The Price of Universal Temporal Reachability

Abstract

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 each vertex represents a player. Players can add or remove time-labeled 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 average distance 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. We show for the shortest distance (minimizing the number of edges) that the price of anarchy can be proportional to the number of vertices, contrasting the constant price conjectured for static networks.

Figures (3)

• Figure 1: A strategy profile with $n=8$. Every edge has an arrow originating from the vertex which has bought the edge. The communication temporal graph contains all the edges, without the arrows which have no meaning for the graph. For high atomic cost $\alpha\geq100$ this strategy profile is a Nash equilibrium when the individual cost is defined using the out-reachability. In particular, vertex $a$ need to buy the edge labeled with $8$ in order to reach its neighbour at the other extremity of that edge. However, this strategy profile is not a Nash equilibrium for any atomic cost $\alpha$ when the individual cost is defined using the in-reachability: vertex $a$ would then rather not buy the two edges labeled $8$ and $11$. In particular, its neighbour at the other extremity of the edge labeled with $8$ can take the edge labeled with $4$ then the edge labeled with $7$ in order to reach vertex $a$.
• Figure 2: Four temporally connected temporal graphs over $n=8$ vertices, whose union graphs are a clique $K_8$ (leftmost), a biclique $K_{2,6}$ (middle-left), a hypercube $Q_3$ (middle-right), and a linear structure called $dF_7$ which is a $7$-fan with one missing edge (rightmost). With respect to the shortest distance, they have diameter $1$, $2$, $3$, and $n-2$, respectively. The arrows on $K_{2,6}$ (middle-left) and the diminished $7$-fan (rightmost) represent two strategy profiles, where vertices buy the arrows from which they originate. For an atomic cost where $\alpha\geq1$, the former strategy profile is a Nash equilibrium. For big enough atomic cost $\alpha\geq3$, both the former strategy and the $Q_3$ (middle-right, with any purchasing configuration) are Nash-equilibria. For high atomic cost $\alpha\geq100$ both former and latter strategy profiles are Nash equilibria together with the hypercube $Q_3$.
• Figure 3: $16$ Optimal labellings for $Q_3$ with foremost temporal distance.

Theorems & Definitions (5)

• Theorem 1
• Lemma 2
• Lemma 3
• Lemma 4
• Lemma 5