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An Empirical Study of Optimization for Maximizing Diffusion in Networks Kiyan Ahmadizadeh Bistra Dilkina, Carla P. Gomes, Ashish Sabharwal Cornell University Institute for Computational Sustainability Diffusion in Networks: Cascades Our

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An Empirical Study of Optimization for Maximizing Diffusion in Networks

Kiyan Ahmadizadeh Bistra Dilkina, Carla P. Gomes, Ashish Sabharwal

Cornell University Institute for Computational Sustainability

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CP2010 Ahmadizadeh, Dilkina, Gomes, Sabharwal 2

Diffusion in Networks: Cascades

 Our diffusion model: cascades  A network: G=(V,E)  Initial set of active nodes S ⊆ V  Diffusion process as local stochastic activation rules of

spread from active nodes to their neighbors

Independent cascade: probability of spread across each edge: pvw ∀ (v,w)∈ E (independent of cascade history)

1 2 5 3 4 6 7

p14 p13 p12

1 2 5 3 4 6 7 1 2 5 3 4 6 7

p12

1 2 5 3 4 6 7

p14 p13

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CP2010 Ahmadizadeh, Dilkina, Gomes, Sabharwal 3

Influencing Cascades

 Assume cascades can only spread to nodes acquired

by some action.

1 2 5 3 4 6 7 1 2 5 3 4 6 7 1 2 5 3 4 6 7 1 2 5 3 4 6 7 1 2 5 3 4 6 7 1 2 5 3 4 6 7 1 2 5 3 4 6 7 1 2 5 3 4 6 7

 

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CP2010 Ahmadizadeh, Dilkina, Gomes, Sabharwal 4

Maximizing Node Activity in Cascades

 A set of actions A = {a1..aL}, a ⊆ A  ai : cost c(ai ), buys nodes Vi ⊆ V. Total budget B.  Time horizon H (discrete).

Typically many years.

 : random variable indicating whether node v

becomes activated in cascade under action set a

at time t

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CP2010 Ahmadizadeh, Dilkina, Gomes, Sabharwal 5

Influencing Cascades: Motivating Examples

 Human Networks: Technology adoption among friends/

peers.

 Social Networks:

 Spread of rumor/news/articles on Facebook, Twitter, or

among blogs/websites.

Targeted-actions (e.g. marketing campaigns) can be chosen to optimize the spread of these phenomena.

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CP2010 Ahmadizadeh, Dilkina, Gomes, Sabharwal 6

Influencing Cascades: Motivating Examples

 Epidemiology: Spread of disease is a

cascade.

 In human networks, or between networks of

households, schools, major cities, etc.

 In agriculture settings.

 Contamination: The spread of toxins /

pollutants within water networks. Mitigation strategies can be chosen to minimize the spread of such phenomena.

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CP2010 Ahmadizadeh, Dilkina, Gomes, Sabharwal 7

Our Application: Species Conservation

 Intuition: Buy land as future

species habitat.

 Nodes: Land patches

suitable as habitat (if conserved).

 Actions: Purchasing a real-

estate parcel (containing a set of land patches).

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CP2010 Ahmadizadeh, Dilkina, Gomes, Sabharwal 8

Our Application: Species Conservation

 Given existing populations in some patches, a limited

budget, and cascade model of species dispersion:

Which real-estate parcels should be purchased as conservation reserves to maximize the expected number

  • f populated patches at the time horizon?

 Target species: the Red-Cockaded Woodpecker

Federally listed rare and endangered species

[USA Fish and Wildlife Service, 2003].

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CP2010 Ahmadizadeh, Dilkina, Gomes, Sabharwal 9

RCW Cascade Model

 Recall spread probabilities: pvw ∀ (v,w)∈ E  Spread probability between pairs of land patches:

Distance.

Suitability score.

 Land patches remain active between time-steps

based on a survival probability.

 Cascade model based on meta-population model

[Walters et al., 2002]

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CP2010 Ahmadizadeh, Dilkina, Gomes, Sabharwal 10

Past Work

 [Kempe et al., 2003] – Initiating cascades.

Limited to choosing start nodes for cascade.

Problem is sub-modular (greedy methods apply).

Sub-modularity does not hold in more general settings.

 [Sheldon et al., 2010] – Single-stage node acquisition

for cascades.

Unrealistic in many planning situations.

 Large planning horizons => multiple rounds of

purchases.

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CP2010 Ahmadizadeh, Dilkina, Gomes, Sabharwal 11

Talk Goals

Study and compare three problem variants

(A) Single-stage up-front budget.

(B) Single-stage split budget.

(C) Two-stage split budget.

Explore the computational difficulty of this problem.

Explore the tradeoffs in solution quality (expected number

  • f active nodes) obtained from these three models.

Informs planners and planning policy makers.

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CP2010 Ahmadizadeh, Dilkina, Gomes, Sabharwal 12

Single-stage Decision Making

Commit to all purchase decisions at t=0.

Decisions not informed by cascade progress (closed loop).

(A) Single-stage Up-front Budget:

Commit to purchases at t=0.

Make purchases at t=0.

Already computationally difficult.

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CP2010 Ahmadizadeh, Dilkina, Gomes, Sabharwal 13

RCW Single-Stage Decision Making

1. Initial conditions. Legend: : active land patch : purchased parcel

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CP2010 Ahmadizadeh, Dilkina, Gomes, Sabharwal 14

RCW Single-Stage Decision Making

  • 2. Purchases made at t=0.

Legend: : active land patch : purchased parcel

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Legend: : active land patch : purchased parcel

CP2010 Ahmadizadeh, Dilkina, Gomes, Sabharwal 15

RCW Single-Stage Decision Making

2. Cascade spreads through purchased patches (t=20).

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CP2010 Ahmadizadeh, Dilkina, Gomes, Sabharwal 16

Single-stage Decision Making

 (B) Single-stage Split Budget:

Purchases in two time-steps with budget split.

Commit to purchase decisions in first time-step

No adjustment for observations on cascade progression.

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CP2010 Ahmadizadeh, Dilkina, Gomes, Sabharwal 17

RCW Single-Stage Decision Making

1. Initial conditions. Legend: : active land patch : purchased parcel : committed decisions

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CP2010 Ahmadizadeh, Dilkina, Gomes, Sabharwal 18

RCW Single-Stage Decision Making

  • 2. Commit to purchases at t=0.

Legend: : active land patch : purchased parcel : committed decisions

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CP2010 Ahmadizadeh, Dilkina, Gomes, Sabharwal 19

RCW Single-Stage Decision Making

  • 3. Cascade spreads through purchased

patches (t=10). Legend: : active land patch : purchased parcel : committed decisions

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CP2010 Ahmadizadeh, Dilkina, Gomes, Sabharwal 20

RCW Single-Stage Decision Making

  • 4. Purchase parcels committed to (t=10).

Legend: : active land patch : purchased parcel : committed decisions

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CP2010 Ahmadizadeh, Dilkina, Gomes, Sabharwal 21

RCW Single-Stage Decision Making

  • 5. Cascade spreads through

purchased patches (t=20). Legend: : active land patch : purchased parcel : committed decisions

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CP2010 Ahmadizadeh, Dilkina, Gomes, Sabharwal 22

Split-Budget in Applications

 Why not adjust based on observations?

Call for proposals, grants, government funding, etc. often require strict, projected budgets.

 Requires making purchase decisions in a single-stage

at t=0.

Little variation in stochastic behavior of cascade.

 First step toward true two-stage model.

Significantly more difficult than single-stage upfront budget.

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CP2010 Ahmadizadeh, Dilkina, Gomes, Sabharwal 23

Two-stage Decision Making

Purchase decisions are made in two time-steps (stages).

(C) Second stage decisions can be informed by the outcome of the first stage (open loop).

 Complete solution specifies

first-stage decisions

second-stage decisions for every possible scenario from the first stage => a “policy tree”

 Goal: Compute first-stage decisions that maximize

expected outcome of second-stage.

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CP2010 Ahmadizadeh, Dilkina, Gomes, Sabharwal 24

RCW Two-Stage Decision Making

1. Initial conditions. Legend: : active land patch : purchased parcel

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CP2010 Ahmadizadeh, Dilkina, Gomes, Sabharwal 25

RCW Two-Stage Decision Making

  • 2. Make purchases at t=0.

Legend: : active land patch : purchased parcel

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CP2010 Ahmadizadeh, Dilkina, Gomes, Sabharwal 26

RCW Two-Stage Decision Making

  • 3. Cascade spreads through purchased

patches (t=10). Legend: : active land patch : purchased parcel

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CP2010 Ahmadizadeh, Dilkina, Gomes, Sabharwal 27

RCW Two-Stage Decision Making

  • 4. Additional purchases made (t=10).

Legend: : active land patch : purchased parcel

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CP2010 Ahmadizadeh, Dilkina, Gomes, Sabharwal 28

RCW Two-Stage Decision Making

  • 5. Cascade spreads through

purchased patches (t=20). Legend: : active land patch : purchased parcel

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CP2010 Ahmadizadeh, Dilkina, Gomes, Sabharwal 29

Search Space Complexity

 Complexity of stochastic optimization illustrated by

scenario tree.

 Goal: Choose the actions that maximize the

expected outcome of stochastic behavior.

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CP2010 Ahmadizadeh, Dilkina, Gomes, Sabharwal 30

Search Space (Single-stage)

Single-stage problems: scenario tree with fan-out linear in scenario space. first-stage actions stochastic realizations max avg

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CP2010 Ahmadizadeh, Dilkina, Gomes, Sabharwal 31

Search Space (Two-stage)

Two-stage problem: scenario tree with quadratic fan-out in scenario space. Largely intractable.

first-stage actions first-stage stochastic realizations second-stage actions second-stage stochastic realizations

max avg max avg

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Solution Methods

CP2010 Ahmadizadeh, Dilkina, Gomes, Sabharwal 32

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CP2010 Ahmadizadeh, Dilkina, Gomes, Sabharwal 33

Stochastic MIP Formulation

Maximizes expected active land patches at time horizon.

Applies to single-stage problems (A) upfront budget and (B) split budget

Deterministic analogue (finite scenario set) => building block for solution procedures.

scenario : cascade realization

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CP2010 Ahmadizadeh, Dilkina, Gomes, Sabharwal 34

Sample Average Approximation

  • Stochastic optimization by solving series of deterministic

analogues [Shapiro, 2003]

  • Sample a set of N scenarios.
  • Optimal solution for one sampled set over-fits to that set.
  • Larger N increases MIP complexity (max 20 scenarios tractable

for RCW).

scenario space

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CP2010 Ahmadizadeh, Dilkina, Gomes, Sabharwal 35

Sample Average Approximation (Single-stage)

Sample a finite set of N cascade scenarios. sample (N scenarios)

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CP2010 Ahmadizadeh, Dilkina, Gomes, Sabharwal 36

Sample Average Approximation (Single-stage)

Maximize the empirical average over this set.

  • ptimize over

sample set (MIP) sample (N scenarios)

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CP2010 Ahmadizadeh, Dilkina, Gomes, Sabharwal 37

Sample Average Approximation (Single-stage)

Evaluate obtained solution s on small set of independent scenarios. s1, o1 evaluate

  • ptimize over

sample set (MIP) sample (N scenarios)

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CP2010 Ahmadizadeh, Dilkina, Gomes, Sabharwal 38

Sample Average Approximation (Single-stage)

Repeat process M times to obtain M candidate solutions. s1, o1 evaluate

  • ptimize over

sample set (MIP) sample (N scenarios) s2, o2 evaluate

  • ptimize over

sample set (MIP) sample (N scenarios) sM, oM evaluate

  • ptimize over

sample set (MIP) sample (N scenarios)

… … …

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CP2010 Ahmadizadeh, Dilkina, Gomes, Sabharwal 39

Sample Average Approximation (Single-stage)

Take candidate solution s* with best evaluation as solution obtained by process. evaluate

  • ptimize over

sample set (MIP) sample (N scenarios) s2, o2 evaluate

  • ptimize over

sample set (MIP) sample (N scenarios) sM, oM evaluate

  • ptimize over

sample set (MIP) sample (N scenarios)

… … …

s1, o1 s* (solution with best evaluation)

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CP2010 Ahmadizadeh, Dilkina, Gomes, Sabharwal 40

Sample Average Approximation (Single-stage)

Evaluate s* on large, independent test set of scenarios (final solution quality). test How good is s* compared to the

  • ptimal solution?

evaluate

  • ptimize over

sample set (MIP) sample (N scenarios) s2, o2 evaluate

  • ptimize over

sample set (MIP) sample (N scenarios) sM, oM evaluate

  • ptimize over

sample set (MIP) sample (N scenarios)

… … …

s1, o1 s* (solution with best evaluation)

  • * (solution quality of s*)
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CP2010 Ahmadizadeh, Dilkina, Gomes, Sabharwal 41

Stochastic optimality guarantees

Expected utility of s* gives a lower bound on true optimum => o* gives a stochastic lower bound on true optimum

E[o] gives an upper bound on the true optimum. => Sampled average of o gives stochastic upper bound

Convergence of bounds guaranteed for increasing sample size N.

number of samples (N)

  • bjective

mean MIP obj s* quality

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CP2010 Ahmadizadeh, Dilkina, Gomes, Sabharwal 42

Two-stage Re-planning with SAA

Purchase decisions made in time-steps 0 and T1 over horizon H. Budgets b1 and b2.

Re-planning approximates solution to (C) Two-Stage Split Budget

Computes set of first-stage decisions.

Nested SAA procedure used to evaluate candidate first-stage decisions.

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CP2010 Ahmadizadeh, Dilkina, Gomes, Sabharwal 43

Two-Stage Re-planning

Obtain M candidate first-stage decisions using SAA for (B) Single-stage split budget re-planning test re-planning evaluate

  • ptimize over

sample set (MIP) sample (N scenarios) s2 re-planning evaluate

  • ptimize over

sample set (MIP) sample (N scenarios) sM re-planning evaluate

  • ptimize over

sample set (MIP) sample (N scenarios)

… … …

s1 s* (solution with best evaluation)

  • * (solution quality of s*)
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CP2010 Ahmadizadeh, Dilkina, Gomes, Sabharwal 44

SAA Re-planning Evaluation

Generate F prefix scenarios, realizing first stage under s

generate prefix scenarios

s (first-stage decisions)

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CP2010 Ahmadizadeh, Dilkina, Gomes, Sabharwal 45

SAA Re-planning Evaluation

For each prefix scenario: SAA single-stage upfront budget for years T1 …. H.

Occupied patches at end of prefix scenario are initial

First-stage purchases available for free

SAA (A) T1…H SAA (A) T1…H SAA (A) T1…H

…

s (first-stage decisions)

generate prefix scenarios

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CP2010 Ahmadizadeh, Dilkina, Gomes, Sabharwal 46

SAA Re-planning Evaluation

Evaluation performance of s: average second-stage performance (qi)

SAA (A) T1…H SAA (A) T1…H SAA (A) T1…H

…

s (first-stage decisions) q1 q2 qF average

generate prefix scenarios

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Experiments

CP2010 Ahmadizadeh, Dilkina, Gomes, Sabharwal 47

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CP2010 Ahmadizadeh, Dilkina, Gomes, Sabharwal 48

Experimental Background

Red-cockaded Woodpecker (RCW) Conservation in North Carolina.

Listed by U.S.A government as rare and endangered [USA Fish and Wildlife Service, 2003].

The Conservation Fund: conserve RCW on North Carolina coast.

Nodes = land patches large enough to be RCW habitat (411 patches).

20 initial territories

H=20 time horizon.

Actions = parcels of land for purchase that contain potential territories (146 parcels).

N = 10 (finite sample set size for forming MIPs)

CPLEX used for MIP solving.

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CP2010 Ahmadizadeh, Dilkina, Gomes, Sabharwal 49

RCW Problem Generator

Instances generated from base map by random perturbation.

Perturbs territory suitability scores around base values.

Randomly choose initial territories in high suitability parcels.

Assign parcel costs with inverse correlation to parcel suitability.

Used to generate large set of maps which we use to study runtime distributions.

Generator available online (C++ Implementation). www.cs.cornell.edu/~kiyan/rcw/generator.htm

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Runtime Distributions

CP2010 Ahmadizadeh, Dilkina, Gomes, Sabharwal 50

(A) Single-stage upfront budget (B) Single-stage split budget 1. Easy-hard-easy pattern 2. Increasing difficulty with survival probability 3. Split budget 10x harder than upfront budget

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Single Instance Difficulty: Power-Law Decay

CP2010 Ahmadizadeh, Dilkina, Gomes, Sabharwal 51

Data: 100,000 MIPs formed from 10 random scenarios on map-30714.

“survivor function”: Fraction of instances unsolved in time t

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Single-stage: upfront vs. split budget

CP2010 Ahmadizadeh, Dilkina, Gomes, Sabharwal 52

  • 1. Close bounds indicate solution close to optimal.
  • 2. Upfront variant obtains higher quality solutions than split.

Upfront (A) and Split (B) UB Upfront (A) LB (best solution) Split (B) LB (best solution)

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Boosting Solution Quality with Re-planning

CP2010 Ahmadizadeh, Dilkina, Gomes, Sabharwal 53

  • 1. Re-planning with observations provides

significant improvements over committing to all decisions upfront.

  • 2. Re-planning can outperform single-stage upfront

budget.

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The Balance in Re-planning

 Re-planning sensitive to balance in budget split:

Benefits with 30-70% split budget with T1=5

Does worse with, e.g., 50-50% split with T1=10, and many

  • ther combinations

Spending too much upfront limits actions available to re- plan in second stage.

Spending too little upfront leaves little variation re-planning can take advantage of.

 Re-planning outperforms either two problem methods

under the correct planning conditions.

 Decision on budget split could be encoded in

  • ptimization problem (future work).

CP2010 Ahmadizadeh, Dilkina, Gomes, Sabharwal 54

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CP2010 Ahmadizadeh, Dilkina, Gomes, Sabharwal 55

Summary and Conclusions

 Presented cascade model for stochastic diffusion in

many interesting networks (conservation, epidemiology,

social networks).

 Extended SAA sampling methodology for stochastic

  • ptimization to a multistage setting for cascades.

 Significant complexity and variation when solving

deterministic analogues of stochastic problems.

Easy-hard-easy patterns, power-law decay

 When decisions are made in multiple stages, re-

planning based on stochastic outcomes can have significant benefit (when budget split is carefully chosen).