The Red Army grew very large indeed during the Russian Civil War, with many sources citing an eventual 5.5 million men in arms. This did not always mean that there were that many boots at the front lines though.
Desertion was prolific throughout the entire war. White cites a total of 2,800,000 soldiers apprehended as deserters in 1919 alone. The number who actually deserted was obviously much higher. And of course desertion rose rapidly in action, so that pre-battle totals might be meaningless.
The "tail" of the army was also massive, so that nine men behind the lines might support one man at the front with rifle in hand. As well as obvious dead-weight, like political workers, the supply system was disastrously bad and used up a lot of men just physically transporting material, even from men supposedly "at the front". Supplies needed to be heavily guarded at all times, just to prevent the rampant corruption and the attacks of greens. Desperately needed infrastructure work was often done by "Labour armies", men in combat units tasked with manual work between combat assignments.
Finally, even for the man with rifle in his hand, as many men might be putting down internal revolts as were in the front lines. As more and more of the periphery was conquered this problem grew considerably, because those new areas taken were not, in general, heavy supporters of the Bolsheviks. So Kolchak's final defeat did not spare many men for other fronts, because enormous peasant rebellions in Siberia required immediate attention or the centre would have starved.
The result was that while the Soviets controlled the majority of the Russian population they were never able to field a great deal more men than their opponents. In fact they never really had enough troops to fight all their fronts at once. When Denikin and Iudenich attacked simultaneously, Lenin was lucky that Kolchak's 1919 offensive had already failed, and that the Poles, Latvians and Estonians did not join in – because that would probably have finished off the Soviet state.
Care is needed when comparing Red totals vs White totals in campaigns and battles. When giving the numbers for a division, the Soviets counted everyone, often referring to them as "eaters", but the Whites counted soldiers at the front line. When we have Red divisions broken down into "bayonets" and "sabres" though we see that the front line soldiers were rarely even 50% of the total, and could be very much less. The Whites seem to have counted staff and artillery separately from the front line, so not in their "bayonet" and "sabre" count. Once we start counting like for like, the Soviet superiority in numbers largely disappears.
On 1 June 1919, the Red Army counted their forces at the fronts as approximately 310,000 bayonets, 35,000 sabres, 8,700 MGs and 1,750 guns. Their estimation of the forces opposing them was 400,000 bayonets, 110,000 sabres, 5,000 MGs and 1,500 guns. There are some anomalies with the counting – Makhno, an implacable opponent in Denikin's rear – did not count as a Soviet, whereas the Freikorps and Estonian national army – currently fighting each other – and all the squabbling Ukrainian factions would seem to be added into the opposition. But the figures cannot be completely discarded.
So while there were local exceptions, the Red Army was not greatly more numerous than its opponents. To explain how they won without large numerical advantages, requires either that the Soviet troops were higher quality than commonly posed, or that its opponents were lower quality or they had strategic advantages. I tend to think all three played a part.
The Osprey Red Army book says that by Winter 1919 that cavalry made up 16% of all Soviet armed forces, but this seems to be wrong. In June 1919 the total of the Soviet fronts shows that about 10% of the "bayonets and sabres" were sabres, but that excludes artillery crews, security troops etc. The figures for the end of 1920 give the proportion on active external fronts as about 18% cavalry (72,000 out of 400,000).
My calculations are that about 10% of the external Soviet Fronts were cavalry throughout the civil war – a bit less early on, a bit more later. The vast expansion of the Red cavalry arm is in all the books, but what is forgotten is that the infantry expanded at the same rate. When the small numbers are mentioned early in the war, the parallel lack of infantry is passed over.
The only exception to this is basically wherever the First Cavalry Army was. But even then the cavalry only reached 15% on the Polish Fronts in 1920 thanks to the presence of the First Horse Army and Gai's KavKor, and that was the priority front at the time. Some Fronts in the south in 1920 reached to nearly 50% mounted – when they contained the cavalry armies.
White, D. D. F. Growth of the Red Army
Old, but relatively sound.