Definition of the Subject
Algebraic models have been proposed to represent structure in social networks . They are usually constructed from a set of operations andrelations defined on network constituents such as paths or walks in the network, or vectors of ties directed to or from individual networkmembers. The algebra represents the relationships among theseconstituents, for example, relations of ordering among all possiblewalks in a multiple network , or relations of overlap and orderingamong profiles of group membership. To date, they have been used torepresent kinship structures [6,9,48], role structures and stabilityin multiple networks [1,5,8,36], states of diffusion processes innetworks [30], informal hierarchy [17], connectivity structures [13]and the structure of membership in affiliationnetworks [16,38]. In several cases, partial algebraic representationshave also been proposed [35,39,40].
Algebraic models have usually been proposed as exact structural representations and,...
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- Social network :
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A social network comprises a set of relationships among members of a set \( { N=\{1,2, \dots, n\} } \) of actors. It can be represented by an \( { n \times n } \) binary array X recording the presence or absence of a social relationship, or tie, between each pair (or orderedpair) of members of \( { N=\{1,2,\dots, n\} } \). If there isa relationship from actor k to actor l, we write \( { X(k,l)=1 } \);otherwise, \( { X(k,l)=0 } \). If the relationship is a property ofa pair of actors, the network is nondirected; if it isa property of an ordered pair, the network is termeddirected. The directed network X may also be regarded asa binary relation R X on the set N with \( { (k,l) \in R_{X} } \) if and only if \( { X(k,l) = 1 } \); equivalently, it may beconstrued as a directed graph with node set N andarc set R X , with an arc from node k tonode l if and only if \( { (k,l) \in R_{X} } \).
- Affiliation network :
-
An affiliation network isan \( { n\times g }\) binary array X recording the membership of each of a set N of actors in a prescribed set G of groups , with\( { X(k,l)=1 } \) if actor k is a member of group l, and \( { X(k,l)=0 } \) otherwise.
- Multiple network :
-
A multiple network is a collection ofnetworks for each of a set of r relations. We let \( { X_{m}(k,l)=1 } \)if the tie from k to l corresponding to the relation oftype m is present; and \( { X_{m}(k\:,l)=0 } \) if the tie isabsent. Nodes k and l are joined by a labeled walk withlabel \( Y_{1}\, Y_{2}\, \dots Y_{j} \) if there is a sequenceof nodes \( k=k_{0}, k_{1}, \dots, k_{j}=l \), forwhich \( Y_{h}(k_{h-1}, k_{h})=1 \) for \( h=1,2,\dots, j \).
- Local network :
-
The (1-)neighborhood of a subset P of actors in a network is defined to be theset \( P \cup\{l\in N\colon X_{m}(k,l)=1 \) forsome \( { k\in P } \) and some relation m}. The q‑neighborhood of P is then definedrecursively as the 1‑neighborhood of the \( { (q-1) }\)‑neighborhood of P. The q‑localnetwork of the subset P of N is the network restricted toits q‑neighborhood.
- Algebra :
-
A ( partially ordered) algebra isa triple [\( { S,F,\leq } \)], where S is a nonempty set ofelements (usually assumed to be finite), F is a specified set ofoperations, \( { f_{\alpha } } \), each mapping a power \( { S^{n(\alpha)} } \)of S into S, for some non‐negative finite integer\( { n(\alpha) } \), and \( { \leq } \) is a partial order on S. Eachoperation f α is assumed to be isotone in each ofits variables: that is, if \( x_{i} \leq y_{i}\, (x_{i}, y_{i} \in\ S;\,i = 1,2, \dots, n(\alpha)) \), then \( f_{\alpha} (x_{1},x_{2}, \dots, x_{n(\alpha)})\leq f_{\alpha}(y_{1},y_{2},\dots, y_{n(\alpha)}) \).A family of algebras is a collection of algebras each having the same set F ofoperations and satisfying a specified set of postulates. Two algebrasbelonging to the same family are termed similar.
- Partial algebra :
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A partial algebra isa triple [\( { S,F,\leq } \)], where S is a nonempty set of elements(usually assumed to be finite), F is a specified set ofpartial operations, \( { f_{\alpha} } \), each mapping somesubset \( { T^{(\alpha)} } \) of \( { S^{n(\alpha)} } \) into S, for somenon‐negative finite integer \( { n(\alpha) } \), and \( { \leq } \) is a partialorder on S. Each partial operation f α is assumed to beisotone in each of its variables: that is, if \( x_{i} \leq y_{i}\,(x_{i}, y_{i}\in S;\,i = 1,2, \dots,n(\alpha)) \), then \( f_{\alpha}(x_{1}, x_{2},\dots, x_{n(\alpha)})\leq f_{\alpha} (y_{1},y_{2} ,\dots, y_{n(\alpha)}) \) provided that both\( (x_{1},x_{2}, \dots, x_{n(\alpha)})\:, (y_{1},y_{2}, \dots, y_{n(\alpha)})\in T^{(\alpha)} \).A family of partial algebras is a collection of algebras each having the same set F of partialoperations defined on the same subsets \( { T^{(\alpha)}} \) of the powersets \( { S^{n(\alpha)} } \).
- Semigroup :
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A (partially ordered) semigroup isan algebra [\( { S,F,\leq } \)] in which F comprises a singlebinary operation f satisfying the associativity condition:
$$f(f(x,y),z)=f(x,f(y,z)) \:.$$ - Lattice :
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A lattice L is an algebra[\( { S,F,\leq } \)] in which F comprises two associative andcommutative binary operations, ∧ and ∨ (termedmeet and join, respectively) satisfying theidentities:
$$x \wedge x =x \:, \quad x \vee x =x \\$$and
$$x \wedge (x \vee y)=x \vee (x \wedge y)=x\:.$$The operations are isotone, so that \( { x \leq y } \) is equivalent tothe pair of conditions:
$$x \wedge y=x \quad \text{and} \quad x \vee y=y \:,$$and the operations of meet and join may be interpreted as the greatestlower bound and least upper bound, respectively. A lattice L is distributive if the identity
$$x \wedge (y \vee z)=(x \wedge y) \vee (x \wedge z)$$holds. A lattice L is modular if, whenever \( { x \leq z } \), then
$$x \vee (y \wedge z)=(x \vee y) \wedge z\:.$$ - Role algebra :
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A role algebra is an algebra[\( { S,F,\leq } \)] in which F comprises a single binarycomposition operation satisfying the condition: \( { s \leq t } \) in Simplies \( { su \leq tu \text{ in }S } \), for any \( { u \in W } \).
- Homomorphism :
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An (isotone) homomorphism froman algebra \( { A = [S,F,\leq] } \) onto a similar algebra \( B =[T,F,\leq] \) is a mapping \( { \phi\colon S \to T } \) such that,
-
(i)
for all \( { f_{\alpha} \in F } \) and \( { x_{i}\in S } \),
$$\phi(f_{\alpha}(x_{1},x_{2}, \dots,x_{n(\alpha)})) \\ =f_{\alpha}(\phi(x_{1})\:,\phi (x_{2}), \dots, \phi(x_{n(\alpha)})) \:; \quad \text{ and}$$ -
(ii)
\( { x \leq y \text{ in } S } \) implies \( { \phi (x) \leq \phi (y)\text{ in } T } \).
The algebra B is termed a (homomorphic) imageof A, and we write \( { B=\phi (A) } \). Each homomorphism ϕ from \( { A =[S,F,\leq] } \) onto \( { B=[T,F,\leq] } \) hasa corresponding binary relation π on S (termed herea π‑relation) in which \( { (x,y) \in \pi } \) ifand only if \( { \phi (y) \leq \phi (x) } \). Theequivalence relation \( { \sigma_{\pi} } \) defined by \( { (x,y) \in \sigma_{\pi} } \) if and only if \( { (x,y) \in \pi } \) and \( { (y,x) \in \pi } \) istermed a congruence relation.
-
(i)
- Homomorphism lattice :
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The homomorphism lattice L(A)of the algebra \( { A =[S,F,\leq] } \) is the collection of allhomomorphisms of A partially ordered by the relation: \( { \phi_{1}\leq \phi_{2} } \) if, for all \( { x, y \in S,\, \phi_{2}(x) \leq\phi_{2}(y) } \) implies \( { \phi_{1}(x) \leq \phi_{1}(y) } \). Thelattice \( { L_{\pi} (A) } \) of π‑relations on A,dual to L(A), has the partial ordering: \( { \pi_{1} \leq \pi_{2} } \)if \( { (x,y) \in \pi_{1} } \) implies \( { (x,y) \in \pi_{2} } \), for any \( { x, y \in S } \).
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Pattison, P. (2009). Social Networks, Algebraic Models for. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_492
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