Optimal random search for a single hidden target

A single target is hidden at a location chosen from a predetermined probability distribution. Then, a searcher must find a second probability distribution from which random search points are sampled such that the target is found in the minimum number of trials. Here it will be shown that if the searcher must get very close to the target to find it, then the best search distribution is proportional to the square root of the target distribution. For a Gaussian target distribution, the optimum search distribution is approximately a Gaussian with a standard deviation that varies inversely with how close the searcher must be to the target to find it. For a network, where the searcher randomly samples nodes and looks for the fixed target along edges, the optimum is to either sample a node with probability proportional to the square root of the out degree plus one or not at all.Comment: 13 pages, 5 figure

Ranking and clustering of nodes in networks with smart teleportation

Random teleportation is a necessary evil for ranking and clustering directed networks based on random walks. Teleportation enables ergodic solutions, but the solutions must necessarily depend on the exact implementation and parametrization of the teleportation. For example, in the commonly used PageRank algorithm, the teleportation rate must trade off a heavily biased solution with a uniform solution. Here we show that teleportation to links rather than nodes enables a much smoother trade-off and effectively more robust results. We also show that, by not recording the teleportation steps of the random walker, we can further reduce the effect of teleportation with dramatic effects on clustering.Comment: 10 pages, 7 figure

Coexistence of opposite opinions in a network with communities

The Majority Rule is applied to a topology that consists of two coupled random networks, thereby mimicking the modular structure observed in social networks. We calculate analytically the asymptotic behaviour of the model and derive a phase diagram that depends on the frequency of random opinion flips and on the inter-connectivity between the two communities. It is shown that three regimes may take place: a disordered regime, where no collective phenomena takes place; a symmetric regime, where the nodes in both communities reach the same average opinion; an asymmetric regime, where the nodes in each community reach an opposite average opinion. The transition from the asymmetric regime to the symmetric regime is shown to be discontinuous.Comment: 14 pages, 4 figure

Map equation for link community

Community structure exists in many real-world networks and has been reported being related to several functional properties of the networks. The conventional approach was partitioning nodes into communities, while some recent studies start partitioning links instead of nodes to find overlapping communities of nodes efficiently. We extended the map equation method, which was originally developed for node communities, to find link communities in networks. This method is tested on various kinds of networks and compared with the metadata of the networks, and the results show that our method can identify the overlapping role of nodes effectively. The advantage of this method is that the node community scheme and link community scheme can be compared quantitatively by measuring the unknown information left in the networks besides the community structure. It can be used to decide quantitatively whether or not the link community scheme should be used instead of the node community scheme. Furthermore, this method can be easily extended to the directed and weighted networks since it is based on the random walk.Comment: 9 pages,5 figure

Organizational Chart Inference

Nowadays, to facilitate the communication and cooperation among employees, a new family of online social networks has been adopted in many companies, which are called the "enterprise social networks" (ESNs). ESNs can provide employees with various professional services to help them deal with daily work issues. Meanwhile, employees in companies are usually organized into different hierarchies according to the relative ranks of their positions. The company internal management structure can be outlined with the organizational chart visually, which is normally confidential to the public out of the privacy and security concerns. In this paper, we want to study the IOC (Inference of Organizational Chart) problem to identify company internal organizational chart based on the heterogeneous online ESN launched in it. IOC is very challenging to address as, to guarantee smooth operations, the internal organizational charts of companies need to meet certain structural requirements (about its depth and width). To solve the IOC problem, a novel unsupervised method Create (ChArT REcovEr) is proposed in this paper, which consists of 3 steps: (1) social stratification of ESN users into different social classes, (2) supervision link inference from managers to subordinates, and (3) consecutive social classes matching to prune the redundant supervision links. Extensive experiments conducted on real-world online ESN dataset demonstrate that Create can perform very well in addressing the IOC problem.Comment: 10 pages, 9 figures, 1 table. The paper is accepted by KDD 201

Power-law distributions in empirical data

Power-law distributions occur in many situations of scientific interest and have significant consequences for our understanding of natural and man-made phenomena. Unfortunately, the detection and characterization of power laws is complicated by the large fluctuations that occur in the tail of the distribution -- the part of the distribution representing large but rare events -- and by the difficulty of identifying the range over which power-law behavior holds. Commonly used methods for analyzing power-law data, such as least-squares fitting, can produce substantially inaccurate estimates of parameters for power-law distributions, and even in cases where such methods return accurate answers they are still unsatisfactory because they give no indication of whether the data obey a power law at all. Here we present a principled statistical framework for discerning and quantifying power-law behavior in empirical data. Our approach combines maximum-likelihood fitting methods with goodness-of-fit tests based on the Kolmogorov-Smirnov statistic and likelihood ratios. We evaluate the effectiveness of the approach with tests on synthetic data and give critical comparisons to previous approaches. We also apply the proposed methods to twenty-four real-world data sets from a range of different disciplines, each of which has been conjectured to follow a power-law distribution. In some cases we find these conjectures to be consistent with the data while in others the power law is ruled out.Comment: 43 pages, 11 figures, 7 tables, 4 appendices; code available at http://www.santafe.edu/~aaronc/powerlaws

Assortative mixing in networks

A network is said to show assortative mixing if the nodes in the network that have many connections tend to be connected to other nodes with many connections. We define a measure of assortative mixing for networks and use it to show that social networks are often assortatively mixed, but that technological and biological networks tend to be disassortative. We propose a model of an assortative network, which we study both analytically and numerically. Within the framework of this model we find that assortative networks tend to percolate more easily than their disassortative counterparts and that they are also more robust to vertex removal.Comment: 5 pages, 1 table, 1 figur

Handling oversampling in dynamic networks using link prediction

Oversampling is a common characteristic of data representing dynamic networks. It introduces noise into representations of dynamic networks, but there has been little work so far to compensate for it. Oversampling can affect the quality of many important algorithmic problems on dynamic networks, including link prediction. Link prediction seeks to predict edges that will be added to the network given previous snapshots. We show that not only does oversampling affect the quality of link prediction, but that we can use link prediction to recover from the effects of oversampling. We also introduce a novel generative model of noise in dynamic networks that represents oversampling. We demonstrate the results of our approach on both synthetic and real-world data.Comment: ECML/PKDD 201

Asymptotic behavior of the Kleinberg model

We study Kleinberg navigation (the search of a target in a d-dimensional lattice, where each site is connected to one other random site at distance r, with probability proportional to r^{-a}) by means of an exact master equation for the process. We show that the asymptotic scaling behavior for the delivery time T to a target at distance L scales as (ln L)^2 when a=d, and otherwise as L^x, with x=(d-a)/(d+1-a) for ad+1. These values of x exceed the rigorous lower-bounds established by Kleinberg. We also address the situation where there is a finite probability for the message to get lost along its way and find short delivery times (conditioned upon arrival) for a wide range of a's