Comment on ``Small-world networks Evidence for a crossover picture''
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小世界效应和无标度-概述说明以及解释
小世界效应和无标度-概述说明以及解释1.引言1.1 概述概述部分提供了关于小世界效应和无标度网络的背景和概要信息。
本节将介绍这两个概念的起源和基本定义，以及它们在网络科学领域的重要性和研究意义。
小世界效应是指在一个具有大量节点的网络中，任意两个节点之间的距离很短，通常只需要经过少数几个中间节点即可到达。
这个现象最早由社会学家斯坦利·米尔格拉姆在1967年的实验中发现，并在1998年由弗兰克和温图拉提出了更为系统的定义。
小世界网络在现实生活中存在广泛，例如社交网络、物流网络和互联网等，这种网络结构具有高效的信息传递和快速的交流特点。
无标度网络是另一个重要的网络拓扑结构，在这种网络中，节点的度数（即与其相连的边的数量）遵循幂律分布。
这意味着有少量的节点具有非常高的度数，而大多数节点的度数相对较低。
这种网络结构的重要性得到了巴拉巴西等学者的广泛研究和关注。
无标度网络具有高度的鲁棒性和抗击毁性，在信息传播、疾病传播和网络攻击等方面具有重要的应用价值。
小世界效应和无标度网络在网络科学领域被视为两个重要的研究课题。
研究人员通过模型构建、实证分析和理论解释等多种方法，探索了这两个概念之间的关系和相互作用。
理解小世界效应和无标度网络的特性和行为规律，有助于我们更好地理解和设计现实世界中的各种网络系统，并且对社会、经济和生物系统等领域的研究有着重要的启示作用。
在接下来的章节中，我们将从不同角度对小世界效应和无标度网络进行深入的研究和分析。
我们将讨论它们的定义、原理、特征，探索它们的影响和应用，并探究它们之间的关系和相互影响。
最后，我们将总结主要观点，评价小世界效应和无标度网络的意义和影响，并提出未来进一步研究的建议。
通过这篇长文的阅读，读者将对小世界效应和无标度网络有一个更全面和深入的了解。
文章结构部分的内容可以按照以下方式编写：1.2 文章结构本文主要分为五个部分：引言、小世界效应、无标度网络、小世界效应和无标度网络的关系以及结论。
复杂网络研究简介
∑d
i> j
ij
d12 = 1
d13 = 1 d 23 = 1
d14 = 2 d 24 = 1 d 34 = 2
d15 = 1 d 25 = 2 d 35 = 2 d 45 = 3
Total = 16 Average:
L = 16 / 10 = 1.6
聚类系数
• 一个网络的聚类系数 C满足:
0<C<1
规则网络
(a) 完全连接;
(b) 最近邻居连接;
(c) 星形连接
规则网络
... ...
(d) Lattice
(z) Layers
随机图理论
• 随机图论 - Erdös and Rényi (1960) • ER 随机图模型统治四十余年…… 直到今天 …… • 当今大量可获取的数据＋高级计算工具，促使人们 重新考虑随机图模型及其方法
“图论之父”
看作4个节点，7条边的 图
路必须有起点和终点。 一次走完所有的桥，不重复，除起点与终点外，其余点必须有偶数 条边，所以七桥问题无解。 1875年, B 与 C 之间新建了一条桥解决了该问题！☺
Euler 对复杂网络的贡献
Euler 开启了数学图论，抽象为顶点与边的集 合 图论是网络研究的基础 网络结构是理解复杂世界的关键
电信网络
(Stephen G. Eick)
美国航空网
世界性的新闻组网络
(Naveen Jamal)
生物网络
人际关系网络
复杂网络概念
• • • • • • 结构复杂：节点数目巨大，网络结构呈现多种不同特征。 节点多样性：同一网络中可能有多种不同的节点。 连接多样性：节点之间的连接权重存在差异，且有可能存在方向性。 网络进化：表现在节点或连接的产生与消失。例如WWW，网页或链 接随时可能出现或断开，导致网络结构不断发生变化。 动力学复杂性：节点集可能属于非线性动力学系统，例如节点状态随 时间发生复杂变化。 多重复杂性融合：即以上多重复杂性相互影响，导致更为难以预料的 结果。例如，设计一个电力供应网络需要考虑此网络的进化过程，其 进化过程决定网络的拓扑结构。当两个节点之间频繁进行能量传输时， 他们之间的连接权重会随之增加，通过不断的学习与记忆逐步改善网 络性能。 复杂网络简而言之即呈现高度复杂性的网络。
高维复杂网络拓扑结构
高维复杂网络拓扑结构随着科技的快速发展和互联网的普及，人们对于网络的研究也越发深入。
除了研究网络中个体节点的连接和行为特征外，研究网络的整体拓扑结构也成为了一个重要的研究领域。
在这个领域中，高维复杂网络拓扑结构成为了一个备受关注的话题。
高维复杂网络拓扑结构是指网络中节点之间的连接关系在高维空间中展现出的结构特征。
传统的网络是基于二维的平面进行研究的，而高维复杂网络则突破了这个限制，可以在更高维度的空间中进行分析和建模。
在高维复杂网络拓扑结构的研究中，有几个常见的模型和算法被广泛应用。
其中，最为著名的就是小世界网络模型和无标度网络模型。
小世界网络模型是由Watts和Strogatz在20世纪90年代提出的，它通过在原有的规则网络上引入一定的随机链接来模拟现实社交网络中存在的短程连接和长程连接。
而无标度网络模型则是由Barabási和Albert在同一时期提出的，它模拟了现实网络中的幂律分布特性，即少数节点拥有大量连接而大部分节点只有少量连接的现象。
除了模型之外，还有一些分析高维复杂网络拓扑结构的算法被广泛使用。
其中，最为重要的算法之一是社区检测算法。
社区检测算法通过将网络划分为不同的社区或群组，能够揭示网络中节点之间的内部组织结构。
这对于我们了解社交网络、生物网络和信息网络等方面具有很大的意义。
在社区检测算法中，常用的方法有基于模块性的算法、谱聚类算法等。
高维复杂网络拓扑结构的研究具有很多重要的应用价值。
首先，通过了解网络的拓扑结构，可以帮助我们理解和预测网络的动态行为。
例如，在疾病传播模型中，研究网络的拓扑结构可以帮助我们了解病毒或信息在网络中的传播路径和速度。
其次，高维复杂网络拓扑结构的研究还可以帮助我们识别网络中的重要节点和关键路径，以及设计更加高效的网络优化算法。
最后，高维复杂网络拓扑结构的研究对于社交网络分析、信号处理和图像处理等领域也具有很大的推动作用。
然而，高维复杂网络拓扑结构的研究也面临一些挑战。
网络发明英文作文
网络发明英文作文英文：In my opinion, the invention of the internet is one of the greatest achievements in human history. It has revolutionized the way we communicate, learn, and even conduct business. The internet has brought people from all over the world closer together, allowing us to share knowledge, ideas, and experiences.One of the most significant benefits of the internet is its ability to connect people. Through social media platforms, we can stay in touch with friends and family regardless of distance. We can also meet new people with similar interests and form communities online. This has opened up a world of opportunities for people who might otherwise feel isolated or alone.Another advantage of the internet is its vast wealth of information. With just a few clicks, we can access anendless supply of knowledge on any topic imaginable. This has made learning more accessible than ever before. Students can take online courses, and professionals can access training and resources from anywhere in the world.The internet has also revolutionized the way we do business. E-commerce has made it possible for companies to reach a global audience and sell products and services online. This has created new opportunities for entrepreneurs and small businesses, allowing them to compete with larger companies on a level playing field.However, the internet is not without its drawbacks. One of the biggest concerns is the issue of online privacy and security. With so much personal information being shared online, there is always a risk of identity theft and other cybercrimes. It is essential to take precautions to protect ourselves and our sensitive information.Overall, I believe that the internet has had a profoundly positive impact on society. It has brought people closer together, made learning more accessible, andcreated new opportunities for businesses and individuals alike.中文：在我看来，互联网的发明是人类历史上最伟大的成就之一。
互联网发明的英语作文
互联网发明的英语作文The advent of the internet has revolutionized the world in a way that few other inventions have. It has brought about a seismic shift in the way we communicate, learn, work, and entertain ourselves. From its humble beginnings as a military project in the 1960s, the internet has exploded into a global network of interconnected computers that spans the globe, connecting billions of people and billions of devices.The internet's impact on communication is profound. It has eliminated the need for physical mail, telegrams, and even phone calls in many cases. With email, social media, and video conferencing, people can stay in touch with friends and family across the globe in real-time, sharing information, photos, and videos instantly. This has not only made communication more convenient but has also broken down barriers, allowing people from different cultures and backgrounds to connect and share their stories.The internet has also transformed the way we learn. With access to vast amounts of information online, people can now educate themselves from anywhere in the world.Online courses, tutorials, and libraries provide a wealth of knowledge that was previously unavailable to many. This has opened up new opportunities for education and has allowed people to pursue their passions and interests, regardless of their location or background.The rise of the internet has also had a profound impact on the way we work. Remote work has become increasingly common, as companies realize that their employees can be as productive working from home as they are in the office. This has provided greater flexibility for workers and has allowed them to maintain a better work-life balance. Additionally, the internet has created new industries and job opportunities, such as web development, digital marketing, and e-commerce, that have driven economic growth and innovation.In terms of entertainment, the internet has revolutionized the way we consume media. Streaming services like Netflix, YouTube, and Spotify have made it possible to access a virtually unlimited library of movies, TV shows, music, and podcasts from anywhere in the world. This has given consumers more choices and has allowed them tocustomize their entertainment experience to their preferences.Moreover, the internet has been a catalyst for social change. It has empowered individuals to share their voices and stories, sparking movements and protests that have led to positive social change. The internet has also been a force for transparency, as it has allowed people to access information that was previously concealed or difficult to obtain, such as government records and corporate reports. However, while the internet has brought many benefits, it has also presented new challenges. Issues such as cyberbullying, privacy concerns, and the spread of misinformation have become increasingly relevant in the digital era. It is important that we continue to develop responsible and ethical uses of the internet to ensure that its benefits are maximized while minimizing its negative impacts.In conclusion, the invention of the internet has been a transformative event in human history. It hasrevolutionized the way we communicate, learn, work, and entertain ourselves, connecting billions of people acrossthe globe. While it has presented new challenges, the internet's potential to bring about positive change and innovation is vast. As we continue to navigate the digital landscape, it is important that we remember the impact that the internet has had on our world and strive to use it responsibly and ethically.**互联网的革命：连接世界**互联网的诞生在人类历史上可谓是一场革命，它改变了我们沟通、学习、工作和娱乐的方式，其影响之广泛和深远，远超过其他许多发明。
关于网络发明的英语作文
关于网络发明的英语作文The Invention of the Internet: A Revolution in Communication.The invention of the internet has revolutionized the way we communicate, learn, work, and entertain ourselves.It has transformed our world into a global village, connecting people from all corners of the earth in waysthat were previously unimaginable.To understand the significance of the internet, one must delve into its humble beginnings. The precursor to the internet was the Advanced Research Projects Agency Network (ARPANET), which was established in the late 1960s by the United States government. Its initial purpose was to create a network that could survive a nuclear attack, ensuringthat military communications would remain intact. However, as the technology evolved, it became apparent that the network had far more potential than just military applications.The early internet was a mere shadow of what it is today. It was limited in terms of speed, bandwidth, and accessibility. Yet, even in its infancy, it demonstrated the potential to revolutionize communication. Email, the first widely used internet application, allowed people to send and receive messages instantaneously, regardless of their geographical location. This was a significant step forward from traditional mail, which could take days or even weeks to deliver a message.As the internet grew and evolved, new technologies and protocols were developed to enhance its capabilities. The invention of the World Wide Web, also known as theinternet's "front end," made it easier for people to access and navigate the vast amount of information available online. With the advent.。
collective dynamics of small-world networks 中文版 duncan J.Watts Steven H.Strogatz
“小世界”网络的集体动力学Duncan J. Watts ， Steven H. Strogatz美国纽约（14853）伊萨卡,康奈尔大学,Kimball学院理论与应用力学系（NATURE | VOL 393 | 4 JUNE 1998 pp.440—442）（译者 王恒山 上海理工大学管理学院、系统工程研究所 上海 200093） ....... ........... ............ ............ ............ ........... .........连结动力系统的网络已被用于为生物振荡器1-4、约瑟夫森结阵列5-6、应激介质7、神经网络8-10、时空博弈11、基因控制网络12以及许多其它自组织系统建立模型。
通常，这些模型的拓扑结构被假定为完全规则网，或完全随机网。
但许多生物网络、技术网络和社会网络却介于这两种极端的网络之间。
在这里我们探究能够直达两种极端网络中间地带的简单网络模型：即通过重新连接规则网络中结点之间的连线来增加该网络的无规则性。
我们发现这些网络能够像规则网那样具有高度的集群性，又像随机网那样具有短的特征路径长度。
由于具有类似小世界现象（即众所周知的六度分离15），我们把它们称为“小世界”网络13-14。
蠕线虫的神经网络、美国西部的电力网络以及电影演员的协作网络已被证明是小世界网络。
具有小世界连接的动力系统模型显示出能增强信号的传播速度、计算能力和同步能力。
尤其是，传染性疾病在小世界网络中传播比在规则网络中要容易得多。
为在规则网络和随机网络之间生成新的网络，我们考虑以下随机重新连线过程（图1）。
从一个有n个顶点以及每个顶点有k条边的环形网络开始，以概率p随机给每条边重新连线。
这一连线过程允许我们在规则网(p=0)和随机网(p=1) 之间生成各种网络，因此就能探查我们还知之甚少的中间区域（0<P<1）网络的特性。
我们用特征路径长度L(p)和集群系数 C(p)来量化这些网络的结构特性，如图2所示。
小世界网络综述
关于小世界网络的文献综述一，小世界在P2P网络方面的研究Small-World模型 (也称 W-S 模型 )是由 W atts和 Strogatz于 1998年在对规则网络和随机网络的研究的基础上提出的。
从本质上说 , W-S模型网络是具有一定随机性的一维规则网络。
W -S模型中定义了两个特征值:(1)特征路径的平均长度 L:它是指能使网络中各个节点相连的最少边长度的平均数 ,即小世界网络的平均距离 ;(2)聚类系数 C:表示近邻节点联系紧密程度的参数。
Scale-F ree网络 ,又称无标度网络。
这类网络中，大多数节点的连接度都不大 ,只有少数节点的连接度很高 ,可以将这些少数节点看成中心节点。
这样的节点一般连接不同的区域, 是重要节点 (或称关键节点 ), 起着簇头的作用。
它们使网络通信范围更广, 可用资源更丰富 , 查询和搜索效率更高。
Barabási和 A lbert (BA)等人研究发现节点的连接具有偏好依附的特性。
因此 ,网络规模随着新节点的加入而增大,但新加入的节点偏向于连接到已存在的具有较大连接度的节点上去。
简要介绍了Small-World模型和Scale-Free模型, 详细介绍了小世界现象在P2P网络中资源搜索以及网络安全方面可能的3个应用点, 并提出了一种基于“小世界现象”的高效的资源搜索策略———关键节点资源搜索法。
该搜索法将中央索引模型和泛洪请求模型相结合, 一方面增强了可伸缩性和容错性, 另一方面避免了消息泛滥, 使得搜索效率明显增强。
二、小世界网络概念方面的研究Watts和Strogatz开创性的提出了小世界网络并给出了WS小世界网络模型。
小世界网络的主要特征就是具有比较小的平均路径长度和比较大的聚类系数。
所谓网络的平均路径长度，是指网络中两个节点之间最短路径的平均值。
聚类系数被用来描述网络的局部特征，它表示网络中两个节点通过各自相邻节点连接在一起的可能性，以及衡量网络中是否存在相对稳定的子系统。
(完整版)小世界网络简介及及MATLAB建模
小世界网络MATLAB建模1.简介小世界网络存在于数学、物理学和社会学中，是一种数学图的模型。
在这种图中大部份的结点不与彼此邻接，但大部份结点可以通过任一其它节点经少数几步就可以产生联系。
若将一个小世界网络中的点代表一个人，而联机代表人与人之间是相互认识的，则这小世界网络可以反映陌生人通过彼此共同认识的人而起来产生联系关系的小世界现象。
在日常生活中，有时你会发现，某些你觉得与你隔得很“遥远”的人，其实与你“很近”。
小世界网络就是对这种现象的数学描述。
用数学中图论的语言来说，小世界网络就是一个由大量顶点构成的图，其中任意两点之间的平均路径长度比顶点数量小得多。
除了社会人际网络以外，小世界网络的例子在生物学、物理学、计算机科学等领域也有出现。
许多经验中的图可以用小世界网络来作为模型。
因特网、公路交通网、神经网络都呈现小世界网络的特征。
小世界网络最早是由邓肯·瓦茨（Duncan Watts）和斯蒂文·斯特罗加茨（Steven Strogatz）在1998年引进的，将高聚合系数和低平均路径长度作为特征，提出了一种新的网络模型，一般就称作瓦茨-斯特罗加茨模型（WS模型），这也是最典型的小世界网络的模型。
由于WS小世界模型构造算法中的随机化过程有可能破坏网络的连通性，纽曼(Newman)和瓦茨(Watts)提出了NW小世界网络模型，该模型是通过用“随机化加边”模式来取代WS小世界网络模型构造中的“随机化重连”。
在考虑网络特征的时候，使用两个特征来衡量网络：特征路径长度和聚合系数。
特征路径长度（characteristic path length）：在网络中，任选两个节点，连同这两个节点的最少边数，定义为这两个节点的路径长度，网络中所有节点对的路径长度的平均值，定义为网络的特征路径长度。
这是网络的全局特征。
聚合系数(clustering coefficient)：假设某个节点有k个边，则这k条边连接的节点之间最多可能存在的边的个数为k(k-1)/2，用实际存在的边数除以最多可能存在的边数得到的分数值，定义为这个节点的聚合系数。
小世界网络
图 5 WS 小世界网络的集聚系数和特征路径长度
NW 模型：Newman 在 WS 小世界模型的基础上，通过捷径额外链接建构了另一种小世界模型，称为 NW 小世界模型[15]。其构建方法为：在规则网络的基础上，按确定平均连接度个数，添加随机选择的点（随机网络），不再拆开规则网络的连接。实际上，NW 小世界模型是规则网络和随机网络的叠加（见图6）。
第二章 小世界网络的特性
Watts和Strogatz的开创性文章引发了研究小世界网络和Watts-Strogatz(WS)模型特性的热潮。通过进一步研究Newman和Watts对WS模型的作了改进，其中边被加入随机选取的两点之间，但却不从规则网中移除。这一模型比原始Watts-Strogatz模型容易分析，因为它不会形成孤立的群集，但是在原始模型中有可能发生。对于足够小的p和大N，这一模型等价于WS模型。接下来我们将总结关于小世界网络的主要特性。
首先,网络的现象涵盖极其广泛,因此,对网络的研究极具意义。其次,复杂网络的研究,在大量网络现象的基础上抽象出两种复杂网络:一种即小世界网络,另一种即无标度网络。这两种网络都同时具有两个基本特征:高平均集聚程度、小的最短路径。对这两种网络的研究，有利于人们理解现实世界中的网络现象。
科学家在研究复杂网络的过程中,通过在规则网络的基础上,断开其中某些顶点的链接,然后导入随机链接其中若干顶点的方法,结果构造出来的网络立刻就具有了小世界的特性。
对于规则环状网格和随机网络之间添加随机再连接的过程，没有改变图中的顶点数或者边数。N 个顶点的环，每个顶点通过随机的边连接到它的 k 个最近的邻居。我们选择一个顶点和它的边，以顺时针方向连接它和它最近的邻居，然后以概率 p 重新连接这条边到一个环上随机选择的顶点，不允许重复，否则我们不动这条边。我们通过沿着环顺时针移动来重复这个过程，在轮流的过程中每个点都要考虑到直到一圈结束。接着再考虑顺时针连接它们第二近邻顶点的边。沿着这个循环操作并且在每一圈以后逐步向远距离的邻居行进，直到原始网格中的每一条边都被考虑过（见图4）。
NW 模型：Newman 在 WS 小世界模型的基础上，通过捷径额外链接建构了另一种小世界模型，称为 NW 小世界模型[15]。其构建方法为：在规则网络的基础上，按确定平均连接度个数，添加随机选择的点（随机网络），不再拆开规则网络的连接。实际上，NW 小世界模型是规则网络和随机网络的叠加（见图6）。
第二章 小世界网络的特性
Watts和Strogatz的开创性文章引发了研究小世界网络和Watts-Strogatz(WS)模型特性的热潮。通过进一步研究Newman和Watts对WS模型的作了改进，其中边被加入随机选取的两点之间，但却不从规则网中移除。这一模型比原始Watts-Strogatz模型容易分析，因为它不会形成孤立的群集，但是在原始模型中有可能发生。对于足够小的p和大N，这一模型等价于WS模型。接下来我们将总结关于小世界网络的主要特性。
首先,网络的现象涵盖极其广泛,因此,对网络的研究极具意义。其次,复杂网络的研究,在大量网络现象的基础上抽象出两种复杂网络:一种即小世界网络,另一种即无标度网络。这两种网络都同时具有两个基本特征:高平均集聚程度、小的最短路径。对这两种网络的研究，有利于人们理解现实世界中的网络现象。
科学家在研究复杂网络的过程中,通过在规则网络的基础上,断开其中某些顶点的链接,然后导入随机链接其中若干顶点的方法,结果构造出来的网络立刻就具有了小世界的特性。
对于规则环状网格和随机网络之间添加随机再连接的过程，没有改变图中的顶点数或者边数。N 个顶点的环，每个顶点通过随机的边连接到它的 k 个最近的邻居。我们选择一个顶点和它的边，以顺时针方向连接它和它最近的邻居，然后以概率 p 重新连接这条边到一个环上随机选择的顶点，不允许重复，否则我们不动这条边。我们通过沿着环顺时针移动来重复这个过程，在轮流的过程中每个点都要考虑到直到一圈结束。接着再考虑顺时针连接它们第二近邻顶点的边。沿着这个循环操作并且在每一圈以后逐步向远距离的邻居行进，直到原始网格中的每一条边都被考虑过（见图4）。
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∗
that F (u∗ ) = u∗ /2, then we obtain, since ℓ(n, 0) ∼ n, that n∗ (p1/2 (n)) = n/u∗ . If n∗ (p) ∼ p−τ , this implies p1/2 (n) ∼ n−1/τ . We show in fig (2) that p1/2 (n) ∼ 1/n (and clearly not ∼ n−3/2 like the results of [1] would imply), meaning that τ = 1: a finite number of rewired links already has a strong influence on ℓ. Again the τ = 2/3 result of [1] is clearly ruled out. It is a pleasure to thank G. Biroli, R. Monasson and M. Weigt for discussions. A. Barrat Laboratoire de Physique Th´orique [*], Universit´ de e e Paris-Sud, 91405 Orsay cedex, France
arXiv:cond-mat/9903323v1 [cond-mat.stat-mech] 22 Mar 1999
Comment on “Small-world networks: Evidence for a crossover picture” In a recent letter, Barth´l´my and Nunes Amaral [1] ee examine the crossover behaviour of networks known as “small-world”. They claim that, for an initial network with n vertices and z links per vertex, each link being rewired according to the procedure of [2] with a probability p, the average distance ℓ between two vertices scales as ℓ(n, p) ∼ n F
n n∗
(1)
where F (u ≪ 1) ∼ u, and F (u ≫ 1) ∼ ln u, and n∗ n∗ ∼ p−τ with τ = 2/3 as p goes to zero. Other quantities can be of interest in small-world networks, and will be discussed in [3]. In this comment however, we concentrate like [1] on ℓ and we show, using analytical arguments and numerical simulations with larger values of n, that: (i) the proposed scaling form ℓ(n, p) ∼ n∗ F (n/n∗ ) seems to be valid, BUT (ii) the value of τ cannot be lower than 1, and therefore the value found in [1] is clearly wrong. The naive argument developed in [1] uses the mean number of rewired links, Nr = pnz/2. According to [1], one could expect that the crossover happens for Nr = O(1), which gives τ = 1 [4]. However they find τ = 2/3. Let us suppose that τ < 1. Then, if we take α such that τ < α < 1, according to eq (1), we obtain that ℓ(n, n−1/α ) ∼ nτ /α F (n1−τ /α ) ∼ nτ /α ln(n1−τ /α ) (2)
10
−2
10
−3
10
−4
100
(n)
1/2
10
−1
z=4 z=6 z=10
FIG. 2. p1/2 (n) such that ℓ(n, p1/2 (n)) = ℓ(n, 0)/2, for z = 4, 6, 10, and values of n ranging from 100 to 11000; the straight line has slope −1.
Unit´ Mixte de Recherche UMR 8627. e [1] M. Barth´l´my and L.A. Nunes Amaral, preprint condee mat/9903108, Phys. Rev. Lett. to appear (Volume 82, Number 15). [2] D.J. Watts and S.H. Strogatz, Nature 393, 440 (1998). [3] A. Barrat, M. Weigt, in preparation; R. Monasson, in preparation. [4] Note that Nr = O(1) gives a finite number of rewired links, and not a “small but finite fraction” like written in [1].
∗
since τ /α < 1 and n1−τ /α ≫ 1 for large n. However, the mean number of rewired links in this case is Nr = n1−1/α z/2, which goes to zero for large n. The immediate conclusion is that a change in the behaviour of ℓ (from ℓ ∼ n to ℓ ∼ nτ /α ln(n)) could occur by the rewiring of a vanishing number of links! This is a physical nonsense, showing that, if n∗ ∼ p−τ , τ cannot be lower than 1. We know present our numerical simulations. The value of n∗ (p) is obtained by studying, at fixed p (we take p = 2k /220 , k = 0, · · · 20), the crossover between ℓ ∼ n at small n to ℓ ∼ ln(n) at large n [1]. For small values of p, it is difficult to reach large enough values of n to accurately determine n∗ , and we think that the underestimation of n∗ given by [1] comes from this problem. We here simulate networks with z = 4, 6, 10 up to sizes n = 11000, and find that n∗ behaves like 1/p for small p (Inset of fig. (1)). We moreover show the collapse of the curves ℓ/n∗ versus n/n∗ in figure (1), for z = 4 and z = 10: note that we obtain the collapse over a much wider rabge than [1]. Besides, we present results for another quantity: at fixed n we evaluate ℓ(n, p) and look for the value p1/2 (n) of p such that ℓ(n, p1/2 (n)) = ℓ(n, 0)/2. This value of p corresponds to the rapid drop in the plot of ℓ versus p at fixed N , and can therefore also be considered as a crossover value. If we note u∗ the number such 1
8
l/n*
n*
100 10 1
−3 −2 −1
6
4
10
10
p
10
1
2
z=4 z=10
0
1
10
n/n*
100
1000
10000
FIG. 1. Data collapse of ℓ(n, p)/n∗ versus n/n∗ for z = 4 and z = 10, for various values of p and n from 100 to 5000. Inset: n∗ versus p for z = 4 (circles), 6 (squares), 10 (crosses); the straight line has slope −1.
that F (u∗ ) = u∗ /2, then we obtain, since ℓ(n, 0) ∼ n, that n∗ (p1/2 (n)) = n/u∗ . If n∗ (p) ∼ p−τ , this implies p1/2 (n) ∼ n−1/τ . We show in fig (2) that p1/2 (n) ∼ 1/n (and clearly not ∼ n−3/2 like the results of [1] would imply), meaning that τ = 1: a finite number of rewired links already has a strong influence on ℓ. Again the τ = 2/3 result of [1] is clearly ruled out. It is a pleasure to thank G. Biroli, R. Monasson and M. Weigt for discussions. A. Barrat Laboratoire de Physique Th´orique [*], Universit´ de e e Paris-Sud, 91405 Orsay cedex, France
arXiv:cond-mat/9903323v1 [cond-mat.stat-mech] 22 Mar 1999
Comment on “Small-world networks: Evidence for a crossover picture” In a recent letter, Barth´l´my and Nunes Amaral [1] ee examine the crossover behaviour of networks known as “small-world”. They claim that, for an initial network with n vertices and z links per vertex, each link being rewired according to the procedure of [2] with a probability p, the average distance ℓ between two vertices scales as ℓ(n, p) ∼ n F
n n∗
(1)
where F (u ≪ 1) ∼ u, and F (u ≫ 1) ∼ ln u, and n∗ n∗ ∼ p−τ with τ = 2/3 as p goes to zero. Other quantities can be of interest in small-world networks, and will be discussed in [3]. In this comment however, we concentrate like [1] on ℓ and we show, using analytical arguments and numerical simulations with larger values of n, that: (i) the proposed scaling form ℓ(n, p) ∼ n∗ F (n/n∗ ) seems to be valid, BUT (ii) the value of τ cannot be lower than 1, and therefore the value found in [1] is clearly wrong. The naive argument developed in [1] uses the mean number of rewired links, Nr = pnz/2. According to [1], one could expect that the crossover happens for Nr = O(1), which gives τ = 1 [4]. However they find τ = 2/3. Let us suppose that τ < 1. Then, if we take α such that τ < α < 1, according to eq (1), we obtain that ℓ(n, n−1/α ) ∼ nτ /α F (n1−τ /α ) ∼ nτ /α ln(n1−τ /α ) (2)
10
−2
10
−3
10
−4
100
(n)
1/2
10
−1
z=4 z=6 z=10
FIG. 2. p1/2 (n) such that ℓ(n, p1/2 (n)) = ℓ(n, 0)/2, for z = 4, 6, 10, and values of n ranging from 100 to 11000; the straight line has slope −1.
Unit´ Mixte de Recherche UMR 8627. e [1] M. Barth´l´my and L.A. Nunes Amaral, preprint condee mat/9903108, Phys. Rev. Lett. to appear (Volume 82, Number 15). [2] D.J. Watts and S.H. Strogatz, Nature 393, 440 (1998). [3] A. Barrat, M. Weigt, in preparation; R. Monasson, in preparation. [4] Note that Nr = O(1) gives a finite number of rewired links, and not a “small but finite fraction” like written in [1].
∗
since τ /α < 1 and n1−τ /α ≫ 1 for large n. However, the mean number of rewired links in this case is Nr = n1−1/α z/2, which goes to zero for large n. The immediate conclusion is that a change in the behaviour of ℓ (from ℓ ∼ n to ℓ ∼ nτ /α ln(n)) could occur by the rewiring of a vanishing number of links! This is a physical nonsense, showing that, if n∗ ∼ p−τ , τ cannot be lower than 1. We know present our numerical simulations. The value of n∗ (p) is obtained by studying, at fixed p (we take p = 2k /220 , k = 0, · · · 20), the crossover between ℓ ∼ n at small n to ℓ ∼ ln(n) at large n [1]. For small values of p, it is difficult to reach large enough values of n to accurately determine n∗ , and we think that the underestimation of n∗ given by [1] comes from this problem. We here simulate networks with z = 4, 6, 10 up to sizes n = 11000, and find that n∗ behaves like 1/p for small p (Inset of fig. (1)). We moreover show the collapse of the curves ℓ/n∗ versus n/n∗ in figure (1), for z = 4 and z = 10: note that we obtain the collapse over a much wider rabge than [1]. Besides, we present results for another quantity: at fixed n we evaluate ℓ(n, p) and look for the value p1/2 (n) of p such that ℓ(n, p1/2 (n)) = ℓ(n, 0)/2. This value of p corresponds to the rapid drop in the plot of ℓ versus p at fixed N , and can therefore also be considered as a crossover value. If we note u∗ the number such 1
8
l/n*
n*
100 10 1
−3 −2 −1
6
4
10
10
p
10
1
2
z=4 z=10
0
1
10
n/n*
100
1000
10000
FIG. 1. Data collapse of ℓ(n, p)/n∗ versus n/n∗ for z = 4 and z = 10, for various values of p and n from 100 to 5000. Inset: n∗ versus p for z = 4 (circles), 6 (squares), 10 (crosses); the straight line has slope −1.