In this chapter, we focus on statistics and measures that quantify a networks structure and characterize how it is organized. These measures have been central to much of network science, and a vast array of material is available to us, spanning across all scales of the network. The measures we discuss include general-purpose measures and those specialized to particular circumstances, which allow us to better get a handle on the network data. Network science has generated a dizzying array of valuable measures over the years. For example, we can measure local structures, motifs, patterns of correlations within the network, clusters and communities, hierarchy, and more. These measures are used for exploratory and confirmatory analyses, which we discussed in the previous chapter. With the measures of this chapter, we can understand the patterns in our networks, and using statistical models, we can put those patterns on a firm foundation.

Most scientists receive training in their domain of expertise but, with the possible exception of computer science, students of science receive little training in computer programming. While software engineering has brought forth sound principles for programming, training in software engineering only translates partially to scientific coding. Simply put, coding for science is not the same as coding for software. This chapter discusses best practices for writing correct, clear, and concise scientific code. We aim to ensure code is readable to others and supports data provenance, not hinders it. We also want the code to be a lasting recording of work performed, helping research reproducibility. Practices to address these concerns that we cover include clear variable names and code comments, favoring simple code, carefully documenting program dependencies and inputs, and using version control and logging. Together, these practices will enable your code to work better and more reliably for yourself and your collaborators.

This chapter introduces state-space descriptions for computational graphs (structures) representing discrete-time LTI systems. They are not only useful in theoretical analysis, but can also be used to derive alternative structures for a transfer function starting from a known structure. The chapter considers systems with possibly multiple inputs and outputs (MIMO systems); systems with a single input and a single output (SISO systems) are special cases. General expressions for the transfer matrix and impulse response matrix are derived in terms of state-space descriptions. The concept of structure minimality is discussed, and related to properties called reachability and observability. It is seen that state-space descriptions give a different perspective on system poles, in terms of the eigenvalues of the state transition matrix. The chapter also revisits IIR digital allpass filters and derives several equivalent structures for them using so-called similarity transformations on state-space descriptions. Specifically, a number of lattice structures are presented for allpass filters. As a practical example of impact, if such a structure is used to implement the second-order allpass filter in a notch filter, then the notch frequency and notch quality can be independently controlled by two separate multipliers.

Network science has exploded in popularity since the late 1990s. But it flows from a long and rich tradition of mathematical and scientific understanding of complex systems. We can no longer imagine the world without evoking networks. And network data is at the heart of it. In this chapter, we set the stage by highlighting network sciences ancestry and the exciting scientific approaches that networks have enabled, followed by a tour of the basic concepts and properties of networks.

This is a detailed chapter on digital filter design. Specific digital filters such as notch and antinotch filters, and sharp-cutoff lowpass filters such as Butterworth filters are discussed in detail. Also discussed are allpass filters and some of their applications, including the implementation of notch and antinotch filters. Computational graphs (structures) for allpass filters are presented. It is explained how continuous-time filters can be transformed into discrete time by using the bilinear transformation. A simple method for the design of linear-phase FIR filters, called the window-based method, is also presented. Examples include the Kaiser window and the Hamming window. A comparative discussion of FIR and IIR filters is given. It is demonstrated how nonlinear-phase filters can create visible phase distortion in images. Towards the end, a detailed discussion of steady-state and transient components of filter outputs is given. The dependence of transient duration on pole position is explained. The chapter concludes with a discussion of spectral factorization.

Much of the power of networks lies in their flexibility. Networks can successfully describe many different kinds of complex systems. These descriptions are useful in part because they allow us to organize data associated with the system in meaningful ways. These associated attributes and their connections to the network are often the key drivers behind new insights. For example, in a social network, these may be demographic features, such as the ages and occupations of members of a firm. In a protein interaction network, gene ontology terms may be gathered by biologists studying the human genome. We can gain insight by collecting data on those features and associating them with the network nodes or links. In this chapter, we study ways to associate data with the network elements, the nodes and links. We describe ways to gather and store these attributes, what analysis we can do using them, and the most crucial questions to ask about these attributes and their interplay with our networks.

Many tools exist to help scientists work computationally. In addition to general-purpose and domain-specific programming languages, a wide assortment of programs exist to accomplish specific tasks. We call attention to a number of tools in this chapter, focusing on good practices when using them, good practices computationally and good practices scientifically. Important computing tools for data scientists include computational notebooks, data pipelines and file transfer tools, UNIX-style operating systems, version control systems, and data backup systems. Of course, the world of computing moves fast, and tools are always coming and going, so we conclude with advice and a brief workflow to guide you through evaluating new tools.

This chapter provides an overview of matrices. Basic matrix operations are introduced first, such as addition, multiplication, transposition, and so on. Determinants and matrix inverses are then defined. The rank and Kruskal rank of matrices are defined and explained. The connection between rank, determinant, and invertibility is elaborated. Eigenvalues and eigenvectors are then reviewed. Many equivalent meanings of singularity (non-invertibility) of matrices are summarized. Unitary matrices are reviewed. Finally, linear equations are discussed. The conditions under which a solution exists and the condition for the solution to be unique are also explained and demonstrated with examples.

Scientists must be ethical and conscientious, always. Data bring with them much promise to improve our understanding of the world around us, and improve our lives within it. But there are risks as well. Scientists must understand the potential harms of their work, and follow norms and standards of conduct to mitigate those concerns. But network data are different. As we discuss in this chapter, network data are some of the most important but also most sensitive data. Before we dive into the data, we discuss the ethics of data science in general and network data in specific. The ethical issues that we face often do not have clear solutions but require thoughtful approaches and understanding complex contexts and difficult circumstances.

In this chapter, we discuss how to represent network data inside a computer, with some examples of computational tasks and the data structures that enable those computations. When working with network data using code, you have many choices of data structures---but which ones are best for our given goals? Writing your own code to process network data can be valuable, yet existing libraries, which feature extensively-tested and efficiently-engineered functionalities, are worth considering as well. Python and R, both excellent programming languages for data science, come well-equipped with third-party libraries for working with network data, and we describe some examples. We also discuss choosing and using typical file formats for storing network data, as many standard formats exist.

Network data, like all data, are imperfect measures of objects of study. There may be missing information or false information. For networks, these measurement errors can lead to missing nodes or links (network elements that exist in reality but are absent from the network data) or spurious nodes or links (nodes or links present in the data but absent in reality). More troubling is that these conditions exist in a continuum, and there is a spectrum of scenarios where nodes or links may exist but not be meaningful in some way. In this chapter, we describe how such errors can appear and affect network data and introduce some ways to handle such errors in the data processing steps. Fixes for errors can lead to different networks, before and after processing, for example, and we must be careful and circumspect in identifying and planning for such errors.

What are the nodes? What are the links? These questions are not the start of your work—the upstream task makes sure of that—but they are an inflection point. Keep them front of mind. Your methods, the paths you take to analyze and interrogate your data, all unfold from the answers (plural!) to these questions. This chapter reflects on where we have gone, where we can go for more, and, perhaps, what the future has in store for data science, networks and network data.

Machine learning has revolutionized many fields, including science, healthcare, and business. It is also widely used in network data analysis. This chapter provides an overview of machine learning methods and how they can be applied to network data. Machine learning can be used to clean, process, and analyze network data, as well as make predictions about networks and network attributes. Methods that transform networks into meaningful representations are especially useful for specific network prediction tasks, such as classifying nodes and predicting links. The challenges of using machine learning with network data include recognizing data leakage and detecting dataset shift. As with all machine learning, effective use of machine learning on networks depends on practicing good data hygiene when evaluating a predictive model’s performance.

Some networks, many in fact, vary with time. They may grow in size, gaining nodes and links. Or they may shrink, losing links and becoming sparser over time. Sitting behind many networks are drivers that change the structure, predictably or not, leading to dynamic networks that exhibit all manner of changes. This chapter focuses on describing and quantifying such dynamic networks, recognizing the challenges that dynamics bring, and finding ways to address those challenges. We show how to represent dynamic networks in different ways, how to devise null models for dynamic networks, and how to compare and contrast dynamical processes running on top of the network against a network structure that is itself dynamic. Dynamic network data also brings practical issues, and we discuss working with date and time data and file formats.