Dynamic networks are graphs in which edges are available only at specific time instants, modeling connections that change over time. The dynamic network creation game studies this setting as a strategic interaction where a set of $n$ vertices represents players. Players can add or remove time-labelled edges in order to minimize their personal cost. This cost has two components: a construction cost, calculated as the number of time instants during which a player maintains edges multiplied by a constant $α$, and a communication cost, defined as the sum of temporal distances to all other vertices in the network. Communication occurs through temporal paths, which are sequences of adjacent edges with strictly increasing time labels and no repeated vertices. For the strict temporal shortest distance, we show a constant bound for the price of anarchy for $α<2$. When $α\geq\frac{1}{8}n^2$, we show Nash equilibria with a communication cost asymptotically higher than $\frac{1}{49}n^3$. We prove that the game admits a best response cycle: there exists a sequence of individually rational strategy changes that returns to the initial strategy profile without ever reaching a Nash equilibrium. This establishes that the dynamic network creation game is not a potential game and that best response dynamics may cycle indefinitely, so convergence to a Nash equilibrium or a socially optimal outcome is not guaranteed in general.

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