human: Given a hyperedge-centered hypergraph: <hypergraph>, where the center hyperedge is induced by one source paper and its member nodes are the cited papers.

Details of the hypergraph:
Center hyperedge (induced by source paper "community detection thresholds and the weak ramanujan property"):
  Abstract: 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.

Includes members:
  Member "spectral redemption in clustering sparse networks":
    also connected through other hyperedges:
      - Hyperedge HE-1 (induced by source paper "approximating spectral clustering via sampling a review")
  Member "graph spectra and the detectability of community structure in networks":
    also connected through other hyperedges:
      - Hyperedge HE-2 (induced by source paper "fragility of spectral clustering for networks with an overlapping structure")
  Member "community detection in the labelled stochastic block model":
    also connected through other hyperedges:
      - Hyperedge HE-3 (induced by source paper "spectral clustering of signed graphs via matrix power means")
  Member "a proof of alon s second eigenvalue conjecture and related problems":
    also connected through other hyperedges:
      - Hyperedge HE-4 (induced by source paper "spectrum preserving short cycle removal on regular graphs")

Question: Please tell me which class the source paper belongs to. The 40 classes are: cs.NA(Numerical Analysis), cs.MM(Multimedia), cs.LO(Logic in Computer Science), cs.CY(Computers and Society), cs.CR(Cryptography and Security), cs.DC(Distributed, Parallel, and Cluster Computing), cs.HC(Human-Computer Interaction), cs.CE(Computational Engineering, Finance, and Science), cs.NI(Networking and Internet Architecture), cs.CC(Computational Complexity), cs.AI(Artificial Intelligence), cs.MA(Multiagent Systems), cs.GL(General Literature), cs.NE(Neural and Evolutionary Computing), cs.SC(Symbolic Computation), cs.AR(Hardware Architecture), cs.CV(Computer Vision and Pattern Recognition), cs.GR(Graphics), cs.ET(Emerging Technologies), cs.SY(Systems and Control), cs.CG(Computational Geometry), cs.OH(Other Computer Science), cs.PL(Programming Languages), cs.SE(Software Engineering), cs.LG(Machine Learning), cs.SD(Sound), cs.SI(Social and Information Networks), cs.RO(Robotics), cs.IT(Information Theory), cs.PF(Performance), cs.CL(Computation and Language), cs.IR(Information Retrieval), cs.MS(Mathematical Software), cs.FL(Formal Languages and Automata Theory), cs.DS(Data Structures and Algorithms), cs.OS(Operating Systems), cs.GT(Computer Science and Game Theory), cs.DB(Databases), cs.DL(Digital Libraries), cs.DM(Discrete Mathematics). Directly output the class name.

gpt: cs.SI(Social and Information Networks)