cascading behavior in networks: part ii · 2018-04-17 · • diffusion in networks • modeling...

Cascading Behavior in Networks: Part II

Assefaw Gebremedhin CptS 591: Elements of Network Science

Assefaw Gebremedhin, CptS 591: Elements of Network Science, http://scads.eecs.wsu.edu

Projects (Class of 2018)

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•  Team 1: Arman Ahmed and Sukhjinder Singh •  Topic: Information Diffusion on Twitter Network

•  Team 2: DeSmet Chance and Christopher Pereyda •  Topic: Sentiment Analysis on Twitter Network

•  Team 3: Valerie Cheathon and Vishal Sonawane •  Topic: Social Network Analysis and Epidemic Models

•  Team 4: Sheng Guan and Hanchao Ma •  Topic: Community Detection on Multi-attribute Time-

varying Graphs •  Team 5: Saurabh Jaiswal and Dharmil Shah

•  Topic: Active Semi-Supervised Learning Using Graph Signals

•  Team 6: Vinit Karanje, Aaron Lightner and Srinivas Vodnala •  Topic: Meso-scale Structure Analysis on Online Social

Networks

•  Team 7: Tony Liu and Zhijie Nie: •  Topic: Efficient Implementation of Label

Propagation Algorithms for Community Detection •  Team 8: Daniel Miller

•  Topic: Analysis of Expression Graphs of Power Devices

•  Team 9: David Sebastian Cardenas •  Topic: Vulnerability Analysis of Software Systems

using Epidemiological Models •  Team 10: Chandan Dhal and Nathan Wendt

•  Topic: Incremental Personalized PageRank •  Team 11: Jinglin Tao and Lusha Wang

•  Topic: Network Clustering Algorithms in Distribution Systems with Renew. Energy Sources

•  Team 12: Ali Tamimi and Amirkhosro Vosughi •  Topic: Air Traffic Network Delay and Flow

Analysis


Overview (previous lecture) •  Diffusion in Networks •  Modeling Diffusion through a Network

•  A networked coordination game •  Cascading Behavior •  Cascading Behavior and Viral Marketing

•  Cascades and Clusters •  Diffusion, Thresholds, and the Role of Weak Ties (Left over topics): •  Extensions of the Basic Cascade Model

•  Heterogeneous Thresholds

•  Knowledge, Thresholds, and Collective Action •  Collective Action and Pluralistic Ignorance •  A model for the effect of knowledge on collective action •  Common knowledge and social institutions

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Extensions of the basic cascade model

• Heterogeneous thresholds

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p ≥ qv = bv/av+bv


Heterogeneous thresholds

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Observations from the previous example

• Diversity in node thresholds plays an important role that interacts in complex ways with the structure of the network. •  E.g. Despite its central position, node 1 would not have succeeded in

converting anyone at all to A were it not for the extremely low threshold on node 3. •  This relates with the observation that, to understand spread of behaviors

in social networks, we need to take into account not just the power of influential nodes but also the extent to which these influential nodes have access to easily influenceable people.

• The notion of clusters as obstacles to cascades can be extended to the case where thresholds are heterogeneous.

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Knowledge, Thresholds, and Collective Action

Consider a related topic that integrates network effects at both the population level and the local network level. •  Collective action and pluralistic ignorance

•  E.g problem of organizing a protest • Model for the effect of knowledge on collective action

•  Effect of structure of underlying social network on individual’s decision making •  Example: next slide

•  Common knowledge and social institutions •  Widely publicized speech •  Super Bowl commercials

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Knowledge, Thresholds and Collective Action

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Threshold k : I will participate if I am sure that at least k people in total participate

1. Cascade capacity 2. Cascades and compatibility

Today’s lecture

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Cascade capacity

•  Motivation: for a given network, we want to answer the question: what is the largest threshold at which any “small” set of initial adopters can cause a complete cascade?

•  To formalize the notion “small” we work with infinite networks, where each node, however, has a finite number of neighbors. •  The cascade capacity of a network is then the largest value of the

threshold q for which some finite set S of early adopters can cause a complete cascade. •  Example: Infinite path

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q = b/(a+b)

Any q < ½ causes cascade. In fact, q = ½ is its cascade capacity.


Example 2: Infinite grid

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Cascade capacity = 3/8

Let q <= 3/8

c, h, i, n

b, d, f, g, j, k, m, o


Some observations from Examples 1 & 2

• Cascade capacity is an intrinsic property of the network. • A network with a large cascade capacity is one in which

cascades happen more easily. • In the grid example, for 3/8 < q < ½, A is the better

technology, but still the structure of the network makes B so entrenched that no finite set of initial adopters of A can cause A to win (failure of social optimality).

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How large can the cascade capacity be?

• Fact: There is no network in which the cascade capacity exceeds ½.

•  Proof idea: • Suppose such a network exists: q > ½, finite set S causes cascade • Show contradiction: argue that nodes stop switching after finite number of steps

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Assefaw Gebremedhin, CptS 591: Elements o Network Science, http://scads.eecs.wsu.edu

How large can cascade capacity be?

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Fact: There is no network with cascade capacity > ½ Proof sketch: •  we track the “interface” between adopters of A and adopters of B. •  show that the size of the interface (starts finite, I0) and strictly decreases in each step •  since each step only results in a finite number of nodes converting to A, the process terminates with only a finite number of nodes adopting A.

•  what happens in one step of the process? Some nodes switch from B to A. This causes the interface to change--some edges leave the interface others join. Consider a node w that switches. Since q>1/2, w must have had more edges to A than B è more edges left the interface than joined


Extending the basic model

Allow people to adopt A and B

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Cascades and compatibilty •  In the basic model we saw so far: •  There are two competing behaviors, A and B •  Payoff comes only from neighbors aligning their behavior: A-A pays off a B-B pays off b A-B pays off 0

• We now extend the model by adding a new strategy, AB (bilingual): •  AB-A: payoff a •  AB-B: payoff b •  AB-AB: payoff max (a,b) •  And choosing AB has associated cost c (maintenance cost)

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Cascades and compatibility: model • Every node in an infinite network starts with B •  A finite set S initially adopts A • Run the model for t = 1, 2, 3, … •  Each node selects a behavior that optimizes its payoff

• How will nodes switch from B to A or AB?

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Example: path graph

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•  Start with all B, a > b (A is better) •  One node switches to A – what happens? •  with just A, B : A spreads completely •  with A, B, AB: Does A spread completely?

•  Let a = 3, b = 2, c = 1

A does not spread very far!


Example: same network, more attractive A

• Let a = 5, b = 3, c = 1

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When does cascade happen in a model that includes the bilingual strategy AB?

• Consider the case of infinite path • Case 1 (A-w-B):

•  Payoff for w: A: a, B:1, AB: a+1-c

• What does node w do in this case?

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a+1-c = 1

a+1-c = a

a = 1

B A

A

AB B

AB


When does cascade happen in a model that includes the bilingual strategy AB?

• Case 2 (AB-w-B): •  Payoff for w: A: a, B:1+1, AB: a+1-c •  Notice now also AB spreads

• What does node w do in this case?

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Combining the two pictures

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Interpretation of cascade region

•  You manufacture the default technology B (b=1), and a new, better technology A (a=1.5) appears. For which values of the bilingual cost c should you expect B to survive? •  Infiltration (low c)

•  Maintaining both technologies is extremely easy, AB spreads wider and wider, people begin dropping B

•  Direct conquest (high c) •  Maintaining both technologies is extremely hard, people

pick the better one (A)

•  Buffer zone (moderate c) •  Maintenance is moderate, People who adopt only A and

people who adopt only B can coexist.

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cascading behavior in networks: part ii · 2018-04-17 · • diffusion in networks • modeling...

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