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Let us first define our model. In alignment with the DK and MT models, here we also have the same set of states (ignorant, spreader, or stifler), which are modeled by associating to each node i three Bernoulli random variables, Xi, Yi, and Zi. If node i does not know the rumor, it is classified as an ignorant (Xi = 1, Yi = Zi = 0). If it knows the rumor and is spreading it, it is called a spreader (Yi = 1, Xi = Zi = 0). However, if it knows the rumor but does not spread it, it is classified as stifler (Zi = 1, Xi = Zi = 0). Note that Xi + Yi + Zi = 1, implying that, for a fixed node, only one variable will be one, while the others will be zero. The spreading evolves through the contact between nodes defined by an undirected network, which is codified by the adjacency matrix A, whose entries Aij are equal to one if there is an edge between nodes i and j and zero otherwise. Our process is defined in continuous-time as a collection of Poisson processes. If the contact is between a spreader and an ignorant, the second node will learn the rumor and become another spreader at rate λ. On the other hand, if the contact happens between a spreader and someone that already knows the rumor (spreader or stifler), then the spreader that initiated the contact will lose interest in the rumor, thus becoming a stifler at a rate α. We also introduce a forgetting mechanism, where each stifler spontaneously becomes ignorant at a rate δ. We assume that only stiflers can forget the rumor as spreaders are actively trying to transmit the information. Denoting the state of the node i as \({({X}_{i},{Y}_{i},{Z}_{i})}_{i}\), the above described local rules are expressed as
A graphical representation of these transitions is presented in Fig. 1. Please see the Supplementary Videos for a visual example of the temporal evolution of the Monte Carlo Simulations for the standard Maki–Thompson model near the critical point and in the supercritical regime.
In this figure, the curly arrows represent transitions driven by contact, while the normal arrow represents a spontaneous transition. To simplify the notation, the states of the nodes are represented in the center of the circles, which show the only Bernoulli random variable that is equal to one.
In this figure, the curly arrows represent transitions driven by contact, while the normal arrow represents a spontaneous transition. To simplify the notation, the states of the nodes are represented in the center of the circles, which show the only Bernoulli random variable that is equal to one.
We remark that, if δ = 0, we recover the original MT model. Notice that δ can be interpreted either as a forgetting mechanism or, for evolving rumors as, individuals who do not know the rumor’s current state that is, someone that has not been informed since the rumor changed. In this case, we could simplify the process by treating an updated rumor as, in fact, a new one. Either way, the essence of the process is captured by the forgetting mechanism (with parameter δ).
In summary, our model combines the “reluctance to tell stale news”, the annihilation by contact, with the eventual forgetting of the rumor. The latter transition assumes that one can only forget a rumor if she/he is not spreading it. The removal mechanism is the central difference between our model and the SIRS model. Namely, in our modified model, the transition from spreaders to stiflers depends on a interaction-based removal mechanism, while in the disease case, this transition is spontaneous. Although these models share similar features, we observed that their differences are enough to generate a different behavior, which is especially clear in the subcritical regime. In this regime, the SIRS has an exponential decay, while our modified rumor model presents a power-law decay to the absorbing state.
From a Markovian point of view, the modified model has a single absorbing state, while the original MT model has infinitely many absorbing states. In other words, by including of the transition \({Z}_{i}\mathop{\to }\limits^{\delta }{X}_{i}\) only the rumor-free state in which all the individuals are ignorant (i.e., Xi = 1 for all i ∈ {1, 2, …N}) traps the dynamics. Note that, for the MT model any state in which Yi = 0 for all i ∈ {1, 2, …N} is absorbing. In this case, as the system size increases, the number of absorbing states also increases. However, from a practical point of view, single absorbing state dynamics are easier to deal with as we can employ the quasi-stationary algorithm42 (for more, see section “Monte Carlo simulations”) and the lifespan method43 (for more, see section “The lifespan method”), allowing for a more precise characterization of the process.
To analyze the model, we first need to define a phase transition and corresponding critical point in our context. As MT and SIR dynamics have infinitely many absorbing states, in this case, the critical point is defined as the parameter that separates two scaling regimes. Before this point, the final number of stiflers (or recovered in the SIR) when the process reaches an absorbing state does not scale with the system size and hence its fraction goes to zero in the thermodynamic limit. After the critical point, the number of stiflers (recovered) scales with the system size. On the other hand, for processes like SIS (Susceptible-Infected-Susceptible) and our modified MT model with δ > 0 where we have a single absorbing state (disease-free state), the critical point is defined as the point where above it we have an active state in the thermodynamic limit, and below it, the system goes to the absorbing state in the thermodynamic limit. Note that our model has infinitely many absorbing states if δ = 0 and a single absorbing state if δ > 0.
To characterize the MT model, we can simulate the process beginning with many different initial conditions and measure the final fraction of stiflers and the time necessary to reach the absorbing state. Note that the initial condition must correspond to a single spreader in an ignorant population to capture the transition. Thus, the initial spreader node changes in the different independent runs of our simulation. Formally, the quantities of interest can be defined as
where \(\inf (\cdot )\) is the infimum and \(\left\langle \cdot \right\rangle\) is the average among simulations. In this context, ρZ measures how far the rumor spreads, while τ measures how fast it spreads. If δ > 0, our modified model has a single absorbing state, and it thus does not make sense to characterize the behavior of the model based on the final state as in Eqs. (2) and (3). Instead, we measure how far the rumor spreads by estimating the density of spreaders, ρY, in the metastable state using the quasi-stationary algorithm42 (for more, see section “Monte Carlo simulations”). Further, the time to reach the absorbing state, τ (see Eq. (3)), diverges exponentially in the supercritical regime when δ > 0. We instead estimate the lifespan of “finite” realizations, τf, using the method proposed in ref. 43 (for more, see section “The lifespan method”).
We remark that both the MT model and our modified version are represented by a continuous-time Markov chain with 3N possible micro-states. Thus, it is theoretically possible to write the infinitesimal generator, leading to a linear system that solves these processes exactly. However, this is not possible in practice due to the prohibitive computational cost. Notice that this is the case of the SIR model or any three-state dynamics. Note also that the concept of phase transitions is only valid in the thermodynamic limit. However, as it is commonly done in complex systems17,18,21,24,26, we also use the term phase transition in finite systems as we observe two different scaling behaviors, as mentioned above. Despite the fact that all our simulations are performed in a finite system, for this theoretical reason, most of our analysis implicitly assumes the thermodynamic limit. The exception here is section “Modified rumor model: Asymptotic analysis”, which follows an individual-based approach.
Our first main result is showing that the MT model has a phase transition. Figure 2 shows the average fraction of stiflers ρZ as a function of λ in the standard MT model with α = 1.0 for random regular networks with different sizes. In this network, all nodes have the same number of neighbors 〈k〉k, and the connections between them are made at random, avoiding both self and multiple connections. We observe that, as the system size increases, for small λ the fraction of stiflers decreases with system size, whereas for large λ it is larger than zero and weakly dependent on the system size (see Fig. 2 for an example). Note that, as we increase the system size, for a small λ regime, i.e., λ < λc, where λc is the critical point, the fraction of stiflers goes to zero. On the other hand, for the larger λ regime, i.e., λ > λc, the curves converge to the value of the fraction in the thermodynamic limit. This behavior suggests a phase transition in the MT model as a function of λ. Complementary to the phase diagram, in Fig. 3, we show the corresponding lifespan. For a continuous phase transition, the peak of lifespan diverges in the thermodynamic limit43,44. By comparing Figs. 2 and 3 we can see that the curves for the order parameter meet at the point in which the lifespan diverges. This behavior is compatible with a continuous phase transition43,44. Moreover, we also observe an unexpected subcritical behavior, where the lifespan increases as λ decreases. For the sake of comparison, in an SIS or SIR dynamics the lifespan would be an increasing function of λ in the subcritical regime. Although these results are in striking contradiction with the mean-field approximations17,18,21,25, we highlight that the mean-field approach neglects correlations between node states which are crucial in rumor models given the contact-driven stifling mechanism. Further, the first-order mean-field approximation is not accurate for rumor models18.
Results for α = 1 and different sizes on a random regular networks with 〈k〉k = 10.
Results for α = 1 and different sizes on a random regular networks with 〈k〉k = 10.
Results for α = 1 and different sizes on random regular networks with 〈k〉k = 10. The dashed line follows τf ~ λ−1.
Results for α = 1 and different sizes on random regular networks with 〈k〉k = 10. The dashed line follows τf ~ λ−1.
To better understand the transition in rumor models, we move to our modified model and concentrate our efforts on the α ≫ δ regime, which includes the MT model. In this case, in the transient period, we can neglect the forgetting transitions and assume that there are only two competing mechanisms, the spreading and the stifling. Thus, for locally tree-like networks and near the absorbing state, we can estimate the expected number of newly informed nodes that result from the initial spread of the rumor to a node of degree k. Note that, at least one spreading event is required to reach the absorbing state from an initial state with one informed node due to the contact-driven stifling process. Formally, we have the following recurrent expression
where qk(i) is an iterative expression that calculates the expected number of newly informed individuals in a process where the seed has k neighbors and, among them i are spreaders. This quantity should be iterated k times and accounts for: (a) the probability of spreading the rumor to one of its k − i available neighbors (ignorants), \(\frac{(k-i)\lambda }{i\alpha +(k-i)\lambda }\); which is multiplied by (b) the probability that the process stops through a stifling event due to one of its i spreader neighbors, \(i\frac{(i+1)\alpha }{(i+1)\alpha +(k-i-1)\lambda }\), (c) or that the process continues, which is encoded in the term qk(i + 1). In the case of the MT model on an infinite tree, the rumor propagation is exactly described by a branching process45. Here, we assume that each node’s degree is sampled from a fixed degree distribution that is independent of the node and that there are no correlations between the degrees of neighboring nodes. By averaging over the degree distribution, the condition that establishes the transition between a phase where the rumor dies out with probability 1 and a phase where there is a non-zero probability of infinite propagation [45 Theorem 1, Chap. I] is
where 〈⋅〉k is the expectation on the degree distribution. We remark that Eq. (5) is not a closed expression, but can be numerically solved.
In other words, Eq. (4) estimates the expected number of newly informed individuals as a result from a single initial spreader event by weighting the probabilities that the rumor stops (accounted by item (a)) or that it continues the spreading (accounted by qk(i + 1)). Moreover, the transition can be obtained from the condition at which the expected number of newly informed nodes is larger than one, i.e., are more likely to spread the information than to stop the spreading. Note also that this approximation is based on the fact that only a single event occurs at a time, which is a property of continuous Markov chains.
In Fig. 4 we compare the solution of q(1) with Monte Carlo critical point estimations for random regular networks with 〈k〉k = 10 for both δ = 0 and δ = 1 (see section “Monte Carlo simulations” for the simulation details and the Supplementary Information, Sec. I, for the individual susceptibility curves and critical point estimations). In the same figure, we also present the naive estimation, considering a first-order approximation, given as \({\lambda }^{* }=\frac{\alpha }{({\langle k\rangle }_{k}-1)}\). Note that this expression accounts for the competition between the spreading processes, with the rate 2(〈k〉k − 1), and the annihilation, with the rate 2α. Moreover, λ* is the rate at which spreading and stifling have the same probability. So, from this approximation, one would expect that, if λ > λ*, the process can spread to a fraction of the population, while if λ < λ* it should be constrained to a finite fraction of nodes. We observe that q(1) seems to be a reasonable approximation as its precision increases with α. On the other hand, λ* might be a reasonable approximation only for small enough α. These solutions suggest that the process can not be reduced to a first-order approximation, and frustrated trials to get to the absorbing state are expected and should be accounted for. Here, a frustrated trial is defined as the scenario in which the system needs one event to be forced into the absorbing state, but a spreading event takes place instead, thus avoiding the absorbing state. A crucial difference between q(1) and the mean-field approaches that fail to predict a phase transition is that q(1) takes into account that the rate at which a spreader turns into a stifler increases with the number of spreader neighbors.
Results for random regular networks with 〈k〉k = 10 and δ = 1 and N = 106. The continuous line expresses the value of λc obtained as a solution of q(1) = 1, from Eq. (5). In contrast, the dashed line represents the naive approximation that accounts only for the probability that the next event is spreading or stifling. In the inset we present the comparison for the low α regime.
Results for random regular networks with 〈k〉k = 10 and δ = 1 and N = 106. The continuous line expresses the value of λc obtained as a solution of q(1) = 1, from Eq. (5). In contrast, the dashed line represents the naive approximation that accounts only for the probability that the next event is spreading or stifling. In the inset we present the comparison for the low α regime.
Typically, heterogeneity plays a major role on dynamical processes. To understand its effects in the phase transition, we analyzed the critical properties of uncorrelated random networks with a power-law degree distribution, P(k) ~ k−γ, generated by the algorithm proposed by Catanzaro et al.46, named uncorrelated configuration model. We remark that, here, the degree correlations can be measure by means of the conditional probability \(P(k^{\prime} | k)\) that a node with degree k is connected with a vertex with degree \(k^{\prime}\). For uncorrelated networks, this probability can be estimated as the probability that any edge points to a vertex with degree \(k^{\prime}\), leading to \({P}_{{{{{\rm{unc}}}}}}(k^{\prime} | k)=k^{\prime} P(k^{\prime} )/{\langle k\rangle }_{k}.\) These networks are important from a numerical point of view because it is possible to test the behavior of dynamical systems whose theoretical solution is obtained only in the absence of correlations. Using the quasi-stationary method (see section “Monte Carlo simulations”), we estimated the critical point and focused our analysis on its behavior as a function of the network size. These results are summarized in Fig. 5 (please see the Supplementary Information, Sec. I, for the individual susceptibility curves and critical point estimations). Importantly, we observe that in power-law networks the threshold seems to vanish for γ < 3.0, and converges to a non-null value for γ > 3.0. This behavior is at odds with the behavior of the SIS model, in which the power-law networks have a vanishing critical point for any value of γ18,47,48. Conversely, it is similar to the SIRS (Susceptible-Infected-Recovered-Susceptible) model49, contact process50, the generalized SIS model with weighted infection rates51, and also modified versions of the SIS model32. In all of these models, the phase transition can be associated with a collective phenomenon that involves the activation of the whole network, whereas the phase transition in the standard SIS model shows an unusual behavior related to mutual reinfection of hubs43,52,53,54.
We plot λc as a function of N and for different values of γ and α, considering δ = 1.
We plot λc as a function of N and for different values of γ and α, considering δ = 1.
Another important result regards the subcritical regime, where the rumor can last for a “very long” time in the small λ regime. To the best of our knowledge, this behavior remained unexplored until now. We briefly mentioned this phenomenon in the analysis of Fig. 3 for the MT model. For our modified model, these behaviors can be observed in Fig. 6, which shows the lifespan τf as a function of the spreading rate λ for an uncorrelated power-law network with P(k) ~ k−γ with γ = 2.25 and N = 106. In the main figure, we show the subcritical regime, whose lifespan follows τf ~ λ−η, while in the inset, we show the peak for τf, which suggests a continuous phase transition. To calculate the lifespan we used the method proposed by Boguñá and collaborators43 (for more, see section “The lifespan method”).
Results for δ = 1, α = 0.5 on an uncorrelated power-law network with P(k) ~ k−γ with γ = 2.25, N = 106. In the main panel, we show a wide range of λ, emphasizing the sub-critical behavior, while in the inset we show the peak that suggests a continuous phase transition. The blue curve (dot dashed line) follows τf ~ λ−1 and the orange curve (dashed line) follows τf ~ λ−0.88, obtained from a fitting of the lifespan obtained using Monte Carlo simulations (the gray curve).
Results for δ = 1, α = 0.5 on an uncorrelated power-law network with P(k) ~ k−γ with γ = 2.25, N = 106. In the main panel, we show a wide range of λ, emphasizing the sub-critical behavior, while in the inset we show the peak that suggests a continuous phase transition. The blue curve (dot dashed line) follows τf ~ λ−1 and the orange curve (dashed line) follows τf ~ λ−0.88, obtained from a fitting of the lifespan obtained using Monte Carlo simulations (the gray curve).
Complementarily, Fig. 7 shows the temporal behavior of the density of spreaders ρ, where we can see that the time to reach the absorbing state is larger for λ = 0.0001 than for λ = 0.01, at odds with exponential decay. Phenomenologically, the transition from the spreader to the stifler depends on at least two individuals who are not ignorant. In the λ ≪ δ regime, the forgetting mechanism reduces the number of stiflers faster than the creation of new spreaders, thus increasing the number of necessary events to reach the absorbing state.
Result for a network with N = 106, γ = 2.25, α = 0.5 and δ = 1 in the regime λ ≪ δ for values of λ near the critical point λc ≈ 0.015.
Result for a network with N = 106, γ = 2.25, α = 0.5 and δ = 1 in the regime λ ≪ δ for values of λ near the critical point λc ≈ 0.015.
The subcritical regime can be characterized in terms of the time to reach the absorbing state, Tabs. We remark that τ in Eq. (3) should converge to Tabs. This quantity can be approximated under the assumption that there is a single spreader and the population is locally tree-like, where nodes have 〈k〉k neighbors. In this case, assuming that node i is a spreader, the shortest sequence of events that will drive the process to the absorbing state consists of: (1) node i spreads the information to node j, (2) either i or j turn into stifler, (3) the other node turns into a stifler, (4) either i or j turn into ignorant, and (5) the other node turns into an ignorant. Since our process is a continuous Markov chain, only one event occurs at a time, the respective probabilities P and expected times 〈τ〉 of these events are
Therefore, the probability and the expected time for reaching the absorbing state through this chain are
From these quantities, we can approximate the average time necessary to reach the absorbing state, \(\left\langle {T}_{{{{{\rm{abs}}}}}}\right\rangle\). In the λ ≪ δ and λ ≪ α regime, \(\left\langle {\tau }_{1}\right\rangle \gg \left\langle {\tau }_{\ell }\right\rangle\), for ℓ = 2, 3, 4, 5. Thus, the time spent in the frustrated trials can be approximated by \(\left\langle {\tau }_{1}\right\rangle\). Here, we call as frustrated trials any sub-chain of events that does not lead to the absorbing state. So, \(\left\langle {T}_{{{{{\rm{abs}}}}}}\right\rangle\) can be approximated by counting the number of times the process fails to reach the absorbing state plus the time it succeed, which is given by
which solves as
and for the regime λ ≪ 1 and λ ≪ α, δ, follows
Note that Eq. (13) is not defined for the standard MT model, i.e., δ = 0. In the regime λ ≪ α, we expect that T ~ λ−1 and only two nodes learn the rumor. It is noteworthy that, in the subcritical regime, the time to reach the absorbing state is dominated by the time the system has to wait before a spreading event happens. In order to reach the absorbing state, whenever we have a single spreader, it first needs to inform a neighbor and, only then, the process will be allowed to reach the absorbing state. Notice that, after the spreading event, the stifling events are much faster than the spreading ones. As the spreading events are the slowest ones in the subcritical regime, the rate of these processes dominates Eq. (13), as shown in Eq. (14). Also, frustrated trials to reach the absorbing state are possible. However, in this case, we are only repeating the above-described process. Figure 8 shows T and T* as a function of λ, where the subcritical behavior is dominated by the time spent on the spreading events. Although our assumptions do not cover heterogeneous networks, T has a similar behavior as the one observed in Fig. 6 for a power-law network, suggesting that our assumptions are reasonable and might be applicable in different substrates.
Results for δ = 1.0 and α = 0.5. The orange curve is the result of Monte Carlo simulations in a random regular network with the same parameters, 〈k〉k = 10 and N = 105.
Results for δ = 1.0 and α = 0.5. The orange curve is the result of Monte Carlo simulations in a random regular network with the same parameters, 〈k〉k = 10 and N = 105.
Note that in our modified model, δ > 0, the rumor might still wander around in the network before die-out. For every frustrated attempt to reach the absorbing state, the spreader individual might be a different one. In this way, although the rumor is fated to disappear, during the transient period, many different individuals might be informed about the rumor before it dies out. This effect is a consequence of the “reluctance to tell stale news” (the annihilation by contact) with the forgetting mechanism. The first mechanism imposes that at least two spreaders are necessary to reach the absorbing state, while the second mechanism enforces a single absorbing state.
From the analytical viewpoint, the analysis of arbitrary networks is much more difficult, and even the proof of the existence of a critical point is very hard or maybe impossible in a first-order approximation. This argument might be of interest to our community as it points to interesting future directions. In a first-order approximation we assume that the probabilities are independent. Denoting \(\left\langle {X}_{i}\right\rangle\), \(\left\langle {Y}_{i}\right\rangle\) and \(\left\langle {Z}_{i}\right\rangle\) by xi, yi, and zi, respectively, we have
Here we follow an asymptotic analysis, considering that \({y}_{i}={y}_{i}^{(1)}{\epsilon }^{c}+O({\epsilon }^{2c})\), \({z}_{i}={z}_{i}^{(1)}{\epsilon }^{k}+O({\epsilon }^{2k})\) and xi ∈ O(1), where ϵ ≪ 1. Moreover, we have to consider a scaling of the parameters as \(\lambda =\tilde{\lambda }{\epsilon }^{m}\) and \(\alpha =\tilde{\alpha }{\epsilon }^{n}\) to be able to balance the equations. Thus, from Eq. (15) in the steady-state and neglecting the higher-order terms, we have
This relation imposes that \(k=n+c+\min (c,k)\) and k = m + c, establishing three different possible regimes: (i) 0 < c < k, where c = m − n and k = 2m − n, (ii) c = k, where m = 0 and n = −k, and (iii) 0 < k < c, where c = −n and k = m − n. Without loss of generality, we also assume δ = 1. In the first regime, we have that \({y}_{i}^{(1)}=\frac{\tilde{\lambda }}{\tilde{\alpha }}\) and hence \({y}_{i} \sim \frac{\lambda }{\alpha }\). For the second and third regimes we have to assume a regular network (i.e., all nodes have the same degree) to obtain a closed-form solution, resulting in \({y}_{i} \sim \frac{\lambda }{\alpha \left(\lambda {{{\Lambda }}}_{\max }+1\right)}\) for the second, and \({y}_{i} \sim \frac{1}{\alpha {{{\Lambda }}}_{\max }}\) for the third regime, where \({{{\Lambda }}}_{\max }\) is the leading eigenvalue of the adjacency matrix. We remark that \({{{\Lambda }}}_{\max } = k\) in regular networks. For the details and derivations of this analysis, see the “Methods” section. A similar asymptotic analysis was made for the SIRS model in the Supplementary Information, Sec. II, for the sake of comparison.
Note that, we can go from one regime to another by controlling the parameters λ and α. However, regardless of the regimes, yi is always positive and larger than zero for any positive non-null rates in the three possible regimes, thus implying that the phase transition is not captured in the first-order approximation. Nonetheless, the transition was observed in our numerical experiments. Thus, we can conclude that the assumptions on the first-order mean-field approach are not enough to capture the essence of rumor models and correlations should be included.
Wenn ich dieses Rezept früher wüsste, würde ich es jeden Tag kochen! Auberginensuppe mit Rippchen! Ich habe noch nie eine so leckere Suppe mit Auberginen gekocht! Sie können nicht sagen, wie lecker es ist! Ich möchte es immer wieder kochen. Die ganze Familie wird dieses Suppenrezept lieben! Wie man Auberginensuppe kocht, heute zeige ich Ihnen ein großartiges Beispiel für ein köstliches Gericht. Suppe mit Rippchen ist immer lecker und erschwinglich! Ein schnelles Rezept für eine Gemüsesuppe, die niemand ablehnen kann, jeder kann kochen. Ein einfaches Suppenrezept, aber mit reichhaltigem Geschmack, wird jetzt dein neuer Liebling in der Küche, Hauptsache alles wie ich und mit Liebe machen!
Rezept und Zutaten:
in 600 g Rippchen schneiden.
5-7 Minuten bei starker Hitze braten.
1,5 Liter Wasser hinzufügen und 20 Minuten kochen.
2 kleine Zwiebeln fein hacken.
1 mittelgroße Aubergine in Würfel geschnitten.
1 Karotte raspeln.
2 mittelgroße Kartoffeln hacken.
Kartoffeln 10 min kochen.
Celera und 1 scharfe Peperoni fein hacken.
2 mittelgroße Tomaten in Würfel schneiden.
2 Knoblauchzehen hinzufügen und 5 Minuten köcheln lassen.
3 EL Sauerrahm, Salz, Pfeffer, Rosmarin und weitere 5 Minuten kochen.
Alles 10 Minuten garen.
etwas Petersilie hacken.
Guten Appetit.
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The PFL regular reason continues this week with 2022 PFL 3, which features women’s lightweights and welterweights in action for the first time this year.
Here’s how to watch PFL 3, with a complete lineup and breakdown by division.
PFL1 at the Esports Stadium Arlington in Arlington, Texas, Wednesday, April 20, 2022. (Cooper Neill / PFL)
PFL 3 takes place Friday at Esports Stadium in Arlington, Texas.
The prelims stream on ESPN+ at 5:30 p.m. ET. The main card also streams on ESPN+ and airs on ESPN2 at 9 p.m. ET.
The entirety of the PFL’s women’s lightweight roster will be in action Friday at PFL 3, including the two-time champion. Kayla Harrison (11-0) will start her quest for a third $1 million PFL title when she takes on Marina Mokhnatkina (6-2). Harrison explored free agency, but ultimately returned to the PFL to go after a third season title. And while all eyes will be on her and that quest, lurking will be former Bellator women’s featherweight champ Julia Budd (16-3), who came to the promotion as a free agent. She’ll meet Genah Fabian (4-2) to open the regular season.
HOLLYWOOD, FL – OCTOBER 27: Ray Cooper III v Magomed Magomedkerimov in a welterweight championship bout during the PFL championships at the Seminole Hard Rock Hotel & Casino on October 27, 2021 in Hollywood, Florida. (Cooper Neill / PFL)
Two-time season champion and $1 million winner Ray Cooper III (24-7-1) will be among the welterweights in action to kick off the season. He’ll take on fight week replacement opponent Carlos Leal (16-3), who stepped in for the unbeaten Magomed Umalatov (11-0), to start his quest for a third title. Also in action will be former UFC and WEC lightweight champ Anthony Pettis (24-12) against Myles Price (11-7) and former Bellator welterweight champ Rory MacDonald (22-8-1) against Brett Cooper (28-15). Both Pettis and MacDonald will be looking for fresh starts after a rough introduction to their PFL careers in 2021.
Kayla Harrison vs. Marina Mokhnatkina
Ray Cooper III vs. Carlos Leal
Anthony Pettis vs. Myles Price
Brett Cooper vs. Rory MacDonald
Magomed Magomedkerimov vs. Joao Zeferino
Julia Budd vs. Genah Fabian
Nikolay Aleksakhin vs. Sadibou Sy
Zamzagul Fayzallanova vs. Larissa Pacheco
Jarrah Al-Silawi vs. Gleison Tibau
Helena Kolesnyk vs. Abigail Montes
Martina Jindrova vs. Vanessa Melo
Prime Video Presents
Jubilee – Official Trailer
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Starring Aditi Rao Hydari, Aparshakti Khurana, Prosenjit Chatterjee, Ram Kapoor, Sidhant Gupta, Wamiqa Gabbi and others
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While renewing one’s driver’s license may seem like a routine task for most Americans, an Arizona man [Mike Palmer] who has been a licensed driver in the Grand Canyon State for four decades is now being denied a renewal because he does not have a Social Security Number – and he doesn’t want one, believing it to be tied to Bible prophecies about the end-times “mark of the beast” . . .
Palmer [a self-described “fundamental Christian”] contacted the state about his situation, and received the following response from Susan Pacheco with Constituent Services at the director’s office of the Motor Vehicle Department:
“Dear Mr. Palmer, Your record and request to apply without a social security number has been reviewed. Unfortunately, we cannot issue a credential without it. You must present a valid social security number in order to obtain a license in the state of Arizona. There is no exception in the state or federal law that would allow you to present any other type of document, such as an affidavit like the one the US State Department accepts for passport issuance. We are bound by the relevant state and federal statutes, Arizona Revised Statutes (A.R.S.) § 28-3158 and 42 United States Code Annotated (U.S.C.A) § 666.”
“Interestingly,” says Palmer, “the federal law that the Arizona MVD cites for authority is USCA § 666, which is a sign to me that I’m right. The Social Security Number IS a precursor to the mark.” (Read more from “Christian Denied Driver’s License, State Cites U.S. Federal Law ‘666’” HERE)
Photo credit: Flickr
Delete Facebook, Delete Twitter, Follow Restoring Liberty and Joe Miller at gab HERE.
Der Yen stürzt gerade richtig ab. Erste Banken wetten bereits gegen die japanische Währung, den Yen. Ist Japan gerade dabei einen globalen Crash der Finanzmärkte auszulösen? In unserer tiefen Recherche sind wir der Sache auf den Grund gegangen und mussten leider feststellen, dass Japan gerade aufzeigt, wohin die Reise des Euros gehen könnte. Seien Sie auf diese Informationen gespannt.
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00:00 Das erwartet Sie im Video
00:19 Der Yen im Sturzflug
03:17 Grund für die Verschuldung
05:01 Wird das System kollabieren?
06:11 Crash auf globaler Ebene möglich?
07:10 Ist unser Euro noch sicher?
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Wie hat Ihnen das Video “JAPANS STAATSPLEITE: Droht jetzt die globale Finanzkrise?” gefallen? Wir würden uns über einen Daumen nach oben, einen Kommentar oder ein Abo unseres YouTube-Kanals freuen.
Vielen Dank und bis zum nächsten Video!
#BankOfJapan #Staatspleite #Finanzcrash
PFL 2022 women’s lightweight champion Larissa Pacheco has requirements for a non-season Kayla Harrison fight.
The Professional Fighters League 2023 season is about to start and one of its 2022 champions is set on winning the tournament again this year. While she prepares for her first bout, women’s lightweight champion Larissa Pacheco still ponders on the possibility of taking on the woman who beat her twice before, Kayla Harrison.
Though Pacheco was able to finally get a win over Harrison in last year’s final, the Brazilian will not have a chance to even the score in 2023’s season, given that Kayla will not be competing in the tournament. For that reason, Larissa believes it is unlikely that a fourth encounter between the two will take place any time soon.
Unless the company is willing to pay Larissa the same purse they agreed to pay Kayla.
“If they offer me a great purse, or at least something equivalent to what they offer her, I would obviously [do it],” Pacheco said in an interview with MMA Fighting. “But am I going to wait for her to decide when she wants to fight? PFL offered me a spot in the season, so I took it.”
“I think this fight won’t happen [in 2023], because I’m stuck in the season,” she continued. “It really depends on the situation, because I would have to abandon the season — and I don’t mean to be cocky, but this season is very advantageous for me – it’s pretty much saying I’ll win another belt. And to drop that for a fight with her and risk my winning streak, they would have to pay me really well, equivalent to what she gets paid, to really make it happen.”
While Larissa does wish to face Harrison again, she prefers to focus on her career and on the chance of winning one million dollars again in this year’s tournament. In fact, with Kayla out of the grand prix, Pacheco believes that her chances of winning again have significantly increased and that is an opportunity the Brazilian just cannot dismiss.
“If I were on the same stage of hers in terms of money and was sure of my contract, relax,” Pacheco said. “Like, ‘I need time for my mind, I need time to assess everything that’s going on and decide which way I wanna go.’ I would possibly do the same thing.
“Kayla has won so many millions of dollars with the PFL and sponsors and everything, so I think she’s happy about that. She wants big names and big fights. Maybe she needs more time to see what’s going on in her life, and all she needs right now are superfights. I’m still very young, I still have a lot to grow. I can’t simply not work just because she doesn’t want [to fight] right now, or because they won’t want to book it right now. I’ll continue working and doing my part.”
Pacheco (19-4) won the 2022 tournament by beating Zamzagul Fayzallanova, Genah Fabian, Olena Kolesnyk and Harrison, with all but the last win coming by way of knockout. Now, she is expected to debut in the 2023 season at PLF 2, against former Bellator featherweight champion Julia Budd.
The April 7 card is scheduled to be healdlined by a heavyweight bout between Ante Delija and UFC veteran Yorgan de Castro.
ATURDAY MORNING CARTOONS ARE BACK – THE ROBONIC STOOGES PARODY SECRET WARS WITH: STOOGE WARS!
AMERICAN MYTHOLOGY PRESENTS – STOOGE WARS #1 MAIN PARODY CVR A SHANOWER
Retail Price: $3.99
Cover: Eric Shanower
Writers: SA Check
Artist: Diego Tapie
Age Rating: All Ages
Color, 32 pages
UPC: 856091006021 40111
On Sale: 9/28/22
CONTENT: Could this be The Robonic Stooges last stand?! The fate of the world hangs in the balance as the mysterious stranger known only as The Overyonder has kidnapped the planet’s greatest heroes and deadliest villains to battle for the ultimate prize! Can Moe, Larry, and Curly pull their ragtag team together? Or will the combined forces of Professor Cluck, PoodleMan, Dr. Crackula, and the rest of the Robo-Stooges’ legendary rogues gallery prove too much for the boys! Tune in to find out! This is the Robonic Stooges event of the year! The Robonic Stooges Present – Stooge Wars comes with three covers – Main Parody Cover by Eric Shanower, Animation Cel cover, and a 100 copy limited edition Century cover!
AMERICAN MYTHOLOGY PRESENTS – STOOGE WARS #1 ANIMATION CEL CVR B
Retail Price: $4.99
Writers: SA Check
Artist: Diego Tapie
Age Rating: All Ages
Color, 32 pages
UPC: 856091006021 40121
On Sale: 9/28/22
ROBONIC STOOGES PRESENT – STOOGE WARS #1 CENTURY 1/100 LTD ED CVR C
Retail Price: $29.99
Cover: Eric Shanower
Writers: SA Check
Artist: Diego Tapie
Age Rating: All Ages
Color, 32 pages
UPC: 856091006021 40131
On Sale: 9/28/22
CONTENT: It’s the Holy Stooge Grail! Only 100 copies were printed of this special edition. Have it slabbed by your favorite grading company!
MIKE BARON’S OUTRAGEOUS ROMP CONTINUES!
FLORIDA MAN #2 MAIN CVR A MARTINS
Retail Price: $3.99
Cover: Elias Martins
Writers: Mike Baron
Artist: Todd Mulrooney, Elias Marins
Age Rating: Teen+
Color, 32 pages
UPC:
On Sale: 8/31/22
CONTENT: After Gary gets out of jail, his luck begins to change. His lottery numbers come in and he can finally give his gal Krystal a big wedding. The happy couple buy a mansion, luxury SUV, state of the art flamethrowers, and much, MUCH more. But the honeymoon is interrupted by the po-po, causing Krystal to go berserk. Luckily her friend gets it all on a viral video launching Krystal’s wrestling career. The hilarious tales continue in this second stupefying issue of Eisner-winner Mike Baron’s Florida Man! Florida Man #2 is available with two covers – Main by Elias Martins and Variant by Barry McClain Jr.
FLORIDA MAN #1 MAIN CVR B MCCLAIN JR VARIANT
Retail Price: $3.99
Cover: Barry McClain Jr
Writers: Mike Baron
Artist: Todd Mulrooney, Elias Marins
Age Rating: Teen+
Color, 48 pages
UPC: 856091006113 00731
On Sale: 8/31/22
REINVENTING CLASSIC HORROR WITH A MODERN TWIST!
AMERICAN MYTHOLOGY MONSTERS VOL 3 #2 CVR A VOKES (MR)
Retail Price: $3.99
Cover: Neil Vokes
Writers: Konstantine Paradais, Glenn Moane, S.A. Check
Artist: EV Cantada, Horacio Domingues, Jorge Pacheco
Age Rating: Mature Readers
B&W, 32 pages
UPC: 856091006588 30211
On Sale: 8/31/22
CONTENT: The series where creators reimagine classic monsters for the 21st century is back for our third volume of original tales of terror! This issue keeps the horror train running with new installments of “Mother Sleepless,” “The Heartbreak,” and “Hell On Board!” Join for another frightening issue of American Mythology Monsters! This issue comes with two covers – Main by legendary comic artist Neil Vokes and a Racy Vampire variant by the Corpse Crew team of Buz Hasson and Ken Haeser!
AMERICAN MYTHOLOGY MONSTERS VOL 3 #2 CVR B RACY VAMPIRE (MR)
Retail Price: $9.99
Cover: Buz Hasson, Ken Haeser
Writers: Konstantine Paradais, Glenn Moane, S.A. Check
Artist: EV Cantada, Horacio Domingues, Jorge Pacheco
Age Rating: Mature Readers
B&W, 32 pages
UPC: 856091006588 30221
On Sale: 8/31/22
CONTENT: Get a peek at the devilish divas of the deep with this Racy Vampire cover! But beware, spotting this siren could cost you your very life!
THE FINAL ISSUE OF TOM HOLLAND’S INCREDIBLE FRIGHT NIGHT REBIRTH!
TOM HOLLANDS FRIGHT NIGHT #4 CVR A MATINEZ
Retail Price: $3.99
Cover: Roy Allan Martinez
Writers: James Kuhoric
Artists: Jason Craig, Neil Vokes
Age Rating: Teen
Color, 32 pages
UPC: 856091006687 00411
On Sale 9/28/22
CONTENT: The final issue of our first series is here! Peter, Amy, and Charlie must once again face the creatures of darkness as their insidious plan is revealed! The resurrection of Jerry Dandrige and there is going to be Hell to pay! But an unsuspecting ally may help the motley crew of vampire killers to escape the mounting horrors with their lives. The official continuation of Tom Holland’s amazing screenplay delivers Fright Night’s unique mix of horror, humor, and heart! This issue comes with four covers – Main by Roy Allan Martinez, Throwback by Neil Vokes, Jerry & Evil by Buz Hasson & Ken Haeser, and a limited edition also featuring the stark Black & White art of Roy Allan Martinez!
TOM HOLLANDS FRIGHT NIGHT #4 CVR B VOKES (C: 0-1-2)
Retail Price: $3.99
Cover: Neil Vokes
Writers: James Kuhoric
Artists: Jason Craig, Neil Vokes
Age Rating: Teen
Color, 32 pages
UPC: 856091006687 00421
On Sale 9/28/22
TOM HOLLANDS FRIGHT NIGHT #1 CVR C HAESER & HASSON (C: 0-1-2)
Retail Price: $3.99
Cover: Buz Hasson, Ken Haeser
Writers: James Kuhoric
Artists: Jason Craig, Neil Vokes
Age Rating: Teen
Color, 32 pages
UPC: 856091006687 00431
On Sale 9/28/22
TOM HOLLANDS FRIGHT NIGHT #4 CVR D VIRGIN ART B&W LTD ED (C: 0-1-2)
Retail Price: $9.99
Cover: Roy Allan Martinez
Writers: James Kuhoric
Artists: Jason Craig, Neil Vokes
Age Rating: Teen
Color, 32 pages
UPC: 856091006687 00441
On Sale 9/28/22
CONTENT: It’s the rare Black & White Virgin art limited edition! This is perfect to slab with your favorite grading company!
ZORRO NEW WORLD #1-4 PHOTO COVER COLLECTOR PACK
Retail Price: $34.99
Age Rating: All Ages
32 pages, 4 Limited Edition Photo Comics in a pack
On Sale: 8/31/22
CONTENT: Did you miss out on American Mythology’s Zorro New World special editions? These super low print run comics featured a special color photo cover from the 1990’s TV series featuring dashing Duncan Regehr as the Masked Avenger, Zorro! This set includes all FOUR of the photo covers and is an incredible discounted value!
ZORRO FLIGHTS #1 MAIN COVER WITH ZORRO SER 2 PROMO CARD
Retail Price: $14.99
Cover: Roy Allan Martinez
Writer: Don McGregor
Artist: Claudio Avella
Age Rating: Teen
Color, 32 pages
On Sale: 8/31/22
CONTENT: Did you miss out on Don McGregor’s return to Zorro! Get a special copy of the first issue with an RRParks Zorro Series 2 Promotional Trading Card in one pack!
ZORRO FLIGHTS #1 AM EXCLUSIVE B&W VARIANT CVR F
Retail Price: $6.99
Cover: Vincenzo Carratu
Writers: Don McGregor
Artists: Vincenzo Carratu
Age Rating: Teen
Color, 32 pages
On Sale: 8/31/22
CONTENT: An awesome rare edition for the American Mythology fanatics! This is the AM Exclusive cover featuring the beautiful black & white Vincenzo Carratu artwork!
OGGY & THE COCKROACHES TPB SIGNED BY SA CHECK
Retail Price: $14.99
Age Rating: All Ages
UPC: 856091006557 00181
On Sale: 8/31/22
CONTENT: Get the Oggy & the Cockroaches TPB signed by writer S.A. Check – no convention trip required!
STARRING SONYA DEVEREAUX SUMMER CAMP MASSACRE #1 COSPLAY CVR C
Retail Price: $14.99
Cover: Photo
Writer: Todd Livingston, Nick Capertankis
Artist: Brendon Fraim, Brian Fraim
Age Rating: Teen
Color, 32 pages
UPC: 856091006441 01971
On Sale: 8/31/22
CONTENT: Get a beautiful cosplay cover featuring Jennifer Goehner as Sonya that was only available as a crowd funder last year! This incredibly rare edition is perfect for getting a convention autograph!