This article from The Diplomat helps explain the historical context from which Graham named the concept:
In Thucydides’ history, human emotion made conflict inevitable, and at several points where peace was possible, emotion propelled it forward. In the beginning (my add: before the beginning of the Peloponnesian War), there is a set of speeches in Sparta debating the possibility of going to war with Athens. Archidamus, the Spartan king, tells the Spartan people not to underestimate the power of Athens and urged that Sparta “must not be hurried into deciding in a day’s brief space a question which concerns many lives and fortunes and many cities, and in which honor is deeply involved – but we must decide calmly.” However, Sthenelaidas, a Spartan ephor, advocated, “Vote, therefore, Spartans, for war, as the honor of Sparta demands.” The Spartans followed Sthenelaidas, which led to a war of honor and fear against the Athenians.
In the seventh year of the Peloponnesian War, at the battle of Pylos, the Athenians won a major victory over Sparta. Because of their loss, Sparta sent envoys to Athens offer a peace treaty. The Spartan envoys enjoined the Athenians to “treat their gains as precarious,” and advised that “if great enmities are ever to be really settled, we think it will be, not by the system of revenge and military success… but when the more fortunate combatant waives his privileges and, guided by gentler feelings, conquers his rival in generosity and accords peace on more moderate conditions than expected.” However, the Athenians, led by Cleon, who Thucydides described as the most violent man in Athens, accused the Spartans of not having right intentions, and made further demands on Sparta for a return of territories that Athens had previously ceded to Sparta before the outbreak of the Peloponnesian War, resulting in a continuation of the war.
Source: http://thediplomat.com/2015/05/the-real ... ides-trap/
From Professor Allison himself in a work sourced below,
More than 2,400 years ago, the Athenian historian Thucydides offered a powerful insight: “It was the rise of Athens, and the fear that this inspired in Sparta, that made war inevitable.” Others identified an array of contributing causes of the Peloponnesian War. But Thucydides went to the heart of the matter, focusing on the inexorable, structural stress caused by a rapid shift in the balance of power between two rivals. Note that Thucydides identified two key drivers of this dynamic: the rising power’s growing entitlement, sense of its importance, and demand for greater say and sway, on the one hand, and the fear, insecurity, and determination to defend the status quo this engenders in the established power, on the other.
One need not be a close student of world affairs to guess that the rise in interest in Allison's work corresponds with the rise of China's power and the growing sense that the full face of The United States of America's power and influence has waxed to it's fullest and is on the wane. Graham has himself written and been interviewed on just that particular aspect of the subject, including a work published in The Atlantic and much quoted again in this December's edition. From Graham's Harvard bio regarding his Atlantic article:
In a major essay for The Atlantic, Graham Allison argues that the best lens for clarifying the dynamics of the U.S.-China relationship is the Thucydides Trap: the structural stress that occurs when a rising power threatens to displace a ruling power. In reviewing the record of the past 500 years, Allison and the Belfer Center’s Thucydides Project have identified 16 cases that reflect this pattern of hegemonic challenge. In 12 of the cases, the rivalry between the rising and ruling powers produced war. Yet as Allison points out, the 4 cases that did not end in bloodshed show that war is not inevitable.
Source: http://belfercenter.ksg.harvard.edu/pub ... _trap.html
Here is a table of the 16 instances he reviewed:

When The Atlantic published it's December edition, the articles it included were written unaware of Trump's election. They included a cautionary interview with one Henry Kissinger that was amended for the online addition to include follow-up material with Kissinger on his take of the election. The original pre-election results included this bit about the problem facing the 45th President:
J.G.: What would you advise the 45th president to do first?
HK: The president should ask, “What are we trying to achieve, even if we must pursue it alone?” and “What are we trying to prevent, even if we must combat it alone?” The answers to these questions are the indispensable aspects of our foreign policy, which ought to form the basis of our strategic decisions.
The world is in chaos. Fundamental upheavals are occurring in many parts of the world simultaneously, most of which are governed by disparate principles. We are therefore faced with two problems: first, how to reduce regional chaos; second, how to create a coherent world order based on agreed-upon principles that are necessary for the operation of the entire system.
J.G.: Crises always intervene before presidents find time to create a coherent world order, no?
HK: Practically all the actors in the Middle East, China, Russia, and to a certain extent Europe are facing major strategic decisions.
J.G.: What are they waiting to do?
HK: To settle some fundamental directions of their policies. China, about the nature of its place in the world. Russia, about the goals of its confrontations. Europe, about its purpose, through a series of elections. America, about giving a meaning to its current turmoil in the aftermath of the election.
...
J.G.: Are Sino–American relations more consequential for U.S. national security than Islamist terrorism?
HK: Islamic terrorism is consequential for the prospects of international order in the short term. Our relations with China will shape international order in the long term. The United States and China will be the world’s most consequential countries. Economically, this is already the case. Yet both nations are having to undergo unprecedented domestic transformations. As a first step forward, we ought to try to develop an understanding of how joint Sino–American action could stabilize the world. At a minimum, we should agree to limit our disagreements; more sophisticatedly, we should identify projects we can undertake together.
J.G.: How should the 45th president make China policy?
HK: After its early years, America was lucky enough not to be threatened with invasion as it developed, not least because we were surrounded by two great oceans. As a consequence, America has conceived of foreign policy as a series of discrete challenges to be addressed as they arise on their merits rather than as part of an overall design. Not until the post–World War II period did we begin to think of foreign policy as a continuous process, even in seemingly tranquil circumstances. For at least 20 years, we forged alliances as a way to put down markers as much as to design a strategy. Henceforth, we must devise a more fluid strategy adjustable to changing circumstance. We must therefore study the histories and cultures of key international actors. We must also be permanently involved in international affairs.
J.G.: Constant engagement with China?
HK: China is an illustration. For most of its history, China also enjoyed isolation. The only exception is the 100 years it was dominated by Western societies. It did not have to continuously engage with the rest of the world, especially outside of Asia. But it was surrounded by relatively smaller nations incapable of disturbing the peace. Until the Xinhai Revolution in 1911, China’s relationships with other countries were managed by the Ministry of Rites, which classified each foreign country as a relative tributary to Beijing. China did not have diplomatic relationships in the Westphalian sense; it did not consider foreign countries equal entities.
J.G.: I think there are countries along its borders that don’t feel they are treated as equal entities.
HK: China is undergoing a tremendous process of domestic change. President Xi Jinping laid out two goals called the “Two 100s”—the 100th anniversary of the Communist Party and the 100th anniversary of the Communist state. The first will be in 2021; the second in 2049. By the time the Chinese reach the second 100, they will be, by their own estimate, the equal of any other country in the contemporary world, and will, by their reckoning, be able to insist on absolute material and strategic equality, including with America. Some Chinese strategists are in effect saying, “If we were in the American position, would we not at least consider preventing another country from reaching equality?” So that is a latent source of tension.
The Chinese internal discussions offer at least two answers. The hard-liners will say, “The Americans are visibly declining. We will win. We can afford to be tough and look at the world with sort of Cold War–ish attitudes.” The other position—apparently that of President Xi—is that confrontation is too dangerous: Cold War with the U.S. would keep China from reaching its economic goals. A conflict with modern weapons might exceed the devastation of the First World War and leave no winners. Hence in the modern period, adversarial countries must become partners and cooperate on a win-win basis.
J.G.: So Xi is a moderate?
HK: President Xi, for his part, has put forward two objectives for China. The first is “Asia for the Asians.” The second is an effort to turn adversaries into partners. In my opinion, we must try to make this second framework the dominant theme of U.S.–China relations. The Chinese view the world very differently than we do. We have to combine our own diplomatic and military capabilities to respond to this reality. But is that possible in the current world, with its weapons of mass destruction and cyber capabilities?
One obstacle is a cultural gap: The basic American attitude is that the normal condition of the world is peaceful, so if there’s a problem, someone is causing it. If we defeat that person or country, everything will become harmonious again.
By contrast, the Chinese do not believe in permanent solutions. To Beijing, a solution is simply an admission ticket to another problem. Thus, the Chinese are more interested in trends. They ask, “Where are you going? What do you think the world will look like in 15 years?”
As a result of this cultural gap, when the American and Chinese presidents meet, there is too often an ambiguous outcome. Progress is made on immediate short-term issues—climate change, some economic concerns. But the basic agenda of developing a common concept for the future is given less priority, in part because of the pressure of time and the impact of the media waiting outside the conference center.
Source: http://www.theatlantic.com/magazine/arc ... 5868/#body
Following the revelation of Trump's surprise election, the following was added:
J.G.: What’s your biggest concern about global stability coming out of this election?
HK: That foreign countries will react with shock. That said, I would like to keep open the possibility that new dialogues could emerge. If Trump says to the American people, “This is my philosophy of foreign policy,” and some of his policies are not identical to our previous policies but share their basic objectives, then continuity is possible.
J.G.: How is China going to react?
HK: I’m fairly confident that China’s reaction will be to study its options. I suspect that will be Russia’s reaction as well.
Source: http://www.theatlantic.com/magazine/arc ... er/505868/
Think about this and reflect on Trump's call with the President of Taiwan on Friday, a move that breaks US "one china policy" that has been in place since Kissinger and Nixon first crafted America's re-engagement with the Center Kingdom. China's response appears to be a public dismissal of a "green" foreign policy team unaware of how things work. Trump's response was to tweet amusement that the US can provide arms to Taiwan but he can't accept a phone call from their President. Correct or otherwise, one can't help but feel that we are in uncharted territory. And history is against our coming out of it without blood being shed unless we can avoid Thycydides' trap. Unfortunately, since we are the existing strong power, it means tamping down on our "fear, insecurity, and determination to defend the status quo", or in other words the very things Trump campaigned on as the core of his foreign policy whatever that may mean.
This is getting long, and will end without satisfying conclusion other than to point out the information since it hasn't apparently been brought up on the forum yet. I'll end with a link to another article from The Atlantic on China and what appears to be it's own retrenchment into national interest and rejection of the West. It was the cover story and titled, "China's Great Leap Backward".
You can guess what it suggests, and again, all written when the US was anticipating the election of Hillary Clinton and a traditional US foreign policy perhaps more Nixon than Obama but one with an informed view of the past and apprehension for the future peace.
http://www.theatlantic.com/magazine/arc ... rd/505817/
Note: Have I ever mentioned that December, for reasons I don't fully understand, feels more Gothic horror than jolly festival to me? That reading scary stories on Christmas Eve became a tradition of mine when I was young enough to have a hard time sleeping on Christmas Eve so finding a book to read was a way to kill time and my earliest memories of this include Poe and Stoker along with Dickens and they felt of a family. Perhaps this is then fitting to the season.
* ETA: I should also note that the reason The Atlantic is cited so often in the OP is that I subscribe to it in paper form. I've had the December edition for a couple of weeks but not had time to really sit and enjoy it until this morning. The two articles on China and the notion that history is betting against us led to more reading but it was the magazine articles the left the initial impression first and formed the germ that became the OP. I had been following the news about Trump's calls and strange, unsettling activities as the First American on the world stage with some chagrin before hand. Reading about concern over China-US relations independent of the Trump factor was just a turbo shot of espresso.
I've thought before that the world was spared an unfortunate conflict between the US and China under Bush due to 9/11. We forget that leading up to that fall, tension with China was escallating with events from captured spy planes to North Korean relations being in the news. One could almost sense the Pentagon was seeking to justify their budgets in a post-Cold War world and locking onto Sino-US relations as the foundation for US military policy. We needed a cold conflict and tension to keep the money rolling in. The terrorists in 2001 provided an actual hot war and we pivoted where the money was easier to justify. But that was perhaps just a postponing of the inevitable when the hawks are in office and the US looks around and realizes the next biggest guy on the block has been currying favors with all our old friends while we've been preoccupied elsewhere.