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Register a new userGenerating Random Networks Without Short Cycles
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Added by
Mohsen Bayati
Publication date
2017
fields
Informatics Engineering
and research's language is
English
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Random graph generation is an important tool for studying large complex networks. Despite abundance of random graph models, constructing models with application-driven constraints is poorly understood. In order to advance state-of-the-art in this area, we focus on random graphs without short cycles as a stylized family of graphs, and propose the RandGraph algorithm for randomly generating them. For any constant k, when m=O(n^{1+1/[2k(k+3)]}), RandGraph generates an asymptotically uniform random graph with n vertices, m edges, and no cycle of length at most k using O(n^2m) operations. We also characterize the approximation error for finite values of n. To the best of our knowledge, this is the first polynomial-time algorithm for the problem. RandGraph works by sequentially adding $m$ edges to an empty graph with n vertices. Recently, such sequential algorithms have been successful for random sampling problems. Our main contributions to this line of research includes introducing a new approach for sequentially approximating edge-specific probabilities at each step of the algorithm, and providing a new method for analyzing such algorithms.
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Motivated by storage applications, we study the following data structure problem: An encoder wishes to store a collection of jointly-distributed files $overline{X}:=(X_1,X_2,ldots, X_n) sim mu$ which are emph{correlated} ($H_mu(overline{X}) ll sum_i H_mu(X_i)$), using as little (expected) memory as possible, such that each individual file $X_i$ can be recovered quickly with few (ideally constant) memory accesses. In the case of independent random files, a dramatic result by Pat (FOCS08) and subsequently by Dodis, Pat and Thorup (STOC10) shows that it is possible to store $overline{X}$ using just a emph{constant} number of extra bits beyond the information-theoretic minimum space, while at the same time decoding each $X_i$ in constant time. However, in the (realistic) case where the files are correlated, much weaker results are known, requiring at least $Omega(n/polylg n)$ extra bits for constant decoding time, even for simple joint distributions $mu$. We focus on the natural case of compressingemph{Markov chains}, i.e., storing a length-$n$ random walk on any (possibly directed) graph $G$. Denoting by $kappa(G,n)$ the number of length-$n$ walks on $G$, we show that there is a succinct data structure storing a random walk using $lg_2 kappa(G,n) + O(lg n)$ bits of space, such that any vertex along the walk can be decoded in $O(1)$ time on a word-RAM. For the harder task of matching the emph{point-wise} optimal space of the walk, i.e., the empirical entropy $sum_{i=1}^{n-1} lg (deg(v_i))$, we present a data structure with $O(1)$ extra bits at the price of $O(lg n)$ decoding time, and show that any improvement on this would lead to an improved solution on the long-standing Dictionary problem. All of our data structures support the emph{online} version of the problem with constant update and query time.
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It is a great challenge to evaluate the network performance of cellular mobile communication systems. In this paper, we propose new spatial spectrum and energy efficiency models for Poisson-Voronoi tessellation (PVT) random cellular networks. To evaluate the user access the network, a Markov chain based wireless channel access model is first proposed for PVT random cellular networks. On that basis, the outage probability and blocking probability of PVT random cellular networks are derived, which can be computed numerically. Furthermore, taking into account the call arrival rate, the path loss exponent and the base station (BS) density in random cellular networks, spatial spectrum and energy efficiency models are proposed and analyzed for PVT random cellular networks. Numerical simulations are conducted to evaluate the network spectrum and energy efficiency in PVT random cellular networks.
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