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

SVD, discrepancy, and regular structure of contingency tables

We will use the factors obtained by correspondence analysis to find biclustering of a contingency table such that the row-column cluster pairs are regular, i.e., they have small discrepancy. In our main theorem, the constant of the so-called volume-regularity is related to the SVD of the normalized contingency table. Our result is applicable to two-way cuts when both the rows and columns are divided into the same number of clusters, thus extending partly the result of Butler estimating the discrepancy of a contingency table by the second largest singular value of the normalized table (one-cluster, rectangular case), and partly a former result of the author for estimating the constant of volume-regularity by the structural eigenvalues and the distances of the corresponding eigen-subspaces of the normalized modularity matrix of an edge-weighted graph (several clusters, symmetric case)

Emergence of slow-switching assemblies in structured neuronal networks

Unraveling the interplay between connectivity and spatio-temporal dynamics in neuronal networks is a key step to advance our understanding of neuronal information processing. Here we investigate how particular features of network connectivity underpin the propensity of neural networks to generate slow-switching assembly (SSA) dynamics, i.e., sustained epochs of increased firing within assemblies of neurons which transition slowly between different assemblies throughout the network. We show that the emergence of SSA activity is linked to spectral properties of the asymmetric synaptic weight matrix. In particular, the leading eigenvalues that dictate the slow dynamics exhibit a gap with respect to the bulk of the spectrum, and the associated Schur vectors exhibit a measure of block-localization on groups of neurons, thus resulting in coherent dynamical activity on those groups. Through simple rate models, we gain analytical understanding of the origin and importance of the spectral gap, and use these insights to develop new network topologies with alternative connectivity paradigms which also display SSA activity. Specifically, SSA dynamics involving excitatory and inhibitory neurons can be achieved by modifying the connectivity patterns between both types of neurons. We also show that SSA activity can occur at multiple timescales reflecting a hierarchy in the connectivity, and demonstrate the emergence of SSA in small-world like networks. Our work provides a step towards understanding how network structure (uncovered through advancements in neuroanatomy and connectomics) can impact on spatio-temporal neural activity and constrain the resulting dynamics.Comment: The first two authors contributed equally -- 18 pages, including supplementary material, 10 Figures + 2 SI Figure