Epidemics on graphs: Thresholds and curing strategies

A. J. GaneshMicrosoft Research,

Cambridge

Thresholds for epidemics on graphs

Joint work with:Laurent Massoulié (Thomson Research) Don Towsley (U. Mass., Amherst)

Model

Topology: undirected, finite graph G=(V,E), connected ;

Xv = 1 if node v V infected,

Xv = 0 if node v healthy = susceptible

SIS epidemic (contact process)

{Xv}vV continuous time Markov process on {0,1}V with jump rates:

Xv : 0 → 1 with rate wv Xw

Xv : 1 → 0 with rate

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

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

Problem description

{Xv}vV 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?

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?

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.

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

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)

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]/[ ]

Coupling proof:

Consider “Branching Random Walk”, i.e. Markov process {Yv}vV

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)

Branching random walk bound:

By “linearity” of Y, dE[Y(t)]/dt = (A I) Y(t),soE[Y(t)] = exp(A I) Y(0) Then use P(X(t) 0) ≤ vV E[Yv(t)]

Probabilistic interpretation:Node v infected at time t there is a node u

infected at time 0 and a pathu = 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

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]

Generalizations Instead of constant infection rate

, can have pairwise infection rates B = (uv)u,vV symmetric

Instead of constant cure rate , can have node-specific cure rates

D = (u)uV

Fast epidemic die out if spectral radius of B−D is negative.

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 )

||

|),(|min|:| S

SSEmSS

m

“perimeter”

“area”

Example: binary tree

m 1 for all m < n/2

Example: lattice

1 = 4, 4 = 2, m 0

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/[2m] .Hence, if m ~ na, survival time T satisfies

E[T] = (exp(na))

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

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

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.

Complete graph

Here, = n1, m = nm

Take m = na, any a < 1

Sharp threshold: fast die-out if / < 1/(n1)exponential survival time if /> 1/(nm)

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) Then ρ ~ ~ dn with high probability.

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

Star network

Spectral radius: = n

Isoperimetric constant: m = 1 for all m

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.

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

Why?

Power laws appear to be widespread in natural and engineered networks

Though some of the evidence is controversial

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)

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)

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)

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, …

Spectral radius of PLRG

Denote by m max. expected degree w1, and by d the average of expected degrees.

Theorem (Chung, Lu and Vu):

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.

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()

Optimal curing strategies

Joint work withChristian Borgs, Jennifer Chayes

and David Wilson (MSR)Amin Saberi (Stanford)

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

Problem formulation

Constant pairwise infection rate Node-specific cure rate u , uV Constraint

uV u(Xt,t) ≤ uV degree(u) Objective: choose u to maximize the

threshold – the minimum value of that results in long-lived epidemics

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

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

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|

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)

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?

Modelling contact tracing

Infection modelled as before

Xv : 0 → 1 with rate w~v Xw

Cure process modified to account for contact tracingXv : 1 → 0 with rate ’ w~v Xw

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

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?

Open problems

Improved bounds for general curing strategies

Analysis of the contact tracing method on general graphs, including power law graphs

Paper available athttp://research.microsoft.com/~ajg