There is great value in being able to provide a brief insightful summary of an elaborate experiment, a complicated theory or a multi-faceted discussion. Suppose one of us gets on an elevator and the only other passenger is a four-star general. She smiles and asks “How is your research going?” This is one of those rare opportunities when a scientist can actually influence policy, resource allocation and the direction of science by being able to communicate effectively. Almost invariably the opportunity is lost because scientists and generals rarely speak the same language. However, there is a trick that one can employ to anticipate this rare event and that is to have something worked out in advance so as not to waste the opportunity. This is the “elevator description” of the most significant research that has been done. In another age it might have been called the “Reader's Digest” version.

That is the position the authors find themselves in now. We believe that we ought to provide a brief but insightful summary of the book you have just completed in a way that conveys the maximum amount of information, but without the mathematics that was necessary to make that information understandable when it was first presented. To accomplish this goal we review the high points of each of the chapters and then attempt to tie them all together into a coherent picture.

In the first chapter we argued that normal statistics do not describe complex webs. Phenomena described by normal statistics can be discussed in terms of averages. The fact that human populations are well represented by normal distributions of heights strongly influences the manufacturing of everything from the size of shoes, shirts and slacks to cars, computers and couches. The relative numbers of shirts in the small, medium and large sizes are determined by our knowledge of the average build of individuals in the population. The distribution of sizes in the manufactured shirts must match the population of buyers or the shirt factory will soon be out of business. Thus, the industrial revolution was poised to embrace the world according to Gauss and the mechanistic society of the last two centuries flourished. But as the connectivity of the various webs within society became more complex the normal distribution receded further and further into the background until it was completely gone from the data, if not from our attempted understanding.

The Italian engineer turned social scientist Vilfredo Pareto was the first investigator to determine that the income in western society followed a law that was fundamentally unfair. He was not making a value judgement about the poor and uneducated or about the rich and pampered; rather, he was interpreting the empirical finding that in 1894 the distribution of income in western societies was not “normal,” but instead the number of people with a given income decreased as a power of the level of income. On bi-logarithmic graph paper this income distribution graphs as a straight-line segment of negative slope and is called an inverse power law. He interpreted his findings as meaning that a stable society has an intrinsic imbalance resulting from its complex nature, with the wealthy having a disproportionate fraction of the available wealth. Since then staggeringly many phenomena from biology, botany, economics, medicine, physics, physiology, psychology, in short every traditional discipline, have been found to involve complex phenomena that manifest inverse power-law behavior. These empirical laws were explained in the last half of the twentieth century as resulting from the complexity of the underlying phenomena.

As the twentieth century closed and the twenty-first century opened, a new understanding of the empirical inverse power laws emerged. This new understanding was based on the connectedness of the elements within the underlying phenomena and the supporting web structure. The idea of networks became pervasive as attention was drawn to society‘s reliance on sewers and the electric grid, cell phones and the Internet, banks and global stock markets, roads and rail lines, and the multitude of other human-engineered webbings that interconnect and support society. In parallel with the studies of social phenomena came new insight into the distribution in size and frequency of earthquakes and volcanic eruptions, global temperature anomalies and solar flares, river tributaries and a variety of other natural phenomena that have eluded exact description by the physical sciences. Moreover, the inverse power laws cropped up in unexpected places such as in heart rates, stride intervals and breathing, letter writing and emails, cities and wars, heart attacks and strokes; the inverse power law is apparently ubiquitous.

In this chapter we explore web dynamics using a master equation in which the rate of change of probability is determined by the probability flux into and out of the state of interest. The master equation captures the interactions among large numbers of elements each with the same internal dynamics; in this case the internal dynamics consist of the switching of a node between two states. A two-state node may be viewed as the possible choices of an individual, say whether or not that person will vote for a particular candidate in an election. This is one of the simplest dynamical webs which has been shown mathematically to result in synchronization under certain well-defined conditions. We focus on the intermittent fluctuations emerging from a phase-transition process that achieves synchronized behavior for the strength of the interaction exceeding a critical value. This model provides a first step towards proving that these intermittent fluctuations, rather than being a nuisance, are important channels of information transmission allowing communication within and between different complex webs. The crucial power-law index μ discussed earlier and there inserted for mathematical convenience is here determined by the web dynamics. This observation on the inverse power-law index leads us to define the network efficiency in a form that might not coincide with earlier definitions proposed through the observation of the network topology.

Both the discrete and the continuous master equation are discussed for modeling web dynamics. One of the most important aspects of the analysis concerns the perturbation of one complex network by another and the transfer of information between complex clusters. A cluster is a network with a uniformity of opinion due to a phase transition to a given state. This modeling strategy is not new, but in fact dates back to Yule [71], who used the master-equation approach, before the approach had been introduced, to obtain an inverse power-law distribution. We investigate whether the cluster’s opinion is robust or whether it can be easily changed by perturbing the way members of the cluster interact with one another.

Complex webs are not all complex in the same way; how the distribution in the number of connections on the Internet is formed by the conscious choices of individuals must be very different in detail from the new connections made by the neurons in the brain during learning, which in turn is very different mechanistically from biological evolution. Therefore different complex webs can be categorized differently, each representation emphasizing a different aspect of complexity. On the other hand, a specific complex web may be categorized in a variety of ways, again depending on what aspect of entanglement is being emphasized. For example, earthquakes are classified according to the magnitude of displacement they produce, in terms of the Richter scale, giving rise to an inverse power-law distribution in a continuous variable measuring the size of local displacement, the Gutenberg—Richter law. Another way to categorize earthquake data is in terms of whether quakes of a given magnitude occur within a given interval of time, the Omori law, giving rise to an inverse power-law distribution in time. The latter perspective, although also continuous, yields the statistics of the occurrence of individual events. Consequently, we have probability densities of continuous variables and those of discrete events, and which representation is selected depends on the purpose of the investigator. An insurance adjustor might be interested in whether an earthquake of magnitude 8.0 is likely to destroy a given building at a particular location while a policy is still in effect.

In the previous chapter we examined examples of hyperbolic and inverse power-law distributions dating back to the beginning of the last century, addressing biodiversity, urban growth and scientific productivity. There was no discussion of physical structure in these examples because they concerned the counting of various quantities such as species, people, cities, words, and scientific publications. It is also interesting to examine the structure of complex physical phenomena, such as the familiar irregularity in the lightning flashes shown in Figure 1.13. The branches of the lightning bolt persist for fractions of a second and then blink out of existence. The impression left is verified in photographs, where the zigzag pattern of the electrical discharge is captured. The time scale for the formation of the individual zigs and zags is on the order of milliseconds and the space scales can be hundreds of meters. So let us turn our attention to webs having time scales of milliseconds, years or even centuries and spatial scales from millimeters to kilometers.

All things happen in space and time, and phenomena localized in space and time are called events. Publish a paper. Run a red light. It rains. The first two identify an occurrence at a specific point in time with an implicit location in space; the third implies a phenomenon extended in time over a confined location in space. But events are mental constructs that we use to delineate ongoing processes so that not everything happens at once. Publishing a paper is the end result of a fairly long process involving getting an idea about a possible research topic, doing the research, knowing when to gather results together into a paper, writing the paper, sending the manuscript to the appropriate journal, reading and responding to the referees’ criticism of the paper, and eventually having the paper accepted for publication.

In the late nineteenth century, it was believed that a continuous function such as those describing physical processes must have a derivative “almost everywhere.” At the same time some mathematicians wondered whether there existed functions that were everywhere continuous, but which did not have a derivative at any point (continuous everywhere but differentiable nowhere). Perhaps you remember our discussion of such strange things from the first chapter. The motivation for considering such pathological functions was initiated by curiosity within mathematics, not in the physical or biological sciences where one might have expected it. In 1872, Karl Weierstrass (1815–1897) gave a lecture to the Berlin Academy in which he presented functions that had the remarkable properties of continuity and non-differentiability. Twenty-six years later, Ludwig Boltzmann (1844–1906), who connected the macroscopic concept of entropy with microscopic dynamics, pointed out that physicists could have invented such functions in order to treat collisions among molecules in gases and fluids. Boltzmann had a great deal of experience thinking about such things as discontinuous changes of particle velocities that occur in kinetic theory and in wondering about their proper mathematical representation. He had spent many years trying to develop a microscopic theory of gases and he was successful in developing such a theory, only to have his colleagues reject his contributions. Although kinetic theory led to acceptable results (and provided a suitable microscopic definition of entropy), it was based on the time-reversible dynamical equations of Newton.

In this chapter we investigate one procedure for describing the dynamics of complex webs when the differential equations of ordinary dynamics are no longer adequate, that is, the webs are fractal. We described some of the essential features of fractal functions earlier, starting from the simple dynamical processes described by functions that are fractal, such as the Weierstrass function, which are continuous everywhere but are nowhere differentiable. This idea of non-differentiability suggests introducing elementary definitions of fractional integrals and fractional derivatives starting from the limits of appropriately defined sums. The relation between fractal functions and the fractional calculus is a deep one. For example, the fractional derivative of a regular function yields a fractal function of dimension determined by the order of the fractional derivative. Thus, changes in time of phenomena that are described by fractal functions are probably best described by fractional equations of motion. In any event, this perspective is the one we developed elsewhere [31] and we find it useful here for discussing some properties of complex webs.

The separation of time scales in complex physical phenomena allows smoothing over the microscopic fluctuations and the construction of differentiable representations of the dynamics on large space scales and long time scales. However, such smoothing is not always possible.

Introduction

Of all the standard clinical imaging techniques ultrasound is by far the least expensive and most portable (including handheld units smaller than a laptop computer), and can acquire continuous images at a real-time frame rate with few or no safety concerns. In addition to morphological and structural information, ultrasound can also measure blood flow in real-time, and produce detailed maps of blood velocity within a given vessel. Ultrasound finds very wide use in obstetrics and gynaecology, due to the lack of ionizing radiation or strong magnetic fields. The real-time nature of the imaging is also important in measuring parameters such as foetal heart function. Ultrasound is used in many cardiovascular applications, being able to detect mitral valve and septal insufficiencies. General imaging applications include liver cysts, aortic aneurysms, and obstructive atherosclerosis in the carotids. Ultrasound imaging is also used very often to guide the path and positioning of a needle in tissue biopsies.

Ultrasound is a mechanical wave, with a frequency for clinical use between 1 and 15 MHz. The speed of sound in tissue is ∼1540 m/s, and so the range of wavelengths of ultrasound in tissue is between ∼0.1 and 1.5 mm. The ultrasound waves are produced by a transducer, as shown in Figure 4.1, which typically has an array of up to 512 individual active sources. In the simplest image acquisition scheme, small subgroups of these elements are fired sequentially to produce parallel ultrasound beams.

Introduction

Of the four major clinical imaging modalities, magnetic resonance imaging (MRI) is the one developed most recently. The first images were acquired in 1973 by Paul Lauterbur, who shared the Nobel Prize for Medicine in 2003 with Peter Mansfield for their shared contribution to the invention and development of MRI. Over 10 million MRI scans are prescribed ever year, and there are more than 4000 scanners currently operational in 2010.

MRI provides a spatial map of the hydrogen nuclei (water and lipid) in different tissues. The image intensity depends upon the number of protons in any spatial location, as well as physical properties of the tissue such as viscosity, stiffness and protein content. In comparison to other imaging modalities, the main advantages of MRI are: (i) no ionizing radiation is required, (ii) the images can be acquired in any two- or three-dimensional plane, (iii) there is excellent soft-tissue contrast, (iv) a spatial resolution of the order of 1 mm or less can be readily achieved, and (v) images are produced with negligible penetration effects. Pathologies in all parts of the body can be diagnosed, with neurological, cardiological, hepatic, nephrological and musculoskeletal applications all being widely used in the clinic. In addition to anatomical information, MR images can be made sensitive to blood flow (angiography) and blood perfusion, water diffusion, and localized functional brain activation.

Introduction

In nuclear medicine scans a very small amount, typically nanogrammes, of radioactive material called a radiotracer is injected intravenously into the patient. The agent then accumulates in specific organs in the body. How much, how rapidly and where this uptake occurs are factors which can determine whether tissue is healthy or diseased and the presence of, for example, tumours. There are three different modalities under the general umbrella of nuclear medicine. The most basic, planar scintigraphy, images the distribution of radioactive material in a single two- dimensional image, analogous to a planar X-ray scan. These types of scan are mostly used for whole-body screening for tumours, particularly bone and metastatic tumours. The most common radiotracers are chemical complexes of technetium (99mTc), an element which emits mono-energetic γ-rays at 140 keV. Various chemical complexes of 99mTc have been designed in order to target different organs in the body. The second type of scan, single photon emission computed tomography (SPECT), produces a series of contiguous two-dimensional images of the distribution of the radiotracer using the same agents as planar scintigraphy. There is, therefore, a direct analogy between planar X-ray/CT and planar scintigraphy/SPECT. A SPECT scan is most commonly used for myocardial perfusion, the so-called ‘nuclear cardiac stress test’. The final method is positron emission tomography (PET). This involves injection of a different type of radiotracer, one which emits positrons (positively charged electrons).

Introduction

X-ray planar radiography is one of the mainstays of a radiology department, providing a first ‘screening’ for both acute injuries and suspected chronic diseases. Planar radiography is widely used to assess the degree of bone fracture in an acute injury, the presence of masses in lung cancer/emphysema and other airway pathologies, the presence of kidney stones, and diseases of the gastrointestinal (GI) tract. Depending upon the results of an X-ray scan, the patient may be referred for a full three-dimensional X-ray computed tomography (CT) scan for more detailed diagnosis.

The basis of both planar radiography and CT is the differential absorption of X-rays by various tissues. For example, bone and small calcifications absorb X-rays much more effectively than soft tissue. X-rays generated from a source are directed towards the patient, as shown in Figure 2.1(a). X-rays which pass through the patient are detected using a solid-state flat panel detector which is placed just below the patient. The detected X-ray energy is first converted into light, then into a voltage and finally is digitized. The digital image represents a two-dimensional projection of the tissues lying between the X-ray source and the detector. In addition to being absorbed, X-rays can also be scattered as they pass through the body, and this gives rise to a background signal which reduces the image contrast. Therefore, an ‘anti-scatter grid’, shown in Figure 2.1(b), is used to ensure that only X-rays that pass directly through the body from source-to-detector are recorded.

Introduction

A clinician making a diagnosis based on medical images looks for a number of different types of indication. These could be changes in shape, for example enlargement or shrinkage of a particular structure, changes in image intensity within that structure compared to normal tissue and/or the appearance of features such as lesions which are normally not seen. A full diagnosis may be based upon information from several different imaging modalities, which can be correlative or additive in terms of their information content.

Every year there are significant engineering advances which lead to improvements in the instrumentation in each of the medical imaging modalities covered in this book. One must be able to assess in a quantitative manner the improvements that are made by such designs. These quantitative measures should also be directly related to the parameters which are important to a clinician for diagnosis. The three most important of these criteria are the spatial resolution, signal-to-noise ratio (SNR) and contrast-to-noise ratio (CNR). For example, Figure 1.1(a) shows a magnetic resonance image with two very small white-matter lesions indicated by the arrows. The spatial resolution in this image is high enough to be able to detect and resolve the two lesions. If the spatial resolution were to have been four times worse, as shown in Figure 1.1(b), then only the larger of the two lesions is now visible. If the image SNR were four times lower, illustrated in Figure 1.1(c), then only the brighter of the two lesions is, barely, visible.

Introduction

In this appendix we provide basic concepts aimed at introducing the formalism of networks. We first introduce graphs and make simple examples, and then discuss the topological properties of networks. Other general presentations of network structure can be found in review articles and books.

A network (or graph in a more mathematical language) is defined as a set of Nvertices (or nodes) connected by links (or edges). Links, and consequently the whole graph, can be either directed (oriented), if a direction is specified as in Fig. A.1a, or undirected (not oriented), if no direction is specified as in Fig. A.1b. More precisely, undirected links are rather bidirectional ones, since they can be traversed in both directions. For this reason an undirected graph can always be thought of as a directed one where each undirected link is replaced by two directed links pointing in opposite directions (see Fig. A.1c). A link in a directed network is said to be reciprocated if another link between the same pair of vertices, but with opposite direction, is there. Therefore, an undirected network can be regarded as a special case of a directed network where all links are reciprocated. The links of a network may also carry a number, referred to as the weight of the edge, representing the strength of the corresponding interaction. In such a case one speaks of a weighted network. In the present appendix we do not consider weighted networks explicitly.