Results on residual Rényi entropy of order statistics and record values

This paper explores properties of the residual Rényi entropy of some ordered random variables. The residual Rényi entropy of the kth order statistic from a continuous distribution function is represented in terms of the residual Rényi entropy of the kth order statistic from uniform distribution. The monotone behavior of the residual Rényi entropy of order statistic under various conditions is discussed. Analogues results for the residual Rényi entropy of record values are also given.     

Fast online graph clustering via Erd?s-Rényi mixture

In the context of graph clustering, we consider the problem of simultaneously estimating both the partition of the graph nodes and the parameters of an underlying mixture of affiliation networks. In numerous applications the rapid increase of data size over time makes classical clustering algorithms too slow because of the high computational cost. In such situations online clustering algorithms are an efficient alternative to classical batch algorithms. We present an original online algorithm for graph clustering based on a Erd?s-Rényi graph mixture. The relevance of the algorithm is illustrated, using both simulated and real data sets. The real data set is a network extracted from the French political blogosphere and presents an interesting community organization.     

Structural Properties of Recurrent Neural Networks

In this article we research the impact of the adaptive learning process of recurrent neural networks (RNN) on the structural properties of the derived graphs. A trained fully connected RNN can be converted to a graph by defining edges between pairs od nodes having significant weights. We measured structural properties of the derived graphs, such as characteristic path lengths, clustering coefficients and degree distributions. The results imply that a trained RNN has significantly larger clustering coefficient than a random network with a comparable connectivity. Besides, the degree distributions show existence of nodes with a large degree or hubs, typical for scale-free networks. We also show analytically and experimentally that this type of degree distribution has increased entropy.     

Probabilistic flooding in stochastic networks: Analysis of global information outreach

This article investigates probabilistic information dissemination in stochastic networks. The following problem is studied: A source node intends to deliver a message to all other network nodes using probabilistic flooding, i.e., each node forwards a received message to all its neighbors with a common network-wide forwarding probability ω. Question is: what is the minimum ω-value each node should use, such that the flooded message is obtained by all nodes with high probability? We first present a generic approach to derive the global outreach probability in arbitrary networks and then focus on Erd?s Rényi graphs (ERGs) and random geometric graphs (RGGs). For ERGs we derive an exact expression. For RGGs we derive an asymptotic expression that represents an approximation for networks with high node density. Both reliable and unreliable links are studied.     

基于复杂网络理论的PPI网络拓扑分析

蛋白质相互作用在生命活动中起核心作用,由蛋白质相互作用构成的PPI网络的拓扑特性分析是后基因组时代最重要的研究课题之一。应用复杂网络理论对DIP数据库中7个物种的8个PPI网络的拓扑结构进行分析与研究。分析结果表明,这些PPI网络具有较小的平均路径长度和较高的聚集系数,其度分布服从幂规律,即p（k）=ak^-r,其中r大于1小于3,a近似等于1±0.5,表现出典型的无标度性,并具有高的异质性。其中平均度大于3.5的5个PPI网络对随机删除不超过10%的顶点都具有很好的鲁棒性,但对有选择的删除2%的高度顶点就开始表现出极弱的抗攻击性。     

A random walk model for infection on graphs: spread of epidemics &; rumours with mobile agents

We address the question of understanding the effect of the underlying network topology on the spread of a virus and the dissemination of information when users are mobile performing independent random walks on a graph. To this end, we propose a simple model of infection that enables to study the coincidence time of two random walkers on an arbitrary graph. By studying the coincidence time of a susceptible and an infected individual both moving in the graph we obtain estimates of the infection probability. The main result of this paper is to pinpoint the impact of the network topology on the infection probability. More precisely, we prove that for homogeneous graphs including regular graphs and the classical Erdős–Rényi model, the coincidence time is inversely proportional to the number of nodes in the graph. We then study the model on power-law graphs, that exhibit heterogeneous connectivity patterns, and show the existence of a phase transition for the coincidence time depending on the parameter of the power-law of the degree distribution. We finally undertake a preliminary analysis for the case with k random walkers and provide upper bounds on the convergence time for both the complete graph and regular graphs.     

An effective and efficient parallel approach for random graph generation over GPUs

The widespread usage of random graphs   has been highlighted in the context of database applications for several years. This because such data structures turn out to be very useful in a large family of database applications ranging from simulation to sampling, from analysis of complex networks to study of randomized algorithms, and so forth. Amongst others, Erd?s–Rényi _Γv,p${\Gamma }_{v,p}$ is the most popular model to obtain and manipulate random graphs. Unfortunately, it has been demonstrated that classical algorithms for generating Erd?s–Rényi based random graphs do not scale well in large instances and, in addition to this, fail to make use of the parallel processing capabilities of modern hardware. Inspired by this main motivation, in this paper we propose and experimentally assess a novel parallel algorithm for generating random graphs under the Erd?s–Rényi model that is designed and implemented in a Graphics Processing Unit (GPU), called PPreZER. We demonstrate the nice amenities due to our solution via a succession of several intermediary algorithms, both sequential and parallel, which show the limitations of classical approaches and the benefits due to the PPreZER algorithm. Finally, our comprehensive experimental assessment and analysis brings to light a relevant average speedup gain of PPreZER over baseline algorithms.     

Flooding in dynamic graphs with arbitrary degree sequence

This paper addresses the flooding problem in dynamic graphs, where flooding is the basic mechanism in which every node becoming aware of a piece of information at step t$t$ forwards this information to all its neighbors at all forthcoming steps t^′>t$t^{\prime }>t$. We show that a technique developed in a previous paper, for analyzing flooding in a Markovian sequence of Erdös–Rényi graphs, is robust enough to be used also in different contexts. We establish this fact by analyzing flooding in a sequence of graphs drawn independently at random according to a model of random graphs with given expected degree sequence. In the prominent case of power-law degree distributions, we prove that flooding takes almost surely O(logn)$O(logn)$ steps even if, almost surely, none of the graphs in the sequence is connected. In the general case of graphs with an arbitrary degree sequence, we prove several upper bounds on the flooding time, which depend on specific properties of the degree sequence.     

Ranking weighted clustering coefficient in large dynamic graphs

Efficiently searching top-k representative vertices is crucial for understanding the structure of large dynamic graphs. Recent studies show that communities formed by a vertex with high local clustering coefficient and its neighbours can achieve enhanced information propagation speed as well as disease transmission speed. However, local clustering coefficient, which measures the cliquishness of a vertex in its local neighbourhood, prefers vertices with small degrees. To remedy this issue, in this paper we propose a new ranking measure, weighted clustering coefficient (WCC) of vertices, by integrating both local clustering coefficient and degree. WCC not only inherits the properties of local clustering coefficient but also approximately measures the density (i.e., average degree) of its neighbourhood subgraph. Thus, vertices with higher WCC are more likely to be representative. We study efficiently computing and monitoring top-k representative vertices based on WCC over large dynamic graphs. To reduce the search space, we propose a series of heuristic upper bounds for WCC to prune a large portion of disqualifying vertices from the search space. We also develop an approximation algorithm by utilizing Flajolet-Martin sketch to trade acceptable accuracy for enhanced efficiency. An efficient incremental algorithm dealing with frequent updates in dynamic graphs is explored as well. Extensive experimental results on a variety of real-life graph datasets demonstrate the efficiency and effectiveness of our approaches.     

基于复杂网络的城市公共交通网络研究

将北京、上海和杭州3个大城市的公共交通网络(常规公交和快速公交)抽象成复杂网络,结合网络图论思想,把公交站点作为节点,站点间的连线作为边,在大量统计数据的基础上,通过Space L和Space P方法研究3大城市的复杂网络特性。统计分析表明,3个城市的公交网络均具有较小的平均路径长度,即典型的小世界特性。其节点的度分布,在Space L方法的描述下具有无标度特性,在Space P方法的描述下具有指数分布特性。通过对Space L和Space P两种描述方法的比较,可以发现对于同样的公交网络,Space P方法描述的网络具有更大的聚类系数和更小的平均路径长度,即具有更强的小世界效应。     

Peer-to-peer information retrieval using shared-content clustering

Peer-to-peer (p2p) networks are used by millions for searching and downloading content. Recently, clustering algorithms were shown to be useful for helping users find content in large networks. Yet, many of these algorithms overlook the fact that p2p networks follow graph models with a power-law node degree distribution. This paper studies the obtained clusters when applying clustering algorithms on power-law graphs and their applicability for finding content. Driven by the observed deficiencies, a simple yet efficient clustering algorithm is proposed, which targets a relaxed optimization of a minimal distance distribution of each cluster with a size balancing scheme. A comparative analysis using a song-similarity graph collected from 1.2 million Gnutella users reveals that commonly used efficiency measures often overlook search and recommendation applicability issues and provide the wrong impression that the resulting clusters are well suited for these tasks. We show that the proposed algorithm performs well on various measures that are well suited for the domain.     

考虑节点亲密度的社交网络谣言传播研究

考虑到真实社交网络中节点间亲密程度对谣言传播的影响,提出一种新的SI2R传播模型,建立谣言传播动力学方程组,研究谣言在无标度网络上的传播特性。该模型中不同节点间谣言传播率的非一致性同时取决于节点度与节点间亲密度,理论分析得到了无标度网络上谣言传播阈值表达式。随后,在BA（Barabási-Albert）无标度网络中就节点亲密度对谣言传播过程的影响进行了仿真实验,并利用Twitter和Live Journal两种真实网络数据集对仿真结果进行验证。研究表明,无标度网络中节点间平均亲密度随网络聚类系数的增大而减小,随着网络中节点间平均亲密度增大,谣言传播最终范围变大。研究还发现,节点间亲密度的存在使无标度网络中存在传播阈值,传播阈值随着节点间平均亲密度增大而减小。     

合作网络局域世界演化模型

根据合作网络中实际合作的局域性特性及项目度对网络的影响,提出一种合作网络局域世界演化模型(CoLW模型)。该模型以完全图为基础层次化构造局域世界,且以项目为基本单元进行网络规模的增长。CoLW模型的节点度服从幂律分布,具有较大的平均聚集系数且网络规模对其影响较小,接近真实合作网络。实验结果表明,CoLW模型可以较好地刻画真实合作网络的拓扑结构与统计特性。     

藏文字同现网络的小世界效应和无标度特性

复杂网络具有自组织、自相似、吸引子、小世界、无标度中部分或全部性质,而语言文字作为人类智慧和文明的结晶,是经过漫长演化形成的复杂网络。该文对藏语诗歌、散文、政治、佛教、教材和口语等六类具有代表性的体裁语料,每类各取15篇共90篇文章构建了97个藏文字同现网络,分析了藏文字同现网络的最短路径长度、聚类系数和度分布,实验数据显示97个藏文字同现网络都具有小世界效应和无标度特性,表明藏文字同现网络都具有小世界效应和无标度特性。     

图书漂流网络模型实证研究

通过收集整理图书漂流(bookcrossing)网站一个月内的图书漂流信息,建立图书与用户的数据库模型,并且构建两者间关系的二分图.从复杂网络的角度分析计算该网络的相关参数,如度分布、聚集系数、平均最短路径、节点项目度、项目大小、点强度及节点兴趣度,得到的图书漂流网络模型同时具有无标度特性和小世界网络的特性.     

A new distribution with decreasing, increasing and upside-down bathtub failure rate

The modeling and analysis of lifetimes is an important aspect of statistical work in a wide variety of scientific and technological fields. For the first time, the so-called generalized exponential geometric distribution is introduced. The new distribution can have a decreasing, increasing and upside-down bathtub failure rate function depending on its parameters. It includes the exponential geometric (Adamidis and Loukas, 1998), the generalized exponential (Gupta and Kundu, 1999) and the extended exponential geometric (Adamidis et al., 2005) distributions as special sub-models. We provide a comprehensive mathematical treatment of the distribution and derive expressions for the moment generating function, characteristic function and rth moment. An expression for Rényi entropy is obtained, and estimation of the stress-strength parameter is discussed. We estimate the parameters by maximum likelihood and obtain the Fisher information matrix. The flexibility of the new model is illustrated in an application to a real data set.     

复杂网络中随机图模型研究

随着复杂网络研究的兴起，随机图成为一种重要复杂网络模型。基于完全图的生成子图的思想，得到了生成随机图的一种新算法，即用去边的方法生成随机图的算法，并用数值实验验证了加边和去边生成的随机图的统计特性（最大度、最小度、聚集系数、平均最短路径和平均度）是相近的，用去边的方法得到的图的度分布曲线在其平均度处达到峰值，随后呈指数下降，这与随机图的度分布是相同的。为了得到稀疏连通的随机图，又提出了一个不去割边的近似随机图生成算法，并从理论上说明了该算法生成的图是连通的，同时通过数值实验验证了图的连通性，并与加边随机图的统计特性进行了比较。     

Rényi entropy rate for Gaussian processes

In this paper, we introduce the definition of the conditional Rényi entropy for continuous random variables and show that the so-called chain rule holds. Then, we use this rule to obtain another relation for getting the rate of Rényi entropy. Using this relation and properties of the Rényi entropy we obtain the Rényi entropy rate for stationary Gaussian processes. Finally, we show that the bound for the Rényi entropy rate is simply the Shannon entropy rate and that the Rényi entropy rate reduces to the Shannon entropy rate as α→1.     

基于中间节点效应的无标度网络演化模型研究

原BA模型以网络中已存在的各个节点与新增节点的连接相互独立为前提。然而,在真实系统中,当网络中一个节点与新增节点连接后,该节点对其邻居节点与新增节点的连接会存在影响。针对该现象,提出了基于中间节点效应的无标度网络演化模型。首先描述与定义了中间节点效应,然后给出了中间节点效应模型的生成算法,并从理论上分析了该模型的度分布情况,最后利用仿真验证了理论分析的正确性,并就度分布、群聚系数、平均路径长度等复杂网络参数与原BA模型进行了对比,结果表明此模型能生成无标度网络并且更符合现实网络的演化过程。     

Evolutionary Dynamics on Scale-Free Interaction Networks

There has been a recent surge of interest in studying dynamical processes, including evolutionary optimization, on scale-free topologies. While various scaling parameters and assortativities have been observed in natural scale-free interaction networks, most previous studies of dynamics on scale-free graphs have employed a graph-generating algorithm that yields a topology with an uncorrelated degree distribution and a fixed scaling parameter. In this paper, we advance the understanding of selective pressure in scale-free networks by systematically investigating takeover times under local uniform selection in scale-free topologies with varying scaling exponents, assortativities, average degrees, and numbers of vertices. We demonstrate why the so-called characteristic path length of a graph is a nonlinear function of both scaling and assortativity. Neither the eigenvalues of the adjacency matrix nor the effective population size was sufficient to account for the variance in takeover times over the parameter space that was explored. Rather, we show that 97% of the variance of logarithmically transformed average takeover times, on all scale-free graphs tested, could be accounted for by a planar function of: 1) the average inverse degree (which captures the effects of scaling) and 2) the logarithm of the population size. Additionally, we show that at low scaling exponents, increasingly positive assortativities increased the variability between experiments on different random graph instances, while increasingly negative assortativities increased the variability between takeover times from different initial conditions on the same graph instances. We explore the mechanisms behind our sometimes counterintuitive findings, and discuss potential implications for evolutionary computation and other relevant disciplines.