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 ; X v = 1 if node v  V infected, X v = 0 if node v healthy = susceptible

4. SIS epidemic (contact process) {X v } v  V continuous time Markov process on {0,1} V with jump rates: X v : 0 → 1 with rate  w  v X w X v : 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 {X v } 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(n a )) 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 {Y v } v  V Y v → Y v + 1 at rate  w~v Y w =  (AY) v Y v → Y v  1 at rate  Y v 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[Y v (t)]

15. Probabilistic interpretation: Node v infected at time t  there is a node u infected at time 0 and a path u = x 0 → x 1 → … → x k = v along which the infection went from u to v. By the union bound, P(X v (t)  0) ≤ (sum over initial infectives u) (sum over k) (sum over paths of length k) (integral s 1 +…+ s k =t) exp(  t)  k ds 1 …ds k

16. Path counting: Number of paths of length k ~  k Therefore, for large t, and any initial condition, P(X v (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 r m /[2  m]. Hence, if m ~ n a, survival time T satisfies E[T] =  (exp(n a ))

22. Coupling proof: Let |X|=  v X v : 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 = n a, 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 p n independent of others Sparse: p n = c log(n)/n, c > 1. Then ρ ≤ c(1+  ) log(n),  ≥ c’ log(n) with high probability, for some c’ < c Dense: d n := np n = Ω(log n) Then ρ ~  ~ d n 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  /  > n a /  n for some a>0, then log(E[T])=  (n a ), 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 (Faloutsos 3, 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 w 1,…,w n : edge (i,j) present w.p. w i w j /  k w k Particular choice: w i = c 1 (c 2 +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 w 1, 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 (X t,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,W c ) ≥  |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 X v : 0 → 1 with rate  w~v X w Cure process modified to account for contact tracing X v : 1 → 0 with rate  ’  w~v X w

48. Contact tracing on the star Fix  =1 Threshold is at  ≈ n 1/3 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