Community detection thresholds and the weak Ramanujan property

Decelle et al.\cite{Decelle11} conjectured the existence of a sharp threshold for community detection in sparse random graphs drawn from the stochastic block model. Mossel et al.\cite{Mossel12} established the negative part of the conjecture, proving impossibility of meaningful detection below the threshold. However the positive part of the conjecture remained elusive so far. Here we solve the positive part of the conjecture. We introduce a modified adjacency matrix $B$ that counts self-avoiding paths of a given length $\ell$ between pairs of nodes and prove that for logarithmic $\ell$, the leading eigenvectors of this modified matrix provide non-trivial detection, thereby settling the conjecture. A key step in the proof consists in establishing a {\em weak Ramanujan property} of matrix $B$. Namely, the spectrum of $B$ consists in two leading eigenvalues $\rho(B)$, $\lambda_2$ and $n-2$ eigenvalues of a lower order $O(n^{\epsilon}\sqrt{\rho(B)})$ for all $\epsilon>0$, $\rho(B)$ denoting $B$'s spectral radius. $d$-regular graphs are Ramanujan when their second eigenvalue verifies $|\lambda|\le 2 \sqrt{d-1}$. Random $d$-regular graphs have a second largest eigenvalue $\lambda$ of $2\sqrt{d-1}+o(1)$ (see Friedman\cite{friedman08}), thus being {\em almost} Ramanujan. Erd\H{o}s-R\'enyi graphs with average degree $d$ at least logarithmic ($d=\Omega(\log n)$) have a second eigenvalue of $O(\sqrt{d})$ (see Feige and Ofek\cite{Feige05}), a slightly weaker version of the Ramanujan property. However this spectrum separation property fails for sparse ($d=O(1)$) Erd\H{o}s-R\'enyi graphs. Our result thus shows that by constructing matrix $B$ through neighborhood expansion, we regularize the original adjacency matrix to eventually recover a weak form of the Ramanujan property

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

Router-level community structure of the Internet Autonomous Systems

The Internet is composed of routing devices connected between them and organized into independent administrative entities: the Autonomous Systems. The existence of different types of Autonomous Systems (like large connectivity providers, Internet Service Providers or universities) together with geographical and economical constraints, turns the Internet into a complex modular and hierarchical network. This organization is reflected in many properties of the Internet topology, like its high degree of clustering and its robustness. In this work, we study the modular structure of the Internet router-level graph in order to assess to what extent the Autonomous Systems satisfy some of the known notions of community structure. We show that the modular structure of the Internet is much richer than what can be captured by the current community detection methods, which are severely affected by resolution limits and by the heterogeneity of the Autonomous Systems. Here we overcome this issue by using a multiresolution detection algorithm combined with a small sample of nodes. We also discuss recent work on community structure in the light of our results

On the Costs and Benefits of Stochasticity in Stream Processing ∗

With the end of clock-frequency scaling, parallelism has emerged as the key driver of chip-performance growth. Yet, several factors undermine efficient simultaneous use of onchip resources, which continue scaling with Moore’s law. These factors are often due to sequential dependencies, as illustrated by Amdahl’s law. Quantifying achievable parallelism can help prevent futile programming efforts and guide innovation toward the most significant challenges. To complement Amdahl’s law, we focus on stream processing and quantify performance losses due to stochastic runtimes. Using spectral theory of random matrices, we derive new analytical results and validate them by numerical simulations. These results allow us to explore unique benefits of stochasticity and show that they outweigh the costs for software streams

A multi-stage neural network approach for coronary 3D reconstruction from uncalibrated X-ray angiography images

Abstract We present a multi-stage neural network approach for 3D reconstruction of coronary artery trees from uncalibrated 2D X-ray angiography images. This method uses several binarized images from different angles to reconstruct a 3D coronary tree without any knowledge of image acquisition parameters. The method consists of a single backbone network and separate stages for vessel centerline and radius reconstruction. The output is an analytical matrix representation of the coronary tree suitable for downstream applications such as hemodynamic modeling of local vessel narrowing (i.e., stenosis). The network was trained using a dataset of synthetic coronary trees from a vessel generator informed by both clinical image data and literature values on coronary anatomy. Our multi-stage network achieved sub-pixel accuracy in reconstructing vessel radius (RMSE = 0.16 ± 0.07 mm) and stenosis radius (MAE = 0.27 ± 0.18 mm), the most important feature used to inform diagnostic decisions. The network also led to 52% and 38% reduction in vessel centerline reconstruction errors compared to a single-stage network and projective geometry-based methods, respectively. Our method demonstrated robustness to overcome challenges such as vessel foreshortening or overlap in the input images. This work is an important step towards automated analysis of anatomic and functional disease severity in the coronary arteries

Analysis of Mechanical Behavior of AL 6063- Cotton Shell Ash and SiC METAL Matrix Composites using Stir Casting

Al-6063 with powder reinforcement consists the superiour stiffness, high strength and better wear resistance to unreinforced base alloy. Those are used for automobile spare parts,components and aircraft body applications. Metal matrix hybrid composites (MMHC) focus mainly on economoic and ecofriendly its trend to good specific strength and hardness application. Aluminium alloy as base material and cotton shell ash (CSA) and silicon carbide (SiC) as reinforcements has better properties. The practial way to create hybrid composite is too difficult in optimum conditons. In this my project work Al 6063-CSA-SiC castings with numerouis volume proportions of CSA and SiC were fabricated, by maintaining the argon environment by the enhance of botttom pouring stir casting method. The hybrid-composites (HC) are fabricated by the reinforcement with 75 μm. We found that with more % of CSA and SiC addition to enhancement in specific strength of the hybrid composite. Surface morphology powder distribution are examined in detail by computerised inverted metallurgical microsope. The fabricated hybrid composite various tests are conducted to determne the mechanical properties of density, porosity, hardness and compression was observed and to analyze the process parameters to their effect of cotton shell ash. Influence of CSA-SIC as a reinforcement of particular emphasis the result was differentiate with unrinforced aluminum alloy