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The God node: to be or not to be in the network.Ideas for social network analysis.

CIEA, Centro Interdisciplinario de Estudios Avanzados, Universidad Nacional de Tres de Febrero, Caseros, Argentina1

e-mail:)[email protected]

1 The author thanks Pablo Jacovkis, Pablo Dominguez Vaselli, Ines Caridi, Carlos Muoz, Victor Bronstein, fortheir useful comments and suggestions.


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The God node: to be or not to be in the network.Ideas for social network analysis.

CIEA, Centro Interdisciplinario de Estudios Avanzados, Universidad Nacional de Tres de Febrero, Caseros, Argentina

e-mail:)[email protected]

Abstract

Modeling enables new methodological approaches in the social sciences. The models thatsimulate social networks and interactions are useful to represent recurring processes in anabstract form, enriching the dominant empiricism in the social sciences. Networks researchallows a methodological and thematic update in the social sciences. In the analysis of realsocial phenomena, as religion, simultaneous connections with a monotheistic God can beunderstood as unidirectional networks of interactions to a central node (Deity). Humanconnections between individuals believing in that Deity arise around that central node. Thisarticle presents a simple description of a religious social network, with simple centralitymeasures (e.g. InDegree). A high InDegree indicates a more influential Deity. A Deityworshiped by only one person does not boost connections with other faithful. Typically, amonotheistic religious network with a central Deity, is centralized. The important issue is not

whether the node Deity exists, but the connections generated around this node. Understoodas thought experiments, these models can be applied to relevant social theories. Thedevelopment of modeling, enables new methodological approaches in the social sciences,enhances the understanding of social phenomena, and will possibly increase the relevance ofthe social sciences.

Keywords: social network analysis, models in social science, sociology of religion


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1 Introduction

This paper addresses the issue of the application of formal and statistical models to social

networks analysis.

Even if we have different measures and concepts in network problems, it seems

necessary to have a more specific link with real social problems, social theory traditions, and

to extend the concept of network to relevant phenomena from the social theory point of view.

Some examples of the application of these models to social network analysis to

religions might be useful to illustrate the potential of the approach, and to suggest some

possible analysis of the specific issue of social networks in religion.

The development of modeling, enables new methodological approaches in the social

sciences, and enhances the understanding of social phenomena. Incorporate modeling

methodologies will probably increase the relevance of the social sciences.

2. Models in social science

The notion of model has played a fundamental role in science and its application can provide

ideas and hypotheses relevant for social science research. Theoretical models are an

abstraction, developed with the application of certain known formal properties (e.g,

probability theory), and not necessarily based on the analysis of empirical observations.

A statistical model of a social phenomenon may clarify a theoretical issue (explanatory

function), or represent a recurring process in abstract form (Farrar, 1997: 73-101), enriching

the dominant systematic empiricism of the social sciences (Willer, 1996: 319-331).


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Different applications of mathematical models to sociology have been developed2

(Rapoport, 1951, pp. 257-277; Farraro, 1997, pp. 73-101; Simon, 1957). Many of the social

scientists involved in these applications, have used elements of the general theories of

systems, games theory, and the information theory (Kemeny, Snell, 1998). In recent years,

they have applied different software (Mathematica, MATLAB, Repast, SWARM) to

modeling diverse phenomena, as living cells, chemical reactions, collisions of black holes,

traffic, the dynamics of the stock market (Castellani, Hafferty, 2009)3.

Some authors suggest more emphasis in the dynamic analysis in social sciences, the use of

formal languages (Hanemann, Riddle, 2005), and computer-aided simulations4.

Ludwig von Bertalanffy emphasized the analysis of open systems and nonlinear

dynamics as a key in the study of systems (Von Bertalanffy, 1968; Luhmann, 1991, Coleman,

1960: 1-149). Coleman (1964), a renowned American sociological theorist, links the modern

idea of rational choice with the older idea of stochastic processes of social contagion. In

general, he suggests relations between social contagion and assumptions of chances of

individual change in a short period of time, a procedure known in the mathematics as

stochastic processes.

Many of these ideas have been used in social networks analysis, and in simulations in

computer of social phenomena (Hummon, Farraro 1995: 79-87; Freeman, 1984: 343-360).

Theories and software applications (Wasserman, 1994) have been developed for analysis and

2 A key paper in the latter work, originally published in 1951 in the American Sociological Review, formalizedthe influential social system theory of Homans (1950). The Homans book has been a continual source ofinsights and data for social network analysts, while Simon's paper has been an exemplar for sociologists whoformalize theories in terms of differential equations.3

... The list of pioneering scholars in this field includes John Holland, Robert Axelrod, Stephan Wolfram andJoshua Epstein, to name a few. It also includes, right up there at the top, the British sociologist, Nigel Gilbert,editor of the international periodical, Journal of Artificial Societies and Social Simulation (e.g., Gilbert andAbbott 2005; Gilbert and Troitzsch 2005) (Castellani, Hafferty, 2009).4 This book has the inmodest goal of reorienting how many social scientist go about building and working withtheory. First, I believe that we would benefit from a shift in substantive focus towards less concern with staticand more concern with dynamics. Second, I believe than social scientists theorethical work would be muchadvanced by the use of powerful and flexible formal languages for expressing theory, as opposed to the currentpractice of using either everyday language or mathematics. Third, I am advocating to the use of computer assisted simulation methods as a useful way for social theorist in the social science Hanemann (Hanemman,1988).


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graphical visualization of social networks (Freeman, 1984; Hanneman, Riddle, 2008; see

also, software Cytoscape5), for the analysis of semantic networks (Izquierdo, Hanneman,

2006), or analysis of the relationship between ego and alter (e.g. software EGO - NET).

Specific data processing software for the analysis of social networks have been developed

(e.g. UCINET, Hanneman, 2008; PAJEK; Igraph, for R software). Also, different kinds of

software for representing complex social networks, and plot relations among members,

friends, groups (http://scouta.com; http://www.digitalismo.com) have been developed in the

last years6. Software Mathematica has incorporated these contents.

Concepts as network density (the number of actual connections on the possible),

social distance or geodesic distance, and the number of links that separate an actor from

others (Hanneman, 1988), homophily (e.g., "love of the same"; tendency of individuals to

associate and bond with similar others), structural holes (absence of ties between two parts

of a network), and the number of links that separate an actor from others (Hanneman &

Riddle, 2005) can be applied to the analysis of social interaction networks7.

These concepts have been also used in the analysis of social capital. Social capital

broadly refers to the resources accumulated through the relationships among people

(Coleman, 1988). Social capital is an elastic term with a variety of definitions in multiple

fields (Adler & Kwon, 2002), conceived of as both a cause and effect (Resnick, 2001;

Williams, 2006). Bourdieu and Wacquant (1992) define social capital as "the sum of the

resources, actual or virtual, that accrue to an individual or a group by virtue of possessing a

durable network of more or less institutionalized relationships of mutual acquaintance and

recognition" (p. 14). The resources from these relationships can differ in form and function.

5 Open source software platform for visualizing complex-networks and integrating these with any type ofattribute data.6 Network theory builds on the assumption that a cause, an effect, or an association between aspects involvesomething that can be conceptualized as a network.7Network has become relevant for Internet and virtual connections analysis: stimulating connections andforging new links at all levels of organization corporate, institutional, national, global and a concern thatsuch connectivity may detract from local interaction (Haythornthwaite, 2011).


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Social networks change over time as relationships are created or abandoned.

Particularly significant changes in social networks may affect one's social capital, as when a

person moves from the geographic location in which their network was formed and thus loses

access to those social resources8

. Also, this type of measurement has specific effects on

individuals. In the case of networks, an individual who locates himself in a sociogram, with

his exchanges and preferences, is not the same before and after he sees himself in the

representation of his social network9.

3. Models of social networks.

Some ideas regarding the application of formalizations and models can be applied to the

analysis of social networks.

Social networks are interactions between nodes, connected with different kind of

social interdependency. In networks we can formalize social interaction occurring

simultaneously, or the dynamics of changes of these networks.

In sociology, interpersonal ties are defined as information-carrying connections

between people. These ties can also be potential; in this case they are usually referred as

latent ties (Haythornthwaite, 2005). For social network analysis, we define possible

connections between nodes, vertices, or elements; network theory builds on the assumption

that a cause, an effect, or an association between aspects involve something that can be

conceptualized as a network (Brandes, Robins, McCRaine, Wasserman, 2013). This type of

analysis requires the conceptual distinction between elements (we assume that elements

8 Putnam (2000) argues that one of the possible causes of decreased social capital in the U.S. is the increase infamilies moving for job reasons; other research has explored the role of the Internet in these transitions(Cummings, Lee, & Kraut, 2006; Wellman et al., 2001).9 Something similar happens with the classification of social classes; if an individual is classified in a certainsocial group according to a certain methodology, and he is informed about that classification, he is not thesame person before and after having been informed of the outcome (Oliva, 2006).


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cannot be subdivided) and relations (similar concepts are used in systems theories, Luhmann,

1990)10.

Related to network analysis, we find in mathematics and computer science, graph

theory. Graphs are mathematical structures used to model pairwise relations between objects.

A "graph" in this context is made up of "vertices" or "nodes" and lines called edges that

connect them (Biggs, Lloyd, Wilson, 1986).

We distinguish the properties of the network (collective property, or emergent

properties), from the properties of individuals (or elements, or nodes) that compose it.

Emergent properties are not directly inferred from the properties of the individual parts.

From these matrices, we can analyze different properties of the connections, such as

the distances between individuals, the density of nodes11, and the direction of the

connections.

Some measures describe the overall network, while others describe the structural

position of a vertex, or a subnet - partition of the network (De Nooy, Mrvar, Batajeli, 2011).

The connections can have different weights according to some kind of quantifying of the

relationships between nodes.

Networks contain elements; the relationships between elements are often also

represented in matrixes. The simplest relation attribute is binary: if a relation is present we

codify 1, and 0 if it is not. This type of matrix is called adjacency matrix. Standard databases

in social science have an analysis unit (usually rows) associated with a variable (usually a

column) and a value. In network analysis, the databases are not focused to analyze attributes

of the actors, but to the relationships between them.

10 From the viewpoint of systems theory, the elements are configured and evolve in an environment, based onthe relationships (Luhmann, 1990), the relationships in a system are constituent of the elements, in other terms.11 On the concept of density of a network, we read in Hanemman "If we measure the ties Among actors withvalues (Strengths, closeness, probabilities, etc..), density is usually we defined as the sum of the values of all tiesdivided by the number of possible ties "(Hanemman, 2008). In other sections, we analyze, for example,weighted data of individuals with varying symbols or resource sharing between them.


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4. Latent networks

A real connection is different from apotential connection. A potential or latent network can

be thought as a set of possible relations -- e.g. the arrangement of social roles in an institution

is as a potential or latent network between the members of the institution --; religions have

potential or latent networks.

In prophetic religions, usually faithful are connected through central nodes (e.g.

deities or gods); a religion, as we will discuss later, is usually also a latent network with a

central deity node.

We can also describe languages as potential networks. An individual who speaks a

language participates in a potential network of semantic conventions. If you are Spanish

speaking, you can potentially interact in different ways with other speakers of that language.

If you are not, you cant, and you are not included in that potential network. Real

connections are time and energy consuming (cf. Giddens, 1987), but latent networks are not,

since they are conventions.

If you own money of a specific currency you can interact in different ways (e.g buy)

in a potential network usually referred as market--, in which all of the nodes have a

potential agreement to exchange goods or services in that currency. If you do not own that

currency, you are not included in this particular potential network12.

Potential o latent networks may be thought as full, or saturated, networks. This means

that if you speak a language A you could potentially communicate with anyone who is A

speaker. In Figure 1 we see a network of four nodes (a, b, c, d) that have all the six possible

12 In a company or organization, everyone has some kind of position. This situation can be described as a latentsaturated network.


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bidirectional connections among them. In this network all nodes are connected13; we call this

a saturated connection -- it has all the possible interactions --.

Imagine now that the lines in Figure 1 correspond to the use of a symbol (one word)

for designating, for example, an object. A person who knows a language participates in a

potential network of semantic conventions. Languages with more speakers, refer to more

generalized potential network connections.

Figure 1

That is, although the four individuals may or may not connect, the connection is

possible, as a member of the network. That is, they have potential or latent connections.

5. To be or not to be in a latent network

The fact that certain individuals (nodes) participate in a network is sometimes taken as

a fact (nodes a, b, c, dparticipate in the network X) and not as a theoretical problem. For

example in network analysis we analyze a group of individuals included in a latent social

network, and the connections established between them. In this analysis the participation or

not in a latent network, is given as a fact, not as a problem. But this is not trivial.

We have to answer the question which is the probability a, b, c, dof an individual to

be or not to be in the latent network?. We assume that all nodes can be classified in a binary

variable: to be in a network (1) or not (0). As in Hamlets famous soliloquy by Shakespeare,

to be or not to be, has also a probability.

13 Represented with software UCINET.


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Applying these type of dichotomous variables to different social phenomena (to name

an object with a certain word14, believe or not in God, and others), we can obtain theoretical

conclusions.

Let us differentiate nodes in and out of the latent network. In Figure 2 individual a, b,

c, are in the latent network (array), but not dand e, and the other individuals (gray area) in the

outside world -- this is similar to the distinction between environment and system in terms of

systemic theory, as in Luhmann (1991) --.

If we may describe individuals in a society as included in a latent network (1) or not

(0), we have a dichotomous variable.

Figure 2

We detect two problems:

a) To be or not in a latent network.

b) Once in the latent network, to interact or not to interact with the other nodes.

Problems a) and b) are different:

a) A simple example would be to participate or not in a social network (regardless of

the connections you make with other members of that network). Once you are a registered in

Facebook or Twitter, you can interact with a set of possible connections (although nobody

interacts with all the possible connections in this social network). This is problem a).

Another example is to refer or not to certain object with a specific and arbitrary word (such as

tree, or rbol); that is for example, being spanish speaking or not.

14 The problem here is, which is the probability of an individual to designate with the arbitrary word tree (orrbol in spanish) an object tree?.


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b) Once connected to Facebook, you establish different real interactions with other

people in that social network. Or if you are English speaking, you communicate with other

people who speak English.

6. Real networks

Upon a potential network, we find real connections. This is what we called before,

problem b). Suppose a network of 7 people, with connections as observed in the matrix of

Table 1.

Table 1

Figure 3 is a visual representation of this matrix.

Figure 3.

From the rows in the matrix, connections go from one individual to other. Individual b

has no connections to other nodes, and has no interactions. But the failure of interacting with

other nodes in a latent network, is different to not being in it. Suppose that among the seven

persons included in the matrix (denoted from a to g) there is a potential network (an

agreement for possible objects interchange using some currency, or to designate objects with

a set of specific words language--). In Figure 3, b is a member of a latent network, but does

not interact with the other members, the situation can be represented in the matrix of Table 1;

b (in Figure 3 the isolated point) does not interact (does not connect to anyone, no one is

connected with him), even if he is included in this latent network. For example we suppose


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seven nodes that are Facebook users but the individual b has no relations or friends in this

social network.

7. The God node

To illustrate the use of these ideas, we apply properties of probability theory to social

network and social systems analysis, and the analysis of real social phenomena, as the

religious networks.

In religion, we classify what is considered sacred (1) or non-sacred (0), in this

example a generic deity. A religion with a monotheistic God can be understood as

unidirectional interactions of the nodes to a central node (God) which form a network. This

central node usually has no direct demonstrable empirical existence; it has a virtual,

subjective, or spiritual, existence. The effects are similar to those of node that has an

empirically verifiable existence. What actually exists without doubt, are the connections

between living believers to that node. These connections involve and create potential

connections between believers, mediated by the central node Deity (or God). Figure 4 is a

graphic representation of a possible religious network.

Figure 4.

In Figure 4, the node d is more religious, regarding his contact with God. The

individual g, which is in the latent network, is not connected with God (this may describe a

situation of a person baptized in a Church or religious institution, who has now converted to

an atheistic, and now has no interaction with God), which also means less connectivity with

the rest of the network. Linton Freeman (one of the authors of UCINET) developed basic


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measures of the centrality of actors based on their degree, and the overall centralization of

graphs.

The data matrix of the network formed by seven believers in a God (represented in

Figure 4) is shown in Table 2.

Table 2

A simple descriptions of previous network with this measures, calculated with software

UCINET is shown in Table 3.

Table 3.

The node "God" has InDegree 10; this indicates a total of 10 targeted relationships

from other nodes to "Deity"; 5 of these relationships come from the node d, this individual

could be considered the most religious. The InDegree measure the influence of God or Deity.

A God worshiped by one person, does not boost connections with other faithful

(NrmOutDeg15 is the normalized out-degree and degree NrmInDeg is the standard input). A

God with many connections is influential, maybe deemed powerful (upon which --typically--

the faithful would like to influence).

In Figure 4, node g is not related to Deity. This central node (Deity) has OutDegree 0,denoting thatDeity has no direct interaction with anyone. Considering thatDeity does not

15 E.g. node d has an outdegree of 5; in this case, we divide the number of outdegree, by the number of nodesminus 1 (here 8-1=7), divided by the maximum outdegree, here 5. So the number here of normalized outdegreeis (5 / 7 / 5) * 100; that is, 14.286.


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interact with itself (we could use other assumptions), we do not include data on the diagonals

of the matrix. Also, we suppose that Deity can not be subdivided.

All these connections in the figures should be thought as a film and not a

photography, since they are dynamic relations and they may change in the future. At the same

time, this has relation with uncertainty and the difference between the past and the future.

There is a theoretical relationship between randomness and time: the past has no random, the

future has random, and this is a difference between the present and the past16 (Oliva, 2011).

Also, we can measure the network centralization; a star network is the most

centralized17. Typically a religious network with a central Deity is centralized. A random

network, as represented in Figure 5 for seven nodes, has different patterns. This kind

network does not describe the usual situation of a religion with a central deity node.

Figure 5.

The beauty of this type of network is that, while the node deity has not a physical

existence, all the other nodes and relationships are strictly empirical, e.g. there are religious

individuals connecting between them. Between two individuals who share the same node

Deity, a potential non direct connection exists, mediated through the central node.

16 This is a big difference between history (which analyzes events with no random), and social sciences like

sociology, usually focuses to contemporary societies and future social events, which have random components.17 The star network is the most centralized or most unequal possible network for any number of actors. In thestar network, all the actors but one have degree of one, and the "star" has degree of the number of actors, lessone. Freeman felt that it would be useful to express the degree of variability in the degrees of actors in ourobserved network as a percentage of that in a star network of the same size. This is how the Freeman graphcentralization measures can be understood: they express the degree of inequality or variance in our network as apercentage of that of a perfect star network of the same size. In the current case, the out-degree graphcentralization is 51% and the in-degree graph centralization 38% of these theoretical maximums. We wouldarrive at the conclusion that there is a substantial amount of concentration or centralization in this wholenetwork. That is, the power of individual actors varies rather substantially, and this means that, overall,positional advantages are rather unequally distributed in this network.


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This type of relation to a Deity requires means of symbolic connection, usually a

language or multiple languages (the Christian God has words in different languages that refer

to it, Dieu, Dios, etc., so religious connections between speakers of different languages is

possible).

From this viewpoint, the important issue is not whether the node Deity exists, but the

connections generated around this node. We cannot launch an empirical theology (so as to

corroborate the existence of a deity), but we know that the social network connected by these

deities are real. The truth of the existence of God is irrelevant from a sociological point of

view: what matters are its social consequences, in this case, the social networks that are

created around these nodes.

The description of a Western and Christian society is the picture of a great latent

and multicultural network. The EastWest Schism (the medieval division of Chalcedonian

Christianity into Eastern Greek- and Western Latin- branches, which later became

commonly known as the Eastern Orthodox Church and the Roman Catholic Church,

respectively) can be thought as splits in these social networks18.

The inclusion of individuals in latent social networks as religions, tend to limit the

likelihood of what is considered socially disruptive or harmful behavior of the individual; as

in the Ten Commandments (Decalogue), a set of biblical principles relating to ethics and

worship, which play a fundamental role in Judaism and Christianity; or the aleyas in the

Quran. There are different social technologies that limit certain unwanted actions. For

example the institutions of religions, define what should be considered sins or non sins.There are usually institutional binary structures of social differentiation as good / bad, in-law

/ outlaw (usually adopting a mode of dummy variables, 0 and 1), used as a social control

18 The EastWest Schism is one of the two schisms to which the term "Great Schism" is applied (theother being the Western Schism).


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technology that organizes a more predictable human behavior and aids to reduce its

contingency.

Predictability is important also in economy. The legend written in the dollars "In God

we trust" can be understood as: we have trustful economic linkage between those individuals

who are connected by the node God (although we do not know if God trusts us). Those who

share connections through the God node, are more trustful for economic exchanges. Both

potential networks tend to overlap. This could be an interpretation of the influence of

protestant ethics in the rise of capitalism, as Weber noted.

When Nietzsche announced that God is dead, he was leaving an extensive network

without a central node, to allow potential exchanges.

These potential networks have also an intergenerational extent, and also include nodes

with individuals who are no longer alive. Populist Latin-American political leaders created

latent networks similar to religious connections, which have long outlived them (e.g.

peronism). The leader builds connections, and nodes that connect through it. For a social

leader, sometimes death is usually a booster of the connections properties; the connections

are not interfered by his decisions in lifetime, and so there is no interference between the

nodes connected by the central node. Everyone may connect through a dead node, but the

dead node will connect with no one. Peronism in Argentina is a network with a central

(node), a not living leader (Pern); the discussion about the kind of leadership and political

decisions is usually overlapped by the convenience of the nodes to be connected. The central

node is a mean of connections between other members of this network, and a source of

political power. This style of political parties aims to replace the lack of economical

influence (for instance people who live in poor households), with political power. Maybe

Hugo Chavez in Venezuela will develop a similar latent network. In some occasions, these

political networks are superimposed to other networks, like religious ones.


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Marx may be also considered as a central node of Marxism, more as an

international and historically extended network. That is why discussion with orthodox

Marxists sounds much as a discussion in religious topics.

In social activity, coordination and star like networks are usually associated with

power. A coordinated group with a central node or leader, is more powerful. The leader has a

role in power construction. In some historical circumstances, the main objective of the

members of an organized group is to be coordinated (adept, affiliates, voters, union members,

churches); in this cases, the ideological or philosophical orientation of the leaders may

become less important, regarding the main objective of the members: to be coordinated, and

to maintain a latent network cohesive in time. In functionalist theory, and its

characterization of role structures that mediate between individual and society, issues related

to coordination and latent networks arise. Why is there a single role of president in a

presidential system (or only one king in a monarchy)? (Parsons, 1951). Possibly, because

only one president (a star like network) allows efficient coordination of actions. In that

sense, this also explains why monotheism may be though as more effective than polytheism.

If there is a central node (one God almighty) is easier to connect the rest of the individuals in

the religious node. In social organization, creation of latent networks is essential. Many of

the structures of this latent networks are hierarchical. Hierarchical structures are widespread

in human social systems (with leadership, hierarchies and central nodes).

The past control of some material resource or property of our dead ancestors,

generates a kind of asymmetrical relationship with them (in the sense that the connection is

not bidirectional; living individuals cannot interact with the dead, but some deceased

continue influencing decisions of the living). This usually occurs in the heritage, for example.

The transfer of the property of material resources owned by an individual when he dies -- at

least in capitalist systems-- , typically follows the pattern of genetic links (e.g., the children


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inherit properties from their parents). The same can be said about power in monarchies:

usually the prince or princess inherits the power of father king or mother queen, and between

them there are only genetic links; usually in European history the procedure of power

heritage was thought in terms of God`s law, that is, the social networks generated by a central

node, God.

Durkheim (1912) understood that religion was the most fundamental social institution

of humankind, and one that gave rise to other social forms. It was the religion that gave

humanity the strongest sense of collective consciousness and provides a number of social

networks from a central node. He defined religion as "a unified system of beliefs and

practices relative to sacred things, that is to say set apart and forbidden, beliefs and practices

which unite into one single moral community, called a church, all those who adhere to them"

(1965 [1912]: 62). In religion we classify sacred (1) o non sacred (0) (Durkheim, 1912)19, a

binary variable that may have a counterpart in to be included or not in a particular latent

network.

These religious connections often tend to overlap with other networks, as languages,

nationalities, economic exchanges that require trustful relations (e.g. credits), symbolic

media20, and even between individuals that share genetic type links: Afro-American

religions, for example.

Relations with the Deity usually has a number of nodes that mediate interactions (e.g.,

in Christian culture, the saints). Perhaps the monotheistic system is more efficient as a latent

network, in a similar form as we usually have only one president. If there is a central node

19 Durkheim (1921) identifies as sacred the ideas that cannot be properly explained, that inspire awe and areconsidered worthy of spiritual respect or devotion. He defined sacred things as collective ideals that have fixedthemselves on material objects... they are only collective forces hypostasized, that is to say, moral forces.20 Chernilo (2002) explains the relationship between symbolic media and coordination: "What are the media?What does really conceptualize the theory of generalized symbolic media? In short, the media are concreteforms of social coordination, they are the most dynamic social coordination constant present in modernsocieties. ... Money, power, love, truth, and the media is the way that social subsystems, first regulate theirinternal functioning, contributing to their own differentiation and, secondly, how they interrelate with each otherto produce coordination between subsystems "(Chernilo, 2002).


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(one God Almighty) is easier to connect the rest of the individuals in the node. In a

polytheistic religion the connection to other nodes can follow different paths (if we have

node A and B connected to God 1 and God 2 at the same time, bridges between A and B

could be through God 1 or God 2) which makes more combinations possible and requires

selection, and complicates the network structure. In a polytheistic religion we could

represent relations between the different gods in a network graph in which each god will be a

node. In Greek mythology, for example, there were twelve gods with linkages including

family and genetic path ties. Also, among gods of a polytheistic system, power issues and

family struggles arise.

In this sense, these networks can be applied to genetic ties connections with a family

organization. In general, these connections are genetic - often have biological hierarchical

order according to the degree of linkage in genes of different individuals (parents, siblings,

uncles, cousins). Once born, genetic links cease to be physical (in the sense that the father

and son are different bodies, although shared genes), to be cultural. From these relationships

are relationships of incest, which necessarily have some kind of genetic linkage, and

foundational of society in terms of Levi Strauss (1969).

From the point of view of religious organizations also need to classify certain

behaviors as desirable or undesirable, in the case of Western religions, for example, a

classification of sin / not sin, or sacred / non sacred.

8. The globalization of religion

Generally speaking, in human history we observe an expanding number of potential

connections between human beings: urban settlement, dynamic density increase -- pointed

out by Durkheim (1893) in The Division of Labour in Society-- , writing, printing, media and


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communication technology, increase the probability of connections21, and the extension of

potential social networks. The growth of the extent of these potential networks, requires

increasing selectivity in interaction, given the same physical time of human life. This issues

are discussed in system theory and cybernetics (Ashby, 1956), and is related to the

discussion on reduction of complexity (Luhmann, 1990), and the emergence of a global

village22, a metaphor by McLuhan (1964).

Connectivity increases, and potential networks become more complex. This allows

new perspectives of phenomena such as population growth, which, simplifying, implies more

potential social connections. Also, this conceptualization is useful in understanding

globalization. Social networks extension can be understood as a part of the global embrace,

referred by McLuhan (1964): Today, after more than a century of electric technology, we

have extended our central nervous system in a global embrace, abolishing both space and

time as far as our planet is concerned. Formal and empirical analysis of social networks

extensions in the global embrace will be useful and necessary. Changes in the extension of

social networks with the new technologies like Internet -- will have important social

consequences23. In religion, globalization and religious integration (in the extreme argument,

amalgamation in a global religion with a single globalized deity as a religious central node)

perhaps would reduce the interreligious and intercultural conflict.

21 Physical objects, such as roads, increase the number of potential connections. The time to move

between two points is related to the probability of connection. Increasing urbanization enables more potential

connections between people settled in short distances. In our modern society physical connectivity is no longeressential to social connectivity; we can use now virtual connections.22 The metaphor by McLuhan (1964) of the emergence of a global village, can be seen in this way: we

have a larger number of potential connections, but the time for possible relationships remains similar to that in avillage life. Although we have more possible connections, we do not live longer, and we do not have time for anunlimited number of connections. Nevertheless, new technologies allow us to choose our connections withgreater selectivity in this global village, and the scenario is different from the classical vil lage.

23 Societies have always been shaped more by the nature of the media by which men communicatethan by the content of the communication. The personal and social consequences of any medium - that is, of anyextension of ourselves - result from the new scale that is introduced into our affairs by each extension ofourselves, or by any new technology (McLuhan, 1964).


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21

Using probability theory, we can calculate some formal characteristics of a network

(in this case, we will describe the state of a network, or emergent properties). Social

networks have different probabilities, as the most widespread languages or religions are most

unlikely.

Methodologically, we assume that all nodes in a latent network N (e.g. a, b, c, d) can

be classified in a dichotomous variable (e.g., node are included or not in latent network N).

Given a theoretical probability to interact, we can calculate a probability of a particular

configuration of interactions in N.

Suppose that we know that to be a member in a latent social network (say, N) is

equally likely (that is, a probability of 1/2). In the case represented by Figure 1, if the four

individuals (a, b, c, d) have a probability of 1/2 to be a member of network N (e.g. a specific

religion), the probability of N is equal to 1 / 16 (0.0625).

With these assumptions, there is a variable that describes the individual (the

probability he is included in a network), and other emergent variable that describes a

characteristic of the network N (the probability to see a group of individuals connected to N).

The likelihood of a particular configuration of a social network (collective property) depends

on the probability of occurrence of an individual's participation in the network. With five

individuals or nodes, the probability of interaction with the network is 1/32 (0.03125); with

seven (a configuration similar to that seen in Figure 4), ceteris paribus, the probability is

1/128, 0.007812. The probability decreases exponentially as you increase the number of

nodes in the network, as shown in Table 4.

Table 4


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Keeping invariant the individual probability of a connection (column C), the larger the

number of individuals or nodes, more unlikely is the saturated configuration. As shown in

column B the probability decreases exponentially. With more individuals in the network (e.g.

a more extended religion), the emergence of a latent saturated connection is a most unlikely

event. These models suggest that it is necessary to have sophisticated social technology to

create extended latent networks.

The conclusion from this type of model is that a network of 5 individuals (0.03125) is

more probable comparing to one of 6 (0.015625) or 7 (0.0078125). The more nodes in a

latent network, the lower the probability of saturated connection. A potential network (a

religion) of a hundred individuals is more unlikely than one of ten nodes.

In society it is necessary for individuals to be included in latent networks (we could

not talk, buy, pray, without this kind of networks). Usually, these social networks are an

intergenerational heritage: they go beyond one generation to the next one, we and receive

currency, money, language, religion from our ancestors.

Modifying the data in column C of Table 4, that is, the individual probability of the

connection, increasing the probability of each individual interaction with the network to 0.9

(instead of 0.5 in column C), the probability of a network of 7 nodes connected increases to

0.478296. With the assumption of the probability 0.9 for one of the seven nodes, we have:

!0!*7

)9.0!*(7)7(

7

hitsP (1)

The result is: P (7 hits) = 0.478296.

Even if in a real social situation we would not be able to calculate the probability of

each individual to be or not in a potential network, we know that a probability has always a

value between 0 and 1. And with some assumptions we can model in a theoretical approach.


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Increasing the number of nodes in potential networks exponentially increases the

number of possible relationships between nodes. If X is a set of n elements, we may

calculate combinations of n objects taken in m different ways. In this case, the element are

nodes, and m=2 is a bidirectional relations (regardless of order and the direction, that is, the

relationship between the element b to a is equal to the relation between a to b). To calculate

the combinations, then:

(2)

)!!*(

!)(

mnm

n

m

nCmn

(2)

If the network has two nodes, there is only 1 possible bidirectional relationship. For 3

nodes, there are 3 possible. For 5 nodes we will have 10 relationships (bidirectional links).

For 6, 15. In Equation 2, if we change m, we could calculate the number of possible triads

(and more complex relations). For a four-node network, there are 4 possible triads; for one of

5, 10, and so on.

We see then that the network complexity grows rapidly as we increase the number of

nodes. Using Pajek, we can visualize a random network with 10 nodes24, 45 bidirectional

links (Figure 6).

Figure 6

When compared to a similar network, but with 20 nodes, we will notice quickly the

change in complexity. The network represented in Figure 7 is a network of 20 nodes (with

190 possible bidirectional relationships).

24 With non-directed relations.


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24

Figure 7

These simple graphs show the rapid increase in the complexity of a network,

considering only the bidirectional links.

In real social situations, it could be difficult to calculate these probabilities. But we

usually have a subjective and qualitative perception - based on the factual probability of the

probability of social events (Oliva, 2011). Therefore, the subjective assessment guides actual

decisions and actions of the actors.

There is also, a subjective perception of probability. That is, the translation into the

subjectivity of the actors is an important aspect of these models. Individuals perceive

likelihood of social phenomenon. It is not necessary for this subjective panorama to have

accurate estimates of probabilities. For example we take care of falling even if we do not

know about the equation that describes gravity acceleration. In a similar way we could

consider that the probability of coordination of networks is something that the individual

perceives without calculations. Many of the actions and strategies of inividuals are based on

qualitative perceptions of the likelihood of possible scenarios.


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25

9. Conclusions

Modeling enables new methodological approaches in the social sciences. The models may be

useful in methodological approaches to social science, just as they are useful in many other

sciences.

The models that simulate social networks and interactions are useful in order to

understand them theoretically. In this sense, these models are explanatory, and represent

recurring processes in the abstract form (Farrar, 1997), enriching the dominant empiricism

in the social sciences (Willer, 1996). In that sense, formalization helps us understand the

nature of the phenomenon25.

Networks research allows a methodological and thematic update in the social sciences

(Wasserman, 1994; Batagelj, Mrvar, 2003). Statistics and models have been used for

inference, analysis and summarization of data on social phenomena, but it has been used to a

lesser extent on the modeling of social theory. The challenge is to apply statistical

formalization to sociological theories and models. The development of the model allows new

methodological approaches in social sciences, even if only they are not directly related to

observable data.

In social life is necessary to create latent networks. In general, the models show that

a greater number of nodes connected, the more unlikely is the latent network. The very low

probability of extended latent networks, makes us think of possible mechanisms to achieve

25 In science, theory simplifies. In fields such as sociology (Willer, 1996: 319-331), empirical researchnever simplifies, because its object is not to theory if not find "findings" and "findings" are typical of aparticular moment in time and space (Willer, 1996).


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them. Thus, this type of model may be useful in understanding social action, language,

power, hierarchies, and religions.

In the approach we used, we start the analysis from two problems: to be or not in a

latent network (Problem A), and once included the network, to interact or not to interact with

the other nodes (Problem B). To participate or not in a social network (e.g. to be registered in

Facebook or Twitter) is an example of Problem A, and what kind of real interactions you

establish with other people in that social network is an example of Problem B.

We can use these concepts in the analysis of real social phenomena, as religion.

Simultaneous connections with a monotheistic God can be understood as

unidirectional networks of interactions to a central node (Deity). Real connections arise

around that central node, between individuals believing in that Deity. The important issue is

not whether the node Deity exists, but the connections generated around this node. The truth

of the existence of God is irrelevant from this viewpoint, and what matters are its social

consequences, in this case, the social networks that are created around these nodes.

In some occasions, these political networks are superimposed to other networks, like

religious ones. Populist Latin-American political leaders created latent networks similar to

religious connections, which have that have long outlived the dead leaders.

From a methodological point of view, we can describe individuals by their significant

network inclusions / exclusion with dichotomous variables: for example, religious distinction

between sacred (1) and non sacred (0).

Even while the specific calculation of these probabilities in a real situation is

complicated or impossible, we could assign probabilities of being or not in a network (a

probability is always a number between 0 and 1), to achieve the calculations of complexity of

social networks.


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These models do not necessarily have to be quantitative and in many cases require

qualitative formulation.

The development of these applications and models is potentially fruitful. These

models of social phenomena, require the transdisciplinarity in knowledge. Sometimes, it is

necessary to overcome some difficulties of mathematization in disciplines such as sociology.

Social scientists often wary of mathematics because they are used to perform real operations

on nature26. Operations on societies in general are more complex, and may be subject to

negative moral evaluations, as the application of experimental methods in the social sciences

is usually considered inappropriate (Marradi, 2007).

Understood as thought experiments, these models can be applied to relevant social

theories. In this sense, the modeling could provide new analytical elements.

The development of modeling, enables new methodological approaches in the social

sciences, and enhance the understanding of social phenomena. Incorporate modeling

methodologies will possibly increase the relevance of the social sciences.

26 Barriga indicates that "the basic orientation of science, in my opinion, should always be open to new ways ofthinking and doing science. Many of the great advances in science, whatever they may be, are due precisely tothe fact that some people had the courage to think outside the box, to expand their ways of thinking beyond whatwe were always told that it should be to do science. Why is it, then, that we hold both our methodologicalparadigms? Why is it so hard for us to recognize the common ground of the two methodological paradigms,which both seek to generate knowledge about social phenomena? (Barriga, 2007) "


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Figures and tables

Fig. 1. A network with four nodes and all possible two-way connections (UCINET).

d e (R)

a b ca

bc

d, e, (R) rest of the individuals in the world.

Fig. 2. A matrix of a potential network with three nodes, and all the other individuals


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Table1. Data matrix of relationship between seven nodes

a b c d e f g

a 1 1 1 1 1 1b

c 1 1 1 1 1 1

d 1 1 1 1 1 1

e 1 1 1 1 1 1

f 1 1 1 1 1 1

g 1 1 1 1 1 1

Fig. 3. A network with 7 nodes with 6 bidirectional connections (UCINET).


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Fig. 4. Representation of a network of 7 individuals and its connections to a non empirical node (God).


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Table 2. Matrix of nodes in a religious network.

a b c d e f g Goda 1

b 1

c 1d 5e 1f 1gGod

Table 3. Centrality measures of matrix of nodes in a religious network.

FREEMAN' S DEGREE CENTRALI TY MEASURES- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

Di agonal val i d? NOModel : ASYMMETRI CI nput dat aset : dei ( D: \ dei )

1 2 3 4 Out Degr ee I nDegr ee Nr mOut Deg Nr mI nDeg - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 4 d 5. 000 0. 000 14. 286 0. 000 1 a 1. 000 0. 000 2. 857 0. 000 3 c 1. 000 0. 000 2. 857 0. 000 2 b 1. 000 0. 000 2. 857 0. 000 5 e 1. 000 0. 000 2. 857 0. 000 6 f 1. 000 0. 000 2. 857 0. 000 7 g 0. 000 0. 000 0. 000 0. 000 8 God 0. 000 10. 000 0. 000 28. 571

DESCRI PTI VE STATI STI CS

1 2 3 4 Out Degr ee I nDegr ee Nr mOut Deg Nr mI nDeg - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 1 Mean 1. 250 1. 250 3. 571 3. 571 2 St d Dev 1. 479 3. 307 4. 226 9. 449 3 Sum 10. 000 10. 000 28. 571 28. 571 4 Var i ance 2. 188 10. 938 17. 857 89. 286 5 SSQ 30. 000 100. 000 244. 898 816. 327 6 MCSSQ 17. 500 87. 500 142. 857 714. 286 7 Euc Norm 5. 477 10. 000 15. 649 28. 571 8 Mi ni mum 0. 000 0. 000 0. 000 0. 000 9 Maxi mum 5. 000 10. 000 14. 286 28. 571

Network Cent r al i zat i on ( Outdegree) = 12. 245%Network Cent r al i zati on ( I ndegr ee) = 28. 571%

Note: For val ued dat a, t he nor mal i zed cent r al i t y may be l arger t han 100. Al so, t he cent ral i zat i on st at i sti c i s di vi ded by t he maxi mum val ue i n t he i nput dat aset .


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Fig. 5. A network with 7 nodes with all possible bidirectional connections (figure made with software UCINET).


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Table 4. Number of nodes in the network, probability of saturated connection, probability of the node to

interact with potential network.

A

Number of nodesin the network

B

Probability ofsaturated

connection

C

Probability of thenode to be in a

potential network1 0.5 0.52 0.25 0.53 0.125 0.54 0.0625 0.55 0.03125 0.56 0.015625 0.57 0.0078125 0.5

Fig. 6. A network with 10 nodes with all possible connections (software Pajek).


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Fig. 7. A network with 20 nodes with all possible connections (software Pajek).


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Acknowledgments

The author thanks Pablo Jacovkis, Pablo Dominguez Vaselli, Ines Caridi, Carlos Muoz,

Victor Bronstein, for their useful comments and suggestions.

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