Mean-field theory has been one of the main approaches traditionally used in the study of phase transitions of physical systems. It dates back to the early 20th century, when it was first applied by Pierre Weiss and others to the analysis of the phenomenon of ferromagnetism. (See, for example, the classical monograph by H. E. Stanley (1971) [264] for a historical account of these developments and an introduction to the field of phase transitions.)

Mean-field theory is usually applied to the analysis of complex systems where the interaction among a large number of individual “particles” proceeds along many dimensions. Under these conditions, the intuitive idea underlying the approach can be simply explained as follows. If the nature of interaction is rich (i.e. highly dimensional), it should be possible to capture the overall behavior of the system through a stylized model of the situation in which the host of effects impinging on each individual entity is replaced by a suitable mean field. In such a mean-field approach, the average description of the system is tailored to a suitable aggregate (or average) of the large number of individual effects exerted by the population at large. The self-referential nature of the exercise is thus apparent: the average state of the system is both an explanatory variable and the variable itself to be explained. This suggests that, in many cases, mean field theory must seek a self-consistent solution. This is why it is also often labeled self-consistent field theory.

THE NEW FIELD OF COMPLEX NETWORKS

In the last decade, the study of complex networks has become a booming field of research with a marked interdisciplinary character. Many different phenomena in the physical, biological, and social worlds can be understood as network based. They build upon some complex (as well as evolving) pattern of bilateral connections among individual entities and the overall performance of the system is largely shaped by the intricate architecture of those connections.

A brief review of alternative domains of application should serve to illustrate the rich diversity of phenomena that are distinctly governed by complex networks. This is the task carried out in Subsection 1.1.1, where the primary aim is to illustrate such diversity with empirical illustrations gathered from a large number of different areas. Next, in Subsection 1.1.2 we elaborate on the idea that, given the nature of the endeavour, a genuinely interdisciplinary approach is well in order in the field of complex networks.

Realms of Application and Empirical Evidence

We may start, as the most tangible, with transportation networks. These include the connections through which modern economies channel the physical movement of all sorts of commodities and signals. Pertaining, for example, to the conveyance of signals, a paradigmatic instance is of course the internet network, the huge mesh of bilateral connections through which bit-codifying electronic impulses across computers are transferred all around the world.

The primary purpose of this monograph has been twofold. Firstly, the objective has been to provide a systematic account of some of the basic models and techniques developed by the modern theory of complex networks. In addition, a second motivation has been to illustrate the broad range of socioeconomic phenomena to which this theory can be naturally applied. In the latter pursuit, our approach is polar to that commonly espoused by classical game theory and economic analysis. It stresses the implications of complexity on the interaction structure, while downplaying the role of incentives in shaping agents' behavior. But, as repeatedly argued, neither of those one-sided perspectives can be judged satisfactory. In general, both complexity and incentive considerations should jointly play a key role in any proper understanding (and thus modeling) of most social phenomena.

To further elaborate on this point, it is useful to recapitulate what are some of the main benefits to be expected from explicitly accounting for complexity in the analysis of socioeconomic environments.

• First, of course, the theoretical approach undoubtedly becomes more realistic since, indeed, we find that so many interesting social problems in the real world are embedded in a complex and ever-changing social network (cf. Chapter 1). It is fitting, therefore, that those problems should also be modeled in ways that respect such underlying complexity.

• […]

The aim of this book is to provide a systematic account of a recent body of theoretical research that lies at the intersection of two fertile strands of literature. One of these strands, the study of complex networks, is a new field that has been developing at a fast pace during the last decade. The other one, social network analysis, has been an active area of research in sociology and economics for quite some time now – only lately, however, has it started to be seriously concerned with the implications of complexity. There is, I believe, much potential in bringing these two approaches together to shed light on network-based phenomena in complex social environments. This monograph is written with the intention of helping both the social scientist and other network researchers in this fascinating endeavor.

For the social scientist, the monograph may be used, inter alia, as a self-contained introduction to some of the main issues and techniques that mark the modern literature on complex networks. Since this literature has largely developed as an outgrowth of statistical physics, some of the powerful methodology being used is often alien to researchers from other disciplines. On the other hand, for the network theorist who lacks an economic background, the present monograph can fulfill a reciprocal role. Specifically, it may serve as an illustration of the questions and concerns that inform the economists' approach to the study of socioeconomic networks.

In this final chapter, we study the interplay of search, diffusion, and play in the formation and ongoing evolution of complex social networks. First, we illustrate how the issue of network formation is addressed by that part of the received economic literature that highlights its strategic dimension. As will be explained in Section 6.1, even though game-theoretic models are concerned with an undoubtedly important aspect of the phenomenon, they also display a significant drawback. Their standard methodology (i.e. the full-rationality paradigm) as well as many of their implicit assumptions (e.g. a largely stable environment) abstract from the inherent complexity that pervades the real world.

In contrast, the approach pursued in the bulk of this chapter adopts a polar standpoint. It stresses the complexity of social networks but, as a contrasting limitation, it largely eschews the strategic concerns that also play an important role in many cases. The analysis ranges from models where local search is the main driving force in network formation (Section 6.2) to those where the focus is on the interaction between network evolution and agents' embedded behavior (Section 6.3). Finally, a brief reconsideration of matters is conducted in Section 6.4, where it is also suggested that a genuine integration of strategic and complexity considerations should be one of the primary objectives of future network analysis.

GAME-THEORETIC MODELS OF NETWORK FORMATION

Here we shall not attempt to provide an extensive description of the large body of literature that studies network formation from a game-theoretic viewpoint.

Suddenly, systems biology is everywhere. What is it? How did it arise? The driving force for its growth is high-throughput (HT) technologies that allow us to enumerate biological components on a large scale. The delineation of the chemical interactions of these components gives rise to reconstructed biochemical reaction networks that underlie various cellular functions. Systems biology is thus not necessarily focused on the components themselves, but on the nature of the links that connect them and the functional states of the networks that result from the assembly of all such links. The stoichiometric matrix represents such links mathematically based on the underlying chemistry, and the properties of this matrix are key to determining the functional states of the biochemical reaction networks that it represents.

The Need for Systems Analysis in Biology

Biological parts lists

During the latter half of the 20th century, biology was strongly influenced by reductionist approaches that focused on the generation of information about individual cellular components, their chemical composition, and often their biological functions. Over the past decade, this process has been greatly accelerated with the emergence of genomics. We now have entire DNA sequences for a growing number of organisms, and we are continually delineating their gene portfolios. Although functional assignment to these genes is presently incomplete, we can expect that we will eventually have assigned and verified function for the majority of genes on selected genomes.

In 1995, the first full genome sequence became available, ushering in the genome era. Since then, a large number of high-throughput technologies have enabled us to define the molecular parts catalogs of cellular components. Although these catalogs are still incomplete, it is now possible to reconstruct, based on this and other information, genome-scale networks of biochemical reactions that take place inside cells. This process of network reconstruction, followed by the synthesis of in silico models describing their functionalities, is the essence of systems biology.

The functions of reconstructed networks are defined by the interconnections of their parts. Since these connections involve chemical reactions, they can be described by a stoichiometric relationship. The stoichiometric matrix, which contains all such relationships in a network, is thus a concise mathematical representation of reconstructed networks. This matrix comprise integers that represent time- and condition-invariant properties of a network. It may therefore be expected to represent a key in the study of the functionalities of complex biochemical reaction networks. Its content and associated information effectively constitute a biochemically, genetically, and genomically structured database.

This book is focused on the stoichiometric matrix. In order to satisfactorily understand the material, good knowledge of linear algebra and of biochemistry is needed. Most of the mathematical concepts and principles, when properly interpreted, have a direct biological and chemical meaning. This text thus tries to relate what might be seen as abstract mathematical quantities to real biological and chemical features.

The stoichiometric matrix and its associated information fundamentally represents a biochemically, genetically, and genomically structured database. It can be used to analyze network properties, and to relate the components of a network and its genetic bases to network or phenotypic functions. In developing biologically meaningful in silico analysis procedures, fundamental characteristics of biology need to be explicitly recognized. Unlike the physicochemical sciences, biology is subject to dual causality or dual causation. Biology is governed not only by the natural laws but also by genetic programs. Thus, while biological functions obey the natural laws, their functions are not predictable by the natural laws alone. Biological systems function and evolve under the confines of the natural laws according to basic biological principles, such as the generation of diversity and natural selection. The natural laws can be described based on physicochemical principles and used to define the constraints under which organisms must operate. How organisms operate within these constraints is a function of their evolutionary history and survival. Survival and its relationship to cellular functions can perhaps be readily understood for simple, single cellular organisms.

Causation in Physics and Biology

Physics

Classically, “cause and effect” is established by formulating mathematical descriptions of conceptual models of fundamental physical phenomena. One example is that of molecular diffusion; see Figure 12.1. The fundamental process underlying diffusion is the random walk process that a collection of molecules undergoes.

The set of chemical reactions that comprise a network can be represented as a set of chemical equations. Embedded in these chemical equations is information about reaction stoichiometry. All this stoichiometric information can be represented in a matrix form; the stoichiometric matrix, denoted by S. Associated with this matrix is additional information about enzyme complex formation, transcript levels, open reading frames, and protein localization. Therefore, once assembled, the stoichiometric matrix represents a biochemically, genetically, and genomically (BIGG) structured database. This database structure represents an interface between high-throughput data and in silico analysis (see Figure 1.4). It allows high-throughput data (often called content) to be put into context. The stoichiometric matrix is the starting point for various mathematical analysis used to determine network properties. Part II of this text will summarize the basic properties of the stoichiometric matrix. Since it is a mathematical object, the treatment is necessarily mathematical. However, S represents biochemistry. We will thus relate the mathematical properties of S to the biochemical and biological properties that it fundamentally represents.

Now that we have come to the end of this text, it is time to ponder what we have done, how far we have come, and what lies ahead. In this chapter I put forth some of my thoughts related to these issues.

Types of questions asked in biology

There are fundamentally three types of questions that are asked in biology: “what,” “how,” and “why.”

What is there?

We have made substantial strides in answering this type of question. We can sequence entire genomes and use bioinformatic analyses to determine what is in a genome. We can expression profile a genome under various conditions. We now have extensive information about genomes, cells, and organisms, and are in a position to continue to generate much more. It is indeed this impressive availability of data that has made biology “data-rich” and has been the driving force for the emergence of systems biology.

How does it work?

Science seeks to generate mechanisms and theories to explain the world around us. Functional genomics tries to assign function to various gene products and segments of a genome. The large number of interactions that needs to be taken into account to explain cellular components has grown substantially with our growing knowledge of cellular components. The drive to reconstruct genome-scale networks and to assess their functional states is a response to this need.