Controlling Spreading Processes in Networks

Transcription

1 Controlling Spreading Processes in Networks (or how to rapidly spread mutants throughout a geographic region) Lorenzo Zino DISMA Weekly Seminar (Seminari Cadenzati - Progetto Eccellenza) July, 08

2 Why to study spreading processes?...because they are ubiquitous!

3 Why networks?...powerful tool to represent interconnected systems!

4 Scientia potestas est Knowledge is power [attributed to F.T. Bacon] Model = Analysis = Control Study of epidemics on networks [Ganesh et al., Proc. INFOCOM, 005; Draief, Physica A, 00; Pastor-Satorras, Rew.Mod.Phys., 05] = control techniques [Borgs et al., Rand.Str.Alg., 00; Drakopoulos et all., IEEE TNSE, 05] Study of opinion dynamics on networks [Frachebourg and Krapivsky, PRE, 99; Montanari and Saberi, PNAS, 00] = optimal influencer placement [Kempe et al., Proc. SIGKDD, 003], control imitative agents [Riehl et al., Proc. CDC, 07]

5 New method to fight mosquito-borne diseases

6 Our goals Model the phenomenon using evolutionary dynamics [Lieberman et al., Nature, 005] with a new approach, which enables to perform analysis and include control Understand how the eradication time and the control cost are influenced by the topology of the geographic region and by the insertion policy used Design and study a control strategy to speed up the eradication of disease-spreading mosquitoes using a feedback control policy

7

8 Geographic region Weighted connected graph G = (V, E, W ): Nodes V = {,..., n} represent locations Undirected links and weights represent (heterogeneous) connections

9 State variables X i (t) {0, } state of node i at time t 0: { if i occupied by mutant at time t X i (t) = Two observables: 0 if i occupied by native species at time t Z(t) = i X i(t) number of locations occupied by mutants B(t) = i j X i(t)( X j (t))w ij boundary between the two species Z(t) = 5 B(t) = 8

10 Spreading mechanism Poisson clocks for activations of undirected links, rates from W {i, j} activates = conflict: mutants win w.p. β > / (evolutionary advantage) Different interpretation link-based voter model [Clifford and Sudbury, Biometrika, 973] = Application in opinion dynamics w.p. β 0 = 0 0 w.p. β

11 External control A target set M(t) V is selected Mutants introduced in m M(t), rate u m (t) > 0 [Harris et al., Nature Biotechnology, 0] Remark Target set and rates can be constant or time-varying Different interpretation fictitious stubborn node s with X s = linked to m M [Acemoglu et al., Mathematics of Operations Research, 03] u 9 u 5

12 Markov jump process X (t) Markov jump process with X (0) = 0. λ + i (x) i i λ i (x) [ λ + i (x) = ( x i ) β ] j W ijx j + u i (t) λ i (x) = x i ( β) j W ij( x j ) X = unique absorbing state; expected eradication time and cost: [ ] τ = E [inf t : X (t) = ] J = E T u(t)dt 0

13 Markov jump process X (t) Markov jump process with X (0) = 0. λ + i (x) i i λ i (x) [ λ + i (x) = ( x i ) β ] j W ijx j + u i (t) λ i (x) = x i ( β) j W ij( x j ) X = unique absorbing state; expected eradication time and cost: [ ] τ = E [inf t : X (t) = ] J = E T u(t)dt given a budget, how long does it take for eradication? 0

14

15 Neighborhood monotone crusade Weighted boundary of W V: B[W] = i W j / W W ij W W 3 4 S 4 5 S 9 5 S B[S] = 9 W W W S

16 Neighborhood monotone crusade Weighted boundary of W V: B[W] = i W j / W W ij Neighborhood monotone crusade from W: S sequence of sets satisfying S 0 = W, and S i = S i {v i }, with v i N Si S i B[S 0 ] = 9 W 5

17 Neighborhood monotone crusade Weighted boundary of W V: B[W] = i W j / W W ij Neighborhood monotone crusade from W: S sequence of sets satisfying S 0 = W, and S i = S i {v i }, with v i N Si S i B[S ] = 5 W 5

18 Neighborhood monotone crusade Weighted boundary of W V: B[W] = i W j / W W ij Neighborhood monotone crusade from W: S sequence of sets satisfying S 0 = W, and S i = S i {v i }, with v i N Si S i B[S ] = 7 W 5

19 Neighborhood monotone crusade Weighted boundary of W V: B[W] = i W j / W W ij Neighborhood monotone crusade from W: S sequence of sets satisfying S 0 = W, and S i = S i {v i }, with v i N Si S i B[S 3 ] = 7 W 5

20 Neighborhood monotone crusade Weighted boundary of W V: B[W] = i W j / W W ij Neighborhood monotone crusade from W: S sequence of sets satisfying S 0 = W, and S i = S i {v i }, with v i N Si S i B[S 4 ] = W 5

21 Neighborhood monotone crusade Weighted boundary of W V: B[W] = i W j / W W ij Neighborhood monotone crusade from W: S sequence of sets satisfying S 0 = W, and S i = S i {v i }, with v i N Si S i B[S 5 ] = 0 W 5

22 Neighborhood monotone crusade Weighted boundary of W V: B[W] = i W j / W W ij Neighborhood monotone crusade from W: S sequence of sets satisfying S 0 = W, and S i = S i {v i }, with v i N Si S i W 5 Ω W all NMCs from W. Lower bound on duration of NMC from W n W τ W := min S Ω W B[S i ] i=0

23 Fundamental limit on performance Theorem I: Fundamental limit on τ and J [working paper] Let U = t 0 M(t) be set of controllable nodes, then τ and J satisfy: τ min µ M J W U µ W τ W, where µ is a probability measure on the subsets of U that belongs to M J := µ : µ W W J W U Remark. The larger is J, the more measures are included in M J, the smaller is the bound on τ trade-off between cost and eradication time Not always easy to compute/estimate τ W and deal with µ

24 Sketch of the proof Main idea: relation between cost 0 T u(t)dt and set of controlled nodes W, in which mutants are introduced Cost = probability measure µ for set of controlled nodes W Stochastic domination techniques: W controlled nodes = W initial condition of Y (t) with β =, u = 0: τ Y τ NMCs are admissible trajectories of Y (t): τ Y Y (0) = W τ W Expected cost = expected value µ = T

25 Example: slow diffusion on rings Comparable examples: W ij = /d avg (d avg average degree) We set W ij = / We use S Ω W = B[S i ] W

26 Example: slow diffusion on rings Comparable examples: W ij = /d avg (d avg average degree) We set W ij = / We use S Ω W = B[S i ] W T = slow diffusion control policy τ J n 800 time 00 time β = 0.7 n 500, β = 0.8 n 500,000

27 Estimation of performance Effective control: C(t) := i ( x i(t))u i (t) Theorem II: Upper bound on τ [IFAC, 07] If during the evolution of the process the following is verified: then it holds τ B(t) + C(t) f (Z(t)), β (β )f (0) + n β f (z) z= Remark. The function f (z) depends on the topology of the network and on the control policy adopted

28 Sketch of the proof Stochastic process Z(t): C(t) βb(t) + C(t) 0... z... n ( β)b(t) Stochastic domination: Z(t) Z(t) C(t) βb(t) 0... z... n ( β)b(t) Discrete-time chain Z(k) (time step at each jump) β 0... z... n β

29 Sketch of the proof (cont d) Process Z(k) is Markov = bound # of entrances E[N 0 ] β β and E[N i ] β Time spent in state Z(t) = z is Poisson random variable with rate µ(t) = λ + (t) λ (t) = B(t) + C(t) f (z) Expected time spent in each entrance in z is E[T z ] /f (z) Bound on # of entrances in z + bound on time spent in z = T L. Zino, G. Como, and F. Fagnani, Fast Diffusion of a Mutant in Controlled Evolutionary Dynamics, IFAC-PapersOnLine, 50(), , 07

30

31 Constant control policies Target set M(t) = M time-invariant, u i (t) = u i To avoid waste, mutants not introduced in nodes already occupied { ui if X u i (t) = i (t) = 0 0 if X i (t) = Expected cost J proportional to M = study eradication time τ u 9 u 5

32 Constant control policy: performance (LB) The bottleneck of G controlled in M is φ M = min M W V B[W]. M S 3 4 S 4 5 S 9 5 S φ M = 3 M S S S

33 Constant control policy: performance (LB) The bottleneck of G controlled in M is φ M = min M W V B[W]. M S 3 4 S 4 5 S 9 5 S φ M = 3 M S S S Corollary: Lower bound on τ under constant control (LB) τ φ M

34 Constant control policy: performance (UB) Minimum conductance profile: φ(z) = min W V, W =z B[W] M S S S 9 5 S φ(3) = M S S S

35 Constant control policy: performance (UB) Minimum conductance profile: φ(z) = min W V, W =z B[W] M S S S 9 5 S φ(3) = M S S S Corollary: Upper bound on τ under constant control (UB) τ n β z= φ(z) + β (β ) T u

36 Example: fast diffusion on expander graphs γ > 0 : φ(z) γ min{z, n z} (complete, ER, small-world,...) We set M = {} and W ij = /d avg Complete Erdős-Rényi

37 Example: fast diffusion on expander graphs γ > 0 : φ(z) γ min{z, n z} (complete, ER, small-world,...) We set M = {} and W ij = /d avg UB + T = fast diffusion: τ = Θ(ln n) 40 time time β = 0.7 n 500, β = 0.8 n 500,000

38 Example: slow diffusion on stochastic block models ER communities linked by few edges (extreme case: barbell) We set M = {} and W ij = /d avg (i, j) E

39 Example: slow diffusion on stochastic block models ER communities linked by few edges (extreme case: barbell) We set M = {} and W ij = /d avg (i, j) E UB + LB = Slow diffusion: τ = Θ(n),000 time time β = 0.7 n 00 β = 0.8 n 500, ,000

40 To sum up... If the topology ensures fast diffusion: If it does not: Improve the control policy we reached our goal! can we speed up the spread?

41 Feedback control policy Assumptions: Target set M(t) = target node m(t), moved during the dynamics Control rate feedback function of the two observables Z(t), B(t) Avoid waste: m m(t) moved so that it is always a node with state 0 = no optimization on m(t) needed ( computationally good!) Contrast slowdowns: velocity of the process proportional to B(t) = u(t) should compensate when B(t) is small { C B(t) if Z(t) n, B(t) < C u(t) = u(z(t), B(t)) = 0 else

42 Feedback control policy: performance Bounds on expected eradication time and expected control cost: [ ] τ = E [inf t : X (t) = ] J = E T u(t)dt Corollary: upper bounds on τ and J under feedback control (UB-F) τ J β (β )C + n β max{φ(z), C} z= β (β ) + {z : φ(z) < C} β 0

43 Application of feedback control policy to SBM C < ( n) = control activates only at bottleneck T+UB-F = fast diffusion: τ Θ(ln n), J Θ() τ β ln n + + β C(β ) J + β β Constant control policy Feedback control policy time time, n 500,000 n 500,000

44 Case Study: Zika in Rwanda Zika Alert in Rwanda since 0 [CDC Centers for Disease Control and Prevention, available online]

45 Case Study: Zika in Rwanda Zika Alert in Rwanda since 0 [CDC Centers for Disease Control and Prevention, available online]

46 Case Study: Zika in Rwanda Location connected within a certain distance. Threshold set to.7 km: max distance traveled by mosquitoes to lay eggs [Bogojevic et al., J.Amer.Mosq.Cont.Ass., 007] Activation rate w = /0. 0 days life-cycle of Aedes aegypti [CDC Centers for Disease Control and Prevention, available online] Parameter Meaning Value n Number of locations W ij Activation rate 0. β Evolutionary advantage 0.53 u Control rate (constant) C Control parameter (feedback).5 t Time unit day

47 Case Study: Zika in Rwanda time,500, Constant vs Feedback cost

48 time Case Study: Zika in Rwanda 0 cost 00 0 constant feedback constant feedback Same cost, performance improved: τ 5%, p-value<< 0.00 L. Zino, G. Como, and F. Fagnani, Controlling Evolutionary Dynamics in Networks: A Case Study, accepted at IFAC-NecSys, Aug 08

49 Conclusion and direction for future research Evolutionary dynamics with heterogeneity and exogenous control Model analyzable = fundamental limit, performance estimation Constant control: in some topologies = slow spread We designed a feedback control policy to speed up the process Validation on case study: Zika in Rwanda = Analyze Robustness (e.g., errors in measurement of B(t)) = Refine the analytical results to improve performance estimation = Plan and study feasibility in real-world applications

50 Thank you for your attention! This work is a joint research project with... Giacomo Como giacomo.como@control.lth.se Fabio Fagnani fabio.fagnani@polito.it