Rumors, consensus and
epidemics on networks
A. J. Ganesh
University of Bristol
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?
• 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.
• 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
A continuous time model
• Identify individuals with nodes of a complete
• 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?
Edge-driven model
• 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.
1 
E[Tk 1  Tk ] 
  
k (n  k ) n  k n  k 
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
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
Graph properties
• The generalized conductance of the nonnegative matrix R is defined as
 ( R) 
minS V
iS , jS c ij
| S || Sc |
• If R is the adjacency matrix, this is closely
related to the isoperimetric constant
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)
• 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
• 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
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
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)
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?
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
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?
The voter model in pictures
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
• 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
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
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)
Duality with coalescing random walks
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
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)
 1
E[Tk 1  Tk ]  n
 
 k 1 k 
• Mean time to consensus bounded by n
– linear in population size for consensus
– logarithmic for rumor spreading
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
 ( A) ( Ac )
  maxA
iA jA i qij
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
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
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
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?
SIS model in pictures
• 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
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
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
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)/()
Lower bound
Generalised isoperimetric constant of G :
| E(S , S c ) |
m  minS V ,|S |m
• 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)
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
• biased random walk, hits m exponentially
many times before hitting 0
• Upper and lower bounds
– match on complete graphs, hypercubes, dense ErdosRenyi 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
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?
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
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
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
How good is this lower bound?
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
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
Spread of influence
• Rumor-spreading and voter models are
• 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
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?
• 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
• Variety of stochastic processes on graphs can
be studied using elementary probabilistic
• Analysis can often be greatly simplified by
choosing the right model
• Often, exact analysis is intractable, but can get
good (?) bounds
• Many applications!

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