Quantifying network theory

Let us concentrate first on the network aspect, with which this chapter originally opened. Another way of approaching its theoretical abstraction is to consider quantifying approaches from the natural sciences, specifically graph- and net-work-theory. Here, interdisciplinary transfer and abstraction is fortunately fac-ilitated by the work of the Hungarian physicist Albert László Barabási, who has summarised this research in a form accessible to a wider academic audience.⁵¹

Network theory emerged from mathematical network analysis and offers both a set of terminological tools for the description of networks and a range of abstract classifications and behavioural models derived from mathematical pat-terns.⁵² In their visualised form, the elements of a network are accordingly de-scribed as nodes, their relations as edges or links. The dataset may also contain additional information about the elements, adding a layer of qualification to the

50 Callon 1991, 134: “[E]very actor contains a hidden but already social being: that agency cannot be dissociated from the relationships between actors.” Italics in the original.

51 Barabási, Albert-László. Linked. The New Science of Networks. Cambridge, MA. 2002. For a self-contained, practical application in the field of cellular biology see Barabási, Albert-László and Oltvai, Zoltan. “Network Biology: Understanding the Cell’s Functional Organization”, in: Nature Reviews Genetics 5 (2004), 101-113.

52 Barabási 2002, 13-24. The original impetus behind network research was the improve-ment of the work done on random networks by the mathematicians Paul Erdős and Alfréd Rényi. In the field of Ancient History, this network-based methodology has been successfully applied by Irad Malkin, who used it to illuminate the dynamics that underpin the development of Panhellenic identity: Malkin, Irad. A Small Greek World.

Networks in the Ancient Mediterranean. Oxford 2011.

quantifiable base data. Generally speaking, quantifying network analysis, consist-ing for instance in the mathematical evaluation of a network’s connectivity or directionality, is more worthwhile and meaningful if the dataset is larger.⁵³ The reason is simply that network analysis offers mathematical tools to describe the configuration and structure of a network, providing points of comparison with other networks.⁵⁴ However, since the networks studied here will never be able to measure up to the standards of data integrity and precision necessary for such network analysis, these mathematical instruments will have no bearing on this study.⁵⁵

Looking beyond mathematical analysis, Barabási also makes us aware of em-pirical studies that suggest that the structure of real-world networks is not purely random, but in fact based on specific regularities that can be expressed in mathematical formulas. The most important of these observations appears to be that real-world networks tend towards centralisation, meaning that the distribu-tion of the number of links between the nodes of a network does not graph as a

53 For the sociological perspective cf. Holzer 2010², 55-63. In practice, empirical network analysis focuses on clearly delimited groups of small to medium sizes, though big data and data aggregation obviously offer opportunities here: Jansen, Dorothea and Diaz-Bone, Rainer. “Netzwerkstrukturen als soziales Kapital”, in: Johannes Weyer (ed.).

Konzepte und Methoden der sozialwissenschaftlichen Netzwerkforschung. Munich 2011², 71-108, here 73f. For a classic practical application in the Social Sciences see for instance Krackhardt, David and Hanson, Jeffrey R. “Informal Networks. The Company behind the Chart”, in: Harvard Business Review (July-August 1993), 104-111.

54 These mathematical analytical tools include the calculation of a network’s density that determines to what degree nodes are connected on average: Denser networks are less likely to contain nodes that monopolise connections. Irad Malkin (2011) draws on the general rule that greater density results in greater network connectivity for his analysis of the small world development of Archaic Greece. Another essential analytical tool consists in measuring the various degrees of centrality possessed by a network’s nodes, i.e. their relative importance in the network. Examples are degree-centrality, i.e. the number of connections per node, and betweenness-centrality, which describes the probability that a path between two given nodes passes through a specific node and thus designates a node’s significance as a mediator (Holzer 2010², 38-48). A third tool measures connectivity patterns, such as circularity or linearity, that can aid, for instance, in identifying social schisms and closed factions.

55 The most important standard is that the network is complete (on this problem see Erlhofer, Sebastian. “Missing Data in der Netzwerkanalyse”, in: Christian Stegbauer (ed.). Netzwerkanalyse und Netzwerktheorie. Ein neues Paradigma in den Sozialwissenschaften.

Wiesbaden 2010², 251-260), a demand that can hardly ever be met in historical studies.

This kind of meta-analysis that draws on mathematical tools therefore seems useful mainly for discourse studies with a limited focus and statistically sound data. Ideally, such a study would dispose of extensive, contemporary materials as points of comparison.

bell curve, but like an exponential function. Networks that show such a dis-tribution are termed scale-free, as their disdis-tribution does not vary with absolute size.⁵⁶ Paired with a growth dynamic, Barabási labels this phenomenon “prefer-ential attachment”, arguing that nodes with more links are more attractive and obviously connective than nodes with fewer, resulting in networks consisting of a small number of nodes with many links and a much larger number of less well-connected nodes. He is able to trace this phenomenon in many different contexts, including the spread of the HI virus, of computer viruses or information, but also in the very infrastructure itself, be it the internet or electricity grids. All these scale-free networks can be expressed using the mathematical parameters identified by Barabási and his colleagues:⁵⁷ their observations suggest that the distribution patterns often approximate 4:1 (the so-called 80-20 rule) and that successful nodes tend to become more successful (the so-called rich-get-richer phenomenon).⁵⁸ It is worth noting that all his examples ultimately derive from human interactions.

The basic tendency towards network centralisation and the other phenomena observed can therefore also apply to human behaviour in interaction – Barabási considers this a “natural” principle.⁵⁹ In an interdisciplinary use as a hermeneutic aid in Ancient History, however, such references to apparently ‘natural’ pheno-mena are of course more problematic, especially since the networks studied here can never be treated in their entirety. Barabási’s investigations do not face this problem, since the networks identified therein are generally scientifically quanti-fiable. In that respect, they can be considered closed and complete, rendering the two principles sufficient for their identification.

56 This conception of network is based on a network’s basic form, the matrix, i.e. a table that charts the links between all entities in the dataset. For an example see e.g. Holzer 2010², 35. Scale-free means that altering the scale of the table does not alter the struc-ture and the matrix shows a relatively static power law distribution no matter how much data is added.

57 On “preferential attachment” see Barabási 2002, 85-92; HI and computer viruses:

Barabási 2002, 123-142; 153f.; the circulation of information: Barabási 2002, 128f.; the structure of the internet: Barabási 2002, 143-153; electricity grid: Barabási 2002, 50;

115f. A scale-free network is characterised by a ratio of connections to nodes that can be expressed as a power law, meaning that a small number of nodes have many connections, while the majority have very few. As a result, the network’s ratio is roughly independent of its absolute size and can be scaled up or down. On scale-free networks see Barabási, Albert-László and Bonabeau, Eric. “Skalenfreie Netze“, in:

Spektrum der Wissenschaft (July 2004), 62–69; Barabási, Albert-László and Albert, Réka.

“Emergence of Scaling in Random Networks”, in: Science 286 (1999), 509-512.

58 80-20 rule: Barabási 2002, 65-78; rich-get-richer phenomenon: Barabási 2002, 79-92.

59 Cf. the analysis of the spread of HIV: Barabási 2002, 123-142.

In the context of complex human interaction networks, however, the problem arises that these are not necessarily scale-free, but are subject to the limits of hu-man ability and connectivity – meaning that if nodes are huhu-man, the maximum amount of social links they can maintain is not unlimited. As a consequence, Barabási’s principles need to be subjected to further scrutiny; in practice, any in-stance of ‘preferential attachment’ will have to be broken down to the strands of interaction that actually produce it. Nevertheless, the results achieved by network analysis in the Natural Sciences do provide a relevant background to this in-vestigation, especially as regards scale-free networks. This will manifest mainly in the terminology adopted and in statements made about the potential dynamics of the reconstructed networks identified. Structural figurations may for example be described as centralised, decentralised or distributed networks.⁶⁰ Centralised net-works are characterised by the existence of a hub, a node that possesses sig-nificantly more links than all the others do. Accordingly such figurations are always likely to be scale-free networks and subject to the regularities identified by Barabási. Decentralised networks possess several such hubs, whereas distributed networks show an even distribution of links across the nodes and lack hubs. The

‘inventor’ of these distinctions, Paul Baran, was in fact concerned with a question of relevance to processes of power, namely the question of maintaining the functionality of communication networks in cases of node failure or hostile attack.

His results were clear: centralised systems were more susceptible to such fallout, since disabling central nodes would single-handedly destroy much of the net-work’s connectivity. Distributed systems did not show this weakness to the same degree as the redundancy of their links was capable of cushioning the impact.

Along with these basic principles, we should also bear the dynamics of ‘pre-ferential attachment’ in mind, as they too can characterise social networks.⁶¹

We are now faced with the question whether it is possible to weld these empirically founded regularities onto a concept of power. It seems to me that directly applying these ‘laws’ to historical social networks is problematic for va-rious reasons, especially in the case of the power processes considered here. Social networks need to be very heavily simplified in order to be able to consider them akin to simple 1/0 circuit boards: as the earlier discussion of the ANT approach has already shown, knowing the number of nodes and the links between them is

60 See Barabási 2002, 144f. with fig. 11.1, and Baran, Paul. On distributed communications:

Introduction to distributed communications networks. 1964, 1f., https://www.rand.org/

content/dam/rand/pubs/research_memoranda/2006/RM3420.pdf (Accessed 21.09.

2017).

61 Cf. Holzer 2010², 33f.; 94f., who draws on Luhmann (1984, 43f.; 60-62) in speaking of the ‘autopoiesis’ of networks. However, the complexity of social networks and the plurality of situationally ‘attractive’ aspects should not be underestimated.

not sufficient for an analysis, as the quality of the links and the plurality of net-works involved are of crucial significance.⁶² This qualitative level is of course due to interactions being embedded in a world order, which is itself produced by an inestimable wealth of interactions.⁶³ Another complicating factor is the obfus-cating effect networks seem to have in life.⁶⁴ This is due to the limited human ability to penetrate complexity: complex socio-political processes are opaque for the actors involved. The more complex these webs of interaction, the greater the significance of the processes of translation identified by Callon, as they limit human perception of contingency, which should be the consequence of such complexity.⁶⁵ The key point is then simply that the network structure of social networks is in itself a cause of societal power processes at the narrative level.

Although it has thus become clear that network science is not directly of use in unravelling power processes, which in turn necessitates further consideration of network approaches to power, we can nevertheless note that structurally con-ceptualising power relationships as networks allows for greater terminological and conceptual precision, which will benefit the study at hand.

A first step in considering such approaches is to assess the merits of social network analysis (SNA), a scientific method that was developed roughly in parallel to network science with the aim of analysing the structures of social networks.⁶⁶ Its focus lies on human relationships, especially those that exceed mere role-play, in that it addresses specific rather than universal interactions.⁶⁷ Applications of

62 Cf. Holzer 2010², 9-11. This may be one of the reasons why the concept of network remains underspecified in Foucault and Mann.

63 Cf. Luhmann 1984, 61. ‘Order’ here describes a state in which the nexus of expecta-tions – or identities – that codifies expected behaviour, operates and adapts relatively smoothly. On ‘normality’ as a symbolic cipher for ‘expected expectation’ see Luhmann 1984, 416.

64 Cf. Barabási 2002, 6-8.

65 The fundamental treatment of contingency used here is Luhmann 1984, 46f.; 152. The sheer number of elements and relationships in the world render it infinitely complex and impose a biological-psychological imperative on any actor to reduce this com-plexity. The aim (in a non-teleological sense) of social order is the reduction of this complexity, which obviously takes the shape of many different figurations. This ge-nerates contingency, defined as the fact that a given selection of elements and relations is neither necessary, nor impossible, but possible in an extremely wide variety of con-figurations. For actors, this establishes ‘risk’ or, perhaps better, uncertainty in acting.

On the societal relevance of power see Luhmann, Niklas. Macht. Stuttgart 1988², 90f.

66 On this see the overview by Holzer 2010², 34-72.

67 Holzer 2010², 11. In practice this means that social network analysis is concerned with firms or groups of friends rather than the many fleeting social contacts of a cashier, for instance. On roles in personal relationships see extensively Goffman, Erving. Wir alle spielen Theater. Zur Selbstdarstellung im Alltag. Munich 2003 (Original: The Presentation

this methodology aim at empirically studying the structures and dynamics of hu-man interaction and at abstracting theoretical patterns from these observations.

Empirical data, compiled into sociomatrices, is visualised as networks and ana-lysed using neutral descriptive terminology and mathematical tools similar to those mentioned above.⁶⁸ Significant theoretical results that have gained wider acclaim are the small-world studies by Stanley Milgram and Mark Granovetter’s strength-of-weak-ties theorem.⁶⁹ The small-world studies showed empirically that the connectivity of social relations is degrees higher than had been previously supposed. The experiments suggested that any given human being is separated from any other by only six links in a global social network, transforming the vast social world into a small world. One should note, however, that the results have been criticised for underestimating the impact of cultural and socio-economic cleavages. Accordingly, the variance in the lengths of the links between individuals can be very substantial.⁷⁰ The strength-of-weak-ties theorem holds that new infor-mation, innovations for instance, predominantly spread via low-intensity rela-tionships rather than close ones, since the former connect social clusters (i.e.

densely enmeshed groups) and thereby increase the potential pool of information, whereas the latter are strongly redundant when it comes to spreading information.

The study of similar figurations has also led to increased scientific interest in

of Self in Everyday Life. New York 1959), who describes the phenomenon of social role-play as a complex process of control determined by collectivised expectation (see p.

217-231).

68 A sociomatrix is basically a table that holds information about nodes and connections with varying degrees of detail. See on this Holzer 2010², 34-36, and the various exam-ples given by Stegbauer, Christian. “Beziehungsnetzwerke im Internet”, in: Johannes Weyer (ed.). Konzepte und Methoden der sozialwissenschaftlichen Netzwerkforschung. Munich 2011², 249-274.

69 Milgram, Stanley. “The Small World Problem”, in: Psychology Today (May 1967), 60-67;

Granovetter, Mark S. “The Strength of Weak Ties”, in: American Journal of Sociology 78 (1973), 1360-1380. On these dynamics see in general Jansen and Diaz-Bone 2011², 76-84; Holzer 2010², 16-22; 63-72; on Granovetter’s observations see also White 2008², 43-45.

70 Criticism was prominently voiced by Kleinfeld, Judith. “The Small World Problem”, in: Society 39 (January-February 2002), 61-66. According to Pierre Bourdieu (Outline of a Theory of Practice. Translated by Richard Nice. Cambridge 1977, 80-85) socio-economic fields tend towards homogeneity and harmony due to the dynamics of habitus-formation; the world within a field is therefore always “smaller” than a world that extends across field boundaries. This obviously constitutes a problem for quan-tifying network analysis.

mediators and brokers, which has revealed the enormous structural significance of such intermediary figures in the social world.⁷¹

What is the value of these abstract results for a data-starved, historical analysis of power relationships? If these are considered processes of communication that spread information, all these theorems are relevant, since the dynamics they iden-tify can apply to the social figurations under discussion here.⁷² That being said, they are usually too general in nature to be of assistance in a detailed analysis of power relationships in the abstract as they are too closely linked to their empirical foundations. Our knowledge of early Hellenistic history is often too good to be content with what we would learn by applying the theoretical statements pro-duced by SNA, but not good enough to perform actual SNA. That is not to say, however, that the theoretical insights of SNA are not valuable in Ancient Studies, for instance where good archaeological material coincides with a dearth of textual sources.⁷³ For the present purposes, SNA is significant more as a heuristic aid that contributes to formulating the network perspective adopted here. This is espec-ially true of the concept of the broker, as such mediators between network clusters can correspond to the OPPs of Callon’s translation model, which similarly hinged on the negotiation of information as a crucial element of power dynamics.⁷⁴

71 See the summary by Holzer 2010², 18-22; 46-48, who also provides an overview of the different kinds of brokerage identified in the Social Sciences.

72 On power as a system of communication see Luhmann 1988², esp. 4-9; cf. also Foucault 1987, 243-247, on power as a double form of subjection: “Schließlich kreisen all diese gegenwärtigen Kämpfe [gegen Macht und Unterwerfung] um dieselbe Frage:

Wer sind wir? Sie weisen die Abstraktionen ab, die ökonomische und ideologische Staatsgewalt, die nicht wissen will, wer wir als Individuen sind, die wissenschaftliche und administrative Inquisition, die bestimmt, wer man sei. Man kann zusammenfassen:

Das Hauptziel dieser Kämpfe ist nicht so sehr der Angriff auf diese oder jene Machtinstitution, Gruppe, Klasse oder Elite, sondern vielmehr auf eine Technik, eine Form von Macht. Diese Form von Macht wird im unmittelbaren Alltagsleben spürbar, welches das Individuum in Kategorien einteilt, ihm seine Individualität aufprägt, es an seine Identität fesselt, ihm ein Gesetz der Wahrheit auferlegt, das es anerkennen muss und das andere an ihm anerkennen müssen. Es ist eine Machtform, die aus Individuen Subjekte macht. Das Wort Subjekt hat einen zweifachen Sinn: vermittels Kontrolle und Abhängigkeit jemandem unterworfen sein und durch Bewusstsein und Selbsterkennt-nis seiner eigenen Identität verhaftet sein. Beide Bedeutungen unterstellen eine Form von Macht, die einen unterwirft und zu jemandes Subjekt macht.” (246f.).

73 On the use of SNA results to make sense of archaeological findings see e.g. Knappett, Carl (ed.). Network Analysis in Archaeology. New Approaches to Regional Interaction. Oxford 2013.

74 Two crucial factors are the exclusivity and redundancy of the network position in question. A simple messenger, for instance, is highly exchangeable and his brokering function hardly exclusive, which generally prevents him from establishing consistent