When a network is too large to study completely, we sample from that network just as we would sample from any large population. The structure of network data, however, is more complicated than that of standard statistical data. The main question is, how can one sample from a network that has nodes and edges? Should we sample the nodes? Or should we sample the edges? The answers to these questions depend upon the complexity of the network. In this chapter, we examine various methods of sampling a network.

Do you think these women are working? One or both of them? What is your assessment based on? If you are like many people, you may be quicker to perceive the women on the right looking at a computer monitor as working than the woman pictured on the left who is feeding her children breakfast. Despite differences in how these two forms of labor are perceived, both are, in fact, work.

Random graphs were introduced by the Hungarian mathematicians Erdős and Rényi (, ), who imposed a probabilistic framework on classical combinatorial graph theory. At the same time, Edgar N. Gilbert () also studied the theoretical properties of random graphs.

Introduction

In the previous chapter, we learnt that pollination results in transferring of pollen grains from the anther to the stigma of a flower. Such a transfer or landing of pollen grains initiates a series of events that involve a continuous exchange of signals between the haploid pollen and the diploid maternal tissue of the pistil. These events and interactions which include pollen selection or rejection, pollen hydration, pollen tube growth, its nourishment and entry into ovules and embryo sac are recognized as the pollen–pistil interactions, (Herrero & Hormaza 1996; Shivanna 2003; Lora et al. 2016). The pollen–pistil interactions result in screening and selection of conspecific/homospecific pollen (of same species) from the heterospecific pollen (pollen of other species) ensuring fertilization only between the conspecific male and female gametes.

In a pistil, while stigma is the landing platform and recipient of the pollen, the style is the conduit for the transfer of non-motile male gametes to the embryo sac seated in the ovules with the help of pollen tubes. The process of delivery of non-motile sperm to the egg via a pollen tube is known as Siphonogamy. It is regarded as a key innovation in the course of evolution of angiosperms that has allowed flowering plants to carry out sexual reproduction on land without the need for water. Flowering plants have an elaborate screening process for selecting the right pollen and the pollen tubes. This system works at different levels in the pistil. The selection of pollen starts at the stigma itself which allows only the compatible pollen grains to germinate while the incompatible ones are rejected. The selected pollen grains then germinate and put forth pollen tubes which grow in the style, where again a competition takes place to select the best mate. The pollen tubes travel at a rate specific to each species to ultimately reach the ovule. A study by Williams (2008) covering about 130 seed plant families and 717 taxa suggested that the time interval between pollination and fertilization ranges between 15 minutes to >12 months in angiosperms. This time interval is referred to as the fertilization interval.

This chapter is not designed for a detailed study of computer architecture. Rather, it is a cursory review of concepts that are useful for understanding the performance issues in parallel programs. Readers may well need to refer to a more detailed treatise on architecture to delve deeper into some of the concepts.

There are two distinct facets of parallel architecture: the structure of the processors, that is, the hardware architecture, and the structure of the programs, that is, the software architecture. The hardware architecture has three major components:

• 1. Computation engine: it carries out program instructions.

• 2. Memory system: it provides ways to store values and recall them later.

• 3. Network: it forms the connections among processors and memory.

An understanding of the organization of each architecture and their interaction with each other is important to write efficient parallel programs. This chapter is an introduction to this topic. Some of these hardware architecture details can be hidden from application programs by well-designed programming frameworks and compilers. Nonetheless, a better understanding of these generally leads to more efficient programs. One must similarly understand the components of the program along with the programming environment. In other words, a programmer must ask:

• 1. How do the multiple processing units operate and interact with each other?

• 2. How is the program organized so it can start and control all processing units? How is it split into cooperating parts and how do parts merge? How do parts cooperate with other parts (or programs)?

One way to view the organization of hardware as well as software is as graphs (see Sections 1.6 and 2.3). Vertices in these graphs represent processors or program components, and edges represent network connection or program communication. Often, implementation simplicity, higher performance, and cost-effectiveness can be achieved with restrictions on the structure of these graphs. The hardware and software architectures are, in principle, independent of each other. In practice, however, certain software organizations are more suited to certain hardware organizations. We will discuss these graphs and their relationship starting in section 2.3.

Another way to categorize the hardware organization was proposed by Flynn and is based on the relationship between the instructions different processors execute at a time. This is popularly known as Flynn’s taxonomy.

SISD: Single Instruction, Single Data

A processor executes program instructions, operating on some input to produce some output. An SISD processor is a serial processor.

Introduction

Angiosperms possess a vast diversity of flowers which serve various purposes for the different groups of living beings, including humans. Due to their color, fragrance, and beauty flowers have always occupied a special place in human lives. Flowers are considered sacred across most cultures and have inspired much artistic expression. Apart from their aesthetic value, flowers possess myriad medicinal properties that further enhance their value to humans. Describing from a botanist's perspective though, the flower is a unit of reproduction in angiosperms. A flower may be defined as a modified determinate shoot system with four distinct whorls, viz. calyx, corolla, androecium and gynoecium arranged on a receptacle. Outer whorls, calyx and corolla are leaf-like structures which are not directly involved in reproduction. The two inner whorls, the androecium and the gynoecium harbor the reproductive organs of the flower and are the ones involved in reproduction. Flowering plants exhibit enormous diversity in size, shape, color, symmetry and the other morphological features (Fig. 2.1). This diversity in floral forms plays a huge role in ensuring pollinator services by different groups of pollinators. The diverse forms of flowers are accompanied by an array of mating strategies and sexual systems in angiosperms.

The timing of flowering in plants is critical for their reproductive success as both late and premature flowering can limit proper seed development. Plants also attempt to realize their reproductive potential by synchronizing their flowering to match pollinator availability. Floral induction is promoted by distinct environmental cues such as photo-period, vernalization and endogenous regulators like phytohormones. These signalling cues are perceived in the leaves and the shoot apical meristem (SAM) for induction of flowering. Plants use genetic machinery to control all events starting from induction of flower to development of different whorls. Research in the last few decades has identified numerous genes which are involved in floral induction, floral meristem formation, and floral organ development. Genes which control floral organ development are called floral organ identity genes. These genes belong to the MADS box gene family and are also known as homeotic genes. The functioning of these genes is explained by the ABCDE model of flower development. This chapter gives an outline of the organization of a flower, sexual system seen in angiosperms and a summary of the components that play important role in the floral induction and floral organ development.

Introduction

Sexual reproduction involves delivery of sperm cells, via the pollen tube, to the egg cell present in the embryo sac, where fertilization occurs and the new sporophyte is formed (Dumas & Mogensen 1993). While the formation of male gametophyte (pollen grains) takes place within the anther, the female gametophyte (embryo sac or megagametophyte) develops within the ovule. Thus, the ovule can be defined as a specialized sporophytic structure within which development of female gametophyte or mega-gametophyte takes place. Ovule is the site for delimitation of megasporocyte, production of a functional megaspore (megasporogenesis) and eventually formation of embryo sac (megagametogenesis). The embryo sac harbors the female gamete or the egg, which subsequently gets fertilized by the male gamete to form an embryo. In angiosperms, apart from the female gametophyte and egg cell development, important reproductive events such as pollen tube attraction and guidance, double fertilization, and embryo and endosperm development all occur within the ovule. The ultimate result of all these events is the formation of seed and therefore, the ovule is also considered the developmental precursor or progenitor of the seed.

Among angiosperms, different modes of female gametophyte ontogeny are seen, leading to different types of female gametophytes. The cells of female gametophyte are very peculiar in their ultrastructure and with the help of electron microscopy, great details of these cells are known. In various species, besides the typical parts of ovule, there are many specialized structures associated with the ovules which aid in pollen tube guidance and facilitate fertilization. All these aspects of structure and development of the angiosperm ovule, female gametophyte and their types have been discussed in the present chapter. The chapter also includes exceptions to these developmental patterns and details of extra ovular structures.

Basic Structure of Ovule

In general, ovules among angiosperms are fundamentally similar in their basic structure, consisting of three major tissues: a nucellus, protective coat(s) or integument(s), and a funiculus. Besides these, a typical ovule also consists of a micropyle, a chalaza and its vascular supply (Fig. 5.1). In angiosperms, the ovules remain enclosed in the ovary, and a stalk like structure through which ovules remain attached to the ovary wall or placenta is known as the funiculus.

Much of what is studied in network analysis derives from the observation that nodes in a graph $\mathcal{G}$ often tend to form “cohesive groups” or, as they are called in social networks, communities, clusters, or modules. This chapter describes statistical methods for identifying such communities using the stochastic blockmodel and an approach based upon the concept of modularity.

Before embarking on the formal study of partialdifferential equations (PDEs), it is essential toequip ourselves with background materials frommultivariable calculus, geometry of curves andsurfaces, and ordinary differential equations (ODEs;including total and simultaneous total differentialequations). Selected ideas from earlier mentionedcourses play an important role in the study of PDEs.For the sake of completeness and to make our textself-contained, we briefly discuss the preliminariesas indicated earlier, which are needed in thesubsequent chapters. The approach adopted in thischapter is somewhat different from the one used inthe rest of the chapters. This chapter isdescriptive in nature, wherein the utilisedarguments are intended towards plausibility andunderstanding rather than adopting traditionalrigorous ways. The preliminaries are divided intofive sections.

1.1 Partial Derivatives and AlliedTopics

PDEs involve at least two independent variables.Consequently, the tools of the calculus of severalvariables are instrumentals in PDEs. In thefollowing lines, we recall some relevant notions andterminologies from differential calculus of two andthree variables.

Vectors: An element of Euclidean spaceℝn is ann-tuple of the formx =(x1, x2,…,xn), which is called ann-vector or simply,a vector. The real numbers x1,x2,…,xn are called components ofthis element. The sum and scalar multiplication inℝn aredefined as component-wise sum and component-wisescalar multiplication. The dot product of twoelements x = (x1, x2,…,xn) and y =(y1,y2,…,yn) isdefined as

The norm or length of a vectorx = (x1, x2,…,xn) is defined as

Thus, A vector of length 1 is called a unit vector.

Standard Basis Vectors: The two vectorse1 = (1,0) and e2 =(0, 1) are called standardbasis vectors of ℝ2. Anyvector u = (a, b) ∈ ℝ2 can bewritten as a linear combination of e1 and e2, so that

Similarly, e1= (1, 0, 0), e2 = (0, 1, 0), and e3 = (0, 0, 1)are called standard basisvectors of ℝ3.

In this and the next three chapters, we study the problem of graph partitioning, which deals with the question of how to divide up the nodes into homogeneous groups (if they exist). This topic is currently receiving the greater part of research activity within the network science community. In this chapter, we describe various methods for graph partitioning using graph cuts, including binary cuts, ratio cuts, normalized cuts, and multiway cuts.

In this book, we develop the theory and methodology for modeling networks. Before we do that, however, we will find it useful to describe examples of the types of real-world networks that exist today. We will be referring to these networks throughout the book. Indeed, there are many different kinds of networks, some small and some big, some simple and some, by the nature of their underlying structure, very complicated.

The inception of interest in plant reproduction is as old as the inception of interest in biology. The science of sexual plant reproduction is more than 400 years old. During all these years there has been an accumulation of information which has greatly enriched our understanding of plant reproduction. Our current knowledge of plant reproduction is a result of continuous efforts of scientists world-wide that transformed it from an observational investigation to an important field of experimental science. This chapter provides a summary of important mile stones in the history of reproductive biology of flowering plants. To maintain conciseness, only significant contributors across the world including India have been mentioned in the chapter without undermining the importance of others whose ideas and concepts have shaped the science of plant reproduction today.

Early Discoveries

The long and venerable history of studies in plant reproduction dates back to seventeenth century with the ideas of European naturalists Rudolf Jacob Camerarius in Germany and Nehemiah Grew in England. Though, Grew first proposed the idea of sexual processes occurring in plants for generation of seeds, to Camerarius must go the principal credit for experimentally establishing the existence of plant sexuality. N. Grew, in an address to the Royal Society of London in 1676, had expressed the view that the stamens are the male organs of a flower and the pollen act as vegetable sperm. He is credited for documenting stamens as the male sex organ of plants in his book The Anatomy of Plants (1682). The experiments of Camemarius were primarily based on the removal of stamens and styles along with isolation of female plants in dioecious species. He provided evidence for the inevitability of both sex organs in seed formation. His publications On the Sex of Plants (1694) and Botanical Works (1697); are landmarks in the history of botany.

In the history of plant reproduction, Adam Zalužanský is a little-known botanist. He was a professor at the University of Prague and his book Methodi herbariae libri tres was published in 1592 and 1604. This book includes a chapter De sexu plantarum in which, almost a century before the work of Camerarius, sexuality of plants was suggested.