Lower threshold ground state energy and testability of minimal balanced cut density

Lov\'asz and his coauthors defined the notion of microcanonical ground state energy $\hat{\mathcal{E}}_\mathbb{a} (G,J)$ -- borrowed from the statistical physics -- for weighted graphs $G$, where $\mathbb{a}$ is a probability distribution on $\{1,...,q\}$ and $J$ is a symmetric $q \times q$ matrix with real entries. We define a new version of the ground state energy, $\hat{\mathcal{E}}^c (G,J)=\inf_{\mathbb{a}\in A_c}\hat{\mathcal{E}}_\mathbb{a} (G,J)$, called lower threshold ground state energy, where $A_c = \{\mathbb{a} :\, a_i\ge c,\,i=1,\dots, q \}$. Both types of energies can be extended for graphons $W$, the limit objects of convergent sequences of simple graphs. In the main result of the paper it is stated that if $0\leq c_1<c_2 \leq 1$, then the convergence of the sequences $(\hat{\mathcal{E}}^{c_2/q} (G_n,J))$ for each $J$ implies convergence of the sequences $(\hat{\mathcal{E}}^{c_1/q} (G_n,J))$ for each $J$. As a byproduct one can derive in a natural way the testability of minimum balanced multiway cut densities, that is one of the fundamental problems of cluster analysis.Comment: 14 page

Analysis about Hungary's attractiveness to investors with particular regard to foreign direct investments

From the beginning of the 1990s Foreign Direct Investments (FDI) inflows have always played an important role in the Hungarian economy. The situation did not change even though the inflows and the stock of FDI have been decreased for the last few years. Hungary as a small open economy depends heavily on foreign capital and foreign direct investments. However foreign capital and foreign direct investments inflows enter the countries under prosperous market, political, economic, social and legal conditions. These factors have a growing significance during the economic and financial crisis. Responding to the challenges of the economic recession more and more countries are seeking to improve their ability to attract capital because the foreign direct investments are defined as a key factor of economic growth. The question is which factors are improving Hungary's ability to attract capital? In the first part of the study1 decisive factors will be revealed contributing to a country's competitiveness and ability to attract capital. In the second part these factors will be analyzed related to the Hungarian economy. In the study we describe some problems of emerging economies such as the existence of the dual economic structure, the phenomenon of stagflation, the high tax burdens and low wages all with regard to Hungary. Furthermore it will be analyzed how the low-wage jobs are promoting Hungary's attractiveness to investors. In the conclusion our proposals will be formulated in order to retain as well as improve Hungary’s attractiveness to investors

Nagyméretű véletlen gráfok statisztikai vizsgálata = Statistical inference on large random graphs

Nagyméretű gráfok struktúrájának feltárására alkalmaztunk és fejlesztettünk ki paraméteres és nemparaméteres statisztikai módszereket. Paraméteres vizsgálatok: az ún. általánosított véletlen gráf modellben és az alpha-beta-modellekben a paraméterek maximum likelihood becslésére EM-algoritmust használtunk. A modellt a Rasch-modell páros gráfokra történő alkalmazásával kiterjesztettük a többklaszteres szituációra. Nemparaméteres vizsgálatok: minimális, maximális és reguláris vágások. A klaszterek számára a normált Laplace ill. modularitás mátrix sajátértékeiből következtettünk, míg maguknak a klasztereknek a megkeresésére a k-közép eljárást alkalmaztuk a csúcsreprezentánsok segítségével. Tételeket bizonyítottunk a vágások, a térfogatregularitás mérőszáma, a spektrális rés és a klaszterek k-varianciája közti összefüggésekre, ha csúcsok száma tart a végtelenbe úgy, hogy nincsen domináns csúcs. Általánosítottuk az ún. Newman-Girvan modularitást, és a normált modularitás mátrix nagy abszolút értékű sajátértékeit és azok előjelét használtuk a klaszterek jellegének megállapítására. Az általánosított véletlen gráfok spektrális karakterizációját adtuk a strukturális sajátértékek és sajátalterek segítségével. Vizsgálatainkat kiterjesztettük súlyozott, irányított gráfokra és kontingenciatáblákra is. Foglalkoztunk továbbá minimális többszempontú vágássűrűségek tesztelhetőségével a Lovász L. és társszerzői által konvergens gráfsorozatoknál használt értelemben. | We applied and developed parametric and nonparametric statistical methods to recover the structure of large graphs. Parametric inference: in the so-called generalized random graph model and alpha- beta-models we applied EM-algorithm for the maximum likelihood estimation of the parameters. We extended the model to the several clusters case via the Rasch-model applied to the bipartite graphs formed by the pairs of the clusters. Nonparametric inference: minimal, maximal, and regular cuts. For the number of clusters, we concluded from the spectra of the Laplacian and modularity matrices, whereas we found the clusters by the k-means algorithm applied for the vertex representatives. We proved theorems for the relations between the multiway cuts, the constant of the volume-regularity, and the spectral gap together with the k-variance of the clusters, when the number of the vertices tends to infinity in such a way that there are no dominant vertices. We generalized the notion of the so-called Newman-Girvan modularity and gave the spectral characterization of the generalized random graphs. We extended our findings to weighted and directed graphs, further, to contingency tables. We also investigated the testability of balanced multiway cut densities, where for the testability we used the definitions of Lovász L. and coauthors in the context of convergent graph sequences