How many protesters in the streets does it take to bring an authoritarian government down? What is more conducive to a democratic revolution's success: the support of the majority or the decisive actions of a minority? Can an active but small group achieve change even if their support by the "passive" majority is very low? And - vice-versa - can a government stay in power even if a majority of the population is against it?
These questions are raised by the success of the celebrated - and feared - "colour revolutions" in Serbia, Georgia and Ukraine , and by the failure of the model to replicate itself in Belarus and Azerbaijan. Many different groups seek answers to them: observers, analysts and political scientists who wish to understand and draw lessons  from these transnational political experiences; civil-society activists and organisations  which attempt to provide resources of thinking and tools of change for people on the ground; and politicians, both those who seek to foment revolution and those who are eager to prevent and stifle it.
This article offers a model which - without aspiring to an exhaustive explanation - proposes a new way of evaluating the factors at work during a revolutionary situation, if not of forecasting its outcome; as well as introducing a different viewpoint and language in understanding the revolutionary process.
A pressure-point formula
The model comprises two elements: the level of popular support for the opposition (dissidents) and the number of people who can be mobilised for action (activists). The first hypothesis suggested is that there is a limit to the number of dissidents that can be transformed into activists even when the opposition behaves in the most effective and decisive manner. The actual proportion of dissidents-turned-activists varies depending on the nature of the regime  in a country, the capacities and abilities of the opposition, as well as historical and temporal  factors. The actual number of dissidents is influenced by these structural conditions.At first sight, this hypothesis is defied by the experience of attempted revolutions in Belarus. The democratic candidates during the presidential elections  in September 2001  and March 2006 received - according to independent assessments, rather than the widely discredited official results - a comparable amount of votes: Uladzimir Hancharyk gathered 27.9% of the votes in 2001, and Alexander Kozulin  and Alexander Milinkevich  jointly received 23.5% of the votes in 2006. At the same time, in 2001, only 2,000-3,000 people turned out to protest against the election result, whereas in 2006, the figure approached 30,000: a substantial difference.
The argument here is not that a certain amount of dissidents inevitably produces a corresponding number of activists; rather, it suggests that there is a "ceiling" to the number of people that can turn up in the streets. In 2006, the Belarusian opposition was more experienced, and prepared for protests well, hence it saw a higher activist turnout. Yet, it is quite unfeasible to expect to see some 100,000 people turn up to protest, let alone higher numbers.
The connection between the numbers of dissidents and of activists is not linear. There are times when the size of the dissident element can produce an explosion of activists, surprising  not only the authorities but the revolutionaries themselves. The evidence of one of the organisers of Ukraine's "orange revolution" in 2004-05, Taras Stetskiv , is telling: "Ten days before 21 November 2004 [the date for the second round of the presidential elections in Ukraine], I asked the leaders of Pora  [the youth movement] how many universities they hoped they would be able to persuade to strike. They said they were hoping for eight at best. On the first day of the protest, all universities in the capital went on strike. Similarly, Pora estimated that they would get some 15,000 students to protest within two days. In fact, 200,000 people turned up on day one." This second example does not refute the model either, however, because the increase in the number of activists became possible only when over half the same number of protestors had gathered.
The model also proposes that the degree of popular pressure on the government depends on the precise numbers of dissidents and activists. General social dissent with the regime affects the members of the administration, and civil servants in particular. The German sociologist Elisabeth Noelle-Neumann  describes the mechanism of this influence with the concept of "the spiral of silence". She explains that people feel uncomfortable when they are isolated from the majority, and have almost an animalistic feel for what the predominant opinion in the society is. In other words, the opinions of the majority influence the views of the rest of the society.
An important pressure-point on the authorities is the size of the public demonstrations by activists and protesters; this has a significant psychological impact on the members of the coercive forces, such as the police. Riot police, who have taken part in clamping down on protests, report feeling vulnerable to the crowd, even if it behaves peacefully. The model indicates that popular pressure on the authorities can reach a critical point, after which the governmental officials lose the will to resist, and that orders to clampdown on protests are effectively disobeyed.
The results of such succumbing to popular pressure are visible in the colour revolutions in Serbia (2000), Georgia (2003-04) and Ukraine (2004-05). Remarkably, the student protests  over the local elections in Belgrade in 1996-97 did not topple the Slobodan Milosevic  regime; the model suggests that this was because the amount of activists was not sustained by a sufficient amount of dissidents. It was the larger numbers of dissidents combined with an approximately similar amount of activists that brought the regime down in 2000 within a matter of days.
The rules of the game
The actual calculation of the functions described earlier - the dependency of the number of activists on the number of dissidents and their pressure on the government - is not straightforward. Yet, three general rules can be drawn from the basic assumptions in the model.
First, there is a certain grey area in the number of dissidents, in which the opposition is incapable of drawing a critical mass of activists and successfully exerting pressure on the government - even if the latter operates in the most skilful and committed way.
During the mass protests in Minsk  after the March 2006 presidential election, the demonstrators regularly repeated a politically clever slogan: "the police is with the people". One of the policemen containing the demonstration rumbled an equally sophisticated retort: "that's right, we are with the people not with you".
Second, there also exists a critical amount of dissidents under which the government crumbles even if the politically organised opposition is weak and helpless. True, the organised minority in that case has to be substantial, but it often emerges spontaneously rather than in the result of the opposition's activity, because of the psychological perception that the activist minority is supported by the mass majority.
Third, there is a zone between these two categories of the number of dissidents and of activists where the outcome is uncertain, where the success of the revolution depends on political skill.
The theory elaborated here fits comfortably into a simple algebraic model which is described in an appendix (see Box 1).
This model may appear too simplistic and impractical, as the actual calculation of the pressure over government hardly yields to quantitative analysis. Yet it is worth noting that the coloured revolutions of the first half of the 2000s have been characterised by both a fast pace and a relatively low level of physical coercion. During the great revolutions in Russia and France, millions engaged in throat-cutting for the sake of their cause. At present, it appears that the actual participants in the revolutionary process are not keen to fight the opponents but rather prefer to wait for the outcome as if from some heavenly calculator. When this calculation is higher than the critical point, the government falls apart: "thou art weighted and thou art found wanting". When the pressure is lower than the crucial point, people leave the central square, where the remaining protesters can save their dignity - but not win.
This article was translated from Russian by Natalia Leshchenko
Box 1 - AppendixLet X be the number of "dissidents", and Y the number of "activists'. Ymax is the maximum amount of activists with a constant X. Then
where A is the positive constant of the size, thousands of people/%.
Let Ymin be the minimal amount of activists that can be mobilised. Then
Ymin=0 if <XO
Ymin=B*(X-XO) if X>XO (2)
where XO is the number of X after which activists mobilise spontaneously even in the absence of organisation from the opposition. B is the positive constant of the size, thousands of people/%.
The amount of pressure on the government can be calculated by the following formula:
- where C and D are the positive constants of the size, thousands of people/% and F is an unlimited value.
Let Fmin be the minimal level of pressure on the government that it can sustain. Then we can see the "revolution line" from the function (3) as the derivative of X and Y
Y=(Fmin/D) -(C/D)*X (4)
If the points X and Y emerge above the revolution line, than the revolution will succeed; if below, it will fail.
The combination of dependencies (1), (2), and (4) gives us a graphic model of revolutions (see picture 1).
The left bottom square limited by the line OX and lines Y=A*X, Y=B*(X-XO) and Y=(Fmin/D) - (C/D)*X is the area of the revolution failure. In any amount of value of X and Y the pressure will not be sufficient enough to cause governmental defeat.
The right top square denoted by Y=A*X, Y= B*(X-X0) и Y=(Fmin/D) - (C/D)*Х is the area of revolution's success.
If the level of mass support for the opposition is low X<X1=Fmin/(A*D+C) (the point where Y=A*X and Y=(Fmin/D) - (C/D)*X cross), then even the most decisive activist minority will not achieve success.
If the mass support for the opposition exceeds the critical point X>X2=(Fmin+B*D*X0)/(C+B*D) (the point where Ymin= B*(X-X0) and Y=(Fmin/D) - (C/D)*X) cross, then the government has no chances to hold on.The interval (X1, X2) is the area for political struggle. It is up to the opposition in this case to try and create popular pressure on the government so that its level of pressure exceeds the critical point. If they do not manage to do so, the revolution fails even if it did have chances of success.