Many different models have been developed in order to study particular features of SOC, such as 1/f noise, non-conservation and anisotropy. In Part II, some of the more important models are introduced and their general properties discussed. At the beginning of each section, the definition and characteristics of each model are catalogued in a box and the exponents listed in a table. Each section is essentially independent, discussing a model in its own right for its particular qualities. Nevertheless, relations to other models are emphasised and a minimal set of common observables (see Sec. 1.3), in particular exponents (Ch. 2), is discussed for each of them.

The attempt to tabulate exhaustively all numerical estimates for exponents of the various models is futile; it is practically impossible to find all published exponents for a model. It is similarly fruitless to draw a clear line between genuine estimates and exponents derived from others using (assumed) scaling relations. Wherever possible, exponents are only listed if the sole underlying assumption is simple scaling as stated in the caption. The tables of exponents therefore serve only to illustrate the variety and sometimes the disparity of results. The exponents are listed in historical order, which often means that the data towards the bottom of the tables are based on more extensive numerics and are thus more reliable. The tables should enable the reader to judge whether a given model displays systematic, robust scaling behaviour or not.

Both of the following models incorporate a form of stochasticity in the relaxation mechanism. In the MANNA Model particles topple to sites randomly chosen among nearest neighbours and in the OSLO Model the local critical slopes are chosen at random. They both display robust scaling and belong to the same enormous universality class, which contains two large classes of ordinary (tuned), non-equilibrium critical phenomena: directed percolation with conserved field (C-DP) and the quenched Edwards-Wilkinson equation (qEW equation). The former is paradigmatically represented by the MANNA Model, the latter by the OSLO Model.

Both models are generally considered to be Abelian, even when they strictly are not. In their original versions, the relaxation of the MANNA Model is non-Abelian and so is the driving in the OSLO Model. This can be perceived as a shortcoming, not only because the BTW Model has been understood in much greater detail by studying its Abelian variant, but also because of the simplification of their implementation, as the final configurations become independent of the order of updates. Nowadays, the MANNA Model and, where the issue arises, the OSLO Model are studied in their Abelian variant. The MANNA Model is currently probably the most intensely studied model of SOC.

In their Abelian form, both models can be described in terms of stochastic equations of motion (Sec. 6.2.1.3 and Sec. 6.3.4.2). These look very different for the two models.

The models discussed in this chapter resemble some of the phenomenology of (naïve) sandpiles. As discussed earlier (Sec. 3.1), it is clear that the physics behind a real relaxing sandpile is much richer than can be captured by the following ‘sandpile models’. Yet, it would be unjust to count that as a shortcoming, because these models were never intended to describe all the physics of a sandpile. The situation is similar to that of the ‘Forest Fire Model’ which is only vaguely reminiscent of forest fires and was, explicitly, not intended to model them. The names of these models should not be taken literally, they merely serve as a sometimes humorous aide-memoire for their setup, similar to Thomson's Plum Pudding Model which is certainly not a model of a plum pudding.

In the following section, the iconic Bak-Tang-Wiesenfeld Model and its hugely important derivative, the Abelian Sandpile Model, are discussed in detail. This is followed by the ZHANG Model, which was intended as a continuous version of the BAK-TANG-WIESENFELD Model. Their common feature is a deterministic, rather than stochastic, relaxation rule. Although a lot of analytical and numerical progress has been made for all three models, their status quo, in particular to what extent they display true scale invariance, remains inconclusive.

The Bak-Tang-Wiesenfeld Model

The publication of the Bak-Tang-Wiesenfeld (BTW) Model (see Box 4.1) (Bak et al., 1987) marks the beginning of the entire field.

When Self-Organised Criticality (SOC) was first introduced in 1987 by Bak, Tang, and Wiesenfeld, it was suggested to be the explanation of the fractal structures surrounding us everywhere in space and time. The very poetic intuitive appeal of the combination of terms self-organisation and criticality, meant that the field gained immediate attention. The excitement was not lowered much by the fact that the claimed 1/f and fractal behaviour were soon realised in reality not to be present in the sandpile model used by the authors to introduce their research agenda. Nor did the lack of power laws in experiments on real piles of sand deter investigators from interpreting pieces of power laws observed in various theoretical models and physical systems as evidence of SOC being essentially everywhere. This led rapidly to a strong polarisation between two camps. On the one side there was the group of researchers who did not worry about the lack of a reasonably precise exclusive definition of the SOC concept and therefore tended to use SOC as synonymous with snippets of power laws, rendering the term fairly meaningless. The other camp maintained that SOC was not to be taken seriously. They arrived at this conclusion through a mixture of factors including the observation that SOC was ill defined, not demonstrated convincingly in models, and absent from experiments on sandpiles. The debate sometimes reflected a reaction in response to bruises received during fierce exchanges at meetings as much as a reaction to scientific evidence.

Dissipation is a major theme in SOC for several reasons. Like every relaxation process, avalanching in a sandpile generally can be seen as a form of dissipation, quite literally so for the sand grains that dissipate potential energy in the BTW Model. In that sense, sandpile models are inherently dissipative. Yet, their dynamics can be expressed in terms of variables, which are conserved under the bulk dynamics (also ‘local’ dynamics), such as the number of slope units in the Abelian BTW Model.

The models described in the present chapter, however, go a step further by obeying dynamical rules without local bulk conservation (also ‘local’ conservation), although all models develop towards a stationary state even in the non-conserved variable, i.e. overall there is asymptotic conservation on average. The observation of scale-free dissipation in turbulence triggered the development of the Forest Fire Model (Sec. 5.1), i.e. it was explicitly designed in a dissipative fashion. The situation is somewhat similar for the OFC Model (Sec. 5.3) which incorporates a dissipation parameter α, whereas in the BS Model (Sec. 5.4) dissipation is a necessary by-product. Both the OFC Model and the BS Model are examples of models driven by extremal dynamics, which consists of identifying the ‘weakest link’ among all sites and starting relaxation from there.

In the light of Hwa and Kardar's (1989a) work (Sec. 9.2.2), which suggested that scale invariant phenomena arise naturally, even generically in the presence of bulk conservation, the existence of non-conservative SOC models is particularly important.

This chapter argues for connecting models of several kinds of macro- and microprocesses as they affect structure and dynamics in the globalization of networks of trade. The purpose is to explore multiple levels of structure, process, and adaptation and to loosen assumptions about determinacy in models of networks and globalization. As do many models of emergence, it questions the notions of inevitability that too often surround studies of globalization. Particularly useful for comparison of cases are the models of “world system” developed by Modelski and Thompson (1996, see Devezas and Modelski, 2008). These focus on national policy-driven innovation and processes of European “evolutionary learning” that began in the 1400s. They put into context the models that focus on core-periphery structure as developed by Braudel (1973), or the “world-system” core-periphery model that for Wallerstein (1974) begins in the 1600s. Study of structures of core-periphery in world systems can benefit from added dimensions, improved measurement of network structure, and understanding the effects of periodic crises in terms of historical dynamics.

An unexpected outcome of this survey for issues of policy is that it develops a deeper historical understanding of how certain kinds of exchange systems develop several kinds of inequalities that are inimical to the concept of fair pricing in the operation of market equilibria, even in the absence of economic oligopolies (monopoly, duopoly) and oligopsonies (monopsony, duopsony). These include longstanding militaristic state-policy domination of international exchange, resultant structural inequality in international trade networks, and cyclical events within polities, that in periods of resource scarcity relative to population, create periods of extreme deflation of wages relative to extremes in elite dominance over wealth-generating property ownership.

Energy security of natural gas supplies in Europe is becoming a key concern. As demand increases, infrastructure development focuses on extending the capacity of the pipeline system. While conventional approaches focus mainly on source dependence, we argue for a network perspective to also consider risks associated with transit countries, by borrowing methods from ecological food web analysis. We develop methods to estimate the exposure and dominance of each country, by using network datasets from the present pipeline system, and future scenarios of 2020 and 2030. We have found that future scenarios will not increase the robustness of the system. Pipeline development to 2030 will shift the relative weight of energy security concerns away from source to transit countries. The dominance of politically unstable countries will increase. The exposure will be slightly redistributed by improving the security of already secure countries, and increasing the exposure of those countries that are already in a vulnerable position.

Introduction

During the first days of 2009 a dispute between Russia and Ukraine led to a closure of major gas pipelines, and the worst dropout of the natural gas supply in Europe so far (Pirani et al., 2009). Supply to 18 countries was disrupted, and some areas with limited reserves and a lack of alternative supply channels were left without heating amidst a bone-chilling cold snap. Initial cuts affected the supplies to Ukrainian consumption (January 1), while deliveries to Europe were reduced drastically on January 6 (e.g. Italy experienced losses of 25% towards its needs and decided to increase imports from Libya, Norway, and The Netherlands; Hungarian consumption was cut off by 40%).

Introduction: applied network science

This volume provides an overview of network science applied to social policy problems. Network science is arguably the most dynamic and interdisciplinary field that has grown up to address problems of an increasingly interconnected world. Social problems transgress disciplinary boundaries, especially with the ever-increasing complexity of our globally interconnected world society. It is natural that with complex problems such as loss in ecological diversity, economic crisis, spread of epidemics, and the safety of our food supply, we turn to a field of research that focuses on explaining complex dynamics, and that is inherently interdisciplinary itself.

Networks have become part of our everyday experience as we routinely use online social network services, we hear reports on the operations of terrorist networks, and we speculate on the six degrees of separation to celebrities and presidents. Less manifestly, we rely on vast and complex infrastructural networks of electric power distribution, Internet data routing, or financial transfers. We only ponder the complexity of these systems when we are faced with avalanche-like dynamics in their collapse, as major blackouts, system stoppages, or financial meltdowns.

With networks on the collective mind, there is ample interest in tools to understand and manage complex network systems of social ties. At the time of writing this introduction (in October 2011), there were eighteen applications available on Facebook to visualize one's social network. The popularity of such software tools shows our fascination with the interesting new perspective that the graphic visualization of friendship ties provides.

Introduction

State agencies responsible for managing various risks in social life issue advisories to the public to prevent and mitigate various hazards. In this chapter we will investigate how information about a common foodborne health hazard, known as Campylobacter, spread once it was delivered to a random sample of individuals in France. The Campylobacter is most commonly found in chicken meat and causes diarrhea, abdominal pain, and fever. The illness normally lasts a week but in rare cases patients can develop an auto-immune disorder, called Guillain-Barré syndrome, that leads to paralysis and can be deadly. Campylobacter, together with Salmonella, is responsible for more that eighty percent of foodborne illnesses in France and strikes over 20,000 people each year. People can take simple steps to avoid infection by cleaning their hands, knives, cutting boards, and other food items touched by raw chicken meat and by cooking the meat thoroughly.

In this chapter we build two different network models to see how the information about Campylobacter diffuses in society, by mapping onto various network structures the data we gathered with three waves of surveys. In these models the spread of information depends on two sets of factors. Firstly, each person has a set of individual properties that influences their propensity to transmit the information to or to receive the information from someone they know. Second, each person is connected to others in ways that also affect transmission. There are three aspects of these social ties that matter.