1. The Effect of Network Topology on the Spread of Epidemics
A. Ganesh, L. Massoulié, D. Towsley
Presented By
VARUN GUPTA
April 16, 2007

2. Theme of Talk Spread of epidemics as a model of network phenomena
Spread of viruses/computer worms
Router failures
Diseases
What is the effect of the network topology on the potency of the epidemic?

3. Susceptible-Infected-Susceptible model
Given a connected undirected graph G=(V,E)
V = {1,2,..,n}
X(t) represents the state of network at time t
X(t) = [X1(t) X2(t) Xn(t)]’
Xi(t) = 1 if node i is infected at time t; 0 otherwise
Infected nodes contaminate neighbors at rate , recover at rate 1
Defines a continuous time Markov process on the set of states {0,1}n
Xi : 0  1 at rate  (j,i)  E Xj
Xi : 1  0 at rate 1

4. [0,0,0] is an absorbing state
Example 2 3
G ≡ 1
[0,0,1]
[1,0,0] 2 1 1
[1,0,1]
[1,1,1]
[0,0,0] 1
[0,1,0] 1 1
Provide more explanation on what rates mean in a CTMC.. Its like probabilities are the normalized rates and the sojourn time is exponential with sum of rates; Also say that epidemic dies out with probability 1 and probability it hasn’t died out by t decays exponentially in t 1
[1,0,0]
[0,0,1]
[0,0,0] is an absorbing state

5. Possible measures of potency
Peak infection
supt [i Xi(t)]
Total infection
i [supt Xi(t)]
Epidemic lifetime
 = inf{ t :  Xi(t) = 0}
More …
This talk

6. Quick vs. Slow die out Interested in E[] as a function of n
Say an epidemic dies out quickly if
An epidemic dies out slowly if

7. Outline Sufficient condition for quick die out
Sufficient condition for slow die out
Conditions for specific network topologies
Hypercubes (Distributed hash tables, P2P)
Complete graphs (BGP router networks)
Erdös-Renyi random graphs
Power law graphs (well…)

8. Definitions
The spectral radius of a graph G is defined to be the spectral radius (largest eigenvalue), (A), of its adjacency matrix A.
The generalized isoperimetric constant of a graph G is defined as:
Intuitively…
Spectral radius is bounded by the minimum and maximum degree.. A measure of how well connected each node is
Generalized isoperimetric constant

9. Sufficient but not necessary conditions
Main Results
Sufficient but not necessary conditions
If b < 1/(A)
If  a>0 and an m=(na) such that >1/(m) , then

10. Proof Outline
Find another (easy to analyze) continuous time Markov process Y such that that Y st X
Upper bound Pr[Y(t)  0]
Gives upper bound on Pr[X(t)  0] and hence on Pr[>t]
Use the above to upper bound E[]
E[] = 0 Pr[>t] dt

11. Proof for sufficient condition for quick die out
Recall: X(t)  {0,1}n
Xi : 0  1 at rate  (j,i)E Xj
Xi : 1  0 at rate 1
Define: Y(t)  Nn
Yi : k  k+1 at rate  (j,i)E Yj
Yi : k  k-1 at rate Yi
Y(t) st X(t) given Y(0)=X(0) (why?)

12. Proof for sufficient condition for quick die out (contd..)
Y(t) st X(t) implies Pr[X(t)0]  Pr[Y(t)0]
Pr[Y(t)  0]  i E[Yi(t)] (why?)
Also,
Hence

13. Proof for sufficient condition for quick die out (contd..)
Implies

14. Proof for sufficient condition for quick die out (contd..)
Finally,

15. Proof for sufficient condition for slow die out
Consider: i Xi(t)
i Xi : x  x+1 at rate E(S,V-S)
i Xi : x  x-1 at rate x
(S = {i  V: Xi(t)=1})
Define: Z(t)  {0,…,m}
Z : z  z+1 at rate (m)z
Z : z  z-1 at rate z
Z(t) st i Xi(t)

16. Proof for sufficient condition for slow die out (contd.)
(m)
(m-2)(m) 
(m-1)(m) 
m-1 m 1 2
m-1 m 1
Lower bound on epidemic lifetime of Z gives a lower bound on that of X
Z(t) st i Xi(t)
Z(t)  m
Downward transition rates of i Xi(t) and Z(t) are identical
For infected set S, |S|m
Upward transition rate for i Xi = E(S,V-S)
Upward transition rate for Z = (m) |S|   E(S,V-S)

17. Proof for sufficient condition for slow die out (contd.)
To apply the corollary, have to find a constant a>0 such that this holds
Let r = 1/(m) 
Theorem: If r<1, then for any X(0) with i Xi(0) > 0
Corollary: If  a>0 and a sequence m=(na) such that r<1 uniformly in n, then
log(E[]) = (na)

18. Proof for sufficient condition for slow die out (contd.)
(m)
(m-2)(m)
(m-1)(m)
m-1 m 1 2
m-1 m 1
Proof of Theorem
r = 1/(m)
Look at embedded discrete time Markov chain for Z (only when events happen)

19. Proof for sufficient condition for slow die out (contd.)
m-1 m 1 1
r/1+r
r/1+r
r/1+r
Proof of Theorem
r = 1/(m)
Look at embedded discrete time Markov chain for Z (only when events happen)

20. Proof for sufficient condition for slow die out (contd.)
m-1 m 1 1
r/1+r
r/1+r
r/1+r
Let q(k) be the probability that the embedded chain enters state m before 0, given it starts at k
Gambler’s ruin:
q(k) = (1-rk)/(1-rm)

21. Proof for sufficient condition for slow die out (contd.)
Let T be the number of steps till absorption at 0
Choose t=r-m+1

22. More Corollaries
Corollary: For a d-regular graph, the epidemic survives for an exponential amount if
 > 2/ ((A)-n-1)
Proof:
Lemma: (G)  2(L)/2
where L is the Laplacian matrix of G (D-A).
For a d-regular graph
2(L) = d-n-1=(A)-n-1

23. Recap Sufficient condition for quick die out: b < 1/(A)
Sufficient condition for slow die out:
> 1/(m)
where m=(na) for some a>0.
Next: Tighter conditions for common graphs

24. Sufficient conditions are tight for this case!
Hypercubes
V = {0,1}log2 n
{i,j}  E iff they differ in only one bit position
Quick die out if
 < 1/log2n
Slow die out if
 > 1/(1-a)log2n
for some 0<a<1
Sufficient conditions are tight for this case!
 = 1/log2 n

25. Sufficient conditions are tight for this case!
Complete Graphs
Sufficient conditions are tight for this case!
Quick die out if
 < 1/(n-1)
Slow die out if
 > 1/(n-m)
when m=na for some a>0

26. Erdös-Renyi random graphs
G(n,p) with d=np >> log n (connected w.h.p)
Quick die out, E[]=O(log n), if
 < (1-u)/d
For 0<u<1
Slow die out, log(E[])=(n), if
 > (1+v)/d
For v>0
Sufficient conditions are tight for this case!

27. Power law graphs (random with power law degree sequence)
Number of nodes with degree k proportional to k-, avg degree d, max degree M
Spectral radius for random power law graphs:

28. Power law graphs (random with power law degree sequence)
Case:  > 2.5
Quick die out if
<(1-u)M-0.5
For some 0<u<1
Slow die out if
>M-0.5
For some 0<<1

29. Power law graphs (random with power law degree sequence)
Case: 2 <  < 2.5
Quick die out if
(A)<(1-u)
For some 0<u<1
Slow die out if
(A)>(1+u) [2/(3-)]2/(-1)
For some u>0

30. Questions