The recycled Kaplansky's game

Motivated by the Nine Men Morris game, the achievement or hypergraph games can be prolonged in the following way. After placing a prescribed number of stones, the players pick some of these up and replace again. We study the effect of this recycling for the k-in-a-row game and some versions of the Kaplansky's game

Intersection of random spanning trees in complex networks

In their previous work, the authors considered the concept of random spanning tree intersection of complex networks (London and Pluhár, in: Cherifi, Mantegna, Rocha, Cherifi, Micciche (eds) Complex networks and their applications XI, Springer, Cham, 2023). A simple formula was derived for the size of the minimum expected intersection of two spanning trees chosen uniformly at random. Monte Carlo experiments were run for real networks. In this paper, we provide a broader context and motivations for the concept, discussing its game theoretic origins, examples, its applications to network optimization problems, and its potential use in quantifying the resilience and modular structure of complex networks

Remarks on the interval number of graphs

The interval number of a graph G is the least natural number t such that G is the intersection graph of sets, each of which is the union of at most t intervals. Here we propose a family of representations for the graph G, which yield the well-known upper bound [1)] , where d is the maximum degree of G. The extremal graphs for even d are also described, and the upper bound on the interval number in terms of the number of edges of G is improved

Dynamic communities and their detection

Overlapping community detection has already become an interesting problem in data mining and also a useful technique in applications. This underlines the importance of following the lifetime of communities in real graphs. Palla et al. developed a promising method, and analyzed community evolution on two large databases [23]. We have followed their footsteps in analyzing large real-world databases and found, that the framework they use to describe the dynamics of communities is insufficient for our data. The method used by Palla et al. is also dependent on a very special community detection algorithm, the clique percolation method, and on its monotonic nature. In this paper we propose an extension of the basic community events described in [23] and a method capable of handling communities found a non-monotonic community detection algorithm. We also report on findings that came from the tests on real social graphs

Approximations of the generalized cascade model

The study of infection processes is an important field of science both from the theoretical and the practical point of view, and has many applications. In this paper we focus on the popular Independent Cascade model and its generalization. Unfortunately the exact computation of infection probabilities is a #P-complete problem [8], so one cannot expect fast exact algorithms. We propose several methods to efficiently compute infection patterns with acceptable accuracy. We will also examine the possibility of substituting the Independent Cascade model with a computationally more tractable model