Bilò, Davide; Friedrich, Tobias; Lenzner, Pascal; Lowski, Stefanie; Melnichenko, Anna Selfish Creation of Social NetworksConference on Artificial Intelligence (AAAI) 2021: 5185–5193
Understanding real-world networks has been a core research endeavor throughout the last two decades. Network Creation Games are a promising approach for this from a game-theoretic perspective. In these games, selfish agents corresponding to nodes in a network strategically decide which links to form to optimize their centrality. Many versions have been introduced and analyzed, but none of them fits to modeling the evolution of social networks. In real-world social networks, connections are often established by recommendations from common acquaintances or by a chain of such recommendations. Thus establishing and maintaining a contact with a friend of a friend is easier than connecting to complete strangers. This explains the high clustering, i.e., the abundance of triangles, in real-world social networks. We propose and analyze a network creation model inspired by real-world social networks. Edges are formed in our model via bilateral consent of both endpoints and the cost for establishing and maintaining an edge is proportional to the distance of the endpoints before establishing the connection. We provide results for generic cost functions, which essentially only must be convex functions in the distance of the endpoints without the respective edge. For this broad class of cost functions, we provide many structural properties of equilibrium networks and prove (almost) tight bounds on the diameter, the Price of Anarchy and the Price of Stability. Moreover, as a proof-of-concept we show via experiments that the created equilibrium networks of our model indeed closely mimics real-world social networks. We observe degree distributions that seem to follow a power-law, high clustering, and low diameters. This can be seen as a promising first step towards game-theoretic network creation models that predict networks featuring all core real-world properties.
Friedemann, Wilhelm; Friedrich, Tobias; Gawendowicz, Hans; Lenzner, Pascal; Melnichenko, Anna; Peters, Jannik; Stephan, Daniel; Vaichenker, Michael Efficiency and Stability in Euclidean Network DesignSymposium on Parallelism in Algorithms and Architectures (SPAA) 2021: 232–242
Network Design problems typically ask for a minimum cost sub-network from a given host network. This classical point-of-view assumes a central authority enforcing the optimum solution. But how should networks be designed to cope with selfish agents that own parts of the network? In this setting, minimum cost networks may be very unstable in that agents will deviate from a proposed solution if this decreases their individual cost. Hence, designed networks should be both efficient in terms of total cost and stable in terms of the agents' willingness to accept the network. We study this novel type of Network Design problem by investigating the creation of \($(\beta,\gamma)$\)-networks, that are in \($\beta$\)-approximate Nash equilibrium and have a total cost of at most \($\gamma$\) times the optimal cost, for the recently proposed Euclidean Generalized Network Creation Game by Bilò et al.SPAA2019. There, \($n$\) agents corresponding to points in Euclidean space create costly edges among themselves to optimize their centrality in the created network. Our main result is a simple \($\mathcal{O(n^2)$\)-time algorithm that computes a \($(\beta,\beta)$\)-network with low \($\beta$\) for any given set of points. Moreover, on integer grid point sets or random point sets our algorithm achieves a low constant \($\beta$\). Besides these results for the Euclidean model, we discuss a generalization of our algorithm to instances with arbitrary, even non-metric, edge lengths. Moreover, in contrast to these algorithmic results, we show that no such positive results are possible when focusing on either optimal networks, i.e., \($(\beta,1)$\)-networks, or perfectly stable networks, i.e., \($(1,\gamma)$\)-networks, as in both cases NP-hard problems arise, there exist instances with very unstable optimal networks, and there are instances for perfectly stable networks with high total cost. Along the way, we significantly improve several results from Bilò et al. and we asymptotically resolve their conjecture about the Price of Anarchy by providing a tight bound.