On Dynamics in Basic Network Creation Games

On the Core of Cost-Revenue Games: Minimum Cost Spanning Tree Games with Revenues

With the growing deployment of Flying Ad hoc Networks (FANETs) in military and civilian applications, constructing a stable and efficient communication backbone has become a critical challenge. This paper tackles the Cluster Head (CH) optimization problem in large-scale and highly dynamic FANETs by formulating it as a Minimum Connected Dominating Set (MCDS) problem. However, since MCDS is NP-complete on general graphs, existing heuristic and exact algorithms suffer from limited coverage, poor connectivity, and high computational cost. To address these issues, we propose TGN-MCDS, a novel algorithm built upon the Temporal Graph Network (TGN) architecture, which leverages graph neural networks for cluster head selection and efficiently learns time-varying network topologies. The algorithm adopts a multi-objective loss function incorporating coverage, connectivity, size control, centrality, edge penalty, temporal smoothness, and information entropy to guide model training. Simulation results demonstrate that TGN-MCDS rapidly achieves near-optimal CH sets with full node coverage and strong connectivity. Compared with Greedy, Integer Linear Programming (ILP), and Branch-and-Bound (BnB) methods, TGN-MCDS produces fewer and more stable CHs, significantly improving cluster stability while maintaining high computational efficiency for real-time operations in large-scale FANETs.

We analyze the structure of equilibria and the price of anarchy in the family of network creation games considered extensively in the past few years, which attempt to unify the network design and network routing problems by modeling both creation and usage costs. In general, the games are played on a host graph, where each node is a selfish independent agent (player) and each edge has a fixed link creation cost~$\alpha$. Together the agents create a network (a subgraph of the host graph) while selfishly minimizing the link creation costs plus the sum of the distances to all other players (usage cost). In this paper, we pursue two important facets of the network creation~game. First, we study extensively a natural version of the game, called the cooperative model, where nodes can collaborate and share the cost of creating any edge in the host graph. We prove the first nontrivial bounds in this model, establishing that the price of anarchy is polylogarithmic in $n$ for all values of~$\alpha$ in complete host graphs. This bound is the first result of this type for any version of the network creation game; most previous general upper bounds are polynomial in~$n$. Interestingly, we also show that equilibrium graphs have polylogarithmic diameter for the most natural range of~$\alpha$ (at most $n \mathop{\rm polylg}\nolimits n$). Second, we study the impact of the natural assumption that the host graph is a general graph, not necessarily complete. This model is a simple example of nonuniform creation costs among the edges (effectively allowing weights of $\alpha$ and~$\infty$). We prove the first assemblage of upper and lower bounds for this context, establishing nontrivial tight bounds for many ranges of~$\alpha$, for both the unilateral and cooperative versions of network creation. In particular, we establish polynomial lower bounds for both versions and many ranges of~$\alpha$, even for this simple nonuniform cost model, which sharply contrasts the conjectured constant bounds for these games in complete (uniform) graphs.

Habitat amount and connectivity in forest planning models: Consequences for profitability and compensation schemes

On the number of irregular assignments on a graph

Symmetry, such as structural symmetry, color symmetry and so on, plays an important role in graph coloring. In this paper, we use structural symmetry and color symmetry to study the characterization for the neighbor-distinguishing index of planar graphs. Let G be a simple graph with no isolated edges. The neighbor-distinguishing edge coloring of G is a proper edge coloring of G such that any two adjacent vertices admit different sets consisting of the colors of their incident edges. The neighbor-distinguishing index χa′(G) of G is the smallest number of colors in such an edge coloring of G. It was conjectured that if G is a connected graph with at least three vertices and G≠C5, then χa′(G)≤Δ+2. In this paper, we show that if G is a planar graph with maximum degree Δ≥13, then Δ≤χa′(G)≤Δ+1, and, further, χa′(G)=Δ+1 if and only if G contains two adjacent vertices of maximum degree.

An innovative business model for rural sub-Saharan Africa electrification

A key question for European energy transition is which forms of renewable energy technologies will play a central role in this process. The recent dynamic growth in offshore wind power together with the vast wind energy potential of the European seas, including the Baltic Sea, make this technology an increasingly attractive and viable option. Considering the high installation and connection costs, government support is considered essential for the development of offshore wind power. The aim of this article is to analyze Poland’s public policy tools, which govern offshore wind farm development, and to present them from a wider geostrategic perspective. Authors identify, classify, and evaluate individual public policy tools with the use of multi-criteria and multi-dimensional methods while explaining their impact on offshore wind development in Poland. The analysis of the individual tools has shown that the currently applied tools give a high probability of achieving public policy objectives. The characteristics of the applied tools prove that vital decisions on offshore wind energy have been made concerning the need for decarbonization but also regarding wider geostrategic calculations. Given the changing security dynamics in the Baltic Sea region, we highlight potential geostrategic risks to the implementation of offshore wind projects.

The economics of offshore wind

We introduce a fully online model of maximum cardinality matching in which all vertices arrive online. On the arrival of a vertex, its incident edges to previously-arrived vertices are revealed. Each vertex has a deadline that is after all its neighbors’ arrivals. If a vertex remains unmatched until its deadline, the algorithm must then irrevocably either match it to an unmatched neighbor, or leave it unmatched. The model generalizes the existing one-sided online model and is motivated by applications including ride-sharing platforms, real-estate agency, etc. We show that the Ranking algorithm by Karp et al. (STOC 1990) is 0.5211-competitive in our fully online model for general graphs. Our analysis brings a novel charging mechanic into the randomized primal dual technique by Devanur et al. (SODA 2013), allowing a vertex other than the two endpoints of a matched edge to share the gain. To our knowledge, this is the first analysis of Ranking that beats 0.5 on general graphs in an online matching problem, a first step towards solving the open problem by Karp et al. (STOC 1990) about the optimality of Ranking on general graphs. If the graph is bipartite, we show that the competitive ratio of Ranking is between 0.5541 and 0.5671. Finally, we prove that the fully online model is strictly harder than the previous model as no online algorithm can be 0.6317 e-competitive in our model even for bipartite graphs.

A new technique for the deep-sea mating of inexpensive connectors

GCPNet: An interpretable Generic Crystal Pattern graph neural Network for predicting material properties.

Given a simple undirected graph G = (V, E) and an integer k < |V|, the Sparsest k-Subgraph problem asks for a set of k vertices which induces the minimum number of edges. As a generalization of the classical independent set problem, Sparsest k-Subgraph is ????-hard and even not approximable unless ?????? in general graphs. Thus, we investigate Sparsest k-Subgraph in graph classes where independent set is polynomial-time solvable, such as subclasses of perfect graphs. Our two main results are the ????-hardness of Sparsest k-Subgraph on chordal graphs, and a greedy 2-approximation algorithm. Finally, we also show how to derive a PTAS for Sparsest k-Subgraph on proper interval graphs.

In this paper we present a correlation between a variation of the list problem coloring in graphs, the (γ, μ)-coloring, and the property of choosability in graphs, resulting in the k-(γ, μ)-choosability. The list coloring problem is a variation of the classical vertex coloring problem, introduced by Erdos et al. in 1979, along with a property very studied in list coloring: the choosability in graphs. In this work, algorithms were developed to determine the k-choosability and the k-(γ, μ)-choosability of a general simple graph, and a greedy heuristic based on DSATUR. In addition, we applied special techniques to prove the property of choosability in some classes of graphs, involving a general proof for simple graphs, which guarantees that if a graph is k-colorable it is also k-(γ, μ)-choosable, being the computational complexity reduced from the class Π <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> -complete to the NP-complete.

Let G be a mixed graph. The eigenvalues and eigenvectors of G are respectively defined to be those of its Laplacian matrix. If G is a simple graph, [M. Fiedler: A property of eigenvectors of nonnegative symmetric matrices and its applications to graph theory, Czechoslovak Math. J. 25 (1975), 619–633] gave a remarkable result on the structure of the eigenvectors of G corresponding to its second smallest eigenvalue (also called the algebraic connectivity of G). For G being a general mixed graph with exactly one nonsingular cycle, using Fiedler’s result, we obtain a similar result on the structure of the eigenvectors of G corresponding to its smallest eigenvalue.