The momentum-shell method

The method discussed in Section 11.3.1 will now be pursued further, in that it will be applied to the full effective Hamiltonian of an O(n) spin system, in which the effective Hamiltonian contains terms that are linear, quadratic, and of fourth order in the spin field. It is in the consideration of the higher order terms in the effective Hamiltonian (higher order than quadratic, that is) that the complications arise. The calculations that will be outlined here are not especially challenging in execution, but we will hint at extensions and generalizations that can become so.

In this chapter, the reader will be introduced to the field-theoretical version of the renormalization group, and to its first effective realization, the ∈ expansion for critical exponents. The approach will be that of the momentum-shell method developed in the previous chapters. The application of the method to the full O(n) Hamiltonian will be more complicated due to the coupling terms, which were neglected before.A straightforward, though somewhat tedious, calculation will lead to a modified set of renormalization equations. These differential equations will then be solved to lowest order in the variable ∈ = 4 – d, where d is the system's spatial dimensionality (three in the cases of interest to us). Using scaling arguments, the critical exponents will be obtained. Their relevance to the self-avoiding random walk will also be discussed.

The techniques to be discussed here are descendants of the original renormalization group method developed by Kenneth Wilson (Wilson, 1971a; Wilson, 1971b; Wilson and Kogut, 1974).

We've now had an introduction to the random walk. There has also been an introduction to the generating function and its utilization in the analysis of the process. Here we investigate some aspects of the random walk, both because they are interesting in their own right and because they further demonstrate the usefulness of the generating function as a theoretical technique in discussing this problem. In particular, we will apply the generating function method to study the problem of “recurrence” in the random walk. That is, we will address the question of whether the walker will ever return to its starting point, and, if so, with what probability.We will find that the spatial dimensions in which the walk takes place play a crucial role in determining this probability. Using an almost identical approach, the number of distinct points visited by the walker will also be determined.

We present the two studies below as a display of the power of the generating function technique in the study of the random walk process. For the skeptic, other examples will be provided in subsequent chapters.

Recurrence

We begin this chapter with a discussion of the question of recurrence of an unrestricted random walk. We will utilize the generating function, and an important statistical identity, to see how the dimensionality of the random walker's environment controls the probability of its revisiting the site from which it has set out. This study complements our earlier discussion and provides evidence for the power of the generating function approach.

A new generating function

A useful – but apparently little known – quantity enables one to obtain some key results with remarkably little effort.

We now have had an introduction to the random walk, and there has been a discussion of some of the most useful methods that can be utilized in the analysis of the process. The unifying theme of this chapter is the introduction of scaling arguments to provide insights into the behavior of the walker under a variety of circumstances. First, we will address in more depth the notion of universality of ordinary random walk statistics. Then, we will discuss the (mathematical) sources of non-Gaussian statistics. Finally, we will develop a few simple but central scaling results by looking once more at the path integral formulation of the random walk. In particular, using simple scaling arguments, we will be able to provide heuristic arguments leading to Flory's formula for the influence of self-avoidance on the spatial extent of a random walker's path.

Universality

Notions of universality play an important role in discussions of the statistics of the random walk.We have already seen a version of universality in Chapter 2, in which it was shown that the statistics of a long walk are Gaussian, regardless of details of the rules governing the walk, as long as the individual steps taken by the walker do not carry it too far. In the remainder of this section, the idea of universality will be developed in a way that will allow us to apply it to the random walk when conditions leading to Gaussian behavior do not apply.

As it turns out, universality in random walk statistics has a close mathematical and conceptual connection to fundamental properties of a system undergoing a special type of phase transition.

Mean field theory and spin wave contributions

Now that we have established the link between self-avoiding random walks and the O(n) model of magnetism, it is appropriate to look again at the magnetic system in the limit n → 0. Of special interest to us is its behavior in the immediate vicinity of the critical point. We will make extensive use of the insights provided by the study of critical phenomena that have emerged over the past three decades. Our initial approach to the problem of the statistical mechanics of the O(n) model will be to utilize the mean field ideas developed by Landau and others. Mean field theory will then be enhanced by the introduction of fluctuations which will be analyzed in a low order spin wave theory. The insights gained by this approach will lead us to a clearer picture of the phase transition as it pertains to the random walk problem. Finally, a full renormalization group calculation will be used to elucidate the scaling properties of this model. This final step in the analysis will be accomplished in Chapter 12.

Outline of the chapter

In this chapter we make use of the connection between the O(n) model and the self-avoiding walk to discuss aspects of the statistics of such a walk. We begin by reviewing the relationship between the spin model described by the O(n) energy function and the self-avoiding walk. Our initial focus will be on self-avoiding walks that are confined to a finite volume of space.

We begin this preface by reporting the results of an experiment. On April 23, 2003, we logged onto INSPEC – the physical science and engineering online literature service – and entered the phrase “random walk.” In response to this query, INSPEC delivered a list of 5010 articles, published between 1967 and that date. We then tried the plural phrase, “random walks,” and were informed of 1966 more papers. Some redundancy no doubt reduces the total number of references we received to a quantity less than the sum of those two figures. Nevertheless, the point has been made. Random walkers pervade science and technology.

Why is this so? Think of a system – by which we mean just about anything – that undergoes a series of relatively small changes and that does so at random. It is more likely than not that important aspects of this system's behavior can be understood in terms of the random walk. The canonical manifestation of the random walk is Brownian motion, the jittering of a small particle as it is knocked about by the molecules in a liquid or a gas. Chitons meandering on a sandy beach in search of food leave a random walker's trail, and the bacteria E. coli execute a random walk as they alternate between purposeful swimming and tumbling. Go to a casino, sit at the roulette wheel and see what kind of luck you have. The height of your pile of chips will follow the rules governing a random walk, although in this case the walk is biased (see Chapter 5), in that, statistically speaking, your collection of chips will inevitably shrink.

The modification of random walk statistics resulting from the imposition of constraints on the walkers will be repeatedly visited in this book. In fact, several chapters will be dedicated to the treatment of walkers that are forbidden to step on points they have previously visited. This kind of constraint represents the influence of a walk on itself. As we will see, calculations of the properties of the self-avoiding random walk require the use of fairly sophisticated techniques. The payoff is the ability to extract the fundamental properties of a model that accurately represents the asymptotic statistics of important physical systems, particularly long linear polymers.

The discussion referred to above will take place in later chapters. Here, we focus on other constraints, which embody the influence of a static environment on a walker. In the first part of this chapter, we will address the effect on a random walker of the requirement that it remain in a specified region of space. Specifically, we will look at walkers confined to a half-space, a linear segment in one dimension and a spherical volume in three dimensions. Then we will look at the case of walkers that are confined to the region outside a spherical volume. This case turns out to be relevant to issues relating to the intake of nutrients and the excretion of wastes by a simple, stationary organism, such as a cell. In fact, the final section of this chapter utilizes random walk statistics to investigate the limits on the size of a cell that survives by ingesting nutrients that diffuse to it from its surroundings.

The Internet is the result of the bold effort of a group of people in the 1960s, who foresaw the great potential of a computer-based communication system to share scientific and research information. While in the early times it was not a user-friendly environment and was only used by a restricted community of computer experts and scientists, nowadays the Internet connects more than one hundred million hosts and keeps growing at a pace unknown in any other communication media. From this perspective, the Internet can be considered as one of the most representative accomplishments of sustained investment in research at both the basic and applied science levels.

The success of the Internet is due to its world-wide broadcasting capability that allows the interaction between individuals without regard for geographic location and distance. The information exchanged between computers is divided into data packets and sent to special devices, called routers, that transfer the packets across the Internet's different networks. Of course a router is not linked to every other router. It just decides on the direction the data packets take. In order to work reliably on a global scale, such a network of networks must be very slightly affected by local or even extensive failures in the network's nodes. This means that if a site is not working properly or it is too slow, data packets can be rerouted, on the spot, somewhere else.

At the large-scale level, the modeling of the Internet focuses on the construction of graphs that reproduce the topological properties observed in the AS and IR level maps. Representing the Internet as a graph implies ignoring the physical features of routers and connections (capacity, bandwidth, etc.), in a effort to gain a more simplified perspective that is still able to reproduce the empirical observations. From this perspective, Internet modeling initially relied on the traditional framework where complex networks with no apparent regularities were described as static random graphs, such as the model of Erdös and Rényi (1959). The graph model of Erdös–Rényi is the simplest conceivable one, characterized by an absolute lack of knowledge of the principles that guide the creation of connections between elements. Lacking any information, the simplest assumption one can make is to connect pairs of vertices at random with a given connection probability p.

Based on the random graph model paradigm, the computer science community has developed models of the Internet to test new communication protocols. The basic idea underlying the use of models to test protocols is that these should be independent (at least in principle) from the network topology. However, it turns out that their performance can be very sensitive to topological details (Tangmunarunkit et al., 2002a; Labovitz, Ahuja, Wattenhofer, and Srinivasan, 2001; Park and Lee, 2001). The use of an inadequate model can lead to the design of protocols that run very efficiently on the model, but perform quite poorly on the real Internet.

The Internet and the World Wide Web (also known as WWW or simply the Web) are often considered as synonyms by non-technical users. This confusion stems from the fact that the WWW is at the origin of the explosion in Internet use, providing a very user-friendly interface to access the almost infinite wealth of information available on the Internet. The WWW, though, is a rather different network in the sense that it is just made from a specific software protocol, which allows access to data scattered on the physical Internet. In other words, it is a virtual network which lives only as a sort of software map linking different data objects. Nevertheless, the Web finds a natural representation as a graph and it is a stunning example of an evolving network. New Web pages appear and disappear at an impressive rate, and the link dynamics is even faster. Indeed, the fact that we are dealing with virtual objects makes Web dynamics almost free from the physical constraints acting on the Internet. Any individual or institution can create at will new Web pages with any number of links to other documents, and each page can be pointed at by an unlimited number of other pages.

The Web is not the only virtual network present on the Internet. Users interactions and new media for information sharing can be mapped as well in a graph-like structure. The graph of e-mail acquaintances of Internet users is a well-defined example of social network hosted by the Internet.

Routers have multiple interfaces, each one corresponding to a physical connection with another peering router. By definition, each interface is associated with a different IP address. As discussed in Chapter 3, path probing discovers router interfaces. In particular each probing path will record a single interface for each traversed router, generally the one from which the packet arrived to the router. It is therefore possible that probes coming from different monitors, even if directed to the same destination, might enter from different interfaces on the same intermediate router. Thus, each time a router is probed from a different perspective, its interfaces are registered as separate routers. A simple example of the difference between the physical router connectivity and the one registered by probing paths is depicted in Figure A1.1. In this example a network of four physical routers are connected through six different interfaces. The graphs resulting from path probes forwarded from opposite directions are different. By merging these different views, a graph with a false connectivity is obtained and the interfaces (router aliases) resolution becomes indispensable to reconstruct the physical router topology.

We have seen in the previous chapter that the graphs representing the physical layout of the large-scale Internet look like a haphazard set of points and lines, with the result that they are of little help in finding any quantitative characterization or hidden pattern underlying the network fabric. The intricate appearance of these graphs, however, corresponds to the large-scale heterogeneity of the Internet and prompts us to the use of a statistical analysis as the proper tool for a useful mathematical characterization of this system. Indeed, in large heterogeneous systems, large-scale regularities cannot be found by looking at local elements or properties. Similarly, the study of a single router connectivity or history will not allow us to understand the behavior of the Internet as a whole. In other words, we must abandon local descriptions in favor of a large-scale statistical characterization, taking into account the aggregate properties of the many interacting units that compose the Internet.

The statistical description of Internet maps finds its natural framework in graph theory and the basic topological measures customarily used in this field. Here we shall focus on some metrics such as the shortest path length, the clustering coefficient, and the degree distribution, which provide a basic and robust characterization of Internet maps. The statistical features of these metrics provide evidence of the small-world and scale-free properties of the Internet. These two properties are prominent concepts in the characterization of complex networks, expressing in concise mathematical terms the hidden regularities of the Internet's structure.

The Internet is a technological infrastructure aimed at favoring data exchange and reachability. The World Wide Web can be used to extract information from distant places with a few mouse clicks, and Internet protocols forward messages to far away computers, efficiently routing them along the intricate network fabric. This extreme efficiency, however, can also work in favor of negative purposes, such as the spreading of computer viruses. Computer viruses have a long history, dating from the 1980s and before, becoming newly and sadly famous after each new bug attack, which eventually causes losses worth millions of dollars in computer equipment and downtime (Suplee, 2000). Their ever-increasing threat has therefore stimulated a growing interest in the scientific community and the economic world, translated in this latter case into the antivirus software business, moving millions of dollars worldwide every year.

Computer virus studies have been carried out for long time, based mainly on an analogy with biological epidemiology (Murray, 1988). In particular most studies have focused on the statistical epidemiology approach, aimed at modeling and forecasting the global incidence and evolution of computer virus epidemics in the computer world. The final goal of this approach is the development of safety and control policies at the large-scale level for the protection of the Internet. Puzzling enough, however, is the observed behavior of computer viruses in the wild, which exhibit peculiar characteristics that are difficult to explain in the usual epidemic spreading framework.

For the majority of people the word “Internet” means access to an e-mail account and the ability to mine data through any one of the most popular public web search engines. The Internet, however, is much more than that. In simple terms, it is a physical system that can be defined as a collection of independently administered computer networks, each one of them (providers, academic and governmental institutions, private companies, etc.) having its own administration, rules, and policies. There is no central authority overseeing the growth of this networks-of-networks, where new connection lines (links) and computers (nodes) are being added on a daily basis. Therefore, while conceived by human design, the Internet can be considered as a prominent example of a self-organized system that combines human associative capabilities and technical skills.

The exponential growth of this network has led many researchers to realize that a scientific understanding of the Internet is necessarily related to the mathematical and physical characterization of its structure. Drawing a map of the Internet's physical architecture is the natural starting point for this enterprise, and various research projects have been devoted to collecting data on Internet nodes and their physical connections. The result of this effort has been the construction of graph-like representations of large portions of the Internet. The statistical analysis of these maps has highlighted, to the surprise of many, a very complex and heterogeneous topology with statistical fluctuations extending over many scale lengths.

The natural framework for a correct mathematical description of complex networks is graph theory. The origins of graph theory can be traced back to the pioneering work of Euler to solve the Königsberg bridges problem (Euler, 1736), and has now reached a maturity in which a wealth of results of immediate applicability are useful for the understanding of real complex networks. In this appendix we shall provide a cursory introduction to the main definitions and concepts of graph theory, useful for the analysis of real networks. The main sources followed are the books by Chartrand and Lesniak (1986), Bollobás (1998), and Bollobás (1985), as well as the review articles by Albert and Barabási (2002), Dorogovtsev and Mendes (2002), and Newman (2003), covering more recent aspects.

Graphs and subgraphs

An undirected graph G is defined by a pair of sets G = (V, E), where V is a non-empty countable set of elements, called vertices or nodes, and E is a set of unordered pairs of different vertices, called edges or links. Throughtout the book a vertex is reffered to by its order i in the set V. The edge (i, j) joins the vertices i and j, which are said to be adjacent or connected. The total number of vertices in the graph (the cardinality of the set V) is denoted as N, the size of the graph.