Related papers: Clusters, currents and Whitehead's algorithm
The community structure of complex networks reveals both their organization and hidden relationships among their constituents. Most community detection methods currently available are not deterministic, and their results typically depend on…
We study the mechanism of formation of synchronized clusters in coupled maps on networks with various connection architectures. The nodes in a cluster are self- synchronized or driven-synchronized, based on the coupling strength and…
We use Gersten's generalization of Whitehead's algorithm to determine whether a given finitely generated subgroup of a free group $F$ is elliptic in an elementary cyclic splitting of $F$. We provide a similar result for all elementary…
Large datasets with interactions between objects are common to numerous scientific fields (i.e. social science, internet, biology...). The interactions naturally define a graph and a common way to explore or summarize such dataset is graph…
We derive an efficient method to perform clustering of nodes in Gaussian graphical models directly from sample data. Nodes are clustered based on the similarity of their network neighborhoods, with edge weights defined by partial…
In a range of scientific coauthorship networks, transitions emerge in degree distributions, correlations between degrees and local clustering coefficients, etc. The existence of those transitions could be regarded as a result of the…
We study the automorphism groups of countable homogeneous directed graphs (and some additional homogeneous structures) from the point of view of topological dynamics. We determine precisely which of these automorphism groups are amenable…
We consider the topological dynamics of the automorphism group of a particular sparse graph M_1 resulting from an ab initio Hrushovski construction. We show that minimal subflows of the flow of linear orders on M_1 have all orbits meagre,…
We study locally closed transformation monoids which contain the automorphism group of the random graph. We show that such a transformation monoid is locally generated by the permutations in the monoid, or contains a constant operation, or…
Most research concerning the influence of network structure on phenomena taking place on the network focus on relationships between global statistics of the network structure and characteristic properties of those phenomena, even though…
This paper introduces a new topological space associated with a nonabelian free group $F_n$ of rank $n$ and a malnormal subgroup system $\mathcal{A}$ of $F_n$, called the space of currents relative to $\mathcal{A}$, which are…
Local graph clustering methods aim to detect small clusters in very large graphs without the need to process the whole graph. They are fundamental and scalable tools for a wide range of tasks such as local community detection, node ranking…
The evolution of random undirected graphs by the clustering attachment (CA) both without node and edge deletion and with uniform node or edge deletion is investigated. Theoretical results are obtained for the CA without node and edge…
Clustering points in a vector space or nodes in a graph is a ubiquitous primitive in statistical data analysis, and it is commonly used for exploratory data analysis. In practice, it is often of interest to "refine" or "improve" a given…
In clustering problems, a central decision-maker is given a complete metric graph over vertices and must provide a clustering of vertices that minimizes some objective function. In fair clustering problems, vertices are endowed with a color…
We propose two spectral algorithms for partitioning nodes in directed graphs respectively with a cyclic and an acyclic pattern of connection between groups of nodes. Our methods are based on the computation of extremal eigenvalues of the…
Unsupervised clustering, also known as natural clustering, stands for the classification of data according to their similarities. Here we study this problem from the perspective of complex networks. Mapping the description of data…
The field theoretical renormalization group equations have many common features with the equations of dynamical systems. In particular, the manner how Callan-Symanzik equation ensures the independence of a theory from its subtraction point…
We develop a refinement of Whitehead's algorithm for primitive words in a free group. We generalize to subgroups, establishing a strengthened version of Whitehead's algorithm for free factors. We make use of these refinements in proving new…
Graph-theoretical approach is used to study cluster formation in mesocsopic systems. Appearance of these clusters are due to discrete resonances which are presented in the form of a multigraph with labeled edges. This presentation allows to…