Fake news is fabricated (mis)information that mimics news media content, and it is propagated through media without organizational processes ( [1] ). Fake news does not go through the same scrutiny as the news from legit news media source. It spreads over social media and has varying degrees of impact on users and society. It can influence the political choices of the users, e.g., [2] discusses fake news articles during the U.S. 2016 elections. It can also impact financial stock markets (e.g., [3] ). The list goes on. Hence it is essential to address growing concerns of fake news and develop an intervention policy.
Based on the empirical database, it is found that fake news propagates differently from real news on OSNs. In [4] , the authors have shown that fake news propagates faster and farther. In [5] , the authors discuss the differences between the propagations using the evidence based on empirical data of (fake and real) posts from Twitter Japan. The above discussions demonstrate a few important facts: a) users could be more attracted to fake news [4] ; b) there are differences between fake and real news propagations. We assume, users can use their cognitive judgement and reasoning capabilities to differentiate between fake and real news items to some extent. It may not yield perfect detection of fake news, but appropriate aggregation of such judgements (a.k.a. collective wisdom) can probably deter the fake news spreading.
Branching processes (BPs) have been used to model content propagation on OSNs (e.g., [6] , [7] ). In general, on an OSN, such as Facebook or Twitter, users share news with their friends. When a friend visits 1 the OSN, he may forward to some/all of his friends, depending on the attractiveness of the news. This spreading is well captured by continuous-time BPs (e.g., [6] , [7] ). Continuous versions allow us to model independent visits of users to OSN.
While mitigating the spread of fake news, we need to minimize the impact on the propagation of authentic news. To this end, we propose a warning-based mechanism, in which: a) every user receives a warning for each news item that he receives; b) the user receives a tag from its sender indicating that the news is fake/real; and c) every user tags the news as fake/real before sharing to his friends, the number of shares should not depend upon the tag. This approach is based on two assumptions: (i) the users have an innate capacity (see [8] ) to identify the fake news (to some extent), which can be significantly accentuated by welldesigned warnings, (ii) the collective wisdom (possible by sharing) (as in [9] ) guided by 'good' warnings can lead towards almost unanimous and correct tagging. The design of warning is based on the judgment (tags) of previous users and on the network's prior knowledge (if any) about the news.
The warning-controlled content propagation can be modelled (only) using population size-dependent continuous-time multi-type BPs. These specialised BPs are studied to a relatively smaller extent (e.g., [10] considers a discrete-time version). To the best of our knowledge, the continuous-time multi-type versions of BPs have not been considered in this context. We derive time-asymptotic proportions of individual populations using stochastic approximation techniques. Such an amalgam of stochastic approximation and branching processes is not seen before.
We define type-1 and type-2 performance measures to quantify the impact of controlled warnings on the propagation of fake and real news. With optimal parameters of a relevant optimization problem, the type-1 performance improves significantly, e.g., only 10% of smart users (20% of average users) mis-tag the fake news as real, when type-2 performance is within 2%. In contrast, in uncontrolled system (where users tag based on system's prior knowledge and their innate capacity), 72% of users mis-tag the fake news. We validate the performance measures using Monte Carlo simulations on network datasets from Twitter ( [11] ).
We consider an OSN with a sufficiently large user base as in Facebook or Twitter. The news posts on the OSN can be fake or real, and any news can be tagged as fake or real by the users. When a user with an unread copy of the news (tagged as fake or real by its sender) visits the OSN, he can view the sender's tag and a system-provided warning. Based on these two pieces of information and the user's cognitive and reasoning capability to recognize the authenticity of the news, the user tags the news as fake or real and forwards the same to his friends. This results in more unread copies of the news tagged as fake or real. This process continues, when another user (with news) visits the OSN. The warnings are designed by the system and depend on the tags of the previous users. This propagation dynamics can be captured by a population size-dependent continuous time multi-type BP (CTMTBP). News Propagation and Branching Process: To capture the above-controlled news propagation dynamics, we model the same using a two-type branching process (X(t), Y (t)). Here, X(t) represents the number of users that have received the news tagged as fake (i.e., the number of unread copies of post with fake-tag, referred to as x-users); Y (t) represents the number of users that have received the news with realtag (referred to as y-users). The dynamics depend upon the underlying news u, which can be fake (i.e., u = F ) or real (u = R). The users visit their timeline independently after an exponentially distributed time with (known) parameter λ.
When any of the users that have received news u with faketag visits the OSN (at time t), and if ω t is the warning at that time, the user tags the news as fake (real) with probability q u F (ω t ) (respectively, 1 − q u F (ω t )) before sharing. We model q u F as a linear function of ω t , i.e., q u F (ω t ) = α u F ω t , where α u F is the sensitivity parameter when the underlying news u is received with fake-tag. Similarly, if a user has received the news with real-tag, he tags it as fake/real with probabilities q u R (ω t ) and (1−q u R (ω t )), respectively. Here again, q u R (ω t ) = α u R ω t . The sensitivity parameters, α u F and α u R , are associated with the user's intrinsic ability to recognize the actuality, i.e., whether the news item u is real or fake. Controlled Warning: The warnings provided by the OSN are based on the responses of the previous users. They are specific to a news item and are generated as follows:
where β(t) := X(t)/(X(t) + Y (t)) is the relative fraction of copies tagged as fake at time t; w and b are the control parameters. Here, w takes any positive value bounded by 1.
A smaller b>0 makes warnings less sensitive to Y (t) (posts with real-tag), and more sensitive to X(t). A small > 0, captures the warning provided by the network, independent of user tags, through some fact-check mechanism. Tagging and Forwarding: When a user with an unread copy of the news tagged as fake/real, reads the news (at time τ ), he forwards it to some/all of his friends based on the attractiveness of the news, represented by η u . This parameter depends on the veracity of the news (u is fake or real). Let F be the number of friends of a typical user of OSN and we assume F to be i.i.d. (independent and identically distributed) across various users. He shares to Bin(F, η u ) among his friends, where Bin(·, ·) is a binomial random variable. Before sharing, he tags the news as fake or real with probabilities, q u F (ω τ ) and (1 − q u F (ω τ )), respectively. With fake-tag, the x-population gets updated; otherwise the y-population gets updated. These shares can be equivalently viewed as the offsprings (of various types) produced in the BP; thus the offsprings produced by x-user, has the following probability distribution (with m f := E[F]):
where ξ xx , ξ xy are the fake (new users that received the news with fake-tag) and real offsprings respectively. Thus the evolution of the system at the transition epoch (of x-user wake-up), τ , is summarized 2 as follows (see (2) ):
where T x F is a flag indicating that the x-user has tagged the news as fake, and τ + := inf{t > τ }, τ − := sup{t < τ } are the times immediately after and before τ respectively.
s., where G t is the sigma-algebra generated by {X(t ), Y (t ); t ≤ t}. If a y-user, one that has received news with real-tag, visits the OSN, he tags it as fake with (conditional) probability q u R (ω t ) and the system evolves in similar fashion:
Here,
s. Under the controlled warnings, our objective is to keep most users informed of the fake news, when u = F . As a first step, we analyze the system for any given w, b. Generator Matrix: The analysis of any BP depends upon its generator matrix, which is computed using the probability generating functions (PGFs) of the offsprings ( [12] ). For any complex vector s = (s x , s y ), from (2), the PGFs equal (by conditioning on F and using the PGF of Bin(·, ·)):
As in [12] , the generator matrix,
By direct computations (see (1), (2), (5)):
Observe that A depends on population sizes Z = (X, Y ); however, more precisely, it depends only on β, the relative fraction, i.e., A(Z) = A(β).
We now proceed to prove a crucial result for the BPs, which plays an important role in performance analysis of Section IV.
We consider a CTMTBP with a special structure as in (1) and (6) . The generator matrix A(Z) depends on the population sizes only via the relative fraction β. We provide the (time) limit of the proportion of the two population types (β(t)), using stochastic approximation (SA) techniques (e.g. [13] ). To this end, we analyse our process at transition epochs, let τ n denote the epoch at which the n th individual wakes up. Define X n := X(τ + n ) and Y n := Y (τ + n ). Then, at τ n , if x-type individual wakes up (see (3)), we obtain
, T x F,n is defined similarly. We have similar transitions for y-wake up. The total population, S n := X n + Y n , progresses (irrespective of the type waking up) as:
Let § n represent the sample mean formed by the i.i.d. sequence of the generated offsprings {ξ n } n plus the initial
Observe that the total population (µ e extinction epoch),
Further, also observe that
Note that the same holds for Y n . By strong law of large numbers, § n → m η − 1 a.s., while S n /n → m η − 1 only in the survival sample paths (i.e., when S n > 0 for all n). Next, we need the following notations for our analysis using SA: let H n denote the indicator that an individual of x-type wakes up at the n th transition epoch, and H c n := 1 − H n . Define Θ n := [ψ n , θ n ] as the ordered pair respectively representing S n /n and X n /n. Let γ n = 1/n, I n := 1 ψn−1>0 . We show that the evolution of Θ n can be captured by the following 2-dimensional stochastic approximation-based scheme:
The analysis is derived using the SA tools of [13] . We use similar notations as in [13] . Define L n := [L ψ n , L θ n ] T , where 
Thus (9) becomes, Θ n = Θ n−1 + γ n L n . The conditional expectation of L n with respect to G n = σ{X k , Y k ; k ≤ n},
Now, the Ordinary Differential Equation (ODE) that can approximate (9) is given by (see [13] ):
We prove that the ODE indeed approximates (9) and derive further results mainly using [13, Theorem 2.2, pp. 131]. Sinceḡ(·) is measurable, the results cannot be applied directly. We provide the required justifications/modifications, identify the attractors (i.e., (ψ * , θ * ) of Theorem 1) and the domain of attraction of the ODE, and finally derive the following result (see Appendix for proof details):
The sequence (ψ n , θ n ) converges a.s. to (ψ * , θ * ) in co-survival sample paths, with ψ * = m f η u − 1 and θ * = β * ψ * , where β * satisfies the fixed-point equation:
Further, (13) has unique solution in (0, 1). In other sample paths, the sequence either converges to (0, 0) (i.e., complete extinction), or (ψ * , ψ * ), or (ψ * , 0) (i.e., only one population explodes).
Remarks: (i) Depending on the irreducibility of the process, the probability that the process converges to (ψ * , ψ * ) or (13) and that of the following fixed point equation: (14) are connected by β * = 1/(1 + v y ). Thus, it is interesting to note that even with population dependency as in (6), the limit proportions are given by the eigenvector. However, the eigenvector is now obtained through a fixed-point equation (14) .
In our warning-based mechanism, we propose users to forward the news by appropriately tagging it; however, some of the users tend to be more reluctant to forward the news after tagging it as fake. One can incorporate this aspect by introducing a reluctance factor η c < 1; if a user tags the news as fake (real), he forwards it to Bin(F, η c η u ) (respectively Bin(F, η u )) among his friends.
We again aim to provide the limit proportions using SA techniques and the initial steps are similar to the case with η c = 1. We now have the following dynamics of our process for x-wake up at transition epoch τ n : ) and T x F,n is defined similarly. Likewise the transitions for y-wake up can be defined. As defined in (7) (3)), we can replace the offsprings distributed according to ξ n,x by those distributed as ξ n,y . Then the resultant sample mean, denoted by § n , dominates § n . The upper bounds of (8) still hold and we get:
Using the convergence of § n , one can proceed as in the proof of Theorem 1; we mention only the required changes here. The modifications in (9)-(11) are:
Thus, (17) can be written compactly as Θ n = Θ n−1 +γ n L n , where L n = [L ψ n , L θ n ] T with:
We have the ODE as in (12) withḡ ψ (Θ),ḡ θ (Θ) as:
The rest of the SA-based details are the same, but the analysis of the ODE (19) is different. We provide the ODEs based analysis to complete the proof of the following result in Appendix:
The sequence (ψ n , θ n ) converges a.s. to (ψ * , θ * ) in co-survival sample paths, with θ * = β * ψ * , where
and β * satisfies the fixed-point equation:
Further, (20) has a unique solution in (0, 1). In the other sample paths, the sequence either converges to (0, 0) (i.e., complete extinction), or (ψ * , ψ * ), or (ψ * , 0) (i.e., only one population explodes).
Remarks: Even with the reluctance factor, the limit proportion converges to a unique limit, the unique zero of (20). The sum population also converges to a unique limit, ψ * , which now depends on β * (unlike the one in Theorem 1). The rest of the paper is dedicated to understand the case with η c = 1 and we plan to study other aspects in the presence of reluctance factor in future.
We aim to control the warnings ω t (through parameters w, b) to mitigate fake news propagation, without significantly affecting the propagation of real news. To this end, we appropriately define type-1 and type-2 performances. The type-1 performance quantifies the effects of controlled warnings on the fake-news propagation, whereas the type-2 performance quantifies the adverse effects on the real-news propagation.
It is important to ensure that most users are informed of the fake news, when u = F . Thus, we define the type-1 performance (Ψ 1 ) as the (time) asymptotic fraction of posts with real-tag Y (t):
By Theorem 1, Ψ 1 = (1 − β * ) u=F . When the underlying news is fake, minimizing Ψ 1 ensures that a minimum number of users are mis-informed about the news being real. In a similar way, when the underling news is real, we define the type-2 performance (Ψ 2 ) as the asymptotic fraction of posts with fake-tag, which is also provided by Theorem 1:
Ensuring Ψ 2 is within a given limit gives an upper bound on the number of real news copies mis-tagged as fake. Both Ψ 1 and Ψ 2 are non-negative fractions bounded by 1. We immediately have following properties with respect to the control parameters, (w, b) (See proof in the Appendix):
Theorem 3. Assume α u F > α u R . When the systems start with same initial state, the type-1 performance (Ψ 1 ) decreases and the type-2 performance (Ψ 2 ) increases, monotonically with increase in w. The same is true for a decrease in b.
This Theorem is useful in optimal design of next section.
In practice, adverse effects on real news propagation cannot be allowed beyond a certain tolerance limit. We thus impose a constraint that Ψ 2 is upper bounded by c, where c > 0 is a design parameter. We consider the following optimization problem 3 for the optimal design:
By Theorem 3, Ψ 2 (0, 1) < Ψ 2 (1, 0). Further (23) simplifies to (Proof in Appendix): Lemma 1. Assume Ψ 2 (0, 1) < c < Ψ 2 (1, 0) (implies nonempty feasible region). Then, the optimal value of (23) is achieved when Ψ 2 = c.
Thus the feasible region reduces to {(w, b) : Ψ 2 (w, b) = c}. Hence, by virtue of Lemma 1 and using (13), we can express the variable b as a function of w (ᾱ R c :
Now, the optimization problem reduces to one dimension:
By differentiating both sides of (13) by w, the partial derivative, ∂β * /∂w, satisfies the following fixed-point equation:
Similarly, one can derive ∂β * /∂b. Observe that the partial derivative (see (24)):
Using these derivatives, we can solve (25) by the following projected gradient descent algorithm:
where [·] A is the projection to set A, and {κ l } is a decreasing sequence of step sizes. We obtain the optimizers, (w * , b(w * )), and the optimal Ψ 1 for several numerical examples using the above method in the following section. Before proceeding further, we discuss some meaningful assumptions, naturally required for the application.
Many a times, users find fake news more attractive ( [4] ); therefore, we assume that η F > η R . When the underlying news u is the same, we assume that α u F > α u R , which indicates that the probability of a user of tagging the news as fake is higher when the sender's tag is fake. We model the intrinsic capability of users to recognize the veracity of the news by assuming α F F > α R F . This assumption indicates that users are more likely to tag fake news as fake, as compared to tagging real news as fake.
We corroborate the results of Theorem 1 using exhaustive Monte-Carlo (MC) simulations for different sets of parameters. In Table I , one can observe that the relative proportions (i.e., X/(X + Y )) well match with β * of Theorem 1. Validation: We validate theoretical results for Ψ 1 and Ψ 2 by MC simulations based on ego-network dataset of Twitter provided by SNAP ( [11] ). It consists of 81, 306 users and 1, 768, 149 (directed) connections among them. We perform simulations at optimal warning parameters. We use the connections of users from the network dataset as friends for 
Performance at Optimal warning: In Figure 1b , we plot optimal type-1 performance as a function of threshold/tolerance value of type-2 performance (see (23)). In the same figure, we also plot results for type-1 performance obtained from MC simulations. One can observe that limiting theoretical results match exactly with MC simulations sample-path wise. From Figure 1b , the control mechanism significantly improves the system performance for all values of c. For an instance, if 2% tolerance is allowed as type-2 performance (i.e., c = 0.02 in (23)), only 10% of the OSN users are mis-informed (and mis-tag) about the fake news being real, when α F F = 0.85. In contrast, for the same settings with no controlled warnings, 72% users mis-tag the fake news as real; here, the users tag the news based on constant warning level, (see (1)) and their intrinsic capacity.
We have considered the problem of mitigating fake news in online social networks without affecting the propagation of authentic ones. We have designed a warning-based control mechanism, where the warning depends on the responses of forwarding users. The spread of fake news is modeled using an appropriate population size-dependent multi-type continuous-time branching process.
Our contributions are two-fold. On one hand, we have results towards the above mentioned specialized branching processes. On the other hand, we proposed a warning based control mechanism that efficiently mitigates fake news propagation, with minimal influence on other propagations.
We derived time asymptotic proportions of individual populations of the size-dependent branching process. These proportions are represented as the solution of a simple fixedpoint equation that depends upon the parameters.
By and large, the literature of branching process considers appropriate scaling of the process that enables convergence to a finite limit, to understand the otherwise transient, exploding process. We adopted a new approach using stochastic approximation tools: a) our first observation is that the embedded chain divided by the number of epochs evolves like a 'sample mean' in survival paths; this leads to a new type of scaling; b) we derived appropriate ODEs to the aforesaid scaled process, whose attractors provide the limit proportions.
Using the new results in branching process, we computed different performance measures of the spreading (under warning) to capture the effectiveness of the control mechanism. We identified structural (monotone) properties of the performance measures.
Finally, we formulated an optimization problem, where the network controller optimizes the type-1 (fake news) performance subject to a constraint on the degradation of the type-2 (authentic news) performance. With 1%, 1.5% and 2% allowed degradation on authentic news, 16%, 88% and 90% users respectively identify the fake news as fake.
Proof of Theorem 1: We will prove the result using [13, Theorem 2.2, pp. 131], asḡ(·) is only measurable. Towards this, we first need to prove (a.s.) equicontinuity of sequence Θ n (t) := Θ n + m(tn+t)−1 i=n
This proof goes through exactly as in the proof of [13, Theorem 2.1, pp. 127] because of the following reasons: the random vector L n is comprised of θ n , ψ n and i.i.d. random variables and by (8) , it suffices to show that sup n E| § n | 2 < ∞, which is trivially true because E[F 2 ] < ∞; further, we exactly have E[L n |F n ] =ḡ(Θ n ) (here β n in [13, Assumption A.2.2] is 0), as well the projection term Z n ≡ 0. Further, {Θ n (0)} n is bounded a.s. by strong law of large numbers as applied to { § n } n .
In Lemma 2, we identify the attractors 4 of (12), with θ * as in (13) . Proof is now completed sample-path wise.
First consider the sample-paths in which ψ n → 0 (i.e., S n → 0). Then clearly, (ψ n , θ n ) → (0, 0). For the sample paths such that (ψ n , θ n ) converges to (ψ * , ψ * ) or (ψ * , 0), there is nothing left to prove. In the remaining sample paths, ψ n → ψ * a.s. (where ψ * = m η − 1 as in (13)). Further, (ψ n , θ n ) visits S δ of Lemma 2 (for any 0 < δ < ψ * ) infinitely often. By applying [13, Theorem 2.2, pp. 131 ] to these sample paths, the sequence converges to (ψ * , θ * ).
Lemma 2. For ODE (12) , (ψ * , θ * ) is asymptotically stable in the sense of Liapunov. For any 0 < δ < ψ * , the set 5
is compact and is in the domain of attraction of (ψ * , θ * ). 4 A set A is said to be Asymptotically stable in the sense of Liapunov, if there exist a neighbourhood (called domain of attraction, D(A)) starting in which the ODE trajectory converges to A as time progresses (e.g., [13] ). 5 Define N δ (ψ * ) := {ψ : |ψ − ψ * | ≤ δ}.
Proof: The ψ-component of (12) has the following solution:
Thus, ψ * = m η = m f η u − 1 is asymptotically stable with (0, ∞) as domain of attraction. For θ component, one needs to substitute solution ψ(t) in its ODE (ḡ θ of (12)) to analyze. Clearly, θ * = ψ * β * , with β * a solution to (13) , is an equilibrium point 6 .
We prove the stability of the above equilibrium point using the ODE corresponding to β = θ/ψ (using (12)):
where, g β (β) :
Using the above two, we have: (i) g β has unique zero, β * , which satisfies (13) ; and (ii) t → β(t) is strictly increasing (derivative, g β (β), strictly positive) when 0 < β(t) < β * , and strictly decreasing when 1 > β(t) > β * . Thus from (27), for any initial condition (ψ 0 , θ 0 ) ∈ S δ , β(t) t→∞ → β * (ψ given by (26)).
Proof of Theorem 2: The proof of Theorem 2 can be done exactly as for Theorem 1, but by now upper bounding § n in place of § n and using (16). However, Lemma 2 needs to proved for η c < 1 with ψ * , θ * , β * as defined in Theorem 2.
To this end, consider the ODE corresponding to β = θ/ψ (using (19), (12) ):
For β * a solution to (20) (i.e., zero of g β (β)), we have ψ * (as in Theorem 2) and θ * = ψ * β * is an equilibrium point. Note that, g β (·) is strict convex function of β when bα u F ≥ α u R , because, second derivative in this case:
is positive. The rest of the proof for bα u F ≥ α u R follows as in Lemma 2.
For bα u F < α u R , consider the third derivative of g β (β): Fig. 2 : The functions g β (·) andġ β (·), when bα u F < α u R and b < η c Now, ... g β (β) is a product of two terms, which has positive first term. Further, it is easy to verify dk(β) dβ = 0, which implies that k(β) is a constant for all values of β. Notice that
for all β ∈ [0, 1). Now, we will analyse the following cases:
• For b = η c : we have ... g β (β) = 0, which implies thaẗ g β (·) is a constant function of β. We have, for all β ∈ [0, 1):
which implies g β (·) is a convex function of β.
• For b > η c : we have ... g β (β) < 0, which impliesg β (β) is a decreasing function of β, and for all β ∈ [0, 1):
Thus,g β (β) > 0 for all β. This implies that g β (β) is a convex function.
Hereafter, the proof follows as in Lemma 2 for the above discussed cases, i.e., b ≥ η c .
• For b < η c : we have ... g β (β) > 0, which implies thaṫ g β (β) is convex function of β. Now, notice thatġ β (·) can not be non-negative for all values of β, as it would imply g β (·) is a non-decreasing function of β, which contradicts the fact that g β (0) = ηcα u
Thus,ġ β (·) will attain negative values as well for some β ∈ [0, 1]. Sinceġ β (·) is a convex function, the set D := {β :ġ β (β) ≤ 0} = [β 1 ,β 2 ] is a connected set, for somẽ β 1 ,β 2 ∈ [0, 1] (as in Figure 2 ). Then, g β (·) is decreasing for β ∈ D and increasing otherwise. From this, we infer g β (·) achieves maximum value atβ 1 and minimum value atβ 2 , and it equals zero at some β ∈ D, i.e., β * in Theorem 2. Further, recall that g β (0) > 0, g β (1) < 0, thus, g β (·) can not have zero in D c . Thus, we have a unique solution for (20). Also clearly, g β (β) > 0 for all 0 < β < β * and g β (β) < 0 for all β * < β < 1. Henceforth, the proof can be completed as in Lemma 2.
Proof of Theorem 3: Consider two systems, with parameters w 1 , w 2 such that w 1 > w 2 and same b. Also, the systems start at the same state, i.e., Z 1 (0) = Z 2 (0). We compare the two systems sample-path wise using appropriate coupling 7 arguments. Until a wake-up event, both the states remain the 7 (i.e., chose the random quantities governing the two evolution in such a way that sample path wise comparison is possible) same, hence can assume the same (say x) user wakes up. At this epoch, t, q u F 1 (ω 1t ) > q u F 2 (ω 2t ), as for i = 1, 2:
We now couple the two flags T x F i , i = 1, 2 as follows: first generate flag T x F 1 and then set T x F 2 = T x F 1 T x F 12 , where flags T x F 1 , T x F 12 equal one with the following probabilities:
. (30) By virtue of this, we have that T x F 1 ≥ T x F 2 a.s (as T x F 12 ≤ 1), i.e., in system 1, it is more likely that a user tags a post as fake in comparison with that in system 2.
If T x F 1 = T x F 2 = 1 or T x F 1 = T x F 2 = 0, one can simply couple the offsprings produced by both systems, i.e., set ξxx 1 (t) = ξxx 2 (t) = Bin(F, ηu), the same realization.
But if T x F 1 = 1 and T x F 2 = 0, i.e., if in system 1 the user declares the news as fake while in system 2 the user declares the news as real, we couple them as: ξ xx 1 (t) = ξ xy 2 (t) = Bin(F, ηu).
Thus we have (as X1(t − ) = X2(t − ), Y1(t − ) = Y2(t − )):
X1(t + ) ≥ X2(t + ) and Y1(t + ) ≤ Y2(t + ) a.s. (32) X1(t + ) + Y1(t + ) = X2(t + ) + Y2(t + ) a.s., and hence, X1(t + ) X1(t + ) + Y1(t + ) ≥ X2(t + ) X2(t + ) + Y2(t + )
Hence again, q u F 1 (ω 1 t + ) > q u F 2 (ω 2 t + ). Further because of (33), by appropriate coupling, either the same type wakes up in both systems after time t, or x-type wakes up in system 1 while y-type wakes in systems 2. In either case the probability of the user tagging news as fake is again bigger in system 1 (as α u F > α u R ). Using similar coupling logic, we will again have (at next wake-up epoch), either both tags are the same, or T x F 1 = 1 and T x F 2 = 0. One can progress in the same manner for all time and the first part is true. From (1), decrease in b (for fixed w) has same effects as increase in w (for fixed b); therefore, the result follows.
Proof of Lemma 1: First observe that solution β * of (13) (for any u) is continuous in (w, b) by Maximum Theorem ([14, Theorem 9.14, pp. 235]) and uniqueness of solution of (13), as β * is the unique minimizer of the following:
(β (1 − q u F (β) + q u R (β)) − q u R (β)) 2 .
Thus Ψ 2 is continuous in (w, b). Consider any point (w * , b * ) such that Ψ 2 (w * , b * ) < c and consider the following cases:
• If Ψ2(w * , b * ) < c < Ψ2(1, b * ): by intermediate value theorem (IVT) applied to continuous mapping, w → ψ 2 (w, b * ), ∃ ε w ≥ 0 such that Ψ2(w * + εw, b * ) = c.
• If not, Ψ2(1, b * ) < c and hence, Ψ2(1, 0) > c > Ψ2(1, b * ). By IVT, ∃ a ε b ≥ 0, such that Ψ2(1, b * − ε b ) = c. In all, ∃ an ε w , ε b ≥ 0 such that Ψ 2 (w * +ε w , b * −ε b ) = c. By Theorem 3, Ψ 1 (w * + ε w , b * − ε b ) ≤ Ψ 1 (w * , b * ). Hence the Lemma follows.