Different approaches to community detection

A precise definition of what constitutes a community in networks has remained elusive. Consequently, network scientists have compared community detection algorithms on benchmark networks with a particular form of community structure and classified them based on the mathematical techniques they employ. However, this comparison can be misleading because apparent similarities in their mathematical machinery can disguise different reasons for why we would want to employ community detection in the first place. Here we provide a focused review of these different motivations that underpin community detection. This problem-driven classification is useful in applied network science, where it is important to select an appropriate algorithm for the given purpose. Moreover, highlighting the different approaches to community detection also delineates the many lines of research and points out open directions and avenues for future research.Comment: 14 pages, 2 figures. Written as a chapter for forthcoming Advances in network clustering and blockmodeling, and based on an extended version of The many facets of community detection in complex networks, Appl. Netw. Sci. 2: 4 (2017) by the same author

Self-adjoint symmetry operators connected with the magnetic Heisenberg ring

We consider symmetry operators a from the group ring C[S_N] which act on the Hilbert space H of the 1D spin-1/2 Heisenberg magnetic ring with N sites. We investigate such symmetry operators a which are self-adjoint (in a sence defined in the paper) and which yield consequently observables of the Heisenberg model. We prove the following results: (i) One can construct a self-adjoint idempotent symmetry operator from every irreducible character of every subgroup of S_N. This leads to a big manifold of observables. In particular every commutation symmetry yields such an idempotent. (ii) The set of all generating idempotents of a minimal right ideal R of C[S_N] contains one and only one idempotent which ist self-adjoint. (iii) Every self-adjoint idempotent e can be decomposed into primitive idempotents e = f_1 + ... + f_k which are also self-adjoint and pairwise orthogonal. We give a computer algorithm for the calculation of such decompositions. Furthermore we present 3 additional algorithms which are helpful for the calculation of self-adjoint operators by means of discrete Fourier transforms of S_N. In our investigations we use computer calculations by means of our Mathematica packages PERMS and HRing.Comment: 13 page

Transition from rotating waves to modulated rotating waves on the sphere

We study non-resonant and resonant Hopf bifurcation of a rotating wave in SO(3)-equivariant reaction-diffusion systems on a sphere. We obtained reduced differential equations on so(3), the characterization of modulated rotating waves obtained by Hopf bifurcation of a rotating wave, as well as results regarding the resonant case. Our main tools are the equivariant center manifold reduction and the theory of Lie groups and Lie algebras, especially for the group SO(3) of all rigid rotations on a sphere

Synchronization Properties of Network Motifs

We address the problem of understanding the variable abundance of 3-node and 4-node subgraphs (motifs) in complex networks from a dynamical point of view. As a criterion in the determination of the functional significance of a n-node subgraph, we propose an analytic method to measure the stability of the synchronous state (SSS) the subgraph displays. We show that, for undirected graphs, the SSS is correlated with the relative abundance, while in directed graphs the correlation exists only for some specific motifs.Comment: 7 pages, 3 figure

Generalized modularity matrices

Various modularity matrices appeared in the recent literature on network analysis and algebraic graph theory. Their purpose is to allow writing as quadratic forms certain combinatorial functions appearing in the framework of graph clustering problems. In this paper we put in evidence certain common traits of various modularity matrices and shed light on their spectral properties that are at the basis of various theoretical results and practical spectral-type algorithms for community detection

Virtual amplitudes and threshold behaviour of hadronic top-quark pair-production cross sections

We present the two-loop virtual amplitudes for the production of a top-quark pair in gluon fusion. The evaluation method is based on a numerical solution of differential equations for master integrals in function of the quark velocity and scattering angle starting from a boundary at high-energy. The results are given for the renormalized infrared finite remainders on a large grid and have recently been used in the calculation of the total cross sections at the next-to-next-to-leading order. For convenience, we also give the known results for the quark annihilation case on the same grid. Outside of the kinematical range covered by the grid, we provide threshold and high-energy expansions. From expansions of the two-loop virtual amplitudes, we determine the threshold behavior of the total cross sections at next-to-next-to-leading order for the quark annihilation and gluon fusion channels including previously unknown constant terms. In our analysis of the quark annihilation channel, we uncover the presence of a velocity enhanced logarithm of Coulombic origin, which was missed in a previous study.Comment: 28 pages, 3 figures, 4 tables, results for the virtual amplitudes attached in Mathematica forma

Chronic Stress Triggers Expression of Immediate Early Genes and Differentially Affects the Expression of AMPA and NMDA Subunits in Dorsal and Ventral Hippocampus of Rats

Indexación: Web of Science; Scopus.Previous studies in rats have demonstrated that chronic restraint stress triggers anhedonia, depressive-like behaviors, anxiety and a reduction in dendritic spine density in hippocampal neurons. In this study, we compared the effect of repeated stress on the expression of α-amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid (AMPA) and N-methyl-D-aspartate (NMDA) receptor subunits in dorsal and ventral hippocampus (VH). Adult male Sprague-Dawley rats were randomly divided into control and stressed groups, and were daily restrained in their motion (2.5 h/day) during 14 days. We found that chronic stress promotes an increase in c-Fos mRNA levels in both hippocampal areas, although it was observed a reduction in the immunoreactivity at pyramidal cell layer. Furthermore, Arc mRNAs levels were increased in both dorsal and VH, accompanied by an increase in Arc immunoreactivity in dendritic hippocampal layers. Furthermore, stress triggered a reduction in PSD-95 and NR1 protein levels in whole extract of dorsal and VH. Moreover, a reduction in NR2A/NR2B ratio was observed only in dorsal pole. In synaptosomal fractions, we detected a rise in NR1 in dorsal hippocampus (DH). By indirect immunofluorescence we found that NR1 subunits rise, especially in neuropil areas of dorsal, but not VH. In relation to AMPA receptor (AMPAR) subunits, chronic stress did not trigger any change, either in dorsal or ventral hippocampal areas. These data suggest that DH is more sensitive than VH to chronic stress exposure, mainly altering the expression of NMDA receptor (NMDAR) subunits, and probably favors changes in the configuration of this receptor that may influence the function of this area.https://www.frontiersin.org/articles/10.3389/fnmol.2017.00244/ful

Enhancing network robustness for malicious attacks

In a recent work [Proc. Natl. Acad. Sci. USA 108, 3838 (2011)], the authors proposed a simple measure for network robustness under malicious attacks on nodes. With a greedy algorithm, they found the optimal structure with respect to this quantity is an onion structure in which high-degree nodes form a core surrounded by rings of nodes with decreasing degree. However, in real networks the failure can also occur in links such as dysfunctional power cables and blocked airlines. Accordingly, complementary to the node-robustness measurement ($R_{n}$), we propose a link-robustness index ($R_{l}$). We show that solely enhancing $R_{n}$ cannot guarantee the improvement of $R_{l}$. Moreover, the structure of $R_{l}$-optimized network is found to be entirely different from that of onion network. In order to design robust networks resistant to more realistic attack condition, we propose a hybrid greedy algorithm which takes both the $R_{n}$ and $R_{l}$ into account. We validate the robustness of our generated networks against malicious attacks mixed with both nodes and links failure. Finally, some economical constraints for swapping the links in real networks are considered and significant improvement in both aspects of robustness are still achieved.Comment: 6 pages, 6 figure

Quantum information analysis of electronic states at different molecular structures

We have studied transition metal clusters from a quantum information theory perspective using the density-matrix renormalization group (DMRG) method. We demonstrate the competition between entanglement and interaction localization. We also discuss the application of the configuration interaction based dynamically extended active space procedure which significantly reduces the effective system size and accelerates the speed of convergence for complicated molecular electronic structures to a great extent. Our results indicate the importance of taking entanglement among molecular orbitals into account in order to devise an optimal orbital ordering and carry out efficient calculations on transition metal clusters. We propose a recipe to perform DMRG calculations in a black-box fashion and we point out the connections of our work to other tensor network state approaches