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**Rumors, consensus and epidemics on networks**

J. Ganesh University of Bristol

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**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?

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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.

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**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

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**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?

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Edge-driven model

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**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.

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**Analysis (cont.) Time to inform all nodes is**

Tn = (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

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**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 informed Mosk-Aoyama & Shah: Bound the time to inform all nodes, based on properties of G or R

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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

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**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)

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**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

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**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

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**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

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**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)

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**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?

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**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

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**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?

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**The voter model in pictures**

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**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

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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

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**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

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**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)

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**Duality with coalescing random walks**

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**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

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**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

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**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

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**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

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**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

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**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

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**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?

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SIS model in pictures

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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

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**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?

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**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 G dE[Y(t)]/dt = (A) EY(t) E[Y(t)] = exp((A) t) Y(0) : spectral radius of A

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**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)/()

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**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)

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**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

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**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

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**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?

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**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

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**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

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**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?

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**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

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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

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**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

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**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?

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**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

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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!

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