Endre Csóka

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Our ability to control complex systems is a fundamental challenge of contemporary science. Recently introduced tools to identify the driver nodes, nodes through which we can achieve full control, predict the existence of multiple control configurations, prompting us to classify each node in a network based on their role in control. Accordingly a node is(More)
We prove that every 3-regular, n-vertex simple graph with sufficiently large girth contains an independent set of size at least 0.4361n. (The best known bound is 0.4352n.) In fact, computer simulation suggests that the bound our method provides is about 0.438n. Our method uses invariant Gaussian processes on the d-regular tree that satisfy the eigenvector(More)
We analytically solve the core percolation problem for complex networks with arbitrary degree distributions. We find that purely scale-free networks have no core for any degree exponents. We show that for undirected networks if core percolation occurs then it is continuous while for directed networks it is discontinuous (and hybrid) if the in- and(More)
Given a set B of finite rooted graphs and a radius r as an input, we prove that it is undecidable to determine whether there exists a sequence (Gi) of finite bounded degree graphs such that the rooted r-radius neighbourhood of a random node of Gi is isomorphic to a rooted graph in B with probability tending to 1. Our proof implies a similar result for the(More)
Consider the problem when we want to construct some structure on a bounded degree graph, e.g. an almost maximum matching, and we want to decide about each edge depending only on its constant radius neighbourhood. We show that the information about the local statistics of the graph does not help here. Namely, if there exists a random local algorithm which(More)
The classical theorem of Vizing states that every graph of maximum degree d admits an edge-coloring with at most d + 1 colors. Furthermore, as it was earlier shown by Kőnig, d colors suffice if the graph is bipartite. We investigate the existence of measurable edge-colorings for graphings. A graphing is an analytic generalization of a bounded-degree graph(More)
We study “positive” graphs that have a nonnegative homomorphism number into every edge-weighted graph (where the edgeweights may be negative). We conjecture that all positive graphs can be obtained by taking two copies of an arbitrary simple graph and gluing them together along an independent set of nodes. We prove the conjecture for various classes of(More)