A paradox in community detection

Recent research has shown that virtually all algorithms aimed at the identification of communities in networks are affected by the same main limitation: the impossibility to detect communities, even when these are well-defined, if the average value of the difference between internal and external node degrees does not exceed a strictly positive value, in literature known as detectability threshold. Here, we counterintuitively show that the value of this threshold is inversely proportional to the intrinsic quality of communities: the detection of well-defined modules is thus more difficult than the identification of ill-defined communities.Comment: 5 pages, 3 figure

Angular Normal Modes of a Circular Coulomb Cluster

We investigate the angular normal modes for small oscillations about an equilibrium of a single-component coulomb cluster confined by a radially symmetric external potential to a circle. The dynamical matrix for this system is a Laplacian symmetrically circulant matrix and this result leads to an analytic solution for the eigenfrequencies of the angular normal modes. We also show the limiting dependence of the largest eigenfrequency for large numbers of particles

A Turbo-Detection Aided Serially Concatenated MPEG-4/TCM Videophone Transceiver

A Turbo-detection aided serially concatenated inner Trellis Coded Modulation (TCM) scheme is combined with four different outer codes, namely with a Reversible Variable Length Code (RVLC), a Non-Systematic Convolutional (NSC) code a Recursive Systematic Convolutional (RSC) code or a Low Density Parity Check (LDPC) code. These four outer constituent codes are comparatively studied in the context of an MPEG4 videophone transceiver. These serially concatenated schemes are also compared to a stand-alone LDPC coded MPEG4 videophone system at the same effective overall coding rate. The performance of the proposed schemes is evaluated when communicating over uncorrelated Rayleigh fading channels. It was found that the serially concatenated TCM-NSC scheme was the most attractive one in terms of coding gain and decoding complexity among all the schemes considered in the context of the MPEG4 videophone transceiver. By contrast, the serially concatenated TCM-RSC scheme was found to attain the highest iteration gain among the schemes considered

Computing Diffusion State Distance using Green's Function and Heat Kernel on Graphs

The diffusion state distance (DSD) was introduced by Cao-Zhang-Park-Daniels-Crovella-Cowen-Hescott [{\em PLoS ONE, 2013}] to capture functional similarity in protein-protein interaction networks. They proved the convergence of DSD for non-bipartite graphs. In this paper, we extend the DSD to bipartite graphs using lazy-random walks and consider the general $L_q$-version of DSD. We discovered the connection between the DSD $L_q$-distance and Green's function, which was studied by Chung and Yau [{\em J. Combinatorial Theory (A), 2000}]. Based on that, we computed the DSD $L_q$-distance for Paths, Cycles, Hypercubes, as well as random graphs $G(n,p)$ and $G(w_1,..., w_n)$. We also examined the DSD distances of two biological networks.Comment: Accepted by the 11th Workshop on Algorithms and Models for the Web Graph (WAW2014

Some Exact Results for Spanning Trees on Lattices

For $n$-vertex, $d$-dimensional lattices $\Lambda$ with $d \ge 2$, the number of spanning trees $N_{ST}(\Lambda)$ grows asymptotically as $\exp(n z_\Lambda)$ in the thermodynamic limit. We present an exact closed-form result for the asymptotic growth constant $z_{bcc(d)}$ for spanning trees on the $d$-dimensional body-centered cubic lattice. We also give an exact integral expression for $z_{fcc}$ on the face-centered cubic lattice and an exact closed-form expression for $z_{488}$ on the $4 \cdot 8 \cdot 8$ lattice.Comment: 7 pages, 1 tabl

Random Vibrational Networks and Renormalization Group

We consider the properties of vibrational dynamics on random networks, with random masses and spring constants. The localization properties of the eigenstates contrast greatly with the Laplacian case on these networks. We introduce several real-space renormalization techniques which can be used to describe this dynamics on general networks, drawing on strong disorder techniques developed for regular lattices. The renormalization group is capable of elucidating the localization properties, and provides, even for specific network instances, a fast approximation technique for determining the spectra which compares well with exact results.Comment: 4 pages, 3 figure

The effect of network structure on phase transitions in queuing networks

Recently, De Martino et al have presented a general framework for the study of transportation phenomena on complex networks. One of their most significant achievements was a deeper understanding of the phase transition from the uncongested to the congested phase at a critical traffic load. In this paper, we also study phase transition in transportation networks using a discrete time random walk model. Our aim is to establish a direct connection between the structure of the graph and the value of the critical traffic load. Applying spectral graph theory, we show that the original results of De Martino et al showing that the critical loading depends only on the degree sequence of the graph -- suggesting that different graphs with the same degree sequence have the same critical loading if all other circumstances are fixed -- is valid only if the graph is dense enough. For sparse graphs, higher order corrections, related to the local structure of the network, appear.Comment: 12 pages, 7 figure

Clustering and the hyperbolic geometry of complex networks

Clustering is a fundamental property of complex networks and it is the mathematical expression of a ubiquitous phenomenon that arises in various types of self-organized networks such as biological networks, computer networks or social networks. In this paper, we consider what is called the global clustering coefficient of random graphs on the hyperbolic plane. This model of random graphs was proposed recently by Krioukov et al. as a mathematical model of complex networks, under the fundamental assumption that hyperbolic geometry underlies the structure of these networks. We give a rigorous analysis of clustering and characterize the global clustering coefficient in terms of the parameters of the model. We show how the global clustering coefficient can be tuned by these parameters and we give an explicit formula for this function.Comment: 51 pages, 1 figur