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Well, feeding cargo into your city will stimulate indrustr/commercial growth, which in turn facilitates job growth and shopping facilities which are a reason why agents want to travel in the first place.
I'm not quite sure in what impact passenger volumes have on city growth, but i can imagine city growth is smaller with 'only' hooking up other cities with minimal passengers movement between those cities.
That because you have 50 years 'extra' ;)
Also because of the fact trains are so slow, and won't make really big profits, you can't really invest in many cargo lines. So probably what you do at 1850 start is connect a few cities and fast-forward a lot which results in a feeling of extra fast growth.
In other words, i doubt the game is hard-coded for extra fast city growth on a certain start date.
The environment rating is largely out of your control. The more a building is surrounded by similar type buildings the better the environment rating. So for instance when all the green stay clumped together they have higher environment rating. Same for commercial and industrial.
Accessibility is how much of something is within a certain range (range being based on how far can they get within a certain amount of time). In Train Fever this time range was based on the '20 minute' rule. So for instance, for a residential building the relevant access that is needed is how many jobs and shopping slots can be reached by people in this building in 20 minutes. The higher this number is relative to the amount of residents in the city determines how quickly that area will grow. Although the 20 minute rule is no longer a strict limit on how far people will go in Transport Fever, I believe it is still strictly determining the range for accessibility purposes.
I never did determine exactly what quality was determined by. But I suspect it is based on the accessibility within average distance of the building. So for instance if a residential building has access to 1000 jobs, but 700 of those jobs are around the 15-20 minute range, while the remaining 300 are in 1-5 minute range, with nothing in between due to the gaps between cities, then it will have a low quality rating for the job accessibility since the majority of them are above the 10 minute mid range. I imagine it works something like that.
With all that in mind it is in fact much quicker to grow similar sized cities in the late game compared to the early game due to 1) the access to personal cars and faster cars as time goes on, both of which extends the range of people and in turn their accessibility. 2) The faster means of transport options you can offer people.
Although accessibility numbers are determined on a per-building basis, I believe the growth factor is determined on the average accessibility of buildings in small neighborhoods which the borders of are hidden to the player. Also I would imagine one neighborhoods growth factor influences surrounding neighborhoods in order for a town to expand outward rather than only expanding where there are already buildings.
You can check some of my Train Fever screenshots from my profile to see some of the shots I took when studying town growth in that game.
Without passenger transportation, your cities won´t grow that much just by supplying goods. Even if you are on excellent level.
I supply three cities with medium to excellent level of all six goods over long periods of time. But they only expanded quite slow because my passenger transportation system was average.
Now the point: I already made huge amounts of money, even on hard difficulty (actually nearly 3 billion credits) and spend time by optimizing lines and testing around.
Now I expanded my passenger system and now my cities are growing very fast. Before, my biggest city was around 2.000 habitants. Now within a few years, it´s at 3.500 and still growing.
I not just expanded tram and bus lines within the city, but connected several cities with train lines and low travel time up to max. 3 min. I even have one route over the whole map which connects 10 or so cities and on this line 17 trains are driving in both directions with 3min travel time. Also important: I have several ways of how cities are connected. My two biggest cities are connected by a train line, a bus line and even a tram line. People love choices and it will help chosing public transportation rather than car.
I didn´t made big studies and aren´t a statistics/ analysis expert and I guess, my advantage was the modern era when best and fastest transportation systems are available. But here is my summary:
1) cargo supply can help making cities bigger and is like a multiplier of city growth
2) you need passenger transportation to make your cities bigger
3) cities grow fast with heavy passenger transportation even without cargo supply
->neighbor cities of my biggest cities are three times bigger than before just by being part of my 10-cities-line.
4) to fasten your growth, you need to connect as much cities as possible, build a divers passenger transportation system and guaranty short travel times.
5) it always helps to supply goods.
6) you don´t need complex inner city-tram and bus lines with stops every 100m and you don´t need to connect every street in a city for good passenger transportation. I made just rough line systems within a city to include all city destricts. The range of every tram/ bus stop is big enough to influence wider areas. Too much stops will unnecessarily increase travel times and therefor also costs.
7) Give people choices and build different ways of transportation. People love choices and it will increase the rate of public transportation.