Nonspecific Networking

A new model of strategic network formation is developed and analyzed, where an agent's investment in links is nonspecific. The model comprises a large class of games which are both potential and super- or submodular games. We obtain comparative statics results for Nash equilibria with respect to investment costs for supermodular as well as submodular networking games. We also study logit-perturbed best-response dynamics for supermodular games with potentials. We find that the associated set of stochastically stable states forms a sublattice of the lattice of Nash equilibria and derive comparative statics results for the smallest and the largest stochastically stable state. Finally, we provide a broad spectrum of applications from social interaction to industrial organization. Models of strategic network formation typically assume that each agent selects his direct links to other agents in which to invest. Nonspecific networking means that an agent cannot select a specific subset of feasible links which he wants to establish or strengthen. Rather, each agent chooses an effort level or intensity of networking. In the simplest case, the agent faces a binary choice: to network or not to network. If an agent increases his networking effort, all direct links to other agents are strengthened to various degrees. We assume that benefits accrue only from direct links. The set of agents or players is finite. Each agent has a finite strategy set consisting of the networking levels to choose from. For any pair of agents, their networking levels determine the individual benefits which they obtain from interacting with each other. An agent derives an aggregate benefit from the pairwise interactions with all others. In addition, the agent incurs networking costs, which are a function of the agent's own networking level. The agent's payoff is his aggregate benefit minus his cost.Network Formation, Potential Games, Supermodular Games

The Strong Perfect Graph Conjecture: 40 years of Attempts, and its Resolution

International audienceThe Strong Perfect Graph Conjecture (SPGC) was certainly one of the most challenging conjectures in graph theory. During more than four decades, numerous attempts were made to solve it, by combinatorial methods, by linear algebraic methods, or by polyhedral methods. The first of these three approaches yielded the first (and to date only) proof of the SPGC; the other two remain promising to consider in attempting an alternative proof. This paper is an unbalanced survey of the attempts to solve the SPGC; unbalanced, because (1) we devote a signicant part of it to the 'primitive graphs and structural faults' paradigm which led to the Strong Perfect Graph Theorem (SPGT); (2) we briefly present the other "direct" attempts, that is, the ones for which results exist showing one (possible) way to the proof; (3) we ignore entirely the "indirect" approaches whose aim was to get more information about the properties and structure of perfect graphs, without a direct impact on the SPGC. Our aim in this paper is to trace the path that led to the proof of the SPGT as completely as possible. Of course, this implies large overlaps with the recent book on perfect graphs [J.L. Ramirez-Alfonsin and B.A. Reed, eds., Perfect Graphs (Wiley & Sons, 2001).], but it also implies a deeper analysis (with additional results) and another viewpoint on the topic

More on discrete convexity

In several recent papers some concepts of convex analysis were extended to discrete sets. This paper is one more step in this direction. It is well known that a local minimum of a convex function is always its global minimum. We study some discrete objects that share this property and provide several examples of convex families related to graphs and to two-person games in normal form

Controlling discrete quantum walks: coins and intitial states

In discrete time, coined quantum walks, the coin degrees of freedom offer the potential for a wider range of controls over the evolution of the walk than are available in the continuous time quantum walk. This paper explores some of the possibilities on regular graphs, and also reports periodic behaviour on small cyclic graphs.Comment: 10 (+epsilon) pages, 10 embedded eps figures, typos corrected, references added and updated, corresponds to published version (except figs 5-9 optimised for b&w printing here

Partitionable graphs arising from near-factorizations of finite groups

AbstractIn 1979, two constructions for making partitionable graphs were introduced in (by Chvátal et al. (Ann. Discrete Math. 21 (1984) 197)). The graphs produced by the second construction are called CGPW graphs. A near-factorization (A,B) of a finite group is roughly speaking a non-trivial factorization of G minus one element into two subsets A and B. Every CGPW graph with n vertices turns out to be a Cayley graph of the cyclic group Zn, with connection set (A−A)⧹{0}, for a near-factorization (A,B) of Zn. Since a counter-example to the Strong Perfect Graph Conjecture would be a partitionable graph (Padberg, Math. Programming 6 (1974) 180), any ‘new’ construction for making partitionable graphs is of interest. In this paper, we investigate the near-factorizations of finite groups in general, and their associated Cayley graphs which are all partitionable. In particular, we show that near-factorizations of the dihedral groups produce every CGPW graph of even order. We present some results about near-factorizations of finite groups which imply that a finite abelian group with a near-factorization (A,B) such that |A|⩽4 must be cyclic (already proved by De Caen et al. (Ars Combin. 29 (1990) 53)). One of these results may be used to speed up exhaustive calculations. At last, we prove that there is no counter-example to the Strong Perfect Graph Conjecture arising from near-factorizations of a finite abelian group of even order