Epidemiologists and social scientists have used the Network Scale-Up Method (NSUM) for over thirty years to estimate the size of a hidden sub-population within a social network. This method involves querying a subset of network nodes about the number of their neighbors belonging to the hidden sub-population. In general, NSUM assumes that the social network topology and the hidden sub-population distribution are well-behaved; hence, the NSUM estimate is close to the actual value. However, bounds on NSUM estimation errors have not been analytically proven. This paper provides analytical bounds on the error incurred by the two most popular NSUM estimators. These bounds assume that the queried nodes accurately provide their degree and the number of neighbors belonging to the hidden sub-population. Our key findings are twofold. First, we show that when an adversary designs the network and places the hidden sub-population, then the estimate can be a factor of Ω(√n) off from the real value (in a network with n nodes). Second, we also prove error bounds when the underlying network is randomly generated, showing that a small constant factor can be achieved with high probability using samples of logarithmic size O(log n). We present improved analytical bounds for Erdős-Rényi and Scale-Free networks. Our theoretical analysis is supported by an extensive set of numerical experiments designed to determine the effect of the sample size on the accuracy of the estimates in both synthetic and real networks.

We present the first sublinear-time algorithm that can compute the edge connectivity λ of a network exactly on distributed message-passing networks (the CONGEST model), as long as the network contains no multi-edge. We present the first sublinear-time algorithm for a distributed message-passing network sto compute its edge connectivity λ exactly in the CONGEST model, as long as there are no parallel edges. Our algorithm takes Õ(n1−1/353D1/353+n1−1/706) time to compute λ and a cut of cardinality λ with high probability, where n and D are the number of nodes and the diameter of the network, respectively, and Õ hides polylogarithmic factors. This running time is sublinear in n (i.e. Õ(n1−є)) whenever D is. Previous sublinear-time distributed algorithms can solve this problem either (i) exactly only when λ=O(n1/8−є) [Thurimella PODC’95; Pritchard, Thurimella, ACM Trans. Algorithms’11; Nanongkai, Su, DISC’14] or (ii) approximately [Ghaffari, Kuhn, DISC’13; Nanongkai, Su, DISC’14]. To achieve this we develop and combine several new techniques. First, we design the first distributed algorithm that can compute a k-edge connectivity certificate for any k=O(n1−є) in time Õ(√nk+D). The previous sublinear-time algorithm can do so only when k=o(√n) [Thurimella PODC’95]. In fact, our algorithm can be turned into the first parallel algorithm with polylogarithmic depth and near-linear work. Previous near-linear work algorithms are essentially sequential and previous polylogarithmic-depth algorithms require Ω(mk) work in the worst case (e.g. [Karger, Motwani, STOC’93]). Second, we show that by combining the recent distributed expander decomposition technique of [Chang, Pettie, Zhang, SODA’19] with techniques from the sequential deterministic edge connectivity algorithm of [Kawarabayashi, Thorup, STOC’15], we can decompose the network into a sublinear number of clusters with small average diameter and without any mincut separating a cluster (except the “trivial” ones). This leads to a simplification of the Kawarabayashi-Thorup framework (except that we are randomized while they are deterministic). This might make this framework more useful in other models of computation. Finally, by extending the tree packing technique from [Karger STOC’96], we can find the minimum cut in time proportional to the number of components. As a byproduct of this technique, we obtain an Õ(n)-time algorithm for computing exact minimum cut for weighted graphs.

In this paper, we propose a 2-Sieve model for the detection of mitosis in breast cancer multispectral images. Multiresolution wavelet features & Gray Level Entropy Matrix (GLEM) features have been computed for each candidate on all the spectral bands. A novel dimensionality selection algorithm has been introduced and its performance compared with other existing algorithms. Data imbalance and data cleaning have been taken care of using classical data mining techniques. Furthermore, a Second Sieve classification is performed to increase the Positive Predictive Value (PPV) with minimal loss in Sensitivity. A final Sensitivity and PPV of 82.35% & 73.04% respectively was achieved over the testing set using the proposed scheme.