Fake news and rumors: a trigger for proliferation or fading away
FFake news and rumors: a trigger for proliferation or fading away
Ahad N. Zehmakan ∗ Department of Computer Science, ETH ZurichSerge GalamCEVIPOF-Centre for Political Research, Sciences Po and CNRS
Abstract
The dynamics of fake news and rumor spreading is investigated using a model with three kindsof agents who are respectively the Seeds, the Agnostics and the Others. While Seeds are the oneswho start spreading the rumor being adamantly convinced of its truth, Agnostics reject any kind ofrumor and do not believe in conspiracy theories. In between, the Others constitute the main partof the community. While Seeds are always Believers and Agnostics are always Indifferents, Otherscan switch between being Believer and Indifferent depending on who they are discussing with. Theunderlying driving dynamics is implemented via local updates of randomly formed groups of agents.In each group, an Other turns into a Believer as soon as m or more Believers are present in the group.However, since some Believers may lose interest in the rumor as time passes by, we add a flippingfixed rate < d < from Believers into Indifferents. Rigorous analysis of the associated dynamicsreveals that switching from m = 1 to m ≥ triggers a drastic qualitative change in the spreadingprocess. When m = 1 even a small group of Believers may manage to convince a large part of thecommunity very quickly. In contrast, for m ≥ , even a substantial fraction of Believers does notprevent the rumor dying out after a few update rounds. Our results provide an explanation on whya given rumor spreads within a social group and not in another, and also why some rumors will notspread in neither groups. Key Words: rumor spreading, bootstrap percolation, Galam model, threshold model.
Last years have witnessed the emergence of the new phenomenon denoted “fake news”, which has becomea worldwide major concern for many actors of political life. In particular, the impact of fake news ontwisting democratic voting outcomes has been claimed repeatedly to explain unexpected voting outcomesas Brexit and Trump victories. Fake news have been also identified during the 2017 presidential Frenchcampaign.Fake news has turned to an important form of social communications, and their spread plays asignificant role in a variety of human affairs. It can have a significant impact on people lives, distortingscientific facts and establishment of conspiracy theories..That has triggered the temptation in many countries to curb actual total and anonymous free speechin Internet by setting up new regulations to hinder the political influence of fake news. However, upto date no solid evidence has been found to demonstrate that fake news have indeed reverse a votingoutcome. It seems, that fake news are spreading among people having already made their political choice.In any case, to fight against fake news phenomena using the implementation of new judicial regulations ∗ Corresponding author; Email Address: [email protected], Postal Address: CAB G 39.3, Institute of Theo-retical Computer Science, ETH Zürich, Universitätstrasse 6, CH-8092 Zürich. a r X i v : . [ c s . D M ] M a y t is of importance to understand the mechanisms at work in their spreading. What eventually mattersis not the production of fake news but their capacity to spread quickly and massively. It is thereforeessential to understand the spreading dynamics.However, before challenging that goal we claim that the novelty of fake news is not their nature andcontent but the support of its spreading, namely Internet and social networks. Indeed, fake news areidentical to the old category of rumors, which were and still are spread through the process of word-of-mouth. Fake news have created a qualitative change of scale with respect to rumors thank to thenew technologies, which have democratized the production and reproduction of information, increasingdrastically the rate at which misinformation can spread. Therefore, control and possible handling tomanipulate information are now major issues in social organizations including economy, politics, defense,fashion, even personal affairs. The issue of fake news (rumors) spreading has become of a strategicimportance at all levels of society.Accordingly, we feel that to reach the core of the phenomenon it is more feasible to focus on thephenomenon of rumor spreading carried on by the word-of-mouth. We address the issue by building amathematical model along an already rich path of contributions from researchers with wide spectrum ofbackgrounds, like political sciences, statistical physics, computer science, and mathematics.In our proposed model, three kinds of agents are considered, the Seeds, the Agnostics and the Others.The Seeds are the initial spreaders of the rumor. They are adamantly convinced the rumor is true andno argument can make them renouncing to it. They trigger the spreading and keep on trying to convinceOthers to believe in the rumor. The
Others constitute the main body of the community. They areeither
Believers , people who are convinced the rumor is true, or
Indifferents , people who do not believethe rumor is true. Indifferents can turn into Believers if given convincing counter arguments. On theother hand, over time Believers might lose their belief and become Indifferents. The
Agnostics are notconcerned about the rumor. In addition, they do not believe in any rumors and reject conspiracy theoriesas a whole. They are always Indifferents to any rumor .The dynamics is studied rigorously using local updates in groups formed randomly through informalsocial gatherings occurring in offices, houses, bars, and restaurants at time break meetings like lunch,happy hours and dinner. Outcomes of those local updates depend on the number m of actual Believersin the group. A Seed contribute to the update as a Believer while an Agnostic acts as an Indifferent.The faith of the spreading is found to depend drastically on the value of m in addition to the actualproportions of Seeds and Agnostics.The rest of the paper is organized as follows. In Section 2, we introduce our model. Section 3 reviewsprior models of rumors propagation with which our model shares some attributes. Finally, in Section 4we provide our results by the rigorous analysis of the model. In m -rumor spreading model , we consider a community with N agents, which includes proportions ofrespectively s of Seeds, bN of Agnostics, and thus (1 − b ) N − s Others. At each discrete-time round t ≥ ,agents gather randomly in fixed rooms of different sizes from 2 to a constant L ≥ , where the numbersof seats in all rooms sum up to N , and then the agents update their opinion as follows: • An Other who is Indifferent and is in a room with at least m ( m ≥ ) Believers becomes a Believer • An Other, who is a Believer, becomes Indifferent independently with a probability < d < . • An Agnostic remains always an Indifferent. • A Seed remains always a Believer.Figure 1 illustrates one round of the process where a black/white square corresponds to a Seed/Agnosticand a black/white circle corresponds to a Believer/Indifferent.Assume that initially N individuals are Believers. We are interested in the sequence N , N , · · · ,where the random variable N t for t ≥ is the number of Believers after round t .2igure 1: One round of the process for m = 2 , N = 16 , b = 0 . , and d = 0 . . Two Indifferents turnBelievers, but one Believer becomes Indifferent for d = 0 . . The value of m is a function of different parameters, like the persuasiveness of the rumor or the tendencyof the community in believing that. We assume that m is a fixed positive integer. Recall that thereare s Seeds, who initiate the rumor. We assume that s is at least as large as m since m Believers areneeded to turn an Indifferent to a Believer. Furthermore, we always suppose that s is a constant whilelet N tend to infinity. That is, Seeds are a very small group. Moreover, b is assumed to be a smallconstant, say . which implies that of the community are Agnostics. We also assume that d is afixed small constant, say . , which means that a Believer loses interest in the rumor after rounds inexpectation and becomes an Indifferent. In the present paper, our main focus is devoted to demonstratehow switching from m = 1 to m ≥ triggers a drastic qualitative change in the spreading process, whichconsequently shed some light on the outcome of some real-world elections. Thus for our purpose, it isrealistic to assume that b ≤ . and d ≤ . . However, it is definitely desirable to analyze the m -rumorspreading model for larger values of d and b . We provide some intuition and analytic explanations inSection 5, but the rigorous analysis of the process in this setting is left for future work.For the sake of simplicity we suppose that all rooms are of size r for some constant r . However, ourresults carry on the general setting with rooms of different sizes by applying basically the same proofideas.Finally, we assume that r > m because otherwise no Indifferent could become a Believer. Note thatfor an Indifferent to become a Believer, it must share a room with at least m Believers which is notpossible for r ≤ m . The m -rumor spreading model defines a Markov chain on state space S = { j : s ≤ j ≤ (1 − b ) n } , wherestate j corresponds to having j Believers. (Note that we rule out states j < s and j > (1 − b ) n sincethere are s Seeds and bN Agnostics). Furthermore, the transition probability P jj (cid:48) , for j, j (cid:48) ∈ S , is theprobability of having j (cid:48) Believers in the next round given there are j Believers in the current round.Assume that we have j Believers for some s ≤ j < (1 − b ) N . With some non-zero probability, in thenext round at least one Indifferent becomes a Believer and all Believers remain unchanged. Thus, thereis a non-zero probability to reach state (1 − b ) N from state j for any s ≤ j ≤ (1 − b ) N . Furthermore,since with some non-zero probability all Believers choose to become Indifferent in the next round, exceptSeeds , there is a non-zero probability to reach state s from state j for any s ≤ j ≤ (1 − b ) N . Therefore,the process eventually reaches state s , where only Seeds believe in the rumor, or it reaches some state j ≥ N/ , where at least half of the community believe in the rumor. We say the rumor dies out in thefirst case and we say it takes over in the second one. We should mention that the values . and . are chosen to make our calculations more straightforward; otherwise,our results hold also for larger values of b and d , say b ≤ . and d ≤ . . However, this does not apply to any value of d and b as we discuss in Section 5. This is trivial since Seeds remain Believers forever by definition. Thus, from now on we will avoid mentioning it. .3 Our contribution Having the m -rumor spreading model in hand, two natural questions arise: what are the conditions forwhich a rumor takes over or dies out? and how fast does this happen? The main goal of the presentpaper is to address these two basic questions.First we consider the m -rumor spreading process for m = 1 and prove that if N ≥ s , then the rumortakes over in O (log N ) rounds asymptotically almost surely (a.a.s.) . This implies that if even initiallyonly Seeds, which are a group of constant size, believe in the rumor, for m = 1 the rumor takes overin logarithmically many rounds. However, by switching from m = 1 to m ≥ a very different pictureemerges. We prove that for m ≥ the rumor dies out in O (log N ) rounds a.a.s. if N ≤ αN for somesufficiently small constant α > . This means if even initially a constant fraction of the community areBelievers, for m ≥ the rumor dies out. Therefore, a drastic change occurs in the behavior of the processby switching from m = 1 to m ≥ , i.e., the process exhibits a threshold behavior.Let us illustrate this threshold behavior by a numerical example. Consider a community of size N = 10 and assume that b = 0 . , d = 0 . , r = 3 , and s = 100 . Let f ( n ) denote the expected numberof Believers in the next round by assuming that there are n Believers in the current round (we willprovide an exact formula for f ( n ) in Section 4, Equation (2)). One can observe in Figure 2 (left) that f ( n ) > n for s = 100 ≤ n ≤ ,
000 = N/ . Therefore, by starting from 100 Believers, in each roundthe number of Believers increases in expectation until there are N/ Believers, i.e., the rumor takes over.In Figure 2 (right), we present the expected percentage of Believers in the t -th round for ≤ t ≤ byassuming that N = 100 . It demonstrates that if initially only . of the community are Believers, theexpected percentage of Believers after 15 rounds is almost equal to . , which implies that we expectthe rumor to take over in less than 15 rounds. Note that the Believers can constitute at most ofthe community, but in that case of them, which is . of the whole community, would turn intoIndifferent in expectation since d = 0 . . This may explain where the value . comes from.Figure 2: (left) Variation of f ( n ) (right) the expected percentage of Believers in round t given N = 100 .Now, we consider the same setting but for m = 2 . Figure 3 (left) illustrates that f ( n ) < n for s = 100 ≤ n ≤ ,
000 = N/ . We also provide the variation of function f ( n ) − n in Figure 3 (middle),where it is easier to spot that f ( n ) < n . Thus, if initially even of the community believe in therumor, we expect the number of Believers decreases in each round until only the Seeds believe in therumor, i.e., it dies out. In Figure 3 (right), the expected percentage of Believers in the t -th round for ≤ t ≤ are drawn by assuming that N = 100 , . This explains that if initially even of thecommunity are Believers, the rumor still dies out in 69 rounds in expectation.Let us briefly discuss the intuition behind such a change in the behavior of m -rumor spreading process.Assume that initially Believers constitute a small fraction of the community. In that case, it is a very We say an event happens asymptotically almost surely whenever it happens with probability − o (1) while we let N tend to infinity. f ( n ) (middle) variation of f ( n ) − n (right) the expected percentage ofBelievers in round t given N = 100 , .unlikely that a room includes two or more Believers. Therefore, if m = 2 , then almost no “new” Believeris generated, i.e., no Indifferent become a Believer. Furthermore, each Believer becomes Indifferent withprobability d . Thus, we expect the number of Believers to decrease by a constant factor. On the otherhand, if m = 1 , then all Believers who are in different rooms can single-handedly turn all the agents intheir room into Believers. Thus, the number of Believers increases by an r factor. (Actually, this is abit smaller since we did not take Agnostics into account.) Note that in this case also a d fraction of theBelievers become Indifferent in expectation, but this is negligible since we assume that d is in order of . but r ≥ . Therefore, the number of Believers increases by a constant factor, in expectation. We analyze the model rigorously applying standard tools and techniques from probability theory. Wecompute the formula for the expected number of Believers as a function of the model parameters and thenumber of Believers in the previous round. This formula turns out to be quite involved. Thus, insteadwe work with suitable upper and lower bounds, which are easier to handle. Building on these bounds,we show that if m = 1 , in expectation the number of Believers increases by a constant factor in eachround until the rumor takes over. To turn such an expectation based argument into an a.a.s. statement,we exploit classical concentration inequalities, like the Chernoff bound and Azuma’s inequality. Sincethe number of Believers increases by a constant factor in each round, the rumor takes over in O (log N ) rounds. We prove that this bounds is asymptotically tight; that is, for some cases a.a.s. the rumor needs Ω(log N ) rounds to take over. A similar argument holds for the case of m ≥ , where in each round thenumber of Believers decreases by a constant factor until the rumor dies out. Within our model, for a given rumor, its success or failure in spreading depends merely on the value of m . The value m = 1 drives viral the rumor while it fades quickly for m ≥ . The actual value of m isexpected to be a function of both the rumor content and the social characteristics of the community inwhich it is launched. Therefore the same rumor may spread in a community and fade away in another.And of two different rumors within a community, one may spread and the other vanishes.In [7] those differences were given an explanation in terms of the existence of different prejudices andcognitive biases, which were activated at ties in even discussing groups with otherwise a local majorityrule update. Here we provide another explanation for this phenomena, based on the value of m in thediscussing group, which does not obey a local majority rule for shifting opinions. It is worth to mentionthat our current approach could in principle be extended to account for a mixture of arguments withdifferent convincing power leading to a combination of groups with m = 1 and m ≥ . The m = 1 caseembodies the promptness of an agent to believe in a given rumor or to the strength of the argument. Thesame agent may requires m ≥ to adopt another rumor as true either due to weaker arguments or thecontent of the rumor. 5he fact that the actual value of m can vary from one social group to another with respect to the samerumor can be illustrated using some recent real cases of differentiated rumor spreading. One significantcase relates to the terrorist attack in Strasbourg, France, on the eve of December 11, 2018. The terrorist,a follower of Islamic State, claimed five person lives and wounded eleven persons, before being shotby the police a few days latter. With the shooting occurring amid the uprising of the Yellow Jackets(Gilets Jaunes) movement, a conspiracy theory emerged at once and spread quite quickly among a specificcommunity of French people, namely the ones identifying with the Yellow Jacket movement, claimingthe attack was set up by the French government to distract support to the on going social movement.At the same time the rumor did not take over among the rest of the French population as shown by thefollowing poll figures. The statement? Evidences are not clear about who committed the attack or theattack was a set up by French government? was found to be agreed on by 42% of yellow jacket activistsand 23% of yellow jacket supporters against only 11% within non supporters of yellow jackets.A similar feature is exhibited with an American case about supposed Russian interference in lastAmerican Presidential election. More specifically, attributing the hacking of Democratic e-mails to Russiawas supported by 87% of Clinton voters against only 20% of Trump voters. As mentioned, the question of how a rumor spreads in a community has been studied extensively indifferent contexts and numerous models have been introduced and investigated, both theoretically andexperimentally, see e.g. [3, 15, 17, 18, 20, 21]. We shortly discuss some results regarding Galam model, DKmodel, and bootstrap percolation model, which are arguably the closest ones to ours.
Consider the relationship network between the people in a community and assume that initially eachagent is a Believer or an Indifferent toward a rumor. In m -bootstrap percolation, an Indifferent becomesa Believer as soon as m of its connections in the network are Believers and remains a Believer forever.The main question in this context is to determine the minimum number of agents who must be Believersinitially to guarantee that the rumor takes over the whole network eventually (that is, all agents becomeBelievers). This has been studied on different network structures, like hypercube [2], lattice [13], randomgraphs [11, 16], and many others. In contrast to the bootstrap percolation model, which assumes thatthe agents are sitting on the nodes of an underlying network, in our model agents gather in groups ofdifferent sizes at random. In this sense, our model is more similar to Galam model, introduced in thenext section. The Galam model was introduced by Galam [6] to describe the spreading of a minority opinion resultingin democratic rejection of social reforms initially favored by a majority. In this model, initially eachagent is positive or negative regarding a reform proposal. Then, in each discrete-time round they aredistributed randomly in groups of different sizes and everyone adopts the most frequent opinion in itsgroup; in case of a tie, negative opinion is chosen. Building on this model, Galam [6] provided someillustrations regarding the output of some real-world elections. Later, several extensions of the modelwere proposed and studied, for example by adding contrarian effect [8], introducing random tie-breakingrule [9, 10], considering three competing opinions [12], and defining the level of activeness [19].
Another well-established rumor spreading model is DK model, introduced by Daley and Kendall [4].There are essentially two differences between DK model and our model. Firstly in DK model, it isassumed that the gatherings are always of size two. Furthermore, a Believer stops spreading the rumoras soon as it encounters another Believer, in DK model. The illustration for such an updating rule is6hat the agent decides that it is no longer “news”, and thus stops spreading it. In that sense, our model isperhaps more similar to the variant introduced by Zhao et al. [22], where a Believer becomes Indifferentindependently with some probability d > . In this section we rigorously analyze the behavior of the m -rumor spreading model. The goal is to proveTheorem 4.1 which is the main contribution of the present paper. Theorem 4.1
In the m -rumor spreading model(i) for m = 1 if N ≥ s , the rumor takes over(ii) for m = 2 if N ≤ αN , the rumor dies outasymptotically almost surely in O (log N ) rounds, where α > is a sufficiently small constant. Theorem 4.1 states that by switching from m = 1 to m ≥ , the process behaves substantiallydifferently. For m = 1 , the rumor takes over if even Others are all Indifferent initially. On the otherhand, for m = 2 the rumor dies out even though a constant fraction of the community believe in therumor already. They both occur in O (log N ) rounds, which is shown to be asymptotically tight.We first set up some basic definitions and state some standard concentration inequalities in Section 4.1.Then, we provide the proof of Theorem 4.1 part (i) and part (ii) respectively in Sections 4.2 and 4.3. Recall that random variable N t for t ≥ denotes the number of Believers at the end of round t . Theconditional random variable N t | N t − = n , called “ N t given N t − = n ”, has probability mass function P r [ N nt = n (cid:48) ] = P r [ N t = n (cid:48) | N t − = n ] , where we shortly write N nt for N t | N t − = n . Furthermore, E [ N nt ] denotes the expected value of random variable N nt .For m = 1 , we show that E [ N nt ] > βn , for some constant β > , if s ≤ n ≤ N/ . Therefore, oneexpects the rumor to take over in logarithmically many rounds. Furthermore for m = 2 , E [ N nt ] < β (cid:48) n ,for some constant β (cid:48) < , if s < n ≤ αN . Thus, by starting from αN Believers we expect the number ofBelievers decreases by a constant factor in each round, i.e., the rumor dies out in a logarithmic numberof rounds. This should intuitively explain why Theorem 4.1 holds. However, to turn this expectationbased argument into a formal proof, we need to show that random variable N nt is sharply concentratedaround its expectation and apply some careful calculations. In the rest of this section, we first computethe value of E [ N nt ] and then provide some basic concentration inequalities, which are later used to provethat N nt is concentrated around E [ N nt ] .To calculate E [ N nt ] , let us first determine the probability that an Indifferent agent I becomes aBeliever in the t -th round given N t − = n . We claim that this probability is equal to N r − (cid:88) j = m (cid:18) nj (cid:19)(cid:18) N − n − r − j − (cid:19) ( r − N − r )! /N ! . (1)There are N ! possibilities to assign N agents to N seats. Let us call each of these assignments a permuta-tion . In Equation (1), the value N ! in the denominator stands for the number of all possible permutationsand the numerator is equal to the number of permutations for which agent I becomes a Believer. Thereare N possibilities for fixing I ’s seat, and the sum corresponds to the number of possibilities to assign theremaining N − agents such that there are at least m Believers in I ’s room. For that, one can chooseexactly j Believers, for some m ≤ j ≤ r − , among n Believers to share the room with agent I , and the Whenever we talk about an Indifferent becoming a Believer, we clearly mean an Other who is an Indifferent; otherwise,an Agnostic never becomes a Believer by definition. r − j − agents among the N − n − Indifferents. After theselection of the r − agents who are in the same room with I , there are ( r − possibilities of assigningthem to the seats and also ( N − r )! possibilities to assign the other agents to the ( N − r ) remaining seats.Each Believer remains a Believer independently with probability − d if it is an Other and withprobability 1 if it is one of the s Seeds. Furthermore, each of the (1 − b ) N − n Indifferents becomes aBeliever with the probability given in Equation (1) (while each of the bN Agnostics remains an Indifferent).Therefore, we have E [ N nt ] = s + ( n − s )(1 − d ) + ((1 − b ) N − n ) ( r − (cid:80) r − j = m (cid:0) nj (cid:1)(cid:0) N − n − r − j − (cid:1)(cid:81) r − j =1 ( N − j ) . (2)The above formula is a bit involved; thus, we sometimes will utilize relatively simpler formulas, whichare easier to handle, as lower and upper bounds in our proofs.Let us recall three basic concentration inequalities which we will apply repeatedly later. Theorem 4.2 (Markov’s inequality [5]) Let X be a non-negative random variable and a > , then P r [ X ≥ a ] ≤ E [ X ] a . Theorem 4.3 (Chernoff bound [5]) Suppose x , · · · , x k are independent Bernoulli random variables tak-ing values in { , } and let X denote their sum, then for ≤ (cid:15) ≤ P r [(1 + (cid:15) ) E [ X ] ≤ X ] ≤ exp( − (cid:15) E [ X ]3 ) and P r [ X ≤ (1 − (cid:15) ) E [ X ]] ≤ exp( − (cid:15) E [ X ]2 ) . Theorem 4.4 (Azuma’s inequality [5]) Let x , · · · , x k be an arbitrary set of random variables and let f be a function satisfying the property that for each i ∈ [ k ] := { , · · · , k } , there is a non-negative c i suchthat | E [ f | x i , · · · , x ] − E [ f | x i − , · · · , x ] | ≤ c i . Then, for constant (cid:15) > P r [(1 + (cid:15) ) E [ f ] < f ] ≤ exp( − (cid:15) E [ f ] c ) = exp( − Θ( E [ f ] c )) and P r [ f < (1 − (cid:15) ) E [ f ]] ≤ exp( − (cid:15) E [ f ] c ) = exp( − Θ( E [ f ] c )) where c := (cid:80) ki =1 c i . Note that although Markov’s inequality does not need any sort of independence, the Chernoff boundrequires X to be the sum of independent random variables. Azuma’a inequality also requires some levelof independence in the sense that to get a reasonable tail bound, the value of c has to be small.We define the random variable N nt as a function of N Bernoulli random variables and apply Azuma’sinequality to attain Corollary 4.5.
Corollary 4.5
In the m -rumor spreading model, for any (cid:15) > P r [(1 + (cid:15) ) E [ N nt ] < N nt ] ≤ exp( − Θ( E [ N nt ] N )) and P r [ N nt < (1 − (cid:15) ) E [ N nt ]] ≤ exp( − Θ( E [ N nt ] N )) . roof. We prove the first inequality, and the proof of the second one is analogous. Consider an arbitrarylabeling from to N on the seats. We define Bernoulli random variable x ni ( t ) for ≤ i ≤ N to be 1 ifand only if the agent assigned to the i -th seat is a Believer given N t − = n . Then, for any ≤ i ≤ N | E [ N nt | x ni ( t ) , · · · , x n ( t )] − E [ N nt | x ni − ( t ) , · · · , x n ( t )] | ≤ L (cid:48) where L (cid:48) is a constant as a function of L . Now by applying Theorem 4.4, we have P r [(1 + (cid:15) ) E [ N nt ] < N nt ] ≤ exp( − Θ( E [ N nt ] (cid:80) Ni =1 L (cid:48) )) = exp( − Θ( E [ N nt ] N )) . (cid:3) m = In this section, we prove that for m = 1 if N ≥ s , then the rumor takes over a.a.s. in O (log N ) rounds.We divide our analysis into three phases. We show that by starting with s Believers, the process reaches astate with at least
Ω(log log log N ) Believers (phase 1), then it reaches a state with at least N / Believers(phase 2), and finally at least half of the community will become Believers (phase 3). Furthermore, eachof these three phases takes O (log N ) rounds. As discussed, the main idea of the proof is to show thatthe number of Believers increases by a constant factor in expectation in each round and then applythe fact that N nt is sharply concentrated around its expectation. However, there are two subtle issueswhich make us to split our analysis into three phases. Firstly, the exact formula for E [ N nt ] is quiteinvolved, see Equation (2). Therefore, we sometimes have to work with relatively simpler lower/upperbounds, which are technically easier to handle, but hold only for a particular range of n . Secondly, theerror probability provided by Corollary 4.5 is of form exp( − Θ( E [ N nt ] N )) , which is equal to a constant for E [ N nt ] = O ( √ N ) . Therefore, for some settings we need to bound the error probability by applying othertechniques. Furthermore, we should mention that the values log log log N and N / are selected in a wayto make the calculations straightforward, otherwise the proof works for some other values as well. Phase 1.
We prove that by starting from N ≥ s , there are at least log c log log N Believers after O (log N ) rounds a.a.s., where c > is a constant to be determined later. Let us first provide Claim 4.6, whose proof is given below.
Claim 4.6
Assume that T = log r log c log log N and ≤ n ≤ log c log log N , then for t (cid:48) ≥ , P r [ N t (cid:48) + T ≥ r T n | N t (cid:48) = n ] ≥ √ log N .
Claim 4.6 implies that from a state with n ≥ Believers, after T rounds the process reaches a state withat least r T n ≥ r T = log c log log N Believers with probability at least / √ log N . The probability that theprocess runs for T log / N rounds and never reaches log c log log N Believers or more is upper-boundedby (1 − √ log N ) log / N ≤ exp( − log / N ) = o (1) where we used − x ≤ exp( − x ) . Thus, for some t ≤ T log / N ≤ log N the process reaches at least log c log log N Believers with probability − o (1) . We assume that all logarithms are to base e , otherwise we point out explicitly. roof of Claim 4.6. Let us first compute
P r [ N nt = rn ] , which is the probability that there are rn Believers in the t -th round given N t − = n , for some n ≤ bN . We define E to be the event that inround t each Believer is assigned to a room with r − Others who are all Indifferents given N t − = n .Furthermore, E is the event that no Believer becomes Indifferent in round t given N t − = n . The numberof Believers will increase to rn in round t if and only if both events E and E occur. That is, P r [ N nt = rn ] = P r [ E ] · P r [ E ] ≥ P r [ E ] · (1 − d ) n (3)where we used that events E and E are independent and P r [ E ] = (1 − d ) n − s ≥ (1 − d ) n . We claim that P r [ E ] = (cid:18) N/rn (cid:19) n ! r n (cid:18) (1 − b ) N − n ( r − n (cid:19) (( r − n )!( N − rn )! /N ! where the value N ! in the denominator stands for the number of all possible permutations and thenumerator is equal to the number of permutations for which the event E occurs. The value (cid:0) N/rn (cid:1) is thenumber of possibilities of selecting the rooms which contain the Believers. Moreover, there are n ! r n waysof placing n Believers in these n rooms such that each of the rooms includes one Believer. We need tochoose ( r − n Indifferents among all (1 − b ) N − n Others who are Indifferents to fill in the remaining ( r − n seats in these rooms. There are (( r − n )! possible assignments of the ( r − n chosen agentsinto the ( r − n seats. Finally, there are ( N − rn )! possibilities to place the remaining N − rn agents.Now, we approximate this probability. Since (cid:0) nk (cid:1) ≥ ( nk ) k by Stirling’s approximation [14] and k ! ≥ ( k/e ) k , we get P r [ E ] ≥ ( Nrn ) n ( ne ) n r n ( (1 − b ) N − n ( r − n ) ) ( r − n ( ( r − ne ) ( r − n (cid:81) rn − j =0 ( N − j ) . We have (1 − b ) N − n ≥ (1 − b ) N for n ≤ bN . Thus, by applying (cid:81) rn − j =0 ( N − j ) ≤ N rn and b ≤ . ,we get P r [ E ] ≥ (1 − b ) ( r − n exp( rn ) ≥ exp( − rn ) . Combining this inequality and Equation (3) yields
P r [ N nt = rn ] ≥ exp( − rn ) · (1 − d ) n ≥ c − n (4)for some constant c > (which is the constant that we promised to determine later).The probability P r [ N t (cid:48) + T ≥ r T n | N t (cid:48) = n ] is equal to T (cid:89) t =1 P r [ N t (cid:48) + t = r t n | N t (cid:48) + t − = r t − n ] ≥ T (cid:89) t =1 c − r t − n ≥ c − r T n T where we applied Equation (4). (To apply Equation (4) we need the inequality r t − n ≤ bN to hold, whichis the case for t ≤ T = log r log c log log N and n ≤ log c log log N .) Now, by plugging in n ≤ log c log log N and T = log r log c log log N implies that P r [ N t (cid:48) + T = r T n | N t (cid:48) = n ] ≥ c − (log c log log N ) ≥ √ log N .
This finishes the proof of Claim 4.6. (cid:3)
Phase 2.
So far we proved that a.a.s. after t rounds, for some t ≤ log N , there are at least n =log c log log N Believers. Now, we show that from a state with at least n Believers, a.a.s. the processreaches a state with at least N / Believers after O (log N ) rounds. In this phase we always assume that n ≤ n ≤ N .Let us define E to be the event that at least n Indifferents become Believers in round t given N t − = n . Furthermore, E is the event that at most n Believers become Indifferents in round t given N t − = n . We first provide lower bounds on the probabilities P r [ E ] and P r [ E ] , respectively in Claim 4.7and Claim 4.8. 10 laim 4.7 P r [ E ] ≥ − exp( − Θ( N )) . Proof.
Assume that there are n Believers. Let R be the set of permutations where at least n Believersshare their room with at least another Believer. Furthermore, define R to be the set of permutationswhere there are at least n Believers who share their room with ( r − Agnostics. We claim that
P r [ E ] ≥ − | R | + | R | N ! . (5)This is true because if a permutation is not in R ∪ R , then there are at least n − ( n + n ) = n Believers who share their room with no Believer but at least one Other who is Indifferent. Therefore,each of these n Believers single-handedly turns an Indifferent into a Believer. That is, at least n Indifferents become Believers.There are (cid:0) nn/ (cid:1) possibilities to choose n Believers. We have (cid:0)
N/rn/ (cid:1) possibilities for the selection of n rooms. Furthermore, ( rn ) n is an upper bound on the number of placements of these n Believersin the rn seats in the selected rooms. Finally, there are ( N − n )! possibilities to place the remainingagents. Clearly, in this way we count each permutation which is in R at least once. Therefore, we have | R | N ! ≤ (cid:0) nn/ (cid:1)(cid:0) N/rn/ (cid:1) ( rn ) n ( N − n )! N ! . By applying (cid:0) nk (cid:1) ≤ ( nek ) k , Stirling’s approximation [14], we get | R | N ! ≤ (20 e ) n ( eNrn ) n ( rn ) n (cid:81) n − j =0 ( N − j ) ≤ (20 e ) n ( eNrn ) n ( rn ) n ( N ) n . In the last step we used the fact that N − j ≥ N for ≤ j ≤ n − . By some simplifications we get | R | N ! ≤ ( (20 e ) × e × r × × ) n ( nN ) n ≤ exp( − Θ( n )) (6)where we used that nN = o (1) for n ≤ N .To bound | R | , we select n Believers, r − n Agnostics, and n rooms. This can be done in (cid:0) n n/ (cid:1)(cid:0) bN r − n/ (cid:1)(cid:0) N/r n/ (cid:1) different ways. There are ( n )! r n possibilities to place the Believers in theserooms such that each of the rooms has exactly one Believer. Finally, there are ( r − n )! possibilities toplace the Agnostics in the remaining r − n seats in these rooms and ( N − rn )! possibilities to placethe remaining agents. In this way, we count a permutation in R at least once. Therefore, we have | R | N ! ≤ (cid:0) n n/ (cid:1)(cid:0) bN r − n/ (cid:1)(cid:0) N/r n/ (cid:1) ( n )! r n ( r − n )!( N − rn )! N ! . Note that N − j ≥ N for ≤ j ≤ rn − and n ≤ N . This implies that ( N − rn )! N ! = 1 (cid:81) rn − j =0 N − j ≤ N ) rn . By applying this inequality and (cid:0) nk (cid:1) ≤ ( nek ) k and n ! ≤ e √ n ( ne ) n , Stirling’s approximation [14], we get | R | N ! ≤ ( e ) n ( (10 e )( bN )8( r − n ) r − n ( eN rn ) n e (cid:113) n ( n e ) n r n e (cid:113) r − n ( r − n e ) r − n ( N ) rn
11y some simplifications, we have | R | N ! ≤ ( 10 e b r − ( 2019 ) r ) n e (cid:114) ( r − n = exp( − Θ( n )) . (7)In the last step, we used that e b r − ( ) r is a constant strictly smaller than 1. This is true because b isa small constant, say b ≤ . (see Section 2.1).To finish the proof, combining Equations (5), (6), and (7) yields P r [ E ] ≥ − exp(Θ( n )) (cid:3) Claim 4.8
P r [ E ] ≥ − exp( − Θ( N )) . Proof.
Define random variable X nt to be the number of Believers who become Indifferent in round t given N t − = n . Consider an arbitrary labeling from to n on the Believers and define Bernoulli randomvariable x i for ≤ i ≤ n to be 1 if and only if the i -th Believer becomes Indifferent. Recall that eachBeliever becomes an Indifferent with probability d independently in round t , except Seeds who remainBelievers. Thus, P r [ x i = 1] ≤ d . Clearly, X nt = (cid:80) ni =1 x i and E [ X nt ] = ( n − s ) d ≤ nd . Note that x i s areindependent; thus, by applying Chernoff bound (see Theorem 4.3) we get P r [ X nt ≥ n ] ≤ P r [ X nt ≥ (1 + (cid:15) ) dn ] ≤ P r [ X tn ≥ (1 + (cid:15) ) E [ X nt ]] ≤ exp( − (cid:15) E [ X nt ]3 ) = exp( − Θ( n )) where we choose the constant (cid:15) > sufficiently small to satisfy (1 + (cid:15) ) d ≤ . This is doable since weassume that d ≤ . , see Section 2.1. Therefore, P r [ E ] = P r [ X nt < n ] = 1 − P r [ X nt ≥ n ] = 1 − exp( − Θ( n )) . (cid:3) If both events E and E occur, then the number of Believers will increase from n to at least n + n − n = (1 + ) n . Since E and E are independent, by applying Claims 4.7 and 4.8 we get P r [ N nt ≥ n ] ≥ P r [ E ∩ E ] = P r [ E ] · P r [ E ] = 1 − exp( − Θ( n )) ≥ − c − n (8)for some constant c > .Equation (8) states that in each round the number of Believers increases by a constant fraction witha probability converging to one. Building on this statement and applying the union bound, we show thata.a.s. in logarithmically many rounds the number of Believers overpasses N .Recall that in phase 1 we proved that there is some t ≤ log N so that N t ≥ n a.a.s. for n =log c log log N . Claim 4.9
Assume that ( ) t − n < N for some positive integer t . Then, given N t ≥ n , we have N t + t ≥ ( ) t n with probability at least − (cid:80) t − t (cid:48) =0 c − ( ) t (cid:48) n . Proof.
Let us define the events A t := { N t ≥ ( 4140 ) t n } t ≥ A (cid:48) t := { N t = ( 4140 ) t n } t ≥ .
12y definition, for ≤ t (cid:48) ≤ t we have P r [ A t | A (cid:48) t (cid:48) ] ≤ P r [ A t | A t (cid:48) ] . We apply this inequality several timeslater.The goal is to prove that for t ≥ P r [ A t | A ] ≥ − t − (cid:88) t (cid:48) =0 c − ( ) t (cid:48) n . We apply proof by induction. As the base case, for t = 1 we have P r [ A | A ] ≥ P r [ A | A (cid:48) ] Eq. (8) ≥ − c − n . As the induction hypothesis (I.H.), assume that
P r [ A t − | A ] ≥ − (cid:80) t − t (cid:48) =0 c − ( ) t (cid:48) n . Now, we have P r [ A t | A ] ≥ P r [ A t | A ∩ A t − ] · P r [ A t − | A ] ≥ P r [ A t | A t − ] · P r [ A t − | A ] ≥ P r [ A t | A (cid:48) t − ] · P r [ A t − | A ] Eq. (8) ≥ (1 − c − ( ) t − n ) · P r [ A t − | A ] I.H. ≥ (1 − c − ( ) t − n )(1 − t − (cid:88) t (cid:48) =0 c − ( ) t (cid:48) n ) ≥ − t − (cid:88) t (cid:48) =0 c − ( ) t (cid:48) n . We need the assumption of ( ) t − n < N to apply Equation (8). (cid:3) We have that ( ) t n ≥ N for t = log N which implies that there exists some t = O (log N ) suchthat ( ) t n ≥ N and ( ) t − n < N . Therefore by applying Claim 4.9, given N t ≥ n , we have N t + t ≥ N with probability at least − t − (cid:88) t (cid:48) =0 c − ( ) t (cid:48) n . To bound this probability, let t (cid:48)(cid:48) = log log c log N . Then, t − (cid:88) t (cid:48) =0 c − ( ) t (cid:48) n ≤ t (cid:48)(cid:48) − (cid:88) t (cid:48) =0 c − ( ) t (cid:48) n + t − (cid:88) t (cid:48) = t (cid:48)(cid:48) c − ( ) t (cid:48) n ≤ t (cid:48)(cid:48) c − n + t c − ( ) t (cid:48)(cid:48) n . Since t = O (log N ) and n = log c log log N , this is upper-bounded by O (log log log N ) c − log c log log N + O (log N )(log N ) − log c log log N = o (1) + o (1) = o (1) . Hence, given N t ≥ n , we have N t + t ≥ N for some t = O (log N ) with probability − o (1) .Overall by phases 1 and 2, starting from s Believers, a.a.s. the process reaches at least N Believersin a logarithmic number of rounds.
Phase 3.
So far we showed that after t + t = O (log N ) rounds a.a.s. there are at least N / Believers.Now, we prove that from a state with at least N / Believers, more than half of the community willbecome Believers (i.e., the rumor takes over) in logarithmically many rounds. We assume that n ≤ N/ in this phase.Let us first lower-bound the value of E [ N nt ] in this setting. By Equation (2), we have E [ N nt ] ≥ (1 − d ) n + ((1 − b ) N − n ) ( r − (cid:80) r − j =1 (cid:0) nj (cid:1)(cid:0) N − n − r − j − (cid:1)(cid:81) r − j =1 ( N − j )
13o simplify this bound, we compute ( r − (cid:80) r − j =1 (cid:0) nj (cid:1)(cid:0) N − n − r − j − (cid:1)(cid:81) r − j =1 ( N − j ) = ( r − (cid:0) N − r − (cid:1) − (cid:0) N − n − r − (cid:1) ) (cid:81) r − j =1 ( N − j ) =( r − (cid:81) r − j =1 ( N − j )( r − (cid:81) r − j =1 ( N − j ) − ( r − (cid:81) r − j =1 ( N − n − j )( r − (cid:81) r − j =1 ( N − j ) = 1 − r − (cid:89) j =1 N − n − jN − j ≥ − N − n − N − ≥ − N − nN = nN . Therefore, E [ N nt ] ≥ (1 − d ) n + ((1 − b ) N − n ) nN = (2 − d − b − nN ) n ≥ ( 32 − d − b ) n where we applied n ≤ N/ in the last step. Recall that we always assume that d and b are small constants,say d ≤ . and b ≤ . . (See Section 2.1) Thus, E [ N nt ] ≥ (1 + δ ) n for some small constant δ > . This implies that in expectation the number of Believers is multiplied by δ in each round. Thus, we expect the rumor to take over in a logarithmic number of rounds. In thefollowing, we prove that this expected behavior occurs a.a.s.By applying the second part of Corollary 4.5 for (cid:15) = δ/ and using (1 − δ )(1 + δ ) ≥ (1 + δ ) , we have P r [ N nt ≤ (1 + δ n ] ≤ P r [ N nt ≤ (1 − δ δ ) n ] ≤ P r [ N nt ≤ (1 − δ E [ N nt ]] ≤ exp( − Θ( E [ N nt ] N )) ≤ exp( − Θ( ((1 + δ ) n ) N )) = exp( − Θ( n N )) . Thus, there exists some constant c > such that P r [ N nt ≥ (1 + δ n ] ≥ − c − n N . (9)Equation (9) states that given N t − = n , for some n = ω ( √ N ) , we have a.a.s. N t ≥ (1 + δ ) n . Usingthis statement and the union bound we prove that a.a.s. after a logarithmic number of round the rumortakes over. Claim 4.10
Assume that (1 + δ ) t − N < N for some positive integer t . Then, given N t + t ≥ N , wehave N t + t + t ≥ (1 + δ ) t N with probability at least − (cid:80) t − t (cid:48) =0 c − (1+ δ ) t (cid:48) N . Proof.
The proof is analogous to the proof of Claim 4.9 by defining the events A t and A (cid:48) t respectively tobe { N t ≥ (1+ δ ) t N } and { N t = (1+ δ ) t N } . Furthermore, we use Equation (9) instead of Equation (8).Note that we need the condition (1 + δ ) t − N < N to apply Equation (9). (cid:3) Since (1 + δ ) t N ≥ N for t = log δ N , there exists some t = O (log N ) such that (1 + δ ) t N ≥ N and (1 + δ ) t − N < N . Thus by applying Claim 4.10, given N t + t ≥ N , we have N t + t + t ≥ N withprobability at least − t − (cid:88) t (cid:48) =0 c − (1+ δ ) t (cid:48) N ≥ − t exp( − Θ( N )) ≥ − O (log N ) exp( − Θ( N )) which is equal to − o (1) . Thus, given N t + t ≥ N a.a.s N t + t + t ≥ N for some t = O (log N ) .Overall, combining phases 1, 2, and 3 implies that by starting from s Believers the rumor takes overa.a.s. in O (log N ) rounds. 14 emark 4.11 The error probability in Equation (9) is of the form exp( − Θ( n N )) , which is bounded bya constant for n = O ( √ N ) . This should explain why in phases 1 and 2, where n ≤ N , we applied adifferent approach. However, our argument works for any choice of N + (cid:15) (cid:48) for small (cid:15) (cid:48) > instead of N . Tightness.
We claim that the upper bound of O (log N ) is tight. Note that starting with s Believers,the number of Believers is multiplied by at most an r factor in each round. Thus, it takes the rumor atleast log r N s = Ω(log N ) to take over. m ≥ In this section, we analyze the m -rumor spreading model for m ≥ . We prove that if even initiallyBelievers constitute a constant fraction of the community (i.e., N = αN for some small constant α > ),the rumor dies out a.a.s. in O (log N ) rounds. We divide our analysis into two phases. In phase 1, weshow that starting from αN Believers, the process reaches a state with less than F ( N, d ) := log d log log N Believers a.a.s. in O (log N ) rounds. Then in phase 2, we prove that from a state with at most F ( N, d ) Believers, all agents will become Indifferents, except Seeds, a.a.s. in logarithmically many rounds. Weshould mention that value of F ( N, d ) has been chosen to make the calculations straightforward. Phase 1.
We show that from a state with at most αN Believers, the process reaches at most F ( N, d ) Believers in logarithmically many rounds a.a.s. In this phase we always assume that F ( N, d ) ≤ n ≤ αN .The main idea is to show that in this setting in each round at most dn Indifferents become Believersand at least dn Believers become Indifferents. Thus, the number of Believers decreases by a constantfactor in each round. To make this argument more formal, let us define E to be the event that at most dn agents switch from Indifferent to Believer in round t given that N t − = n . Furthermore, let E be theevent that the number of Believers who become Indifferents is at least dn in round t given N t − = n .We first bound the probabilities P r [ E ] and P r [ E ] respectively in Claim 4.12 and Claim 4.13. Buildingon them, we provide a lower bound on the probability P r [ N nt ≥ (1 − d ) n ] . Claim 4.12
P r [ E ] ≥ − exp( − Θ( n )) . Proof.
We claim that
P r [ E ] ≤ (cid:0) Nrdn r (cid:1) (cid:0) n dn r (cid:1) ( dn r )!2 dn r ( r ( r − dn r ( N − dn r )! N ! where E is the complement of event E . The numerator is an upper bound on the number of permutationswhich result in the generation of at least dn new Believers (Indifferents who become Believers) out ofall possible N ! permutations. Note that in such permutations, there must exist dn r rooms such that eachof them contains at least two Believers, otherwise less than dn Indifferents become Believers for m ≥ .There are (cid:0) Nrdn r (cid:1) possibilities to select dn r rooms and (cid:0) n dn r (cid:1) ways to choose dn r Believers. We claim that thenumber of possibilities to assign to each of these rooms exactly two of the Believers is equal to ( dn r )! / dn r .This is true because the number of choices to pair dn r agents is ( dn r )!( dn r )! 2 dn r (see e.g. [1]) and there are ( dn r )! ways of distributing these pairs into the rooms; thus, the number of all possibilities is equal to ( dn r )!( dn r )! 2 dn r ( dn r )! = ( dn r )!2 dn r . Finally, there are r ( r − possible placements for two Believers in a room of size r and there are ( N − dn r )! possible placements for the remaining agents. In this way, we clearly count all our desired permutationsat least once (we are actually over-counting, but this is fine since we need an upper bound).15y (cid:0) nk (cid:1) ≤ ( nek ) k , Stirling’s approximation [14], we get P r [ E ] ≤ ( eNdn ) dn r ( erd ) dn r ( dn r ) dn r r dn r dn r (cid:81) dn r − j =0 ( N − j ) ≤ ( eNdn ) dn r e dn r n dn r r dn r ( N ) dn r =( 2 e r d ) dn r ( nN ) dn r ≤ ( 2 e r d ) dn r α dn r = exp( − Θ( n )) . where we used n/N ≤ α . Furthermore, we applied twice that α is a sufficiently small constant. Firstly, wehave ( N − j ) ≥ N for ≤ j ≤ dn r − by utilizing dn r − ≤ N , which is true for α ≤ r d = r d . Secondly,we applied α < d e r in the last step. Thus, we need constant α to be smaller than min( r d , d e r ) ,which holds since we assume α to be a sufficiently small constant. (cid:3) Claim 4.13
P r [ E ] ≥ − exp( − Θ( n )) . Proof.
Define random variable X nt to be the number of Believers who switch into Indifferents in round t given N t − = n . Recall that each Believer becomes Indifferent with probability d independently, exceptSeeds. Consider an arbitrary labeling from to n on the Believers and define Bernoulli random variable x i for ≤ i ≤ n to be 1 if and only if the i -th Believer becomes Indifferent. We know that P r [ x i = 1] is equal to d if the i -the agent is Other and 0 if it is a Seed. Clearly, X nt = (cid:80) ni =1 x i , which implies that E [ X nt ] = ( n − s ) d . Thus, dn ≤ E [ X tn ] by using sd ≤ n , which is true since s is a small constant and n ≥ F ( N, d ) = log d log log N . Notice that x i s are independent; thus, by applying Chernoff bound (seeTheorem 4.3) we have P r [ E ] = P r [ 34 dn ≤ X nt ] = P r [(1 −
16 ) 910 dn ≤ X nt ] ≥ P r [(1 −
16 ) E [ X nt ] ≤ X nt ] =1 − P r [ X nt < (1 −
16 ) E [ X nt ]] ≥ − exp( − ( ) E [ X nt ]2 ) ≥ − exp( − ( ) dn which is equal to − exp( − Θ( n )) . (cid:3) If both events E and E occur, at most dn Indifferents become Believers and at least dn Believersbecome Indifferents, which implies that the number of Believers decreases from n to (1 − d ) n . Since,events E and E are independent, applying Claims 4.12 and 4.13 yields P r [ N nt ≤ (1 − d n ] ≥ P r [ E ∩ E ] = P r [ E ] · P r [ E ] ≥ (1 − exp( − Θ( n ))) (1 − exp( − Θ( n ))) = 1 − exp( − Θ( n )) . Therefore, there exists some constant c > such that for F ( N, d ) ≤ n ≤ αn , we have P r [ N nt ≤ (1 − d n ] ≥ − c − n . (10)This implies that if in round t − there are n Believers a.a.s. in the t -th round there are at most (1 − d ) n Believers. Building on this statement and applying the union bound, we prove that from N ≤ αN , thenumber of Believers decreases to F ( N, d ) a.a.s. in O (log N ) rounds. To make this argument more formallet us provide Claim 4.14. Claim 4.14
Assume that (1 − d ) t − αN ≥ F ( N, d ) for some positive integer t . Then, P r [ N t ≤ (1 − d t αN | N ≤ αN ] ≥ − t − (cid:88) t (cid:48) =0 c − (1 − d ) t (cid:48) αN . roof. We apply an inductive argument similar to the one in the proof of Claim 4.9. Let us define theevents A t := { N t ≤ (1 − d t αN } t ≥ A (cid:48) t := { N t = (1 − d t αN } t ≥ . Note that by definition, for ≤ t (cid:48) ≤ t we have P r [ A t | A (cid:48) t (cid:48) ] ≤ P r [ A t | A t (cid:48) ] , which we are going to utilizeseveral times later.Our goal is to prove that for t ≥ P r [ A t | A ] ≥ − t − (cid:88) t (cid:48) =0 c − (1 − d ) t (cid:48) αN . Now, we do induction on t . As the base case, for t = 1 P r [ A | A ] ≥ P r [ A | A (cid:48) ] Eq. (10) ≥ − c − αN . As the induction hypothesis (I.H.), assume that
P r [ A t − | A ] ≥ − (cid:80) t − t (cid:48) =0 c − (1 − d ) t (cid:48) αN . Now, we have P r [ A t | A ] ≥ P r [ A t | A ∩ A t − ] · P r [ A t − | A ] ≥ P r [ A t | A t − ] · P r [ A t − | A ] ≥ P r [ A t | A (cid:48) t − ] · P r [ A t − | A ] Eq. (10) ≥ (1 − c − (1 − d ) t − αN ) · P r [ A t − | A ] I.H. ≥ (1 − c − (1 − d ) t − αN )(1 − t − (cid:88) t (cid:48) =0 c − (1 − d ) t (cid:48) αN ) ≥ − t − (cid:88) t (cid:48) =0 c − (1 − d ) t (cid:48) αN . Notice that we need the condition (1 − d ) t − αN ≥ F ( N, d ) to apply Equation (10). (cid:3) Now, by applying Claim 4.14 for a suitable choice of t , we finish the proof of phase 1. Let us define T := (cid:100) log (1 − d ) F ( N, d ) αN (cid:101) . We have that (1 − d ) T − αN ≥ F ( N, d ) because (1 − d T − ≥ (1 − d log (1 − d F ( N,d ) αN = F ( N, d ) αN . (11)Thus by Claim 4.14, given N ≤ αN , we have N T ≤ (1 − d T αN ≤ F ( N, d ) with probability at least − T − (cid:88) t (cid:48) =0 c − (1 − d ) t (cid:48) αN . We show that this probability is equal to − o (1) and T = O (log N ) . Therefore, there will exist at most F ( N, d ) = log d log log N Believers a.a.s. after O (log N ) rounds.Since (1 − d ) = ( − d ) − , T = (cid:100) log (1 − d ) F ( N, d ) αN (cid:101) = (cid:100) log − d αNF ( N, d ) (cid:101) ≤ (cid:100) log − d N (cid:101) = O (log N ) .
17t remains to show that (cid:80) T − t (cid:48) =0 c − (1 − d ) t (cid:48) αN = o (1) . We have T − (cid:88) t (cid:48) =0 c − (1 − d ) t (cid:48) αN ≤ T − (cid:88) t (cid:48) =0 − d ) t (cid:48) αN = 1(1 − d ) T − αN T − (cid:88) t (cid:48) =0 − d ) t (cid:48) − ( T − Eq. (11) ≤ F ( N, d ) T − (cid:88) t (cid:48) =0 (1 − d t (cid:48) ≤ F ( N, d ) ∞ (cid:88) t (cid:48) =0 (1 − d t (cid:48) = 2 F ( N, d ) d = o (1) . In the second-to-last step we used that the geometric series (cid:80) ∞ t (cid:48) =0 (1 − d ) t (cid:48) is equal to − (1 − d/ = d . Phase 2.
So far we proved that if N ≤ αN , then N t ≤ F ( N, d ) = log d log log N a.a.s. for some t = O (log N ) . In this phase, we show that from a state with at most F ( N, d ) Believers, there is noBeliever, except Seeds, in a logarithmic number of rounds a.a.s.Let us first prove Claim 4.15, which asserts if in some state the number of Believers is at most F ( N, d ) ,their number does not overpass F ( N, d ) a.a.s. during the next log N rounds. Claim 4.15
Given N t ≤ F ( N, d ) , we have N t + t ≤ F ( N, d ) for ≤ t ≤ log N a.a.s. Proof.
Let us define for ≤ t ≤ log N the event B t := { N t + t ≤ F ( N, d ) } . We want to prove that P r [ (cid:86) log Nt =1 B t | B ] = 1 − o (1) . We have P r [ log N (cid:94) t =1 B t | B ] = log N (cid:89) t =1 P r [ B t | t − (cid:94) t (cid:48) =0 B t (cid:48) ] where the right-hand side is a telescoping product. We prove that P r [ B t | t − (cid:94) t (cid:48) =0 B t (cid:48) ] ≥ − √ N . (12)Therefore,
P r [ log N (cid:94) t =1 B t | B ] ≥ (1 − √ N ) log N ≥ − log N √ N = 1 − o (1) . It remains to prove that Equation (12) holds. Let B t be the complement of B t . We claim that P r [ B t | N t + t − ≤ F ( N, d )] ≤ Nr (cid:0) F ( N,d )2 (cid:1)(cid:0) r (cid:1) ( N − N ! . Given N t + t − ≤ F ( N, d ) , for event B t to happen (i.e., N t + t > F ( N, d ) ), at least two Believers mustshare a room in round t + t , to generate a new Believer. The numerator is an upper bound on thenumber of permutations in which at least two Believers share a room, from all N ! possible permutations.There are N/r possibilities to select a room and at most (cid:0) F ( N,d )2 (cid:1) possibilities to choose two Believers.There are (cid:0) r (cid:1) ways to place the two Believers in the room and ( N − possibilities to place the remaining N − agents. We count each permutation with at least two Believers in a room at least once in this way.We might actually count a permutation several times, but it does not matter since we are interested inan upper bound.Simplifying the right-hand side gives us P r [ B t | N t + t − ≤ F ( N, d )] ≤ F ( N, d ) rN − which is smaller than √ N since F ( N, d ) = log d log log N . Thus, we get P r [ B t | N t + t − ≤ F ( N, d )] ≥ − √ N . By applying the definition of B t , we have P r [ B t | t − (cid:94) t (cid:48) =0 B t (cid:48) ] ≥ P r [ B t | N t + t − ≤ F ( N, d )] ≥ − √ N (cid:3) By Claim 4.15, given N t ≤ F ( N, d ) , we have a.a.s. N t + t ≤ F ( N, d ) for ≤ t ≤ log N , that is, for log N rounds the number of Believers will not exceed F ( N, d ) . Furthermore, in each round a Believerbecomes Indifferent independently with probability d , except Seeds. Thus, in each of these log N roundsthe probability that all Believers in that round (whose number is at most F ( N, d ) ) become Indifferentsis at least d log d log log N = 1log log N .
The probability that this does not occur in any of these log N rounds is upper-bounded by (1 − N ) log N ≤ exp( − log N log log N ) = o (1) where we used − x ≤ exp( − x ) . Thus, a.a.s. after at most t + log N = O (log N ) rounds, the rumor diesout. Tightness.
We claim that the upper bound of O (log N ) is tight. That is, if the process starts with αN Believers, it takes the rumor at least a logarithmic number of rounds to die out. Assume that N = αN and no new Believer is generated during the process, i.e., no Indifferent becomes a Believer. Even underthis assumption, in each round in expectation a d fraction of the Believers will become Indifferents. Thusin expectation, it takes the process logarithmically many rounds to reach the state where all agents areIndifferents, except Seeds. To turn this argument into an a.a.s. statement, we can apply Chernoff bound(see Theorem 4.3). Assume X nt denote the number of Believers who become Indifferent in the t -th round,given N t − = n . Since each Believer becomes Indifferent independently with probability d , except s Seeds, then E [ X nt ] = ( n − s ) d ≤ nd . Thus, for a small constant (cid:15) > we have P r [(1 + (cid:15) ) nd ≤ X nt ] ≤ P r [(1 + (cid:15) ) E [ X nt ] ≤ X nt ] ≤ exp( − Θ( n )) . Thus, if there are n = ω (1) Believers in round t − , a.a.s. in round t at most (1 + (cid:15) ) d fraction of themwill become Indifferents. Now, applying an inductive argument similar to the one from Claim 4.14, it iseasy to show that starting from N = αN , a.a.s. it takes the rumor Ω(log N ) rounds to die out. As we discussed, for the purpose of the present paper it is realistic to assume that parameters b and d are relatively small constants, say b ≤ . and d ≤ . . Furthermore, since the behavior of the processis a function of several parameters, it is sensible to fix some parameters to study the impact of the othersin a more transparent set up. The range that the value of a fixed parameter is chosen from can be setaccording to the potential application of the model.Nevertheless, it would be interesting to study the m -rumor spreading model for larger values of d and b in future work. In this section, we try to illustrate the behavior of the process in this setting buildingon some approximate arguments. However, the rigorous analysis is left for future work.We first introduce some notations. Let the density of Believers be the ratio of the number of Believersto N . Furthermore, let f ( P ) denote the expected number of Believers in the next round by assumingthat the density of Believers in the current round is equal to P , i.e., there are P N
Believers.Let us first consider the case of m = 1 and r = 2 . Note that each Believer becomes Indifferentindependently with probability d . Thus in expectation a d fraction of Believers will become Indifferents.(Actually, there are s Seeds who remain Believers regardless of the choice of d . However, since s is a smallconstant, this can be ignored for our rough argument here.) Furthermore, Others who are Indifferentsconstitute a (1 − b − P ) fraction of the community. Each of them shares its room with a Believer, andconsequently will become a Believer, with probability P . Therefore, we can “approximate” the value of f ( P ) to be f ( P ) = (1 − d ) P + (1 − b − P ) P. f ( P ) has a unique fixed point − b − d for P ∈ (0 , − b ) . More accurately, f ( P ) < P for P ∈ (0 , − b − d ) and P < f ( P ) for P ∈ (1 − b − d, − b ) . See Figure 4 (left). Therefore, we expectthe process to converge to a state with Believer density almost − b − d . It is easy to check that this isconsistent with our results in Theorem 4.1 part (i), which holds for b ≤ . and d ≤ . .Figure 4: (left) function f ( P ) has a unique fixed point for P ∈ (0 , − b ) (right) function f ( P ) has twofixed points P and P for P ∈ (0 , − b ) .Now, we consider the case of m = 2 and r = 3 . Note that each Believer becomes Indifferent with prob-ability d . (We are again skipping Seeds) Furthermore, an Other who is Indifferent will become Believerif it shares its room with two Believers. Thus, by an argument similar to above, we can “approximate” f ( P ) to be f ( P ) = (1 − d ) P + (1 − b − P ) P . (This is just an approximation since we assume that each seat is occupied by a Believer with probability P independently , which is not true.)Function f ( P ) in this setting has two fixed points for P ∈ (0 , − b ) . More precisely, f ( P ) < P for P ∈ (0 , P ) , f ( P ) > P for P ∈ ( P , P ) and f ( P ) < P for P ∈ ( P , − b ) , where P = (1 − b ) + (cid:112) (1 − b ) − d , P = (1 − b ) + (cid:112) (1 − b ) − d . Therefore, if the initial density of Believers is less than P , we expect the rumor to die out. However, ifthe initial density is larger than P , we expect the process to converge to a state with Believer density ofalmost P . See Figure 4 (right). We have shown that within the m -rumor spreading model, switching from m = 1 to m ≥ triggers adrastic qualitative change in the spreading process. More precisely, when m = 1 even a small group ofBelievers manage to convince a large part of the community very quickly, but for m ≥ even a substantialfraction of Believers may not prevent the rumor from dying out after a few update rounds. Our resultsshed a light on why a given rumor spreads within a social group and not in another as noted in Section 2.5regarding some recent real cases of rumor spreading.At this stage it is worth stressing that our model like other models, is not the reality. However, thecommon aim is to grasp some essential traits of the reality despite making crude approximations. Wehope our findings shed some additional light on the understanding of rumor spreading phenomena. Acknowledgment.
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