓ.

• They converge to a limit q1,∞ and q2,∞.

• The expected cascade size is given by

S =∑

k

pk

k1∑

l=0

k2∑

j=0

F ((l, j),k)

(

k1l

)

ql1,∞(1− q1,∞)k1−l

×

(

k2j

)

qj2,∞(1− q2,∞)k2−j .

where q1,∞ and q2,∞ correspond to the stable fixed point.

Expected Size of Global Cascades: S


Simulation results


0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

Average Degree (z1 = z2)

FractionalSize

Content C1 with c = 0.25

0.85 0.9 0.95 1 1.050

0.1

0.2

0.3

0.4

Pt r ig -Anlys

Pt r ig -Expt.

S -Anlys

S -Expt.

0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

Average Degree (z1 = z2)

FractionalSize

Content C2 with c = 1.0

0.55 0.6 0.65 0.70

0.05

0.1

0.15

0.2

Ptr ig – Analysis

Ptr ig – Expt.

S – Analysis

S – Expt.

Figure 5: n = 5× 105, τ is fixed at τ⋆ = 0.18, and F and W are ER with

mean degrees z1 and z2, respectively.

• Excellent agreement between analysis and simulations!

• Content parameter c affects the range, probability, and size of

cascades ⇒ Distinguish between link types!


0 1 2 3 40

0.2

0.4

0.6

0.8

1

Content Parameter (c)

FractionalSize

z1 = 1.5, z2 = 5.5

S – Expt.

S – Analysis

Ptrig – Analysis

(a)

0 1 2 3 40

0.05

0.1

0.15

0.2

0.25


FractionalSize

z1 = z2 = 0.7

S – Expt.

S – Analysis

Ptrig – Analysis

(b)

0 1 2 3 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9


FractionalSize

z1 = 6.0, z2 = 1.5

S – Expt.

S – Analysis

Ptrig – Analysis

(c)

• All parameters except c are fixed, and the variation of Ptrig

and S are observed.

• Content parameter c can dramatically change the dynamics

of complex contagions over the same network.

• Depending on z1, z2, the range of c that favors global cascades

changes significantly.


Summarizing . . .

• 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.


Ideas for future work

• Checking the validity of the content-dependent threshold model

in real-world networks.

• Estimating content parameter c in real cascade processes.

• Expanding the analysis results to networks that have

clustering, assortativity, etc.

• Revisiting the influence maximization∗ problem for

multiplex networks under the content-dependent threshold rule.

⋄ Deriving different marketing strategies for different

products.

∗ Kempe, Kleinberg, Tardos, KDD 2003.


Thanks!

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


A few quotations

∗ The notion of perceived proportion of active neighbors is

used first by Mark Granovetter in his seminal paper “Threshold

Models of Collective Behavior,” Am. J. Sociol. 83, 1978, pp. 1429.

“By ‘social structure’ I mean here only that the influence of any

given person has on one’s behavior may depend upon the

relationship. Take a simple case, where the influence of friends is

twice that of strangers. . . . What we may then call the

‘perceived proportion of rioters’ . . .”

∗ Jon Kleinberg, “Cascading Behavior in Networks,” 2007.

“It would be nice to express the notion that an individual will

adopt a behavior when, for example, two of her relatives and

three of her co-workers do so.”


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

dynamical processes on ‘coupled’ complex networks...

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