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Computational Complexity pp 2925–2939Cite as

Social Networks, Algebraic Models for

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Article Outline

Glossary

Definition of the Subject

Introduction

Algebras from Networks

Algebraic Structure

Algebraic Analysis

Algebraic and Network Mappings

Examples

Future Directions

Bibliography

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Abbreviations

Social network :

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 :

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 \(f_\alpha (x_1 ,x_2 , \ldots ,x_{n(\alpha )} ) \le f_\alpha (y_1 ,y_2 , \ldots ,y_{n(\alpha )} )\) 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 :

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 :

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:

$$ \begin{aligned}& x \wedge x =x \:, \\ & x \vee x =x\end{aligned}$$

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 :

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 :

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,

  1. (i)

    for all \( f_\alpha \in F \) and \( { x_{i}\in S } \),

    $$\phi(f_{\alpha}(x_{1},x_{2}, \dots,x_{n(\alpha)})) \\ =f_{\alpha}(\phi(x_{1})\:,\phi (x_{2}), \dots, \phi(x_{n(\alpha)})) \:; \quad \text{ and}$$
  2. (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.

Homomorphism lattice :

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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Authors and Affiliations

  1. Department of Psychology, University of Melbourne, Parkville, Australia

    Philippa Pattison

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  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Pattison, P. (2012). Social Networks, Algebraic Models for. In: Meyers, R. (eds) Computational Complexity. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1800-9_180

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