Correction: Basic Network Creation Games

We prove a previously stated but incorrectly proved theorem: there is a diameter-3 graph in which replacing any edge {v, w} of the graph with {v, w′}, for any vertex w′, does not decrease the total sum of distances from v to all other nodes (a property called sum equilibrium). Theorem 5 in [1] states that there exists a diameter-3 sum equilibrium graph, that is, an undirected graph such that, for every edge {v, w} and every node w′, replacing edge {v, w} with {v, w′} does not decrease the total sum of distances from v to all other nodes (and thus no vertex v has incentive to swap an incident edge). In this short note, we observe an error in the original construction and proof, but present a different example that is indeed a diameter-3 sum equilibrium graph, thereby restoring the theorem. First we describe why Figure 3 of [1] is not in sum equilibrium. Specifically, vertex d1 has an incentive to replace the edge {d1, c1,1} with {d1, c2,1}, as the total distance is 27 in the first case and 26 in the last. The original proof ignores that c2,1 is a neighbor of c1,1 and, hence, Lemma 8 of [1] implies that the distance from d1 to c1,1 increases by 1 and not by 2 as claimed. Figure 1 below presents a diameter-3 sum equilibrium graph G (which is also simpler than the original construction). In this instance, vertices v2, v4, v5, and v7 have local diameter 2 so, by Lemma 6 of [1], they have no incentive to swap any edge. (Lemma 6 states that a vertex of local diameter 2 never has incentive to swap an incident edge, as the number of distance-1 neighbors remains fixed, and thus the number of nodes at distance≥ 2 remains fixed, so keeping their distances equal to 2 is optimal.) Among the remaining vertices, by symmetry, it suffices to prove that v1 and v3 do not have an incentive to swap edges. Consider vertex vi for i ∈ {1, 3}. Let G−i be the graph obtained by removing vertex vi and its incident edges; refer to Figure 2. The sum of distances for vi in G is 13. Because vi has degree 2, the smallest possible sum of distances for vi is 12, which can be obtained if vi were connected to two vertices that form a dominating set in G−i. (A dominating set of cardinality larger than ∗Schools of Mathematics and Computer Science, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, 69978, Israel; and IAS, Princeton, NJ 08540, USA; [email protected]. Supported in part by a USA Israeli BSF grant, by a grant from the Israel Science Foundation, by an ERC Advanced Grant, and by the Hermann Minkowski Minerva Center for Geometry at Tel Aviv University. †MIT Computer Science and Artificial Intelligence Laboratory, 32 Vassar St., Cambridge, MA 02139, USA, [email protected]. Supported in part by NSF grant CCF-1161626 and DARPA/AFOSR grant FA9550-12-1-0423. ‡Computer Science Department, University of Maryland, College Park, MD 20742; and AT&T Labs — Research, 180 Park Ave., Florham Park, NJ 07932, USA; [email protected]. Supported in part by NSF CAREER award 1053605, NSF grant CCF-1161626, ONR YIP award N000141110662, and DARPA/AFOSR grant FA9550-12-1-0423. §Computer Engineering & Informatics Department, University of Patras, 26504, Rio, Greece, kanellop@ceid. upatras.gr ¶Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA 02139, USA; and Akamai Technologies; [email protected]