Community detection or cluster detection in networks is a well-studied, albeit hard, problem. Given the scale and complexity of modern day social networks, detecting ``reasonable'' communities is an even harder problem. Since the first use of k-means algorithm in 1960s, many community detection algorithms have been invented - most of which are developed with specific goals in mind and the idea of detecting ``meaningful'' communities varies widely from one algorithm to another. With the increasing number of community detection algorithms, there has been an advent of a number of evaluation measures and objective functions such as modularity and internal density. In this paper we divide methods of measurements in to two categories, according to whether they rely on ground-truth or not. Our work is aiming to answer whether these general used objective functions are well consistent with the real performance of community detection algorithms across a number of homogeneous and heterogeneous networks. Seven representative algorithms are compared under various performance metrics, and on various real world social networks.

We start by defining a pruning process involving sellers on one side and buyers on the other. The goal is to quickly select a subset of the sellers so that the products that these sellers bring to the market has small cost ratio, i.e., the ratio of the total cost of the selected sellers' products to amount that interested buyers are willing to pay. As modeled here, the pruning process can be used to speed up distributed implementations of greedy algorithms (e.g., for minimum dominating set, facility location, etc). We present a randomized instance of the pruning process that, for any positive k, runs in O(k) communication rounds with O(log N)-sized messages, yielding a cost ratio of O(Nc/k). Here N is the product of the number of sellers and number of buyers and c is a small constant. Using this O(k)-round pruning algorithm as the basis, we derive several simple, greedy, O(k)-round distributed approximation algorithms for MDS and facility location (both metric and non-metric versions). Our algorithms achieve optimal approximation ratios in polylogarithmic rounds and shave a "logarithmic factor" off the best, known, approximation factor, typically achieved using LP-rounding techniques.

In this paper we present fast, distributed approximation algorithms for the metric facility location problem in the CONGEST model, where message sizes are bounded by O(log N) bits, N being the network size. We first show how to obtain a 7-approximation in O(log m + log n) rounds via the primal-dual method; here m is the number of facilities and n is the number of clients. Subsequently, we generalize this to a k-round algorithm, that for every constant k, yields an approximation factor of O(m2/√k ∙ n3/√k). These results answer a question posed by Moscibroda and Wattenhofer (PODC 2005). Our techniques are based on the primal-dual algorithm due to Jain and Vazirani (JACM 2001) and a rapid randomized sparsification of graphs due to Gfeller and Vicari (PODC 2007). These results complement the results of Moscibroda and Wattenhofer (PODC 2005) for non-metric facility location and extend the results of Gehweiler et al. (SPAA 2006) for uniform metric facility location.

This paper presents a distributed algorithm for wireless ad-hoc networks that runs in polylogarithmic number of rounds in the size of the network and constructs a lightweight, linear size, (1+ε)-spanner for any given ε> 0. A wireless network is modeled by a d-dimensional α-quasi unit ball graph (α-UBG), which is a higher dimensional generalization of the standard unit disk graph (UDG) model. The d-dimensional α-UBG model goes beyond the unrealistic “flat world ” as-sumption of UDGs and also takes into account transmission errors, fading signal strength, and physical obstructions. The main result in the paper is this: for any fixed ε> 0, 0 < α ≤ 1, and d ≥ 2 there is a distributed algorithm run-ning in O(log n·log ∗ n) communication rounds on an n-node, d-dimensional α-UBG G that computes a (1+ε)-spanner G′ of G with maximum degree Δ(G′) = O(1) and total weight w(G′) = O(w(MST (G)). This result is motivated by the topology control problem in wireless ad-hoc networks and improves on existing topology control algorithms along sev-eral dimensions. The technical contributions of the paper include a new, sequential, greedy algorithm with relaxed edge ordering and lazy updating, and clustering techniques for filtering out unnecessary edges.