Trellis-Based Equalization for Sparse ISI Channels Revisited

Sparse intersymbol-interference (ISI) channels are encountered in a variety of high-data-rate communication systems. Such channels have a large channel memory length, but only a small number of significant channel coefficients. In this paper, trellis-based equalization of sparse ISI channels is revisited. Due to the large channel memory length, the complexity of maximum-likelihood detection, e.g., by means of the Viterbi algorithm (VA), is normally prohibitive. In the first part of the paper, a unified framework based on factor graphs is presented for complexity reduction without loss of optimality. In this new context, two known reduced-complexity algorithms for sparse ISI channels are recapitulated: The multi-trellis VA (M-VA) and the parallel-trellis VA (P-VA). It is shown that the M-VA, although claimed, does not lead to a reduced computational complexity. The P-VA, on the other hand, leads to a significant complexity reduction, but can only be applied for a certain class of sparse channels. In the second part of the paper, a unified approach is investigated to tackle general sparse channels: It is shown that the use of a linear filter at the receiver renders the application of standard reduced-state trellis-based equalizer algorithms feasible, without significant loss of optimality. Numerical results verify the efficiency of the proposed receiver structure.Comment: To be presented at the 2005 IEEE Int. Symp. Inform. Theory (ISIT 2005), September 4-9, 2005, Adelaide, Australi

Spectral redemption: clustering sparse networks

Spectral algorithms are classic approaches to clustering and community detection in networks. However, for sparse networks the standard versions of these algorithms are suboptimal, in some cases completely failing to detect communities even when other algorithms such as belief propagation can do so. Here we introduce a new class of spectral algorithms based on a non-backtracking walk on the directed edges of the graph. The spectrum of this operator is much better-behaved than that of the adjacency matrix or other commonly used matrices, maintaining a strong separation between the bulk eigenvalues and the eigenvalues relevant to community structure even in the sparse case. We show that our algorithm is optimal for graphs generated by the stochastic block model, detecting communities all the way down to the theoretical limit. We also show the spectrum of the non-backtracking operator for some real-world networks, illustrating its advantages over traditional spectral clustering.Comment: 11 pages, 6 figures. Clarified to what extent our claims are rigorous, and to what extent they are conjectures; also added an interpretation of the eigenvectors of the 2n-dimensional version of the non-backtracking matri

Fragmenting networks by targeting collective influencers at a mesoscopic level

A practical approach to protecting networks against epidemic processes such as spreading of infectious diseases, malware, and harmful viral information is to remove some influential nodes beforehand to fragment the network into small components. Because determining the optimal order to remove nodes is a computationally hard problem, various approximate algorithms have been proposed to efficiently fragment networks by sequential node removal. Morone and Makse proposed an algorithm employing the non-backtracking matrix of given networks, which outperforms various existing algorithms. In fact, many empirical networks have community structure, compromising the assumption of local tree-like structure on which the original algorithm is based. We develop an immunization algorithm by synergistically combining the Morone-Makse algorithm and coarse graining of the network in which we regard a community as a supernode. In this way, we aim to identify nodes that connect different communities at a reasonable computational cost. The proposed algorithm works more efficiently than the Morone-Makse and other algorithms on networks with community structure.Comment: 5 figures, 3 tables, and SI include

Magnetic pattern at supergranulation scale: the Void Size Distribution

The large-scale magnetic pattern of the quiet sun is dominated by the magnetic network. This network, created by photospheric magnetic fields swept into convective downflows, delineates the boundaries of large scale cells of overturning plasma and exhibits voids in magnetic organization. Such voids include internetwork fields, a mixed-polarity sparse field that populate the inner part of network cells. To single out voids and to quantify their intrinsic pattern a fast circle packing based algorithm is applied to 511 SOHO/MDI high resolution magnetograms acquired during the outstanding solar activity minimum between 23 and 24 cycles. The computed Void Distribution Function shows a quasi-exponential decay behavior in the range 10-60 Mm. The lack of distinct flow scales in such a range corroborates the hypothesis of multi-scale motion flows at the solar surface. In addition to the quasi-exponential decay we have found that the voids reveal departure from a simple exponential decay around 35 Mm.Comment: 6 pages, 8 figures, to appear in Astronomy and Astrophysic