The horse in the ancient and medieval world was a vehicle. The number of horses that a sedentary state had often showed the power of that state. Horses, one of the most versatile animals on the planet require maintenance. Such maintenance has stables and entire infrastructure behind it.
There is also prestige of horse warfare. Being on top of these animals with sword or bow was always seen as a more glamorous The Roman Empire under Augustus was still a heavily infantry-oriented empire while the cavalry was growing in importance after engagements with the Parthians in the East with their horse archers.
By the time of the Byzantine Empire, the army had shifted away from soldiers fighting on the ground towards a more cavalry based army. This was in response to the Germanic migrations and invasions, which had nomadic groups that relied heavily on cavalry as well as the Sassanid Persian Empire.
The Manpower of Byzantine Cavalry in the Byzantine Empire from 400 AD to 1400 AD.
As you can see here, there are two heights to the usage of cavalry in the Byzantine Empire.
The first height was around 400 AD and the second height was around 900 AD, the time of the Byzantine efforts to regain their territories.
The Byzantine Empire shows that the Roman state, while brittle at times, was highly adaptable to many enemies and rivals in the Middle Ages.
Byzantine-Empire.com 