R语言统计分析网络数据实战指南

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《Springer Statistical Analysis of Network Data with R》(使用R)是由Eric D. Kolaczyk和Gábor Csárdi合著的一本专著，属于Springer的UseR!系列之一，该系列旨在介绍如何使用R语言进行特定领域的数据分析。这本书出版于2014年，针对的是网络数据的统计分析，是对于那些想要深入理解和应用R在处理复杂网络结构、理解网络属性及其相关分析方法的读者来说，一本非常有价值的参考资料。作者们在书中结合丰富的实证案例，讲解了如何利用R的强大功能来探索网络数据的特性，包括节点间的连接模式、网络的度分布、社区结构、路径分析、中心性测量以及网络动力学等方面的内容。这不仅包括基本的数据预处理和可视化，还涵盖了网络统计模型的建立，如随机图模型、度关联性和复杂网络的度分布分析等。书中特别强调了R语言的优势，因为它不仅是一个广泛使用的开源工具，而且提供了大量的包（如igraph、networkD3等）来支持网络数据的处理。通过阅读这本书，读者不仅可以学习到R的基本语法和网络分析函数，还能了解到如何利用R进行高级分析，如混合模型、空间网络分析和动态网络建模。与本书一同提及的其他UseR!系列书籍，如Albert的《Bayesian Computation with R》、Bivand/Pebesma/Gómez-Rubio的《Applied Spatial Data Analysis with R》等，展示了R在不同领域数据分析中的广泛应用。这些书共同构成了一个丰富的学习资源库，为R用户提供了多样化的学习路径。《Statistical Analysis of Network Data with R》是一本实用的指南，适合对网络数据分析感兴趣的R用户，无论是初级还是进阶学习者，都能从中收获关于网络数据处理和分析的重要技能和理论知识。无论是研究者、数据科学家还是工程师，这本书都是扩展其在R环境下网络分析能力的宝贵资源。

2 1 Introduction

number of key areas. For example, in mathematics the formal underpinnings were

systematically laid, with K

onig [92] cited as the ﬁrst key architect. The theory of

electrical circuits has always had a substantial network component, going back to

work of Kirchoff, and similarly the study of molecular structure in chemistry, going

back to Cayley. As the ﬁelds of operations research and computer science grew

during the mid-1900s, networks were incorporated in a major fashion in problems

involving transportation, allocation, and the like. And similarly during that time

period, a small subset of sociologists, taking a particularly quantitative view towards

the topic of social structure, began developing the use of networks in characterizing

interactions within social groups.

More recently—starting, perhaps, in the early to mid-1990s—there has been an

explosion of interest in networks and network-based approaches to modeling and

analysis of complex systems. Much of the impetus for this growth derives from

work by researchers in two particular areas of science: statistical physics and com-

puter science. To the former can be attributed a seminal role in encouraging what

has now become a pervasive emphasis across the sciences on understanding how

the interacting behaviors of constituent parts of a whole system lead to collective

behavior and systems-level properties or outcomes. Indeed the term complex system

was coined by statistical physicists, and a network-based perspective has become

central to the analysis of complex systems. To the latter can be attributed much of

the theory and methodology for conceptualizing, storing, manipulating, and doing

computations with networks and related data, particularly in ways that enable efﬁ-

cient handling of the often massive quantities of such data. Moreover, information

networks (e.g, the World Wide Web) and related social media applications (e.g.,

Twitter), the development of which computer scientists have played a key role, are

examples of some of the most studied of complex systems (arguably reﬂecting our

continued fascination with studying ourselves!).

More broadly, a network-based perspective recently has been found to be useful

in the study of complex systems across a diverse range of application areas. These

areas include computational biology (e.g., studying systems of interacting genes,

proteins, chemical compounds, or organisms), engineering (e.g., establishing how

best to design and deploy a network of sensing devices), ﬁnance (e.g., studying the

interplay among, say, the world’s banks as part of the global economy), marketing

(e.g., assessing the extent to which product adoption can be induced as a type of

‘contagion’), neuroscience (e.g., exploring patterns of voltage dynamics in the brain

associated with epileptic seizures), political science (e.g., studying how voting pref-

erences in a group evolve in the face of various internal and external forces), and

public health (e.g., studying the spread of infectious disease in a population, and

how best to control that spread).

In general, two important contributing factors to the phenomenal growth of

interest in networks are (i) an increasing tendency towards a systems-level perspec-

tive in the sciences, away from the reductionism that characterized much of the

previous century, and (ii) an accompanying facility for high-throughput data collec-

tion, storage, and management. The quintessential example is perhaps that of the

changes in biology over the past 10–20years, during which the complete mapping

1.2 Types of Network Analysis 3

of the human genome, a triumph of computational biology in and of itself, has now

paved the way for ﬁelds like systems biology to be pursued aggressively, wherein a

detailed understanding is sought of how the components of the human body, at the

genetic level and higher, work together.

Ultimately, the study of systems such as those just described is accompanied by

measurement and, accordingly, the need for statistical analysis. The focus of this

book is on how to use tools in R to do statistical analysis of network data. More

speciﬁcally, we aim to present tools for performing what are arguably a core set of

analyses of measurements that are either of or from a system conceptualized as a

network.

1.2 Types of Network Analysis

Network data are collected daily in a host of different areas. Each area, naturally,

has its own unique questions and problems under study. Nevertheless, from a sta-

tistical perspective, there is a methodological foundation that has emerged, com-

posed of tasks and tools that are each common to some non-trivial subset of research

areas involved with network science. Furthermore, it is possible—and indeed quite

useful—to categorize many of the various tasks faced in the analysis of network

data across different domains according to a statistical taxonomy. It is along the

lines of such a taxonomy that this book is organized, progressing from descriptive

methods to methods of modeling and inference, with the latter conveniently sepa-

rated into two sub-areas, corresponding to the modeling and inference of networks

themselves versus processes on networks. We illustrate with some examples.

1.2.1 Visualizing and Characterizing Networks

Descriptive analysis of data (i.e., as opposed to statistical modeling and inference)

typically is one of the ﬁrst topics encountered in a standard introductory course in

statistics. Similarly, the visualization and numerical characterization of a network

usually is one of the ﬁrst steps in network analysis. Indeed, in practice, descriptive

analyses arguably constitute the majority of network analyses published.

Consider the network in Fig. 1.1. Shown is a visualization

of part of a dataset

on collaborative working relationships among members of a New England law ﬁrm,

collected by Lazega [98]. These data were collected for the purpose of studying

cooperation among social actors in an organization, through the exchange of various

types of resources among them. The organization observed was a law ﬁrm, consist-

ing of over 70 lawyers (roughly half partners and the other half associates) in three

ofﬁces located in three different cities. Relational data reﬂecting resource exchange

The R code for generating this visualization is provided in Chap. 3.

1.2 Types of Network Analysis 5

afﬁnity for binding or of genes and their regulatory relationships. As a result, the

visualization of large networks is a separate challenge of its own. We will look at

tools for network visualization in Chap. 3.

Characterization of network data through numerical summaries is another

important aspect of descriptive analysis for networks. However, unlike in an int-

roductory statistics course, where the types of summaries encountered typically

are just simple measures of center (e.g., mean, median, etc.) and dispersion (e.g.,

standard deviation, range, etc.) for a set of real numbers, summary measures for

networks necessarily seek to capture characteristics of a graph. For example, for

thelawyerdatainFig.1.1, it is natural to ask to what extent two lawyers that

both work with a third lawyer are likely to work with each other as well. This

notion corresponds to the social network concept of transitivity and can be captured

numerically through an enumeration of the proportion of vertex triples that form

triangles (i.e., all three vertex pairs are connected by edges), typically summarized

in a so-called clustering coefﬁcient. In this case, the relevant characterization is an

explicit summary of network structure. On the other hand, given the two types of

lawyers represented in these data (i.e., corporate and litigation), it is also natural

to ask to what extent lawyers of each type collaborate with those of same versus

different types. This notion corresponds to another social network concept—that of

assortativity—and can be quantiﬁed by a type of correlation statistic (the so-called

assortativity coefﬁcient), in which labels of connected pairs of vertices are com-

pared. In this case, the focus is on an attribute associated with network vertices (i.e.,

lawyer practice) and the network structure plays a comparatively more implicit role.

There are a host of network summaries available, with more being deﬁned reg-

ularly. A variety of such measures are covered in Chap. 4. These range from char-

acterization of properties associated with individual vertices or edges to properties

of subgraphs to properties of the graph as a whole. The trick to employing net-

work characterizations in a useful fashion generally is in matching the question(s) of

interest in an underlying complex system with an appropriate summary measure(s)

of the corresponding network.

1.2.2 Network Modeling and Inference

Beyond asking what an observed network looks like and characterizing its structure,

at a more fundamental level we may be interested in understanding how it may have

arisen. That is, we can conceive of the network as having resulted from some under-

lying processes associated with the complex system of interest to us and ask what

are the essential aspects of these processes. In addition, the actual manner in which

the network was obtained, i.e., the corresponding measurement and construction

process, may well be important to take into consideration. Such concerns provide

the impetus for network modeling and associated tools of statistical inference.

Network modeling has received much attention. Broadly speaking, there are

two classes of network models: mathematical and statistical. By ‘mathematical

6 1 Introduction

models’ we mean models speciﬁed through (typically) simple probabilistic rules

for generating a network, where often the rules are deﬁned in an attempt to capture

a particular mechanism or principle (e.g., ‘the rich get richer’). In contrast, by ‘sta-

tistical models’ we mean models (often probabilistic as well) speciﬁed at least in

part with the intention that they be ﬁt to observed data—allowing, for example, the

evaluation of the explanatory power of certain variables on edge formation in the

network—and that the ﬁt be effected and assessed using formal principles of statis-

tical inference. While certainly there is some overlap between these two classes of

models, the relevant literatures nevertheless are largely distinct.

The simplest example of a mathematical network model is one in which edges

are assigned randomly to pairs of vertices based on the result of a collection of

independent and identically distributed coin tosses—one toss for each vertex pair.

Corresponding to a variant of the famous Erd

os–R

enyi formulation of a random

graph, this model has been studied extensively since the 1960s. Its strength is in

the fact not only that its properties are so well understood (e.g., in terms of how,

for example, cohesive structure emerges as a function of the probability of an edge)

but also in the role it plays as a standard against which to compare other, more

complicated models.

From the statistical perspective, however, such mathematical models generally

are too simple to be a good match to real network data. Nevertheless, they are not

only useful in allowing for formal insight to be gained into how speciﬁc mechanisms

of edge formation may affect network structure, but they also are used commonly

in deﬁning null classes of networks against which to assess the ‘signiﬁcance’ of

structural characteristics found in an observed network. For example, in Fig. 1.1,

the clustering coefﬁcient for the network of lawyer collaboration turns out to be just

slightly less than 0.40. Is this value large or not? It is difﬁcult to say without a frame

of reference. One way of imposing such a frame of reference is to deﬁne a simple

but comparable class of networks (e.g., all graphs with the same numbers of vertices

and edges) and look at the distribution of the clustering coefﬁcient across all such

graphs. In the spirit of a permutation test and similar data re-use methods, examining

where our observed value of 0.40 falls within this distribution gives us a sense of

how unusual it is in comparison to values for networks across the speciﬁed class.

This approach has been used not only for examining clustering in social networks

but also, for example, in identifying recurring sub-graph patterns (aka ‘motifs’) in

gene regulatory networks and functional connectivity networks in neuroscience. We

will explore the use of mathematical network models for such purposes in Chap. 5.

There are a variety of statistical network models that have been offered in the

literature, many of which parallel well-known model classes in classical statis-

tics. For example, exponential random graph models are analogous to generalized

linear models, being based on an exponential family form. Similarly, latent net-

work models, in specifying that edges may arise at least in part from an unmeasured

(and possibly unknown) variable(s), directly parallel the use of latent variables in

hierarchical modeling. Finally, stochastic block models may be viewed as a form of

mixture model. Nevertheless, importantly, the speciﬁcation of such models and their

ﬁtting typically are decidedly less standard, given the usually high-dimensional and

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