There are several ways to pass on knowledge. You can use long-form methods like books and classes, medium-form methods like speeches or articles, or short-term methods like maxims or proverbs. One of the maxims that has resounded is that the best defense is a good offense. Widely attributed to several historical figures such as George Washington, Carl von Clausewitz and Jack Dempsey, this philosophy has realistically been around for millennia. This is not only a philosophy for personal action - this philosophy can steer geopolitical strategy for a nation. The desire to be safe is an innate part of living, and this need extends from the individual to the nation - a person is made safe from external threats by the ability of their nation to defend against them.
A stronger nation is more able to protect its people, and if this strength comes at the expense of people not part of its nation - too bad. This is not a system ideal for human rights or fairness. This is a system ideal for survival in a cruel, antagonistic world filled with enemies whose motivations are impossible to discern. It was in this world that the empires of old rose, conquered, prospered, and fell. It was from this world that we live in today. And in this world, there maintains a constant - it is better to be an imperial than a colonist.
If you were to make a Mount Rushmore of empires, it is first important to distinguish the eras that they existed in. The influence that technology and doctrine has on warfare is paramount, but how rapidly technology advanced and how vital it was to have adapted doctrine was not a constant. For instance, if you pitched Alexander the Great's army from around 300 BC against Edward the Black Prince's army from around 1350 AD, Alexander the Great would probably trash Edward's forces - the principles of warfare did not drastically change in that span of time. This changed with the idea of combined arms, often attributed to the Spanish tercios of the sixteenth century. Corresponding with the Renaissance, technology started to advance faster, and doctrine likewise had to adapt - this culminated in the Napoleonic corps system, which was a logistic necessity because of the levee en masse France used to counter a Europe united against its revolution.
Keeping in mind the context of each empire's location and technology, it is thereby necessary to distinguish between eras. Similar to how baseball has a deadball era and a liveball era, empires had a deadball era of their own - the era before mass mechanization. If a date had to be set, it might be best to set it during the 1899 Hague Convention, when global powers agreed to restrictions on the methodology of warfare, including dropping bombs from balloons, using chemical weapons, and firing bullets that we would recognize as hollow-tipped. The 19th century leadup to the Hague convention was one gradual realization that war would never be the same - from Napoleonic corps to Crimean hospitals to the Siege of Petersburg to the Prussian blitz on Paris, the methodology to win a war increased in complexity and scale.
Taking this into account, it seems wrong to compare the might of the Mongol Empire to the Third Reich - both were playing in different metaphorical sandboxes, with different metaphorical toys. As such, when compiling the premier empires, there must be a distinction between the pre-1899 deadball and the post-1899 liveball. The deadball era featured the shattering but short-lived empires such as Alexander's Empire and the Mongol Empire, and the longer-lasting and influential empires of Rome, Han China, and Britain. The liveball era also featured the British Empire, but one definitively on the decline - if the First World War did not occur, perhaps the British Empire may have continued in perpetuity. But that cataclysm was the beginning of the end for the old empires, and saw the rise of the new ones - the brief Third Reich, the lumbering Soviet Union, and the United States of America, which was ultimately victorious in the First World War, the Second World War, and the Cold War.
It was during the Cold War that the world was more or less split between two empires - the First World led by the United States of America and the Second World led by the Soviet Union, with unaligned states such as Brazil and India being considered part of the Third World.. But the empires of the Cold War were not empires in the traditional sense of the deadball era. They would better be understood as opposing world orders rather than empires - though the Soviet Union would crush anti-communist revolts after the Second World War, it didn't directly assume control of the areas they revolted in. Rather, governments of similar ideology would be installed and maintained - similar to how the United States attempted to install and ensure anti-communist governments in South America, Africa and Southeast Asia, with inconsistent results. The power of an order is based more on the soft power of culture, investment and the threat of police action rather than the hard power of invasion, pillage and conquest that an empire would utilize.
The fall of the Soviet Union meant the dominance of the American Order, with the dominant international agencies such as the United Nations, the World Bank and the World Health Organization generally acting in accordance with American grand strategy. The United Nations, while never entirely in America's corner because of the setup of the Security Council, would almost never act directly against America's interests. And the spread of NATO eastward across Europe brought greater security for members and more unease for Russia. Russia, the major successor state of the Soviet Union, would try and reinstate an order of its own with the Collective Security Treaty Organization, but it doesn't have the institutions, the membership, the money, or the power projection that the United States does.
But that does not mean the United States is the only country able to set up a world order. China's imperial past long involved non-Chinese tributary states, and eastern Asia is frequently referred to as the Sinosphere for the immensity of Chinese influence in military history, cultural influence, linguistic formation and trade routes in the region. China is very much a country on the ascent - free trade allowed foreign companies to produce in China, which encouraged rapid modernization and led to urbanization. Military technology that is not independently developed is often stolen from the United States, and administrative reforms have streamlined the Central Committee's ability to balance the benefits of capitalism and socialism. America's Order is slowly but surely starting to fray, and China is extremely eager to step into whatever vacuum may develop - not only to increase their own power, but to limit the power that possible opponents may be able to wield against China.
China's ambition can be seen in its Belt and Road initiative. An echo of the legendary Silk Road, the Belt and Road initiative is China's blueprint for its world order. Trade agreements and infrastructure construction are the foundation for this world order, and Chinese investments in Pakistan, Nigeria, Australia, and elsewhere are not only ways to build China's long-term wealth - foreign investment is a way to make the invested countries dependent on China economically. Pakistan is of particular importance - China and India recently agreed to a cessation of conflict in the Himalayas, but they are still have severe strategic opponents. India is trying to have the industrial renaissance that China had, but it has been held back by political turmoil, racial riots and intermittent war with Pakistan, with Kashmir being a particularly tumultuous issue.
Complicating the issue is the Dalai Lama - when China invaded and occupied Tibet, the Dalai Lama fled to India. India's refusal to hand him over to China ensured a level of animosity between the two nations, because the Dalai Lama would always be a rallying point for Tibetian separatists and Buddhist opponents of Communist control in China. Chinese control of the Panchen Lama, who chooses the next Dalai Lama, means that this casus belli may expire with the current Dalai Lama's death. But what will not expire is Chinese control over most rivers in South and Eastern Asia - Chinese control of the Tibetan plateau enables it to set up hydroelectric plants all along the riverways before they reach other countries. This can generate a great amount of power - it can also disrupt the environment, economy, and livelihood of countless people. Water wars have been predicted as a new source of conflict in this century, and it may be in Arunachal Pradesh, a disputed state of India, that a war involving three nuclear powers and three billion people may start.
Be it an empire or an order, the projection of power is necessary to maintain it. China is willing to create and order and increasingly able to maintain it; the United States is wavering in commitment to its order and is becoming increasingly demoralized with quagmires abroad and a lack of political leadership at home. North Korea may be the catalyst for the definitive fall of the American Order and the rise of the Chinese Order; it could also be a red herring that will end with Chinese invasion of North Korea followed by an installation of a Chinese-friendly government that doesn't have a throbbing desire for nuclear weapons; such an action would simultaneously make South Korea and Japan safer while also make them more cautious of Chinese invasion and bring them further into the American Order.
In the end, an Order is based on the vision of its central member. An America that lacks a leader with a realistic vision of the world it tries to lead is more prone to have its Order disintegrate. You may disagree with American adventurism abroad, and the results of our recent wars would justify your belief. But the Order we have, the Order we maintain, is to avoid the chaos that would benefit those who would like to see us weak. If chaos is a ladder, it is in our interest to keep that ladder locked up.