Just a Few Seeds More: Value of Network Information for...
Transcript of Just a Few Seeds More: Value of Network Information for...
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Just a Few Seeds More:Value of Network Information for Diffusion
Virtual Market Design Seminar
Mohammad Akbarpour, Stanford
Suraj Malladi, Stanford
Amin Saberi, Stanford
June 2020
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Optimal Seeding
• Fix: • Social network with n people• Diffusion model
• For example: Once informed, you inform your friends with some probability.
• Budget for informing S “seeds” initially
• Who are the optimal S individuals to seeds? (from 𝑛𝑛𝑆𝑆 options)
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Sometimes Easy, Often (Very) Hard
• With 3 seeds:
Thanks to Ozan Candogan for this network visualization of their paper
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Optimal Seeding
• Fix: • Social network with n people• Diffusion model
• For example, SIR: Once informed, you inform your friends with probability c.• Budget for informing S “seeds” initially
• Who are the optimal S individuals to seeds? (from 𝑛𝑛𝑆𝑆 options)
• The problem is NP-Complete (Kempe-Kleinberg-Tardos (2003))
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Network Seeding: Many Applications
• Diffusion of microfinance [Banerjee et al, 2013, …]• Spread of new technologies [Beaman et al, 2018, …]• HIV prevention information [Wilder et al, 2017, …]• Word-of-mouth marketing [Domingos, 2005, …]• Spread of political ideas [Lazarsfield et al, 1944, …]• …
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Heuristic Solutions for Optimal Seeding
• Eigenvector centrality (Cai et al 2015)• Diffusion centrality (Banerjee et al 2013)• K-shell index (Kitsak et al 2010)• Discounted Degree (Chen et al 2009)• Diffusion-Based Detection (Wang et al 2010, Jackson-Storms 2017)• Others (Narayanam et al 2010), (Leskovec et al 2007), (Jiang et al 2011),
(Zhou et al 2014)...
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Heuristic Solutions for Optimal Seeding
Kempe-Kleinberg-Tardos (2003)
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What Have We Learned?
[Network targeting] has important implications for policy makers to pick the right people to inform in order to ensure that a new idea or product or piece of information reaches the maximum number of people.
Banerjee-Chandrasekhar-Duflo-Jackson, Diffusion of Microfinance, (2013)
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What Have We Learned?
Targeting individuals who are more central in the village network for this intervention can make a significant difference in the size of the multipliers achieved.
Cai-Janvry-Sadoulet, Social Networks and the Decision to Insure (2015)
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What Have We Learned?
Theory-driven targeting using detailed social network data can increase technology adoption relative to the status quo approach to agricultural extension services.
Beaman-BenYishay-Magruder-Mobarak, Can Network Theory-based Targeting Increase Technology Adoption? (2020)
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Computationally Feasible, But Require…
• Eigenvector centrality (Cai et al 2015)• Diffusion centrality (Banarjee et al 2013)• K-shell index (Kitsak et al 2010)• Discounted Degree (Chen et al 2009)• Diffusion-Based Detection (Wang et al 2010, Jackson-Storms 2017)• Others (Narayanam et al 2010), (Leskovec et al 2007), (Jiang et al 2011), (Zhou et
al 2014)...
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Computationally Feasible, But Require…
• Eigenvector centrality (Cai et al 2015)• Diffusion centrality (Banarjee et al 2013)• K-shell index (Kitsak et al 2010)• Discounted Degree (Chen et al 2009)• Diffusion-Based Detection (Wang et al 2010, Jackson-Storms 2017)• Others (Narayanam et al 2010), (Leskovec et al 2007), (Jiang et al 2011), (Zhou et
al 2014)...
Banerjee-Chandrasekhar-Duflo-Jackson (ReStud, 2020)
Breza-Chandrasekhar-McCormick-Pan (Forthcoming, AER)
Network Data
Expensive ImperfectWhich network we’re talking about?
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Quantifying the Value of Network Data: A Simulation
Diffusion of Microfinance [Banerjee-Chandrasekhan-Duflo-Jackson, 2013]:
• Network data• Model of diffusion:
• Once informed: participate or not participate with prob. p• Participating agents communicate with prob. 𝒄𝒄𝒑𝒑• Non-participating agents communicate with prob. 𝒄𝒄𝒏𝒏• The whole diffusion process stops after T periods
• Structural estimates of parameters
Compare: The heuristic suggested (“diffusion-centrality”) with random seeding.
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Quantifying the Value of Network Data: A Simulation
Diffusion of Microfinance [Banerjee-Chandrasekhan-Duflo-Jackson, Science, 2013]:
• Network data• Model of diffusion:
• Once informed: participate or not participate with prob. 0.24• Participating agents communicate with prob. 0.55• Non-participating agents communicate with prob. 0.05• The whole diffusion process stops after 7 periods
• Structural estimates of parameters
The heuristic suggested based on experiment is “diffusion-centrality”
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Network Targeting vs. “Naïve” Expanded Outreach
0
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1 2 3 4 5 6
Expe
cted
Diff
usio
n
Number of Seeds
Microfinance Diffusion in “All Inclusive Villages” Networks
Diffusion-Central Random
Network targeting “significantly” helps diffusion
Random with 3 extra seedswins!
asd
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DistributionsPe
rcen
tage
of o
bser
vatio
ns
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8Percentage of informed nodes
0.0
0.2
0.4
0.6
0
.81.
0
RAND with 1 seed
OPT with 1 seed
OMN with 1 seed
RAND with 5 seeds
RAND with 3 seeds
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This Paper: Value of Network Data and Analysis
For what values of 𝒙𝒙 does random seeding beat optimal?
Exploit the network:Inform 𝒔𝒔 agents
optimally
Ignore the network:Inform 𝒔𝒔 + 𝒙𝒙 agents
randomly
Quantifies the value of network data + computational power using a policy-relevant measure.
VS.
Result: In viral diffusion, for a wide class of diffusion models, random with “a few” extra seeds outperforms the optimum. (and yes, even for power-law networks)
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Outline1. Model
2. Main Theorem & Proof ideas
3. Power-Law and Real-world Networks
4. Limitations: Towards Guiding Empirical Research1. Diffusion model2. Speed of diffusion3. Diffusion minimization
5. Concluding Remarks
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Model
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Model: Basics
• 𝑁𝑁 = {1, … ,𝑛𝑛} is the set of agents (or nodes)
• 𝐺𝐺 = (𝑁𝑁,𝐸𝐸) is the social network• 𝐸𝐸 ⊆ 𝑁𝑁2
• 𝑖𝑖𝑖𝑖 ∈ 𝐸𝐸 if agents 𝑖𝑖 and 𝑖𝑖 are linked (or friends, neighbors)• Consider undirected networks: if 𝑖𝑖𝑖𝑖 ∈ 𝐸𝐸 ↔ 𝑖𝑖𝑖𝑖 ∈ 𝐸𝐸 (study directed in paper)
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The Intellectual Exercise
Random Heuristic with imperfect network data
Heuristic with perfect network data
Optimum with perfect network data
Omniscient
Practically relevant gap
Theoretical bound we prove
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Distributions
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Network Model: Inhomogeneous Random Networks (IRN)• Each node 𝑖𝑖 has some type 𝜃𝜃𝑖𝑖 ∈ 𝜏𝜏 = {1, 2, … , 𝑟𝑟}• Kernel function is a symmetric function 𝑘𝑘: 𝜏𝜏2 → [0,𝑛𝑛]• Each type 𝜃𝜃𝑖𝑖 and 𝜃𝜃𝑗𝑗 are linked with probability 0 ≤ 𝑘𝑘 𝜃𝜃𝑖𝑖 ,𝜃𝜃𝑗𝑗 /𝑛𝑛 ≤ 1
• Let 𝑘𝑘𝑖𝑖𝑗𝑗 be the expected number of type 𝜃𝜃𝑗𝑗 friends of a 𝜃𝜃𝑖𝑖 node
• Let 𝐓𝐓𝑘𝑘 = 𝑘𝑘𝑖𝑖𝑗𝑗 𝑖𝑖,𝑗𝑗∈𝜏𝜏and consider its largest eigenvalue:
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Main Theorem
Theorem. Let 𝑠𝑠 = 𝑜𝑜( 𝑛𝑛log(𝑛𝑛)
). If ||𝐓𝐓𝑘𝑘|| > 1𝑐𝑐
, then 𝛼𝛼 > 0 and for any 𝑥𝑥:
lim𝑛𝑛→∞
𝐇𝐇(RAND, 𝑠𝑠 + 𝑥𝑥)𝐇𝐇(OMN, 𝑠𝑠) = 1 − 1 − 𝛼𝛼 𝑠𝑠+𝑥𝑥
If ||𝐓𝐓𝑘𝑘|| < 1𝑐𝑐
, then lim𝑛𝑛→∞
𝐇𝐇 OMN, 𝑠𝑠 = 0
Define: 𝐇𝐇(𝑓𝑓, 𝑠𝑠) ≝ 𝔼𝔼G[𝐡𝐡(𝐺𝐺, 𝑠𝑠,𝑓𝑓)]• Expected fraction of infected nodes given seeding strategy with s seeds, drawing an IRN
Let 𝛼𝛼 = lim𝑛𝑛→∞
𝐇𝐇(OMN, 1)
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A Corollary: Single-type IRN
Erdős–Rényi random graph is a special case of IRN with 𝒌𝒌 𝜽𝜽𝒊𝒊,𝜽𝜽𝒋𝒋 = 𝒅𝒅 for all types.
• 𝒅𝒅 is the expected number of friends (degree) of a node
Definition. In an Erdős–Rényi random graph on 𝒏𝒏 nodes and parameter 𝒅𝒅, link exists between any two nodes independently with probability 𝒅𝒅/𝒏𝒏
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Value of Network Data: Erdős–Rényi (𝒏𝒏,𝒅𝒅/𝒏𝒏)
Theorem 1. Let 𝑠𝑠 = 𝑜𝑜( 𝑛𝑛log(𝑛𝑛)
). If 𝑑𝑑𝑐𝑐 > 1, then for any 𝑥𝑥:
lim𝑛𝑛→∞
𝐇𝐇(RAND, 𝑠𝑠 + 𝑥𝑥)𝐇𝐇(OMN, 𝑠𝑠)
= 1 − 1 − 𝛼𝛼 𝑠𝑠+𝑥𝑥
If 𝑑𝑑𝑐𝑐 ≤ 1 , then:lim𝑛𝑛→∞
𝐇𝐇 OMN, 𝑠𝑠 = 0
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Proof Ideas(it’s not about friendship, it’s about communication)
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Proof Ideas: Communication Network
If any agent in a connected component is informed, all others are.• OMNICIENT seeding strategy picks top s component sizes. • RAND picks them with prob. proportional to their size
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Communication Network Components’ Sizes
Regime 1: if ||𝐓𝐓𝑘𝑘|| > 1𝑐𝑐
Regime 2: if ||𝐓𝐓𝑘𝑘|| < 1𝑐𝑐
𝛼𝛼 ⋅ 𝑛𝑛𝑂𝑂(log 𝑛𝑛 )
𝑂𝑂(log 𝑛𝑛 )
Chance random hits the giant component: 1 − 1 − 𝛼𝛼 𝑠𝑠+𝑥𝑥 Even omniscient cannot do much
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Some Lessons for Corona (March 4th, 2020)
• In ER, if 𝑐𝑐𝑑𝑑 = 2, then giant component is 79% of the network!
• ||𝐓𝐓𝑘𝑘|| > 1𝑐𝑐
is the condition for the pandemic to go viral.• LHS is only a function of social network structure• RHS is only a function of the virus, hand-washing, etc
• If the condition holds, just a few (random) seeds are enough!
• Power-law networks makes it even easier for the condition to be satisfied!
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(Un)Directed Communication
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Directed Communication
OMN will pick these in addition to those in the giant component.
We prove: These paths are of length 𝑶𝑶(𝒍𝒍𝒍𝒍𝒍𝒍(𝒏𝒏))
(To the best of our knowledge, previous known bound is 𝑛𝑛, which
is too generous for our purposes)
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Power-law Networks(What about @TaylorSwift or @LeoMessi?)
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Chung-Lu (2002) Networks
• Node 𝑖𝑖 expected degree is 𝑤𝑤𝑖𝑖
• A special case of IRN !
Definition. Fix a sequence 𝑤𝑤 = 𝑤𝑤1, … ,𝑤𝑤𝑛𝑛 ∈ R𝑛𝑛. A Chung-Lu (undirected) network on 𝑛𝑛 nodes, 𝐶𝐶𝐶𝐶(𝑛𝑛,𝑤𝑤), is generated by including each edge {𝑖𝑖, 𝑖𝑖}independently with probability 𝑝𝑝𝑖𝑖𝑗𝑗 = min{𝑤𝑤𝑖𝑖𝑤𝑤𝑗𝑗
∑𝑘𝑘 𝑤𝑤𝑘𝑘, 1}.
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Random vs. OMN: Power-law Networks
• Let 𝐹𝐹(𝑥𝑥) = 1 – (𝑑𝑑/𝑥𝑥)𝑏𝑏 be weight distribution in a Chung-Lu graph, for 𝑏𝑏 > 1.
Theorem. Let 𝑠𝑠 = 𝑜𝑜( 𝑛𝑛log(𝑛𝑛)
). If either 𝑏𝑏 ∈ (1, 2], or if 𝑏𝑏 > 2 and 𝑑𝑑𝑐𝑐 > (𝑏𝑏 −1)(𝑏𝑏 − 2), then for any 𝑥𝑥 :
lim𝑛𝑛→∞
𝐇𝐇(RAND, 𝑠𝑠 + 𝑥𝑥)𝐇𝐇(OMN, 𝑠𝑠)
= 1 − 1 − 𝛼𝛼 𝑠𝑠+𝑥𝑥
If 𝑏𝑏 > 2 and 𝑑𝑑𝑐𝑐 < (𝑏𝑏 − 1)(𝑏𝑏 − 2), then:
lim𝑛𝑛→∞
𝐇𝐇 OMN, 𝑠𝑠 = 0
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Communication Network Components’ Sizes
𝑏𝑏 ∈ (1, 2], or if 𝑏𝑏 > 2 and 𝑑𝑑𝑐𝑐 > (𝑏𝑏 − 1)(𝑏𝑏 − 2) If 𝑏𝑏 > 2 and 𝑑𝑑𝑐𝑐 < (𝑏𝑏 − 1)(𝑏𝑏 − 2)
𝛼𝛼 ⋅ 𝑛𝑛𝑂𝑂(log 𝑛𝑛 )
𝑂𝑂(log 𝑛𝑛 )
Random with superconstant extra seeds hits the giant component Even omniscient cannot do much
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With Equal Seeds, Random Performs PoorlyCommunication probability = 0.51 central node, 𝑛𝑛 = 1000 leaves, 1 seed
Optimal seedingDiffusion ≃ 500
Random seedingDiffusion ≃ 250
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With Additional Seeds, Random Catches upCommunication probability = 0.51 central node, 𝑛𝑛 = 1000 leaves, 1 seed
Optimal seedingDiffusion ≃ 500
Random seedingDiffusion ≃ 250
Diffusion ≃ n2
(1 − 12
x) → 𝑛𝑛
2as 𝑥𝑥 grows very quickly!
Random seeding with 𝑥𝑥 extra seeds
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Real-world Networks: Facebook Subnetwork
Facebook subnetwork data from http://snap.stanford.edu/data/
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LimitationsDiffusion Model
Speed of diffusionDiffusion minimization
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Beyond Simple SIR Diffusion
• Main insight holds for some more complex models:
Directed communication (theoretical results) Microfinance [Banerjee et al, 2013]: Participants vs. Non-participantsWeather insurance [Cai et al, 2015]: Linear probability model Diffusion games [Sadler, 2018]: Agents “decide” to adopt or not
• But not for all:
Threshold model of diffusion Limited capacity to listen …
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Alternative Diffusion Model: Weather Insurance
• Cai-Janvry-Sadoulet (2015)• Diffusion of information in weather insurance programs in Chinese villages
• Your chance of adoption increases linearly by the # of your friends
• We repeat a similar exercise on their model and network data.
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Simulations: Weather insurance
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Limitation: “Threshold” Model
• Suppose agents adopt only if a certain fraction of their friends do
• Then it is important to pick agents in “clusters”, so random performs poorly
• Jackson & Storms (2018) formalized this
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Limitation: Limited Capacity Model
• Suppose agents (regardless of how many links they have) don’t listen to more than a certain number of them.
• @Taylor_Swift and @Leo_Messi do not listen to all 100M links they have!
• Then results may fail.
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Speed of Diffusion
• Clearly, results will not go through if you care about first period diffusion
• But that’s not really “viral” diffusion…
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• A k-level random network and a bounded diffusion process that ends in T≥1
• Similar result will not hold true for power law graphs• Remember the star example and T=1
Speed of Diffusion
Theorem. Let s be a non-negative integer.𝐇𝐇(RAND, 𝑠𝑠 + 𝑙𝑙𝑜𝑜𝑙𝑙(𝑛𝑛))/𝐇𝐇(OMN, 𝑠𝑠) → 1 as 𝑛𝑛 → ∞.
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Speed of Diffusion in Microfinance Data
0
2
4
6
8
10
12
14
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
EXTR
ASE
EDS
NEE
DED
NUMBER OF SEEDS
EXTRA SEEDS NEEDED BY RANDOM TO BEAT DIFFUSION-CENTRAL IN SPEED
T=1 PERIOD T=2 PERIODS T=3 PERIODS T=4 PERIODS
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“Vaccination”: Network Can Matter a lot
Consider the “minimization” problem: Some agent initially infected, and goal is to curb spread of infection through vaccinations
Optimal vaccinationNo vaccination Random vaccinationof several agents
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Outline1. Model
• Diffusion model, Network model
2. Value of network information• Random vs. Optimum
3. Proof ideas
4. Generalizations1. Network model: Power-law, Clustering, Real-world data2. Objective: Speed of diffusion3. Alternative diffusion models & Limitations
5. Concluding remarks
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Networks Can Matter
“Our results identify conditions under which network optimization is not very important.... If an analyst believes (or finds out) that employing a complicated algorithm that accounts for the network structure will yield large gains, then their environment must depart materially from the setting studied here.”
A’-Li-Oveisgharan, JPE (2020)
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Some Questions to Ask Before Seeding
• What is the underlying diffusion process?
• Do you hope the diffusion is going to become viral?
• Do you particularly care about what happens in the first 1-2 periods?
• Do you want to maximize or minimize diffusion?
• And more!
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Statistical vs. Economic Significance
In addition to statistical significance, also report:
is a useful and easily interpretable information about the economic significance of the results.
Extra seeds required by the a network-agnostic seeding strategy to get to the 1 − 𝛿𝛿 % of the network-guided heuristic
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A Statistic to Report
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QUESTIONS?
Illustration by Harriet Lee-Merrion
Thank you!
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A “Micro-view” at Diffusion Functions
Number of neighbors informed
Prob
. of g
ettin
g in
form
ed