Rumors, consensus and epidemics on networks

A. J. GaneshUniversity 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?

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.

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

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?

Edge-driven model

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.

knknknk

TTE kk111

)(1][ 1

Analysis (cont.)

• Time to inform all nodes isTn = (TnTn1)+(Tn1Tn2)+...+(T2T1)+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,jV : 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 informedMosk-Aoyama & Shah: • Bound the time to inform all nodes, based on

properties of G or R

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

||||1min

)( ,

c

SjSi ijVS

SSn

rR

c

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 iS(k), jS(k) rij

• Time to inform one more node is stochastically dominated by Exp(k(nk)(R)/n)

• Implies that mean time to inform all nodes is bounded by 2log(n) / (R)

Examples

• G=Kn ; rij =1/n for all i,jV(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] = n1

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

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 matrixDo 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 possibleHow 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

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

Final state: complete graph case

• Each direction equally likely to be chosen.• So, 01 and 10 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, ij

• Define qii = ji 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(k1) directed edges between occupied nodes, rate 1/n on each edge

• Tk1 Tk Exp(k(k1)/n)

• Mean time to consensus bounded by n– linear in population size for consensus– logarithmic for rumor spreading

kknTTE kk

111][ 1

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

ijiAjAi

c

A qAA

)()(max

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 1p, 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

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

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?

Branching random walks

• Yi(t) : # of individuals at node i at time t+1 at rate ji Yj(t)1 at rate Yi(t)

• A : adjacency matrix of graph GdE[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 :

• 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)

|||),(|min ||, S

SSE c

mSVSm

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

Remarks

• Upper and lower bounds – 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

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 pij i : union bound

• (pA) es , where s is the initial infective• If p < 1, implies that mean number of nodes ever

infected (n)/(1p)• Upper bound can be improved to 1/(1p) 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

nodes:

How good is this lower bound?

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

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 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

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?

Results

• G=(V,E) is the d-regular random graph• Bernoulli initial condition : each node in state

1, with probability p, independent of othersTheorem (Balogh and Pittel) : Suppose 1<k<d-1.

There is a p*(0,1) such thatp>p*+ : all nodes eventually in state 1 whpp<p* : the fraction of nodes in state 0 tends to

a non-zero constant whp

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!