HALTING VIRUSES IN SCALEFREE NETWORKS

1
HALTING VIRUSES IN SCALE-FREE NETWORKS

• Zoltán Deszó and Albert-László Barabási

H. Gül Çalikli 2004800069 Bogaziçi
University Department of Computer Engineering
2
INTRODUCTION

• Diffusion Processes of Practical Interest

3
INTRODUCTION

• topology of scale-free nws deviate from those
  of
• random nws and regular lattices
• differences btw. topologies impact nws
• robustness
• attack tolerance
• diffusion dynamics of viruses

4
INTRODUCTION
Internet e-mail network ? responsible for
spread of computer viruses ? They have
scale-free topology exhibiting power-law degree
distribution
P(k) k ? 2 ? 3

• Social networks responsible for spread of
  diseases such as AIDS also exhibit a scale-free
  network structure

5
INTRODUCTION

• Susceptible-Infected-Susceptible (SIS) Model
• Simple model used to study generic features of
  virus spreading
• The model consists of
• Nodes
• represent individuals
• an individual can be
• healthy, or
• infected
• Links
• represent connections btw. individuals along
  which virus can spread

6
INTRODUCTION SIS Model (contd)

• In each time step t
• ? probability with which a healthy node is
  infected
• healthy node should be connected to at least one
  infected node in order to become infected
• d probability with which an infected node is
  cured
• THUS, effective spreading rate ? for the
  virus is formularized as

? ? / d
7
INTRODUCTION SIS Model (contd)

• CASE 1 SIS Model placed on a regular lattice
  or random network
• CASE 2 SIS Model placed on a scale-free
  network

8
INTRODUCTION SIS Model (contd)

• CASE 1
• Given ? effective spreading rate of virus
• ?c critical threshold for effective
  spreading rate of virus

9
INTRODUCTION SIS Model (contd)

• CASE 2
• Pastor-Satorras Vespignani (2001)
• ? 3 ? ?c 0 (i.e. epidemic threshold
  vanishes)
• even weakly infectious viruses can prevail.
• ?c 0 is a consequence of HUBs.

10
INTRODUCTION

• In this paper authors show that
• random distribution of cures in a scale-free nw.
  is ineffective in eradicating an epidemic
• curing hubs with higher probability than less
  linked nodes can restore the epidemic threshold.
• hub-biased policies ? most cost effective

11
CURING THE HUBS

• ?c 0 in a scale-free nw. ? rooted in infinite
  variance of degree distribution P(k)
• To retore a finite epidemic threshold we need to
  induce finite variance
• Origin of infinite variance ? in tail of P(k)
  dominated by hubs.
• Curing all hubs with degree ko will restore
  finite variance and thus nonzero epidemic
  threshold.

? 8
?c 0
variance of P(k)
12
CURING THE HUBS

• scale free nw. nodes with k k0 all healthy?
  finite ?c

We dont have detailed nw. maps. Thus we cant
effectively identify hubs
13
CURING THE HUBS

• THUS, no curing method is expected to succeed in
  finding all hubs with degree k0
• Such a biased policy might miss some hubs and
  identify nodes with less links as hubs.

14
CURING THE HUBS

• GOAL investigate effect of incomplete
  information about hubs
• ASSUME The likelihood of identifying and
  administering a cure to an infected node in a
  given time frame depends on the nodes degree as
  k a
• a characterizes the policys ability to find the
  hubs.
• a 0 ? random cure distribution
• a 8 ? an optimal policy treating all hubs
  with degree k0

15
CURING THE HUBS

• within the framework of SIS model
• each node infected w/ probability ?
• each infected node cured w/ probability
• d d0 ka
• having healed node become susceptible to disease
  again ? a node can get multiple cures during
  simulation
• spreading rate of virus ? ? ? / d0

16
MEAN-FIELD THEORY

• SIMULATION
• 1) place nodes on a scale-free nw.
• 2) initially infect half of them
• 3) after a transient regime, system reaches a
  steady state characterized by constant ?
• ? avg. density of infected nodes

17
CURING THE HUBS
SIMULATION RESULTS
Numerical simulations indicate that we have
non-zero ? before the limit a 8 (even for a
1)
18
MEAN-FIELD THEORY
?k(t)
?
1-?k(t)
k
?(?)
-
?t
d0 ka
?k(t)


eqn 1
?t ?k(t) time evolution of ?k(t)
?k(t) density of infected nodes with
connectivity k
eqn 2
d0 ka ?k(t) probability that an infected node is
cured
d0 ka probability that a node with k links will
be selected for the cure
? 1-?k(t) k ?(?) probability that a healthy
node is infected
? infection rate
1-?k(t) density of healthy nodes with k links
k number of links
?(?) probability that a given link points to an
infected node
back2
back1
19
MEAN-FIELD THEORY

• using ? ? / d0
• Stationary condition ? ?t ?k(t) 0
• Solving the eqn. below
• we find stationary density ?k

eqn 3
back1
back2
20
MEAN-FIELD THEORY

• Connectivity distribution P(k) for scale-free nw
• To generate scale free nw. (Barabasi Albert
  1999)
• Start w/ m0 nodes
• At every time step, add a new node w/ m m0
  edges linking the new node to m different
  vertices alredy present in the nw.

HOW TO DERIVE
back3
back1
back2
21
MEAN-FIELD THEORY

• avg. density of infected nodes ? ?(?)

eqn 5
22
MEAN-FIELD THEORY

23
MEAN-FIELD THEORY

• solve ?(?) for the case a 1
• solve for ? in eqn 6 for the case a 1
• substitude eqn 8 into eqn 7 for ? 1 ?(?)
• solve eqn 8 for ?(?) insert obtained eqn 10
  into eqn 9

?c(a 1) 1
24
MEAN-FIELD THEORY

• To determine ?c as a function of a ? solve ?(?)
  0
• solve eqn 5 with ?(?) 0 and obtain eqn 11
• eqn 11 indicates that,
• ?c is non-zero for any positive value of a ? any
  policy biased towards curing hubs can retore
  nonzero ?c
• Policies w/ larger a are expected to be more
  likely in eradicating a virus due to larger ?c

• for a 0 ? ?c 0
• for a 0 ? ?c 1

25
MEAN-FIELD THEORY

• Excellent agreement btw. simulations
  analytical
• prediction

Results of numerical simulations based on SIS
model
?c
?c a m a-1
a
26
COST EFFECTIVENESS

• Policies that obtain the largest effect with the
  smallest number of administered cures are more
  desirable due to cost effectiveness
• To address cost effectiveness of a policy
  targeting hubs ?
• Cost number of cures administered in a time step
  per node for different values of a
• Increasing the policys bias towards hubs by
  allowing higher values of a decreases rapidly the
  number of cures

27
COST EFFECTIVENESS
Cost
a
28
CONCLUSIONS

• Targeting the more connected infected nodes can
  restore ?c ? eradication of a virus made possible
• Even polices w/ small a can restore nonzero ?c
• ?c decreases w/ a ? policy becomes more effective
  in identifying and curing hubs of a scale free
  nw.
• Higher chances in eradicating a virus
• Biased policy is less expensive then random
  immunization as well as being more effective

29
Thank YouAny Questions
30
Map of Internet Colored by IP Addresses (William
W. Cheswick)
Back