1. Rumors, consensus and epidemics on networks
J. Ganesh
University of Bristol

2. Rumor spreading Population of size n
One person knows a rumor at time 0
Time is discrete
In each time step, each person who knows the rumor chooses another person at random and informs them of it
How long before everyone knows the rumor?

3. Motivation
Simple model for diffusion of information, spread of infection etc., over social networks
Basis of information dissemination algorithms in large-scale distributed systems
Primitive of algorithms for distributed computing, distributed consensus etc.

4. Results With high probability, all n people learn the rumor in time
log2n + log n + o(log n) : Frieze and Grimmett
log2n + log n + O(1) : Pittel
Intuition:
in early stages, number of informed people doubles in each time step
in late stages, number uninformed decreases by factor of 1/e in each time step

5. A continuous time model
Identify individuals with nodes of a complete graph
Associate mutually independent unit rate Poisson processes, one with each node
If a node is informed, then, at points of the associated Poisson process, it informs a randomly chosen node
How long till all nodes are informed?

6. Edge-driven model

7. Analysis Tk : first time that k nodes know the rumor
Number of edges between informed and uninformed nodes : k(n-k)
Time to inform one more node is minimum of k(n-k) independent Exp(1) r.v.s.

8. Analysis (cont.) Time to inform all nodes is
Tn = (TnTn1)+(Tn1Tn2)+...+(T2T1)+T1
So E[Tn]  2 log n
Similar calculations show variance < 2/3
Chebyshev’s inequality implies
Tn = 2 log n + O(1)
in probability

9. Rumor spread on networks
G=(V,E) : directed, strongly connected graph
R = (rij), i,jV : contact rate matrix
Model:
node i contacts node j at the points of a Poisson process of rate rij
informs node j at this time if node i is informed
Mosk-Aoyama & Shah:
Bound the time to inform all nodes, based on properties of G or R

10. Graph properties
The generalized conductance of the non-negative matrix R is defined as
If R is the adjacency matrix, this is closely related to the isoperimetric constant

11. Analysis of rumor spreading
Tk : first time that k nodes are informed
S(k) : set of informed nodes at this time
Total contact rate of uninformed nodes by informed nodes is iS(k), jS(k) rij
Time to inform one more node is stochastically dominated by Exp(k(nk)(R)/n)
Implies that mean time to inform all nodes is bounded by 2log(n) / (R)

12. Examples G=Kn ; rij =1/n for all i,jV
(R)=1, bound is 2 log n, E[T]  2 log n,
G is the star on n nodes, rij =
1/n, if i is the hub and j a leaf,
1, if i is a leaf and j the hub
(R)1/n, bound is 2n log n, E[T]  n log n
G is the cycle on n nodes, rij=1/2 for all (i,j)E
(R)=4/n, bound is (n log n)/2, E[T] = n1

13. Remarks Rumor spreading is
fast on the complete graph, expander graphs
slow on the cycle, grids, geometric random graphs
Can it be speeded up by passing the rumor to random contacts rather than neighbors?
Not obvious: sampling random contacts takes time
Dimakis, Sarwate, Wainwright: Geographic gossip
Benezit, Dimakis, Thiran, Vetterli: Randomized path averaging

14. Other models: stifling
Stop spreading rumors when they are stale
Nodes may be uninformed (U), spreaders (I) or stiflers (S)
U+I = I+I; I+I = I+S; I+S = S+S
Rumor only reaches a fraction of population, rather than all nodes
Daley & Kendall, Maki & Thomson

15. Other models: push-pull
Discrete time model:
Push is effective in early stages
In late stages, Pull is much better
Say fraction  of nodes is uninformed at time t
Push:  e uninformed at time t+1
Pull :  2 uninformed at time t+1
Exploited by Karp et al. to reduce number of messages required, from nlog(n) to nloglog(n)

16. Consensus: de Groot model
n agents, initial opinions xi(0), i=1,...,n
Discrete time
Agents update opinions according to
xi(t+1) = j pij xj(t),
where P is a stochastic matrix
Do all agents reach consensus on a value?
If so, what is the consensus value, and how long does it take?

17. Results for de Groot model
Recursion is x(t+1)=Px(t)
Reaching consensus means x(t) c1 as t, where c is a constant and 1 is the all-1 vector
This is guaranteed for all initial conditions if and only if P is irreducible and aperiodic
Consensus value is x(0), where  is the unique invariant distribution of P
Time to reach consensus is determined by the spectral gap of P

18. Consensus: the voter model
n agents, with opinions in {0,1}
Agent i contacts agents j at the points of a Poisson(qij) process, and adopts its opinion
Once all agents have the same opinion, no further change is possible
How long does it take to reach consensus?
What is the probability that the consensus value is 1?

19. The voter model in pictures

20. Voter model on the complete graph
All agents can contact all other agents.
They do so at equal rates: qij = 1/n for all i,j
Equivalently,
each undirected edge is activated at rate 2/n,
and then oriented at random
agent at tail of arrow copies agent at head

21. Motivation
Voter model on complete graph is same as Moran model in population genetics
also used to model cultural transmission, and
competition between products or technologies, especially with network externalities
Consensus is important in distributed systems and algorithms
and in collective decision making in biology

22. Final state: complete graph case
Each direction equally likely to be chosen.
So, 01 and 10 are equally likely transitions. Hence,
number of 1s is a martingale.
P(consensus value is 1) = initial fraction of 1s

23. Final state: general case
Contact rates qij, ij
Define qii = ji qij
Assume Q is an irreducible rate matrix
Then it has unique invariant distribution 
Hassin and Peleg:
 X(t) is a martingale. Therefore,
P(consensus value is 1) =  X(0)

24. Duality with coalescing random walks

25. Coalescing random walks
Initially a single particle at each site
Particles perform random walks according to rate matrix Q, but
if particle moves from node i to node j and j is occupied, it coalesces with the particle there
random walks are independent between coalescence events
When there is a single particle left, consensus has been reached

26. Coalescence time: complete graph
Tk : time when k particles remain. Tn=0.
At Tk, have k(k1) directed edges between occupied nodes, rate 1/n on each edge
Tk1  Tk  Exp(k(k1)/n)
Mean time to consensus bounded by n
linear in population size for consensus
logarithmic for rumor spreading

27. Coalescence time: general graphs
Suppose Q is the generator of a reversible random walk, with invariant distribution 
Aldous and Fill : Mean coalescence time of two independent random walks started at any nodes i, j bounded by n log 4, where

28. General graphs (continued)
Example: G is a connected, undirected graph and qij = 1{(i,j)E}
Then, i = 1/n for all i
  n/4 since there is always at least one edge between any A and Ac
Mean coalescence time of any two random walks bounded by n2 log(2)/2

29. Consensus time on general graphs
Even-Dar and Shapira: For Q as above,
use Markov’s inequality to bound the probability that two random walks haven’t coalesced
then union bound to bound the probability that there is some random walk that hasn’t coalesced with a specific one, say one starting at i
implies that, with high probability, consensus reached within O(n3 log n) time

30. Open problem: Evolving voter model
Graph G, nodes in state 0 or 1
Pick a discordant edge at random and orient it at random
with probability 1p, caller copies called node
with probability p, it rewires to a random node with same current state
Simulations show critical value of p,
below which network reaches consensus,
and above which it fragments

31. Epidemics: SIS model Graph G=(V,E) on n nodes, undirected
Each node in one of two states, {S,I}
Nodes change state independently,
S I at rate  (# of infected neighbours)
I S at rate 
How long is it until all nodes are in state S?

32. SIS model in pictures

33. Motivation
Models spread of certain diseases, and certain kinds of malware (SIR model better for others)
Propagation of faults
Models persistence of data in peer-to-peer / cloud networks
Can be used to model diffusion of certain technologies or behaviours

34. Upper bound: branching random walk
Infected individuals initially placed on graph
Each individual gives birth to offspring at rate  at each neighboring node, dies at rate 
How long does it take for the population to die out?

35. Branching random walks
Yi(t) : # of individuals at node i at time t
+1 at rate ji Yj(t)
1 at rate Yi(t)
A : adjacency matrix of graph G
dE[Y(t)]/dt = (A) EY(t)
E[Y(t)] = exp((A) t) Y(0)
 : spectral radius of A

36. Upper bound on epidemic lifetime
G., Massoulie, Towsley: Epidemic stochastically bounded by branching random walk
therefore, so is epidemic lifetime
If , then E[Y(t)]  0
By Markov’s inequality, P(|Y(t)|1)  0
Implies that mean epidemic lifetime is bounded by log(n)/()

37. Lower bound Generalised isoperimetric constant of G :
S(t) : set of infected nodes at time t
If S(t)  m, then
rate of infecting new nodes  mS(t)
rate of recovery of infected nodes = S(t)

38. Lower bound on epidemic lifetime
G., Massoulie, Towsley
If m   , then epidemic lifetime is exponential in m , because
when # of infected nodes is less than m, new nodes are infected faster than infected nodes recover
biased random walk, hits m exponentially many times before hitting 0

39. Remarks Upper and lower bounds Results imply that epidemic lifetime is
match on complete graphs, hypercubes, dense Erdos-Renyi and random regular graphs
separated by big gap on cycles, grids etc.
Gap is also big on scale-free random graphs, but can handle them by focusing on high-degree stars
Results imply that epidemic lifetime is
logarithmic in population size for small infection rates,
exponential in population size for large infection rates

40. Epidemics: SIR model G=(V,E) : n nodes, undirected, connected
Each node in one of three states, {S,I,R}
Nodes change state independently,
S I at rate  (# of infected neighbours)
I R at rate  , or after random time with specified distribution
How many nodes are ever infected?

41. SIR model description Single initial infective
p : probability that a node which becomes infected ever tries to infect a given neighbor
p =  E(length of infectious period)
insensitive to distribution of infectious period
i : probability that node i is ever infected

42. Upper bound on epidemic sizes
Draief, G., Massoulie
j  pij i : union bound
(pA)  es , where s is the initial infective
If p < 1, implies that mean number of nodes ever infected  (n)/(1p)
Upper bound can be improved to 1/(1p) if graph is regular
Matching lower bounds in some cases – star, Erdos-Renyi random graphs

43. Lower bound on epidemic sizes
Bandyopadhyay and Sajadi
Consider any BFS spanning tree T of G
Epidemic on G stochastically dominates epidemic on T
Hence, i  pd(i,s), d(,) – graph distance
Implies lower bound on mean number of infected nodes:
How good is this lower bound?

44. Results Gn : sequence of graphs indexed by |V|
sn : infection source in Gn
Xn : mean number of infected nodes
LBn : lower bound based on BFS spanning tree
Theorem: If there is an (log n) sequence of neighborhoods of sn in which Gn is a tree, then there is a pc>0 such that, for p<pc, Xn/LBn  1

45. Results (continued)
Theorem: Suppose there is a deterministic or random rooted tree (T,s) such that (Gn,sn)  (T,s) in the sense of local weak convergence. Suppose the maximum node degree in all Gn is bounded uniformly by , and p < 1. Then, Xn  LBn  0

46. Spread of influence Rumor-spreading and voter models are simplistic
What if node is only influenced if some number, or some fraction, of neighbors have a different opinion?
Can be motivated by best response dynamics in network games

47. Bootstrap percolation
Connected, undirected graph G=(V,E)
Initial states of nodes in {0,1}
Node changes state from 0 to 1 if at least k of its neighbors are in state 1
Nodes don’t change from 1 to 0
Can we guarantee that all nodes will eventually be in state 1?

48. Results G=(V,E) is the d-regular random graph
Bernoulli initial condition : each node in state 1, with probability p, independent of others
Theorem (Balogh and Pittel) : Suppose 1<k<d-1. There is a p*(0,1) such that
p>p*+ : all nodes eventually in state 1 whp
p<p* : the fraction of nodes in state 0 tends to a non-zero constant whp

49. Conclusions
Variety of stochastic processes on graphs can be studied using elementary probabilistic tools
Analysis can often be greatly simplified by choosing the right model
Often, exact analysis is intractable, but can get good (?) bounds
Many applications!