HAMBERGER _ Kinship Network Analysis

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    Kinship Network Analysis

    K l a u s H a m b e r g e r , M i c h a e l H o u s e m a n , a n dD o u g l a s R . W h i t e

    35

    KINSHIP IN A NETWORK PERSPECTIVE

    Kinship, like language, is a structure, not a sub-stance.1 The distinctive features of kinship networksreside less in how their constitutive ties be theybiological, jural, ritual, symbolic, or whatever aredefined and established than in the way theseties are organized. Kinship network theoryis thus not just another application of generalnetwork theoretic methods to a particular socialdomain but a specific branch of social networktheory in itself, defined by its own axioms anddescribed by its own theorems.

    Kinship networks are characterized by theinterplay of three fundamental principles: filia-tion, marriage, and gender. We ordinarily repre-sentfiliation by a set of arcs (descent arcs) that aredirected from parents to children, and marriageby a set of undirected edges (marriage edges)between spouses (for alternative representationsof kinship networks without edges, see below).Kinship networks thus are mixed graphs, contain-ing both arcs and edges. Genderis usually takeninto account by a partitioning of the vertex set(the gender partition), usually into two or threedisjoint classes (male, female, and possiblyunknown sex).

    The characteristic features of kinship networkscan be described in terms of cyclicity. While kin-ship networks do not contain oriented cycles(nobody can be his or her own descendant2), theymay contain cycles (where arc direction does notmatter): people may marry persons with whomthey are already linked by kinship or affinity.Now, such cyclic configurations occur not justrandomly but in ways that are informative aboutthe self-organizing behavior of the network. Somekinds of relatives hardly ever marry, while other

    kinds of kinship ties between spouses may beoverrepresented. Marriage rules and prohibitions,but also residential organization, social morpho-logy, and so forth, affect the relative frequenciesof different types of cycles in a kinship network.Analyzing the distribution of cycles thereforeis the key to kinship network classification andinterpretation.

    Paths and cycles in kinship networksKinship network theory thus rests on a theory ofcyclic configurations. We call a path an alternat-ing sequence of vertices and lines (edges or arcsof whatever direction), where every vertex is inci-dent with the lines that precede and follow it in thesequence, and all vertices are distinct. If, by con-trast, the first and the last vertices are identical (allothers being distinct), we obtain a cycle. A path issaid to be closedby a line connecting its first andlast vertices, so that adding that line turns thepath into a cycle. A path or cycle is called orientedif all lines are arcs oriented in the same direction.3A weakly acyclic network is one that contains no

    oriented cycles and an acyclic network containsno cycles whatsoever.

    Alternatively to their definition as (open orclosed) sequences of vertices and lines, paths andcycles are also often defined as the graphs madeup of these vertices and lines (in this perspective,a cycle is a connected graph where every vertexhas degree 2, a path a connected graph where twovertices have degree 1, and all others degree 2).There is, however, a crucial difference betweenthe two concepts. If we define a path as asequence, the starting point matters. A path ABC

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    THE SAGE HANDBOOK OF SOCIAL NETWORK ANALYSIS534

    is distinguished from its inverse CBA, or to takea kinship example, the path fathers wifesdaughter (FWD) is distinguished from its inverse,mothers husbands son (MHS). If, by contrast,

    we define a path as a graph, ABC and CBA areonly two different notations for one and the samemathematical object for the kinship case, thismeans that FWD and MHS are two different waysto express the same kinship chain.

    The ambiguity is less severe in the case ofcycles, where, by convention, graph theorists treatstarting and ending points as irrelevant: thesequences ABCA and BACB are considered to beidentical. Nevertheless, in kinship network analy-sis we often need to distinguish them. In an ego-centric perspective, marrying ones fatherswifes daughter (FWD) is different from marry-ing ones sons wifes mother (SWM). By con-

    trast, in a socio-centric perspective as it isrequired if, for example, we want to count allcycles of a given type in a kinship network wehave to treat FWD and SWM marriages as twodifferent aspects of one and the same configura-tion. In other words, we should define paths andcycles as sequences when adopting an ego-centricview and as graphs when adopting a socio-centricview. In order to avoid any ambiguity, we shallreserve the terms path and cycle to sequencesand use the terms chain and circuit when wespeak of the corresponding graphs.4 A chain isthus a graph made up by the vertices and lines ofa single path, and a circuitis a graph made up by

    the vertices and lines of a single cycle (alternativedefinitions in terms of degrees and connectednessare given below).

    A path is an alternating sequence of vertices andlines (arcs or edges), where each vertex is inci-dent to the preceding and the succeeding line,the vertices preceding and succeeding a line areits endpoints, and all vertices are distinct.

    A cycle is a sequence of vertices and lines sharingthe properties of a path, except that the first andthe last vertex are identical (all other verticesbeing distinct).

    A path or cycle is called oriented if all its lines are

    arcs oriented in the same direction.

    A weakly acyclicgraph is one that contains no ori-ented cycles and an acyclic graph contains nocycles whatsoever.

    A chain is a graph whose vertices and arcs form a

    single path (alternatively, it may be defined as aconnected graph where two vertices have degree1 and all others have degree 2).

    A circuit is a graph whose vertices and arcs forma single cycle (alternatively, it may be definedas a connected graph where every vertex hasdegree 2).

    The first and the last vertices of a path are calledendpoints. Any line connecting the endpoints ofa path is said to closeit; the lines addition to thepath transforms the latter into a cycle.

    Armed with these basic concepts, we can nowdescribe kinship networks entirely in structural

    terms, without making statements regarding thenature of the relations involved. Characterizingkinship networks by weakly acyclic filiation, acy-clic gendered descent, nonoccurrence of certaintypes of circuits, and so on is not equivalent tostating that no one can be engendered by his or herown offspring, that it takes a man and a woman toproduce children, or that self-organization isbrought about by incest prohibitions that preventpeople in certain kinship relations from havingsex with each other. Filiation does not necessarilyinvolve a biological tie, procreation is not the onlybasis for parenthood, and marriage prohibitionsare not defined everywhere in terms of sexual

    relations. Kinship network structures may havebiological explanations, but these are culturallydetermined, variable from one society to another,and far from universal.

    Biology is the privileged model for kinshipinasmuch as it affords a universally intelligiblecode for expressing the latters fundamental rela-tions: procreation as a model for filiation, sex as amodel for marriage, and so forth. However, beinga model for a relationship does not mean being theessence of this relationship. There are many socialnetworks that are clearly kinship networks, bothaccording to structural criteria and in the minds ofthose concerned, but where filiation, marriage, and

    gender cannot be defined exclusively in reference

    Figure 35.1 Paths and cycles in kinship networks

    b

    a c a c a c a c

    Path Cycle Oriented path Oriented cycle

    bb b

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    KINSHIP NETWORK ANALYSIS 535

    to biological givens. Kinship is a type of socialstructure, and if people all over the world havechosen to express its basic relations in a biologicalidiom, there are numerous cases where these

    relations are defined independently of, and some-times even in opposition to, biological relation-ships. Defining kinship in network terms recognizesthis fact.

    NETWORK REPRESENTATIONSOF KINSHIP

    Ore-graph representation

    What we have described in our introductory state-

    ment is the most conventional representation ofa kinship network, where vertices representindividuals, arcs represent filial ties, and edgesrepresent marriages. Such networks are calledOre-graphs.5 If not specified otherwise, kinshipnetworks are treated in this article as Ore-graphs.Their characteristic feature is that they are weaklyacyclic.6 As a consequence, kinship networks con-tain a directed generational hierarchy.

    Gender in Ore-graphs is represented by a parti-tion of the vertex set. This gender partition givesrise to an analogous partition of arcs, according tothe gender of the vertex from which they origi-nate. The subgraph produced by arcs originating

    from parental vertices of the same gender can betermed a descent graph. Cycles in descent graphsare ruled out as soon as we impose the conditionofunique descent, that is, filial arcs for only oneparent of each gender: one father and one mother.Under this condition, descent graphs are acyclic,and their connected components are trees (thewell-known unilinear descent trees of lineages).This condition is closely related to the conditionof heterosexual marriage, according to which amarriage edge can only link vertices from a differ-ent gender. We speak of a standard kinshipnetwork if these two conditions unique descentand heterosexual marriage are satisfied. This

    allows for the possibility of nonstandard kinshipnetworks, which, because they admit multipledescent7 and homosexual marriage, require a morecomplex analysis. In this chapter, we will restrictourselves to discussing standard kinship networks.

    In many cases, marriage is the correlate, some-times even the condition or equivalent of havingchildren in common. It is therefore useful to dis-tinguish, as a particular category of standardkinship networks, those networks that meetthe condition of married co-parents: a standardkinship network will be called canonical if thepresence of descent arcs from two parent vertices

    to the same child vertex necessarily implies theexistence of a marriage edge between the parentvertices.

    A kinship network is a mixed graph G(V, E,A, ~), where Vis a set of vertices, called individualvertices, E is a set of edges, called marriageedges, A is a set of arcs (directed from parentsto children), called descent arcs, and ~ is anequivalence relation on V partitioning it into ndisjoint classes Vi (i = 1, , n), called genders(usually n = 2 for {male,female}), with thefollowing property:

    1 the network is weakly acyclic

    The descent graphfor the ith gender is the subgraphof a kinship network produced by all descent arcsspringing from individuals of gender i. They are

    called agnatic for male gender and uterine forfemale gender.

    A kinship network is regularif

    2 [unique descent] no vertex in a subgraphG(V, Ai) (the descent graph for the ith gender)has indegree higher than 1 (which rules outcycles in descent graphs), where Ai is thesubset of arcs with origin in Vi. Descent graphsof regular kinship networks are acyclic.

    A regular kinship network is standardif

    3 [heterosexual marriage] the subgraphsG(Vi, E) are empty.

    A standard kinship network is canonicalif

    4 [married co-parents] any two vertices thatare arc-adjacent to the same vertex8 are edge-adjacent to each other.

    Every kinship network G(V, E, A, ~) is par-tially ordered on V, as is any weakly acyclicgraph. This is the generational partial order rela-tion. In addition, a kinship network may beordered on V by assigning an arbitrary uniqueidentity number to each individual (this is impor-

    tant for computation issues, but also for datastorage in general, see below).Two individuals linked by a descent arc are

    calledparentand childwith respect to each other.Two individuals linked by an oriented path ofdescent arcs are called ascendantand descendantwith respect to each other. Two individuals linkedby a marriage edge are called the spouses of eachother. Two individuals are called co-parentsif they are parents of the same child (arc-adjacentto the same vertex), siblings if they are childrenof the same parent (arc-adjacent from the samevertex), and co-spouses if they are spouses

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    of the same spouse (edge-adjacent to the samevertex).

    We can reformulate conditions (1)(4) asfollows:

    1 In a kinship network, no individual can be his orher own ascendant or descendant.

    2 In a regular kinship network, no individual hasmore than one parent of each gender.

    3 In a standard kinship network, parents areregular and spouses are always of differentgender.

    4 In a canonical kinship network, spouses arestandard and all co-parents are spouses.

    Simple digraph representations

    There are a variety of ways of representing stand-ard kinship networks as digraphs. In addition toan Ore-graph, two of these, P-graphs (White andJorion, 1992; Harary and White, 2001, also seeWhite, this volume) and bipartite P-graphs(proposed by White, implemented by Batagelj andMrvar, 2004), are represented in Figure 35.3where letters indicate individuals (a marriedcouple, wife A and husband B, who have a daugh-ter C and a son D).9 These are all isomorphic oncearc-labels for P-graphs are included.

    P-graphsP-graphs10 represent couples as vertices and indi-viduals as individually and gender-labeled lines.As these individuals are at once born of one

    couple and may become partners in another, thearcs that represent them run from the coupleformed by an individual and his or her spouse tothe couple formed by his or her parents. Unmarriedindividuals are treated like couples.

    P-graphs have the advantages of incorporatingfewer lines and vertices, allowing marriage cyclesto be more easily detected. An individual whomarries several times is represented by severallines that are numbered with individual identitynumbers (IDs) or can be given names. The way todistinguish two lines representing the same indi-vidual from those representing two same-genderfull siblings is by either the line IDs or vectorsfor individual male or female IDs for eachvertex. Apart from these two differences, P-graphsshare the structural properties of Ore-graphs,such as weak acyclicity and generational partialordering.

    Bipartite P-graphsBipartite P-graphs are two-mode networkswhere individuals and couples are represented by

    Figure 35.2 Types of kinship networks

    Kinship network Regular kinshipnetwork

    Standard kinshipnetwork

    Canonical kinshipnetwork

    Figure 35.3 Graph representations of kinship networks

    A B

    C D

    A B

    C D

    A B

    C

    Ore-graph P-graph Bipartite P-graph

    D

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    KINSHIP NETWORK ANALYSIS 537

    vertices. There are therefore no marriage edges,but there are arcs that run from individuals tocouples and from couples to individuals.

    Bipartite P-graphs are weakly acyclic and,

    because marriages are represented by vertices, it ispossible to represent sibling relations even if thesiblings parents are unknown. In addition, mar-riages can be easily partitioned, for example,according to marriage dates.

    KINSHIP PATHS, KINSHIP RELATIONS,AND MATRIMONIAL CIRCUITS

    Kinship paths and relations

    The most general definition of a kinship relationrelies on the existence of a path linking one indi-vidual to another in a kinship network. If weabstract from the individual identity of vertices,kinship paths can be characterized by genericproperties such as the genderof vertices and thedirection of lines. This abstract form of a kinshippath is an elementarykinshiprelation. Elementarykinship relations thus can be considered as abstractkinship paths (where vertex identity does notmatter as long as vertex gender, line direction, andvertex-line incidence are preserved). Whateverproperty we may state of a path without referenceto individual vertices, we may consider as a

    property of the corresponding elementary kinshiprelation.We shall call a simple kinship relation one that

    connects Ego and Alter by a single line. By con-trast, we shall call a relation compoundif it con-nects Ego and Alter by several consecutive lines.Compound relations can be defined by the compo-sition of simple relations. For instance, father isa simple kinship relation, while mothers fatheris a compound one.

    Elementary kinship relations are defined by acomplete specification of the gender and thedirection pattern of a single kinship path connect-ing Ego and Alter. Complexkinship relations are

    obtained by combining elementary relations bymeans of one or more logical operations (and,or, not, etc.). If the logical connective is or,we obtain a disjunctive (or classificatory) rela-tion (e.g., uterine sibling is defined as mothersson or mothers daughter). If the connective isand, we obtain a conjunctive (or multiple)relation (e.g., full brother is defined asmothers son andfathers son). If the connectiveis not, we obtain a residual relation (e.g.,nonagnatic kin).

    Finally, we call a kinship relation mixed ifproperties of vertices and lines other than gender

    and direction enter into its definition; for example,the definition of widow requires a supplemen-tary partition of vertices (alive vs. dead), the defi-nition of elder brother makes reference to a

    partial order relation defined on filial arcs, andso forth.

    A kinship path is a path in a kinship network. Thefirst vertex of a kinship path is called Ego; thelast one is Alter.

    The direction of a line in a kinship path is 0(horizontal) if it is a marriage edge, 1(descending) if it is an arc directed to thesuccessor, and +1 (ascending) if it is inverselydirected.

    Two kinship paths are isomorphic if there is abijection between them that preserves thegender of vertices and the sequence and direc-

    tion of lines.An elementary kinship relation corresponds to amaximal set of isomorphic paths in a kinshipnetwork. Any of these paths representsthe kin-ship relation. Any invariant property of thesepaths beginning with the gender and directionsequence can be considered as a property ofthe elementary relation.

    A complex kinship relationis any relation that can beobtained by logical junction of several elemen-tary kinship relations.

    The notation of kinship paths and relations

    The conventional notation of kinship relationsuses capital letters for indicating direction and thegender of Alter (the gender of Ego must be indi-cated by additional signs such as [male Ego]or [female Ego] placed before the initial letter):ascending arcs are F(ather) and M(other), descend-ing arcs are S(on) and D(aughter), marriage edgesare H(usband) and W(ife), plus supplementaryletters for B(rother) and Z (sister) relations.Examples of this are MBD (mothers brothersdaughter, a matrilateral cross-cousin), ZH(sisters husband, a brother in-law), and FWS(fathers wifes son, a stepbrother).

    This conventional notation, a simple abbrevia-

    tion of English kinship terminology, has theadvantage of being easy to learn and to apply, butit is hardly the best tool for analysis. It may evenobscure the structural similarities and the distinc-tive properties of kinship relations.

    In this respect it can be contrasted with thealternative, positional notation developed byBarry (2004). Here, a kinship relation is repre-sented by a sequence of letters specifying vertexlabels (gender) and diacritical signs, which indi-cate the presence of a marriage edge (the pointor full stop .) and the apical position ofa vertex (the parentheses ()). All letters not

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    THE SAGE HANDBOOK OF SOCIAL NETWORK ANALYSIS538

    separated by a point represent vertices connectedby arcs, vertices in apical position (i.e., verticesthat are not a neighbors children) are put intoparentheses and, by convention, all arcs to the left

    of an apical vertex have ascending direction, allarcs to its right have descending direction, and themarriage point implies change of direction.

    Consider, for example, the kinship path linkinga male Ego (1) to a female Alter (5) who is Egospaternal sisters husbands daughter (Figure 35.4).In positional notation, this relation is written as1 (2) 3 . (4) 5, where 2 is Egos and 3s father, 4is Egos sisters husband and 5 is the lattersdaughter.

    By abstracting the concrete identity of the indi-viduals concerned (represented by their number)and retaining their gender only (H for male and Ffor female11), one obtains the kinship relation in

    question: H(H)F.(H)F.12

    See Table 35.1 for anexample of how standard and positional notationsare used.

    The principle of positional notation also appliesto P-graphs. In P-graph notation, with labels forarcs rather than for vertices, the ZHD relation ofFigure 35.4 is written HfH.hf, where H and F(solid and dotted lines) give the parents of a maleand a female, respectively,13 relation inversesh=H-1, f=F-1 give sons and daughters, and mar-riages are thus written fH and hF, respectively. Thefull stop . in H.h identifies the same individualin a different couple a distinction that is neces-sary to clarify that 5 is the child of 4 but not of 3.

    Positional notation has a number of importantadvantages. It incorporates the gender of Ego aswell as that of Alter, and it provides a clear repre-sentation of the structural properties of kinshiprelations, which is not radically changed by sym-metry transformations. Thus, for example, a manwho marries his HF()HF is his wifes FH()FH(in P-graph version: HFhf and FHfh), whereasin conventional notation, a man who marrieshis MBD is his wifes FZS. Allowing for a

    homogeneous representation of kinship paths(with individual numbers), of kinship relations(with gender letters), and of kinship relationclasses (with gender variables), positional nota-

    tion may be used not only as a means of notationbut also as a classificatory or programming tool.

    The classification of kinship relationsIn order to compare kinship relations, and toanalyze the ways they combine so as to give riseto particular network structures, these relationsmay be classified according to different criteria.We restrict ourselves here to some basic defini-tions (for a more extensive treatment, seeHamberger and Daillant, 2008 and Hambergerforthcoming).

    A kinship relation is linearif it is oriented (we speakof oriented paths but of linear relations).Any linear kinship relation can be represented by a

    characteristic number

    where is the degree of the relation (see note15) and i is the gender number (0 = male,1 = female) of the ith individual in ascendingdirection (starting with Ego i =0), for example,= 1 for male Ego, = 3 for F/S, = 5 for M/S, = 7 for FF/SS, = 13 for MM/DS, and so on.14Characteristic numbers impose an order on all linear

    kinship relations.A kinship relation is canonicalif it contains no link-ing children positions (i.e., if the kinship pathdoes not pass through parental triads as definedbelow).

    A canonical kinship relation is consanguineousif itcontains no marriage edge.

    A consanguineous component of a kinship networkis a maximal set of individuals linked to eachother by consanguineous paths (an individual

    Figure 35.4 A kinship relation in Ore-graph and P-graph representation

    1

    2

    3 4

    5

    1 3 4 4

    5

    Ore-graph P-graph

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    KINSHIP NETWORK ANALYSIS 539

    without consanguineous kin constitutes a consan-guineous component in itself).

    The length of a kinship relation is the numberof arcs and edges it contains. The depth of akinship relation is the length of the longestlinear kinship relation it contains. The orderof akinship relation is the number of consanguine-ous components it contains.15 The kinship rela-

    tion represented by the path in Figure 35.4, forexample, has a length of 4, a depth of 2, andan order of 2.

    Matrimonial circuits

    The concept of a matrimonial circuitReal-world kinship networks grow in two ways.On the one hand, new individuals are born, gener-ally being assigned a father and a mother frombirth. In other words, a new vertex emerges

    together with two arcs linking it to its parents,who are, at the moment of birth, the individualsonly direct neighbors in the network. If we assumethat the parents are already linked by a marriageedge (i.e., we are dealing with a canonical kinshipnetwork), the emergence of a new child vertexdoes not create a chain between individuals whoare not already linked by a shorter chain. Its

    impact on global structure may thus be said to bemarginal: it enlarges the network but does notalter its connectivity.

    On the other hand, kinship networks also growby marriage, that is, by the creation of new edgesbetween two vertices, each of which can alreadybe linked to a neighborhood of other vertices byseveral different lines going in all directions. Thenew marriage edge creates new connectionsbetween all these neighbors. In this way, marriagechanges social structure. But at the same time, theway in which social structure would be changedby a potential marriage influences the marriage

    Table 35.1 Matrimonial census

    111 marriages (2.06%) involving 211 (1.41%) individuals (105 men, 106 women) in 114 circuits

    of 37 different types (average frequency 3.08) of order 1 and depth 3

    ID Standard Positional P-Graph Marriages Circuits % Circuits

    1 FBD HH()HF HHhf 3 3 2.63

    2 FZD HH()FF HHff 5 5 4.39

    3 FFSD HH(H)HF HHH.hhf 4 4 3.51

    4 FFDD HH(H)FF HHH.hff 10 10 8.77

    5 MBD HF()HF HFhf 9 9 7.89

    6 MZD HF()FF HFff 2 2 1.75

    7 MFSD HF(H)HF HFH.hhf 13 13 11.4

    8 MFDD HF(H)FF HFH.hff 2 2 1.75

    372 marriages (6.89%) involving 667 (4.47%) individuals (339 men, 328 women) in 267 circuits

    of 152 different types (average frequency 1.76) of order 2 and depth 2

    46 FWFSD H(H).F(H)HF HH.hfH.hhf 6 3 1.12

    47 FWFDD H(H).F(H)FF HH.hFH.hff 2 1 0.37

    48 FWFFSD H(H).FH(H)HF HH.hFHH.hhf 2 1 0.37

    49 MHBD H(F).H()HF HF.fHhf 2 1 0.37

    50 BW H()H.(F) HhF 20 10 3.75

    51 BWMD H()H.F(F)F HhFF.ff 2 1 0.37

    62 marriages (1.15%) involving 112 (0.75%) individuals (53 men, 59 women) in 23 circuits

    of 16 different types (average frequency 1.44) of order 3 and depth 1

    190 FWBWZ H(H).F()H.F()F HH.hFhFf 6 2 8.7

    191 FWZHD H(H).F()F.(H)F HH.hFfH.hf 6 2 8.7

    192 FWFSWFD H(H).F(H)H.F(H)F HH.hFH.hhFH.hf 3 1 4.35

    193 FWFDHD H(H).F(H)F.(H)F HH.hFH.hfH.hf 6 2 8.7

    194 BWBWFD H()H.F()H.F(H)F HhFhFH.hf 3 1 4.35...

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    choice itself, be it explicitly (through marriagerules and incest prohibitions), implicitly (by virtueof preferences or strategies), directly (by takinginto account kinship ties between potential

    spouses), or indirectly (by taking into accountother factors that are in turn correlated withkinship). The probability of two potential partnersbecoming a couple thus depends upon the natureof the kinship chains between them. Now, themost direct way to study this dependency is tolook for those kinship chains that actually connectmarriage partners in other words, to look forcircuits.

    It should be clear from the preceding remarksthat we are not interested in just any type of cir-cuit, but in those circuits that contain at least onemarriage edge. However, this restriction is notenough. Every triangle formed by a couple and

    their child aparentaltriangle is a circuit con-taining a marriage edge, and yet the couples rela-tion to their common child is hardly a condition ofmarriage that we wish to consider. More gener-ally, we want to exclude all circuits containingparents together with their children parentaltriads. We can therefore define the types of cir-cuits of interest to us as those that contain at leastone marriage edge and no parental triads. But infact, the second condition implies the first: asdescent is weakly acyclic, the only way to form acircuit in a kinship network without passingthrough a parental triad is to pass through a mar-riage edge. We thus arrive at a simple definition of

    a matrimonial circuit:

    A parental triadis a graph formed by three verticesand arcs pointing from two of them to the third(i.e., by parents and their child). If the parents arejoined by a marriage edge, the resulting circuitconstitutes a parental triangle.

    A matrimonial circuit is a circuit that does notcontain a parental triad. Alternatively, it can bedefined as a connected subgraph where everyvertex has degree 2 but no vertex has arc-indegree 2. Because of the weak acyclicity ofdescent (condition 1 in the definition of kinshipnetworks), this definition implies that a matri-

    monial circuit necessarily contains at least onemarriage edge. In P-graph representation (wheremarriages are vertices and not edges), everycircuit is a matrimonial circuit.

    Any maximal connected consanguineous chain withina circuit is called an archof the circuit.

    A matrimonial path is a kinship path that passesthrough all vertices of a matrimonial circuit aswell as through all of its lines except the closingmarriage edge, which links the first and the lastvertex of the path. For a matrimonial circuit con-taining nmarriage edges, there are 2ndifferentmatrimonial paths.

    If the kinship network is ordered (e.g., by arbitraryidentity numbers), there is a unique rule of select-ing, for any matrimonial circuit, a characteristicpath: it is the matrimonial path that has the

    lowest possible Ego and (if there are two suchpaths) the lowest possible Alter.

    A matrimonial circuit type is a class of isomorphicmatrimonial circuits. Any matrimonial circuit typecan be represented as a complex kinship rela-tion formed by a marriage relation and anotherelementary kinship relation. We can define aunique rule for selecting, among these relations,the characteristic relationof a matrimonial circuittype: for example, the relation that begins withthe longest sequence of ascending arcs and (ifthere are several sequences of equal length) withthe lowest characteristic number.

    Circuit inclusion and ringsMatrimonial circuits have been defined in a mostgeneral manner as circuits that do not pass throughparental triads. However, there may be situationswhere we might want to reduce our analysis toonly some of these circuits namely, those thathave no shortcut linking two of its vertices.Consider, for example, a marriage with the daugh-ter of the maternal uncles wife (MBWD). Now, ifthis woman is at the same time the maternaluncles daughter (MBD), many anthropologistswould consider it improper to count her as an

    MBWD and would want to distinguish suchmarriages from marriages with true MBWDs(stepdaughters rather than daughters of maternaluncles). There is no a priori answer to the questionof whether or not to count circuits that containshortcuts of this kind. The choice depends bothon the ethnographical context16 and on the type ofcircuit in question.17 In either case, it is useful todistinguish circuits that contain shortcuts fromcircuits without shortcuts that link their vertices.The latter are called rings (White, 2004).18

    We say that a circuit A includes another circuitB if all vertices of the circuit B form part ofthe circuit A. This may also be stated by saying

    that B forms part of the subgraph induced bythe vertices of A, that is, the graph constitutedby these vertices and all of the lines thatconnect them in the global network (if a circuitA contains all the vertices of circuit B, the sub-graph induced by these vertices also includes thelines of B). An induced circuit or ring can thusbe defined as a circuit that is its own inducedsubgraph.19

    The subgraph of a graph G inducedby a vertex setV is the maximal subgraph of G having V as itsvertex set (see Harary, 1969: 11). The subgraph

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    KINSHIP NETWORK ANALYSIS 541

    induced by a circuit in the kinship network G isthe subgraph of G induced by the vertices of thecircuit. We also call it briefly the induced sub-graph of the circuit.

    A circuit A includesa circuit B if every vertex of B isalso a vertex of A (i.e., if B lies in the inducedsubgraph of A).

    An induced circuitor ring is a circuit that does notinclude any other circuit (i.e., a circuit that isits own induced subgraph). Alternatively, it canbe defined as a circuit such that no two verticesof the circuit are connected by a line that is notitself part of the circuit.

    Circuit intersection and compositionThe definition of an induced circuit (or ring)rules out circuits whose vertices are connected by

    extra lines. It does not, however, exclude circuitswhose vertices are connected by extra chains(consisting of more than one line). Consider, forexample, Figure 35.5, in which a man marries hisMMBDD, while his own mother is his fathersFFZD.20

    The chains 1-8-9-5 and 2-8-9-5 shorten theouter circuit 1-2-3-4-5-6-7-1 (of type MMBDD),forming two inner rings of type FFZD: 8-9-5-4-3-2-8 and 1-8-9-5-6-7-1. But note that the outercircuit also constitutes a ring: as the vertices 8 and9 do not belong to it, neither of the inner rings isincluded in it.21 However, it intersects with thetwo inner rings in the sense that it has one or more

    lines in common with them (for a further discus-sion of circuit intersection, see below). Moreover,the entire outer ring can be composed from thetwo inner rings and the parental triangle 1-2-8-1by taking their union and deleting all lines thatform part of more than one circuit (an operationcalled circuit union). A circuit that in this mannercan be entirely decomposed into circuits shorter

    than itself is called reducible; if it cannot, it iscalled irreducible. By definition, every irreduciblecircuit is also a ring.

    It should be stressed that reducible circuits are

    not necessarily sociologically less relevant thanirreducible circuits. If a Fulani man, for instance,marries an FFBSD who is at the same time anMBD (due to an FBD marriage between the hus-bands parents), the apparent cross-cousin mar-riage MBD may simply be a by-product ofsuccessive parallel-cousin marriages (FBD andFFBSD): it is then the longer and not the shorterring that matters for marriage decisions. Nor doesthe formation of a reducible circuit presuppose theprevious formation of some irreducible circuit(see note 17).

    The study of irreducible matrimonial circuits isclosely related to the idea of a cycle basis in gen-

    eral graph theory. A cycle basis for a graph is aminimal set of circuits from which all circuits ofa graph can be composed by a circuit union.Different sets of circuits can constitute a cyclebasis. However, the number of circuits in thebasis (also called the circuit rank or cyclomaticnumberof the graph) is invariant: we can computeit from the numbers of its arcs, edges, and compo-nents by means of a simple formula (see below).In order to find a cycle basis for a kinshipnetwork, it is reasonable to concentrate on irre-ducible circuits. Note, however, that the cyclebasis may well be smaller than the set of irreduc-ible circuits: if a man marries three sisters, we

    have three irreducible circuits, but the circuitrank is only two (as two circuits are sufficient tocompose the third).

    Two circuits intersect if they have lines in common.They intersect matrimonially if they have mar-riage edges in common.

    The unionof two circuits is the graph that has theunion of their line (vertex) set as its line (vertex)set. The circuit unionof two intersecting circuitsconsists in taking their union and deleting thelines they have in common.

    A circuit is irreducible if it cannot be composed bya circuit union from a set of circuits that are all

    shorter than itself.A cycle basisof a graph is a minimal set of circuitswhose union contains all the circuits of thegraph.

    The cyclomatic numberor circuit rankof a graph isthe number of circuits constituting its cycle basis(which is equivalent to the number of lines onehas to remove from a graph to make it acyclic).For a graph with e lines, v vertices, and c compo-nents, it is calculated as

    = e v + cFigure 35.5 Circuit intersection andcomposition

    1

    2

    3

    4

    5

    6

    7

    8

    9

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    Counting circuits: The matrimonialcensus

    A matrimonial census provides an exhaustive list

    of all matrimonial circuits of specified propertiesin a given kinship network, and it counts theoccurrences of every distinct circuit type (whichmay or may not be aggregated into broaderclasses).

    A circuit search usually has to be restricted to alimited set of circuit types. Even if the totalnumber of circuits in a finite network is not infi-nite, it is usually so high that an unbounded searchexceeds the capacities of the most advanced per-sonal computers. For exploratory analysis, it isrecommended to restrict matrimonial circuitsearch by criteria that are as neutral as possible,such as maximal order and depth of circuits. Do

    not just count first-cousin marriages in order todecide whether you are dealing with generalizedexchange, arab marriage, and so on a look athigher degree consanguineous marriages, and atmarriages between affines, may change the entirepicture! Kinship structures constitute a whole andcan only be understood if one considers them assuch; they cannot be characterized by the fre-quency of this or that circuit type but only by therelative proportions and the interdependency ofthese frequencies. A matrimonial census musttherefore be comprehensive in order to provide thebasis for further analysis and interpretation, evenif subsequent circuit searches may be more

    restricted and refined. It is clear that, even forsmall networks, such a task cannot be accom-plished without computer support.

    Circuit searches can be undertaken on theentire network or restricted to certain subsets ofvertices. This restriction does not imply that allvertices of the matrimonial circuit have to belongto the subset (if, for example, we are interested inconsanguineous marriages concluded betweenpeople born after 1800, we, of course, allow con-sanguineous chains to pass through ancestors bornbefore 1800). Subsets may be defined accordingto exogenous criteria recorded for the individu-als in the network (dates of birth, death, or

    marriage, residence, occupation, etc.) but alsoaccording to endogenous criteria derivingfrom the kinship network itself (e.g., sibling groupsize, number of known ascendants, number ofspouses, etc.). Such restrictions are not only con-venient for comparative analysis and tests of rep-resentativity, but they may also be helpful indetermining the optimal network to work with(see below).

    Table 35.1 presents excerpts from a matrimo-nial circuit census produced by the software Puck(infra) of a kinship network collected among theWatchi of Togo,22 restricted to circuits of order

    1 and depth 3, circuits of order 2 and depth 2, andcircuits of order 3 and depth 1.

    NETWORK REPRESENTATIONS OFCIRCUIT STRUCTURES

    The set of matrimonial circuits thus obtained canin turn be studied with genuine network-analytictools.

    One of these tools is to construct the networkthat the circuits compose; this gives us a subnet-work of the original kinship network. A secondtool consists of constructing the network of thestructural relations between the circuits them-selves; this gives us a second order network in

    which the circuits represent the vertices and theirstructural interrelations are represented by lines.In the following section, we shall discuss one

    network of the first type, the matrimonial net-work, and one of the second type, the circuitintersection network.

    Networks derived from circuit sets:The matrimonial network

    A matrimonial networkis a subgraph of a kinshipnetwork resulting from the union of a set of mat-rimonial circuits, as, for instance, the circuits

    found by the matrimonial census in Table 35.1.Note that this is not equivalent to the subgraphinducedby this set: for an arc or edge to be in thesubgraph, it is not enough that each of its end-points is in some circuit the arc or edge mustitself be part of a circuit. The matrimonial networkderived from a set of matrimonial circuits found ina kinship network is thus simply the network com-posed of these circuits. It consists, in other words,of the matrimonially interesting regions of theoriginal kinship network. The components of thematrimonial network (which we call matrimonialcomponents) are connected subnetworks of matri-monial circuits, which may be studied from vari-

    ous perspectives. On the one hand, we maysuppose that the frequent occurrence of particularmatrimonial patterns is correlated with other prop-erties of the network region concerned (e.g., socialclass, geographical region, or historical period);we may then apply several partitions to the net-work in order to evaluate the degree to whichpartition clusters correspond to matrimonial com-ponents. On the other hand, we may interpret thedensity of circuits as an effect of self-reinforcingsocial mechanisms (behavior transmission, imita-tion, or the presence of rules) or as a simple net-work effect (circuits combining to compose other

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    KINSHIP NETWORK ANALYSIS 543

    circuits) that we did not consider when definingthe criteria for our initial circuit search.

    The concept of a matrimonial network is alsomeaningful in and of itself, independent of any

    particular circuit set. Even in cases where it is notpossible to precisely identify all matrimonial cir-cuits (without limits of size) that may exist in akinship network, it is possible to determine whichpart of the network is composed of matrimonialcircuits (of whatever length). The result thelargest possible matrimonial network is thenucleus of the kinship network, the union of allexisting circuits in the network (see Grange andHouseman, 2008).

    The concept of the nucleus can be more strictlydelineated by introducing the concept of a matri-monial bicomponent, that is, a maximal subgraphin which every two vertices form part of a matri-

    monial circuit (note that this is a stricter conditionthan that of simply forming part of a circuit, as isrequired for the more general notion of a bicom-ponent). The nucleus is simply the union of allmatrimonial bicomponents.23

    As a result, matrimonial components (consist-ing only of circuits) are closely related to matri-

    monial bicomponents: both are line-biconnected(two distinct line series link each vertex to everyother), but matrimonial bicomponents have theadditional feature of being vertex-biconnected as

    well (the two interconnecting line series never runthrough the same vertex).

    In the kinship network represented inFigure 35.6, for example, the shaded individualsand their interconnections within the bold bounda-ries constitute the nucleus, which is composed oftwo matrimonial components (A and B) and threematrimonial bicomponents (1, 2, and 3), two ofwhich overlap (one individual is included in both1 and 2).

    Given a set of circuits R in a kinship network K,the matrimonial network derived from R is thesubgraph of K resulting from the union of the

    circuits of R. In other words, it is a subgraph inwhich every line belongs to some matrimonialcircuit of R. The components of a matrimonialnetwork are called the matrimonial componentsof K with respect to R. Matrimonial componentsare line-biconnected but not necessarily vertex-biconnected.

    Figure 35.6 Matrimonial components and bicomponents

    Matrimonialcomponent B

    Matrimonial

    bicomponent 2 Matrimonial

    bicomponent 3Matrimonial

    bicomponent 1

    Matrimonialcomponent A

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    A matrimonial bicomponent is a subgraph of K inwhich every pair of vertices (no matter howdistant) belongs to a matrimonial circuit (henceno pair of vertices can be separated by removal

    of a single intermediary vertex). In P-graph rep-resentation, every bicomponent is a matrimonialbicomponent.

    The nucleus of a kinship network is the networkresulting from the union of all matrimonialcircuits in K. The nucleus is equivalent to theunion of all matrimonial bicomponents of thekinship network. The largest of these matrimo-nial bicomponents constitutes the kernel of thekinship network.

    Networks of circuits: The circuit

    intersection networkCircuit graphs are tools for analyzing the interde-pendence of matrimonial circuits. A simple exam-ple is the circuit intersection network, wherecircuit interdependence is measured by the fre-quency of shared marriage edges.

    It is often the case that a given marriage formspart of two or several matrimonial circuits. Suchoverdetermination poses problems of sociologicalinterpretation: if a man has married a woman whois at the same time his fathers sisters daughter,his maternal aunt, and his sister-in-law, should wesay that he married his cousin, his aunt, or his

    sister-in-law? And how shall we interpret such asituation if, for instance, it is considered to be verygood to marry ones cousin but bad to marry onesaunt? Under such circumstances, it is clearlyinsufficient to simply count the frequency ofmaternal aunts that are spouses and affirm thepresence of a high number of bad marriages,without considering the fraction of such auntswho are at the same time cousins and thus repre-sent good spouses. But there is a further reasonfor wanting to determine the frequencies of circuitintersection: if a fathers sisters daughter is atthe same time a maternal aunt, such a configura-tion mathematically implies that the husbandsfather has married his sisters daughter. We shalltherefore necessarily find a high number of niecemarriages in our network, and if we did not yetsearch for them, such a finding will prompt us todo so.

    Circuit intersection networks are an easy andintuitive way to approach these questions. In thesenetworks, vertices represent circuit types, the size(or vector value) of vertices represents circuitfrequencies, and the values of the lines connectingtwo vertices represent the number of marriagesthat are simultaneously part of a circuit of the twocorresponding types.

    The matrimonial intersection of two circuit typesA and B is the set of all marriage edges belong-ing simultaneously to a circuit of type A and to acircuit of type B.

    The circuit intersection network corresponding toa set of circuit types T is a valued graph G,such that

    1 each vertex of G corresponds to a circuittype of T, and the value (size) of the vertexis proportional to the frequency of the cor-responding circuit type

    2 each edge between two vertices correspondsto a nonempty matrimonial intersection ofthe corresponding circuit types, and theweights of the edges correspond to the sizesof the intersections.

    ALLIANCE NETWORKS

    All matrimonial structures hitherto discussed havebeen defined in terms of filial and marriage rela-tions between individuals. But kinship also has todo with relations between groups derived fromrelations between individuals. One convenienttool of analyzing these relations is the alliancenetwork.

    The alliance networkcorresponding to a givenkinship network (real or simulated) is a network

    composed of vertices representing groups of indi-viduals and arcs representing marriage frequen-cies between the groups, where the value of an arcfrom A to B indicates the number of marriages ofa woman of A with a man of B. One can think ofalliance networks as resulting from shrinkingthe kinship network with respect to some parti-tion, after having transformed marriage edges intoarcs that point in the husbands direction, that is,from the group of wife-givers to the group ofwife-takers. The clusters of the partition arethose collectivities related by the marriagesbetween their respective members. The partitionused may be exogenous to kinship (e.g., if we

    are dealing with residential units, professionalcategories, or social classes), but it may also bederived from the kinship network itself. Thus, forexample, clusters may represent the componentsof the agnatic or uterine subnetwork, that is,lineages within the kinship network.

    Alliance networks no longer have the particu-larities of kinship networks. They are simplevalued digraphs. Circuits in alliance networksmay be called connubial circuits in order to distin-guish them from matrimonial circuits in kinshipnetworks, and the connubial circuit structure of analliance network can be represented by its matrix

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    (the alliance matrix). In exogamous systems, thealliance matrix will have only zero values in itsdiagonal; in a perfect system of balanced bilateralmarriage alliances (designated by Lvi-Strauss

    [1949 (1967)] as restricted exchange) it will besymmetric; in a system having a single directedHamiltonian cycle (generalized exchange), itwill be asymmetric and contain only one nonzerovalue in each line and column. Note, however, thatreal-world alliance structures are not as clear-cut.On the one hand, several circuit patterns may besuperimposed, such that, for example, the result-ing cumulative pattern may appear symmetricaleven if partial patterns are asymmetric. On theother hand, the structure of alliance networks ishighly sensitive to the definition of wife-givingand wife-taking groups, connubial circuit struc-tures being in general not continuous with respect

    to changes in the level of group aggregation. Thus,what may appear to be endogamous unions on onelevel may be exogamous on another, with combi-nations of endogamy and balanced exchange onone level giving rise to asymmetrical patterns onanother (for an example, see Gabail and Kyburz,2008).

    The alliance networkcorresponding to a partitionedkinship network K is a digraph G where

    1 each vertex of G corresponds to a partitioncluster of K

    2 each arc between two vertices of G corre-

    sponding to two clusters A and B correspondsto the existence of a marriage of a womanof A with a man of B, and the weight ofthe arcs corresponds to the number of suchmarriages.

    A connubial circuit is a circuit in an alliancenetwork.

    APPLICATION ISSUES

    Data collection and savingOrientations of data collectionOf course, the way kinship data collection is ori-ented depends largely on what one wants to find.However, in order to minimize biases and to allowthe data to be used by others, there are somecriteria that every corpus of kinship data shouldfulfill, regardless of the specific purpose of thecollection:

    Document the dates and informant(s) for everypart of the data set; this is not only useful for

    checking possible errors and biases but may bean interesting datum in itself, for instance, toappraise genealogical memory.

    Document missing data; indicate if the spouses

    and children you have noted for a given individ-ual are, according to your knowledge, complete.

    In case of contradictory data, keep recordsregarding alternative items of information (andtheir origin) and on the reasons for your choices.

    Try to avoid biases from the start by always fol-lowing both male andfemale lines; do not recordonly those kinship ties that are easily given butmake an effort to search those that are missing,even if this may be costly in the case of an exten-sive matrimonial area.

    If you are interested in the kinship relationsbetween two individuals, do not be content toask how they are related but try to establish their

    entire pedigrees within given bounds you willsurely find quite a number of additional ties yourinformants did not mention spontaneously!

    Try to determine the relative order of births andmarriages (in particular in the case of polygamy),even if absolute dates are not available. In inter-preting marriages with affines or overlappingmatrimonial circuits, this may turn out to be veryimportant.

    Keep an account of the research method usedand specify the purpose for which the corpus(or part of the corpus) is established.

    Storing data without a computerData are not only a result but also a means of datacollection. They should be easily accessible inorder to guide your research and to cross-checkinformant answers. When dealing with archives,this is often fairly simple: you can take a computerwith you. But in many fieldwork situations this isnot possible. However, noting kinship by handcan be extremely fast and efficient, if some basicprinciples are observed.

    Always use a compact medium, such as a note-book. Do not use filesheets or loose papers. Youcannot easily use them during interviews, and

    there is a high risk of losing some of them. Separate graphics and text. A good method isto use a notebook with the left page for draw-ing genealogies, the right page for listing theindividuals and their properties, and numbersfor identifying these individuals (if numbers getlarge, it is recommended to use, in addition,initial letters to prevent identification problemsin case of numbering errors).

    Attribute an identity number to each individualand never attribute that number to anotherindividual. If you have duplicates, make a linkfrom the redundant to the original number but

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    do not re-assign it. Gaps in the series of numbersdo not cause any damage, but ambiguities inidentity numbers cause much damage and areextremely difficult to detect.

    Do not use identity numbers as codes. Identitynumbers serve to identify individuals and noth-ing else (except, perhaps, to recall the order inwhich you have entered them and to documentthe history of your corpus). If you want to conveyinformation on individuals gender, clan affilia-tion, residence, and so on, do not use identitynumbers for that purpose.

    Never forget to make copies and store themin different places. This holds for all data, butespecially for kinship data, due to the networkproperties of kinship: one lost notebook mayrender all other notebooks useless.

    Data interpretation

    It is necessary to know the biases, gaps, and limitsof a genealogical corpus. In some cases, this maylead one to conclude that certain analyses simplycannot reasonably be made. For instance, it makeslittle sense to search for consanguineous marriagecircuits in a corpus with very shallow genealogies.Here, as in all cases, knowledge of the basicqualities of the corpus is essential for interpretingthe findings.

    In particular it is important to keep in mind thatkinship networks are virtually infinite. Everycorpus is necessarily only a part of a larger whole,and incomplete both with regard to individualsand with regard to the links that connect them.Choices pertaining to the delimitation and compo-sition of genealogical corpuses arise not onlyduring fieldwork but also prior to analysis asa means of reducing the collected data to ameaningful core.

    Choosing bases and boundariesThe usefulness of exogenous reductions bringingsociological, geographical, and demographical

    criteria into play is fairly self-evident. However,the value of endogenous reductions based onstructural features of the network itself is perhapsless so, and depends on what one is trying to show.For example, one cannot first eliminate all unmar-ried individuals if one later wants to comparematrimonial circuit frequencies with those of kin-ship relations (e.g., White, 1999). This applies notonly to endogenous reductions but also to endog-enous augmentations of the network, such as thecreation of fictive individuals in order to preserveinformation on full siblingship in those caseswhere one or both parents are unknown.

    A related problem concerns the delimitation ofthe network to be used for analytic purposes. Tobegin with, it is important that results relating tounconnected components of the matrimonial net-

    work not be combined indiscriminately, as eachcomponent represents an autonomous matrimo-nial universe whose patterning may not obey thesame rules. This also applies, to a lesser degree, tomatrimonial bicomponents. However, even whenanalysis is limited to the largest matrimonialbicomponent (kernel) of the kinship network,additional restrictions are often necessary. Becausebicomponents contain matrimonial circuits of anylength, depth, and order, including those thatincorporate very distant ties whose sociologicalrelevance is questionable, it is helpful to confineanalysis to subnetworks formed by matrimonialcircuits of a certain maximal depth or order. It is

    not always easy to determine the optimal criteriafor such restricted matrimonial networks, choicesbeing largely guided by two contrary principles.On the one hand, the resulting subnetwork shouldbe as large as possible so as to be sufficientlyrepresentative of the wider kinship network fromwhich it is drawn. On the other hand, in order toavoid redundancies and to keep the number ofcircuits and circuit types at manageable propor-tions, it is essential that the latter be kept to aminimum.

    Determining representativity and significance

    Once a delimited data set has been chosen andfirst results obtained, another difficulty arises: towhat extent do the regularities observed in thecorpus provide information on the organization ofthe real social network? This, of course, is a gen-eral problem in network analysis; however,because issues of incompleteness are so omnipres-ent in genealogical research, questions regardingthe representativity of kinship networks areparticularly pressing.

    One central question concerns the extent towhich a corpuss boundaries correspond to endog-enously definable subnetworks of the real kinshipnetwork. Another, related question concerns the

    over- and underrepresentation of persons belong-ing to certain kinship categories. These issuesapply not only to the individuals (vertices) thatmake up the corpus, but to the relations (lines)between them. Kinship networks are often biasedin that they favor some types of relations overothers (agnatic over uterine relations, for example,for a population with patrilocal residence). Itmight be possible to adjust ones findings so as toeliminate such biases. However, these biases maythemselves be a function of matrimonial behavior:kinship ties that form part of matrimonial circuitsmay be more easily remembered than others.

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    The stronger this correlation between networkstructure and collective memory, the more diffi-cult it is to do away with the biases concerned.

    The problem of network biases is directly

    related to issues of significance: are the observedregularities indicative of a behavioral pattern, orare they simply due to chance? In order to treatthis question by means of a comparison withrandom kinship networks, it is necessary to simu-late a network that not only reproduces the demo-graphic features of the concerned population butalso the biases of the corpus itself. Because ofthis, random permutation (see White, 1999; White,this volume) or explicit simulation of the datacollection process (virtual fieldwork) may provemore useful than conventional demographicsimulation.

    Interpretation of resultsIf interesting regularities appear in analysis forwhich the known norms and institutions of thesociety do not provide a straightforward account,one is immediately tempted to take them as indi-cators of some hidden norm or institution. Themost important thing in kinship data analysis isto resist this temptation. Before formulating asociological hypothesis, one must always checkthe following:

    First, does the interesting structural featuresimply reflect a bias of the corpus? (e.g., high

    incidences of patrilateral marriages in a corpuswhere uterine genealogies are shallow). Second, is the structural feature, if it does not

    correspond to any known rule, the result of acombinationof known rules? (e.g., in a societywhere uterine nieces are preferred spouses andmaternal aunts are avoided, patrilateral cross-cousins have a greater chance of being at thesame time maternal aunts and therefore avoided,even if no rule states that they should be avoidedas such).

    Finally, remember that a sociological hypothesisis not validated by the simple fact that no otherinterpretation can account for the structural fea-

    ture observed. Sociological hypotheses often canbe validated only in the field.

    Resources

    A wide variety of commercial software (BrothersKeeper, Family Tree Maker, Legacy, Kith andKin, etc.) and noncommercial programs (PersonalAncestral File, Gramps, etc.) exist for entering,storing, and outputting genealogical data. Someof them are particularly flexible and are usedextensively by social scientists: Gnatique is

    used by many French historians; Alliance Project,developed by S. Sugito and S. Kubota in Japan(Sugito, 2004), is widely used by Australiananthropologists; Kinship Editor, written by

    M. Fischer, a key element of the European Kinshipand Social Security (KASS) project, allows bothfor systematic data collection and the modeling ofkinship phenomena such as terminological usage.There are also additional software tools developedby demographers or historians for managingrecords and calculating demographic and othervariables on the basis of empirical genealogicaldata (e.g., CASOAR [Hainsworth and Bardet,1981]). None of these programs, however, allowfor an in-depth analysis of kinship networks assuch.

    Such analyses may be undertaken using generalpurpose social network analysis tools. Perhaps the

    software most used in this way is the networkanalysis and visualization program Pajek, devel-oped by V. Batagelj and A. Mrvar (de Nooy et al.,2005; Batagelj and Mrvar, 2004, 2008; Mrvar andBatagelj, 2004). A number of kinship-centeredmacros have been developed for Pajek by theauthors and by others (e.g., Tip4Pajek pack byK. Hamberger).

    Finally, certain programs have been specifi-cally developed for the analysis of kinship net-works (including matrimonial circuit censuses).First attempts, such as Gen-Par by Selz (1987; seealso Hritier, 1974) and Pgraph by White (1997;White et al., 1999), have since given way to more

    flexible and easier-to-use software such as Genosby Barry (2004) and, more recently, Puck byHamberger et al. (2009).

    An open depository of kinship networks fromhistorical and anthropological sources, controlledby a scientific board, is hosted at the site of thekinsources project (http://kinsource.net).

    NOTES

    1 We are grateful to Isabelle Daillant and

    Vladimir Batagelj as well as to the editors Peter

    Carrington and John Scott for helpful comments anddiscussions.

    2 We are talking of individuals and not of classes.

    An individual may well belong to ones parents

    parents marriage class, as in Australian alternating

    generation models.

    3 In digraphs, the notions path and cycle are

    often restricted to oriented paths and cycles, whereas

    the terms semipath and semicycle are used if

    arcs are not consistently oriented. Because we are

    talking of mixed graphs (containing edges as well as

    arcs), we use path and cycle as the general

    terms (as in the case of undirected graphs) and

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    THE SAGE HANDBOOK OF SOCIAL NETWORK ANALYSIS548

    specify them as oriented if all of their lines are arcs

    pointing in the same direction.

    4 The terms chain and circuit are equivalent

    to path graph and cycle graph in graph-

    theoretic literature. Note, however, that these termsare also sometimes used with a different meaning,

    namely, as synonyms for walk (open and closed,

    respectively).

    5 Named after the Scandinavian mathematician

    Oystein Ore (1960). The representation of marriages

    by edges has been introduced by the computer

    program Pajek (see Batagelj and Mrvar, 2008).

    6 Note that the weakly acyclic property of kinship

    networks holds for remarriage cycles because a cycle

    consisting entirely of edges is not an oriented cycle.

    7 Kinship networks that incorporate adoptive,

    godparent, or co-genitor relations often have to

    allow for the possibility of more than one parent of a

    given gender (multiple descent).8 A vertex x is arc-adjacent to another vertex y if

    the arc goes from x to y. It is arc-adjacent from y ifthe arc goes from y to x.

    9 The upper and lower vertices of the P-graph

    and bipartite P-graph are reversed in Figure 35.3

    compared to the Ore-graph. This is because their

    arrows run in reverse, from children to parents, usu-

    ally as a unique function mapping a child to a unique

    parental couple. The inverse of this function gives the

    relation of parents to children.

    10 The P stands for parent (French for

    kinship).

    11 From the French homme (man) and femme

    (woman).12 This positional notation may also be used,

    within limits, to represent complex kinship relations,

    by leaving the parentheses empty (or by putting two

    letters in it) if the relation entails kinship paths that

    pass through apical ancestors of both genders. For

    instance, the MBD (mothers brothers daughter)

    relation is represented as HF()HF, the ZH (sisters

    husband) relation as H()F.H.

    13 Original P-graph notation (White and Jorion,

    1992) used letters G and F (from the French garon

    (boy) and fille (girl). In its present version, P-graph

    notation uses the same letters as positional Ore-

    graph notation (but applies them to lines while Ore-

    graph notation applies them to vertices).

    14 This is a variant of the ahnentafel genealogical

    numbering system.

    15 In the case of consanguineous kinship rela-

    tions, length is also called roman (or civil) degree,

    whereas depth is also called german (or canonic)

    degree (these terms derive from the history of

    European kinship). In the case of linear kinship rela-

    tions, roman and german degrees are identical, and

    we can simply speak of its degree.

    16 For example, in the case of a Dravidian kinship

    network from southern India, where cross-cousin

    marriages may be considered as redoublings of

    alliance relations rather than as consanguineous mar-

    riages (Dumont, [1953] 1975), counting all MBWDs,

    whether they are or are not also MBDs, would make

    good sense.

    17 For instance, if two brothers marry two sisters,one brother dies, and the remaining brother marries

    the widow, the fact that the longer circuit (of type

    BWZ) includes two shorter ones (of types BW and

    WZ) does not in the least make it sociologically less

    relevant: the BWZ marriage clearly precedes the BW

    (and WZ) marriage (we are grateful to Isabelle

    Daillant for this example).

    18 The distinction between circuits and rings

    (induced circuits) is a rather recent one. Much of

    what has been said regarding matrimonial rings in

    Hamberger et al. (2004) refers to matrimonial circuits

    in general, while rings in White (2004) refers toinduced circuits.

    19 Note, however, that this restriction may notmean the same thing in P-graph and in Ore-graph

    representation. The subgraph induced by the vertices

    of a circuit changes meaning according to whether

    vertices are defined as individuals (Ore-graph) or as

    marriages (P-graph). For instance, a marriage with

    an MMBDD who is at the same time an MBD consti-

    tutes a ring in Ore-graph but not in P-graph repre-

    sentation. If the context is not clear, one should

    therefore speak of Ore-rings and P-rings to avoid

    ambiguities.

    20 This example holds in a P-graph as well as in

    Ore-graph representation: in both cases, the outer

    circuit is a ring. In general, Ore-rings are not always

    P-rings (see note 19).21 This is equally true in P-graph representation,

    where the two individuals 8 and 9 are represented by

    two lines forming an extra chain between the verti-

    ces (representing marriages) of the outer circuit.

    22 See http://kinsource.net/kinsrc/bin/view/

    KinSources/Watchi.

    23 Every line of a matrimonial bicomponent is by

    definition in some matrimonial circuit. On the other

    hand, every matrimonial circuit is either a matrimo-

    nial bicomponent in itself or forms part of some

    larger matrimonial bicomponent. The concept of the

    nucleus is a restriction of the definition of the core byWhite and Jorion (1996; cf. Houseman and White,

    1996) as the union of all matrimonial bicomponents

    and their single-link connections (equating with

    2-core as defined by Seidman, 1983). Neither is

    necessarily connected.

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    Software

    Alliance Project: http://study.hs.sugiyama-u.ac.jp/e/Brothers Keeper: http://www.bkwin.org/Family Tree Maker: http://www.familytreemaker.com/Gnatique: http://www.cdip.com/Gramps: http://gramps-project.org/wiki/index.php?title=

    Main_Page

    Kinship Editor: http://era.anthropology.ac.uk/Kinship/Kith and Kin: http://www.spansoft.org/Legacy: http://www.legacyfamilytree.com/Pajek: http://pajek.imfm.si/doku.phpPersonal Ancestral File: http://www.familysearch.org/eng/paf/Pgraph: http://eclectic.ss.uci.edu/~drwhite/pgraph/pgraph.

    html

    Puck: http://kintip.net/Tip4Pajek: http://kintip.net/, http://intersci.ss.uci.edu/wiki/