Epidemics on graphs: Thresholds and curing strategies

1. Epidemics on graphs: Thresholds and curing strategies A. J. Ganesh Microsoft Research, Cambridge

2. Thresholds for epidemics on graphs Joint work with: Laurent Massoulié (Thomson Research) Don Towsley (U. Mass., Amherst)

3. Model • Topology: undirected, finite graph G=(V,E),connected ; • Xv = 1 if node v V infected, Xv = 0if nodevhealthy = susceptible

4. SIS epidemic (contact process) • {Xv}vV continuous time Markov process on {0,1}Vwith jump rates: • Xv : 0 → 1 with rate  wv Xw • Xv : 1 → 0 with rate 

5. Motivation • SIS epidemic model relevant to some biological epidemics • … and also to some kinds of computer viruses and worms • Cascading failures e.g. BGP router crashes, electrical power networks

6. Some weaknesses of the model • SIS is not always the right model – SIR is better suited to many applications • Model ignores latency/incubation periods • Markovian assumption • Focus is on understanding impact of network topology • … simplicity allows us to get the right qualitative intuition

7. Problem description • {Xv}vV Markov process on {0,1}V • Unique absorbing state at 0 • All other states communicate, 0 is reachable. • Epidemic eventually dies out • Define T = time to absorption • How does T depend on , and G?

8. Infinite lattices • Infinite d-dimensional lattice: fix =1 • There is a c > 0 such that: • if  < c, then epidemic dies out with probability 1 • if  > c, then epidemic has positive probability of surviving forever • Is there a signature of the phase transition in finite graphs?

9. Finite lattices • Durrett & Liu, Durrett & Schonmann • 1-D lattice on n nodes. fix =1 • T = time to absorption • if  < c, then E[T] = O(log n) from any initial condition • if  > c, then E[T] = (exp(na)) for some a > 0, and any non-zero initial condition.

10. Outline of rest of talk • General conditions for fast epidemic die-out: based on spectral radius • General conditions for long survival of epidemic: based on a generalisation of the isoperimetric constant • Special cases: star, power-law graphs • Optimal curing strategies

11. General graphs: fast recovery • G=(V,E): arbitrary connected graph • n=|V|: number of nodes • A: adjacency matrix of G • :spectral radius of G = largest eigenvalue of A • Phase transition at =  claimed by Wang, Chakrabarti, Wang & Faloutsos (SRDS 2003)

12. Fast die-out and spectral radius Let  be the spectral radius of the graph’s adjacency matrix, A, and n=|V| . Theorem: For any initial condition, P(X(t)  0) ≤ n exp(()t) Hence, when  < , survival time T satisfies: E(T) ≤ [log(n)+1]/[ ]

13. Coupling proof: Consider “Branching Random Walk”, i.e. Markov process {Yv}vV • Yv→Yv + 1 at rate  w~v Yw =  (AY)v • Yv → Yv  1 at rate  Yv Processes can be coupled so that, for all t, X(t) ≤ Y(t)

14. Branching random walk bound: By “linearity” of Y, dE[Y(t)]/dt = (A  I) Y(t), so E[Y(t)] = exp(A  I) Y(0) Then use P(X(t)  0) ≤vV E[Yv(t)]

15. Probabilistic interpretation: Node v infected at time t  there is a node u infected at time 0 and a path u = x0→ x1 → … → xk = v along which the infection went from u to v. By the union bound, P(Xv(t) 0) ≤ (sum over initial infectives u) (sum over k) (sum over paths of length k) (integral s1+…+ sk=t)exp(t) k ds1…dsk

16. Path counting: Number of paths of length k ~ k Therefore, for large t, and any initial condition, P(Xv(t) 0) ≤ n (sum over k)exp(-t) (  t)k /k! = n exp [( - )t]

17. Generalizations • Instead of constant infection rate , can have pairwise infection rates B = (uv)u,vV symmetric • Instead of constant cure rate , can have node-specific cure rates D = (u)uV • Fast epidemic die out if spectral radius of B−D is negative.

18. Slow die-out: Generalized isoperimetric constant Graph isoperimetric constant: n/2 related to “spectral gap”,  of random walk on graph (in particular, n/2 ≤/2 ) “perimeter” “area”

19. Example: binary tree • m  1 for all m < n/2

20. Example: lattice • 1 = 4,4 = 2,m  0

21. Slow die-out and isoperimetric constant Suppose that for some m ≤ n/2, r := m/ > 1 Theorem: With positive probability, epidemics survive for time at least rm/[2m] . Hence, if m ~ na, survival time T satisfies E[T] = (exp(na))

22. Coupling proof: Let |X|=v Xv : number of infected nodes Suppose |X| = k. Then, no matter where the k infected nodes are located, the number of edges from them incident on susceptible nodes is at least k Therefore, a new node becomes infected at rate at least k Infected nodes are cured at rate k

23. Proof (continued): |X| dominates process Z on {0,…,m} with transition rates: z → z + 1 at rate z, z → z  1 at rate  z If  > , then mean time for Z to hit 0 is exponential in m – gambler’s ruin problem

24. Summary: Two bounds: (/) < m (slow die-out), or (/) > (fast die-out) If  ≈ m then we have a sharp threshold. Otherwise, can’t say in general.

25. Complete graph Here,  = n1, m = nm Take m = na, any a < 1 Sharp threshold: fast die-out if / < 1/(n1) exponential survival time if / > 1/(nm)

26. Erdős-Rényi random graph • Edge between each pair of nodes present with probability pn independent of others • Sparse: pn = c log(n)/n, c > 1. • Then ρ ≤c(1+) log(n), ≥ c’ log(n) with high probability, for some c’ < c • Dense: dn := npn = Ω(log n) • Thenρ ~  ~ dn with high probability.

27. 1-dimensional lattice • =2, m = 2/m • Implied bounds on threshold c are: • Spectral radius: c > 0.5 • Isoperimetric constant: c <  • Known that 1.5 ≤ c ≤ 2 • Neither of the general bounds is tight in this case

28. Star network Spectral radius:  = n Isoperimetric constant: m = 1 for all m

29. Epidemic on the star Theorem: For arbitrary constant c>0, if / < c/n, then E[T]=O(log(n)), i.e., epidemic dies out fast. If / > na/n for some a>0, then log(E[T])=(na),i.e., epidemic survives long.

30. Power law graphs • Power law graph with exponent : • number of vertices with degree k is proportional to k . • Differs from classical random graphs • number of vertices with degree k decays exponentially in k

31. Why? • Power laws appear to be widespread in natural and engineered networks • Though some of the evidence is controversial

32. Instances of power laws • Internet AS graph with =2.1 (Faloutsos3, 1999) • Artifact of traceroute? (Lakhina, Byers, Crovella, Xie) • Distribution of hyperlinks on web pages (Barabasi, Albert and Jeong) • Number of sexual partners (Liljeros et al.) • Lognormal? (Kault)

33. Generative models for power laws • Preferential attachment in graphs (Barabasi and Albert) • Earlier examples • Distribution of species in genera (Yule) • Distribution of income, city sizes etc. (Simon) • (See survey paper by Mitzenmacher)

34. Epidemics on power-law graphs • Zero epidemic threshold claimed by Pastor-Satorras and Vespignani • Based on mean-field models • Rigorous analysis by Berger, Borgs, Chayes and Saberi (2005)

35. Power-law random graph model (Chung and Lu) • Random graph with expected degrees w1,…,wn: edge (i,j) present w.p. wi wj / k wk • Particular choice: wi = c1(c2+i)−1/(−1) • Other models proposed by Barabasi and Albert, Bollobas and Riordan, Cooper and Frieze, Norros, …

36. Spectral radius of PLRG Denote by m max. expected degree w1, and by d the average of expected degrees. Theorem (Chung, Lu and Vu):

37. Epidemic on PLRG, >2.5 Epidemics on full graph live longer than on subgraph. Hence, looking at star induced by highest degree node: slow die-out for / > m-1/2 Spectral radius condition: fast die-out for / < m-1/2 Thresholds differ by m ; same gap as for star.

38. Epidemic on PLRG, 2 <  < 2.5 Consider N highest degree nodes, for suitable N: induced subgraph contains E.R. graph, with isoperimetric constant  = F()  Gap between thresholds  and  : a constant factor, F()

39. Optimal curing strategies Joint work with Christian Borgs, Jennifer Chayes and David Wilson (MSR) Amin Saberi (Stanford)

40. Optimal curing strategy • Problem: Suppose there is a fixed total cure rate D – how should this be allocated between the nodes? • Static vs. dynamic schemes

41. Problem formulation • Constant pairwise infection rate  • Node-specific cure rate u , uV • Constraint uV u(Xt,t) ≤ uV degree(u) • Objective: choose u to maximize the threshold – the minimum value of  that results in long-lived epidemics

42. A static scheme: cure proportional to degree • Take u = degree(u),  is constant Theorem: If 1, then E[T] = O(log n), for any initial condition Idea of proof: Mean number infected by a node before it is cured is smaller than 1 – subcritical branching process

43. Can we do better? • Can’t say on general graphs • On expander graphs, no scheme can perform more than a constant factor better • even a dynamic one with full information on current epidemic state • Scaling law: total cure rate needs to grow as fast as total number of edges

44. Expander graphs Definition: A graph G is said to be an (,) expander if, for any subset of nodes W with |W| ≤ |V|, the number of edges between W and its complement satisfies E(W,Wc) ≥ |W|

45. Limitations of arbitrary curing strategies Theorem: For any adapted curing strategy with the total cure rate being bounded by the number of edges, and for arbitrary >0, • If  > (1+)d/(), where d is the mean node degree • Then log E[T] = Ω(n log n)

46. Related methods: Contact tracing • In addition to treating infected individuals, identify and treat their contacts (who they may have infected, or who may have infected them) • Used in practice • To what extent does it help?

47. Modelling contact tracing • Infection modelled as before Xv : 0 → 1 with rate  w~v Xw • Cure process modified to account for contact tracing Xv : 1 → 0 with rate  + ’w~v Xw

48. Contact tracing on the star • Fix =1 • Threshold is at  ≈ n1/3 for contact tracing • compared to  ≈ √n for constant cure rate • and  = 1 for curing proportional to node degree

49. Open problems • Extension to other graph models, e.g., small world networks • Better models needed for social networks • Conditional on long survival of the epidemic, what fraction of nodes is infected? Which nodes?

50. Open problems • Improved bounds for general curing strategies • Analysis of the contact tracing method on general graphs, including power law graphs • Paper available at http://research.microsoft.com/~ajg