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Relations & Combinations: Applying Matrix Algebra David Knoke University of Minnesota POLNET

Relations & Combinations: Applying Matrix Algebra David Knoke University of Minnesota POLNET Universiteit van Tilburg June 20, 2007. It All Began with Sociograms.

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Relations & Combinations: Applying Matrix Algebra David Knoke University of Minnesota POLNET

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  1. Relations & Combinations: Applying Matrix AlgebraDavid Knoke University of Minnesota POLNET Universiteit van Tilburg June 20, 2007

  2. It All Began with Sociograms Although origins of network analysis lie in 1920s, Jacob Moreno (1934) pioneered social network analysis for “psychodrama” therapy. Moreno used sociomatrices and hand-drawn sociograms to show boys’ and girls’ likes and dislikes of classmates as directed graphs (digraphs).

  3. Basic Matrix Elements Asociomatrix(adjacency matrix), designated by a boldface capital letter such as X, is the most common matrix form for presenting the social network information in agraphG . (Aone-mode sociomatrixis a square g X g array of the graph’s N nodes, where both the rows and the columns are displayed in the identical sequential order.) In digraphs, the senders are in the rows and the receivers are in the columns. The g2cell elements, denoted by lowercase xij, are the values of the L lines in the graph, with a distinct value for every <i,j> ordered nodal pair. Row actor i sends a relation to receiver in column j. In graphs of nondirected lines, all pairwise cells have equal values, xij = xji (i.e., the matrix is symmetric). But, a digraph’s matrix may have pairwise values are not equal. For example, in a binary matrix if i sends to jbut j does not send to i, then xij = 1 and xji = 0. If a pair reciprocate ties, both cells have identical values. A single row or column of a matrix is a called vector.

  4. Here’s a nondirected one-mode graph and its matrix: HARRY ● SALLY● ● TOM DICK● ● BETTY BDHST BETTY 0 1 1 0 0 DICK 1 0 1 1 0 HARRY 0 1 0 0 1 SALLY 0 0 1 0 0 TOM 0 0 0 0 0

  5. Affiliation Networks An affiliation network consists of two-mode data, different sets connected by relations between but not within each set. If the two sets are “actors” and “events,” elements within each mode are indirectly tied, via common links to the other mode. Familiar examples of affiliation networks include: persons belonging to voluntary associations; social movement activists participating in protest events; firms creating strategic alliances; nations signing trade and military treaties. • Formally, a pair of elementary sets connected by a (0-1 binary or ordinal) relation: • Set N of g nodes (“actors”): N = {n1, n2, ..… ng} • Set M of h nodes (“events”): M = {m1, m2, … mh} • L nondirected lines join the gxh ordered pairs of nodes <ni, mj> An affiliation network can be displayed either as a bipartite graph, or as a gxhaffiliation matrix (A) whose i,j entry indicators whether actor i participated in event j. Its hxgtranspose matrix (A’) shows whether event j attracted actor i.

  6. Duality of Persons & Groups Ronald Breiger’s (1974) classic article on the duality of persons and groups discussed: (1) actor-actor connections occurring through their co-membership or co-attendance at the same events; and (2) event-event connections via the overlap or interlocks with shared actors. • These two dual networks can be created by either pre- or post-multiplying an affiliation network and its transpose to create two one-mode matrices: • AA’ is a gxg symmetrical matrix; its main diagonal entries show the number events in which an actor is affiliated; its off-diagonal elements are the number of events in which a row & column pair jointly participated. • A’A is an hxh symmetrical matrix whose main diagonal entries show the number actors participating in the row event; its off-diagonal elements are the number of actors affiliated with a particular pair events. Both dual matrices may be analyzed as one-mode networks, measuring such properties as size, density, reachability, and cohesion. Interpretations of co-memberships must recognize that entities are indirectly connected, and that the specific identities of those indirect paths cannot be known from the dual matrix (e.g., we know the number of events a pair attended but not which events).

  7. Affiliations in the GIS Consider this 1998 affiliation matrix (A) from the Global Info Sector project, 10 U.S. computer/software firms participating as partners in 54 alliances. A small portion of the bipartite graph appears below. 111111111122222222223333333333444444444455555 123456789012345678901234567890123456789012345678901234 ------------------------------------------------------ 1 APPLE 000000001000001000000001000000000000000000000000000000 2 COMPAQ 100011100110000101000000000110000110011100001100110000 3 DELL 100000000000000000000000000000000100010000000010100000 4 HP 000100010000010000001101101011100100000000010000100110 5 IBM 110000000000100010001010000010011110110000001001101000 6 INTEL 100011101101000100110010000000000001000000100101001000 7 MICROSOFT 011111100011011101110100101111111001001100001110110000 8 NETSCAPE 000000000000000000000000000000000000000011101000000001 9 ORACLE 000000010000000000000000010000000000000011001000100010 10 SUN 001000000000100010000000010000000000100000010000000101 Apple ● Compaq ● Dell ● HP ● IBM ● INTEL ● ● SA #1 ● SA #2 ● SA #3 ● SA #4 ● SA #5 ● SA #6

  8. How to Multiply Matrices Two matrices can be multiplied only if they’re conformable: the number of columns in Matrix 1 must equal the number of rows in Matrix 2. Resulting matrix dimensions = # rows of Matrix 1; # columns of Matrix 2. A row-column cell value in the resulting matrix is the sum of products of the elements in the corresponding Matrix1 row times Matrix 2 column. X Y To illustrate: Multiply 5x5 matrix X by 5x2 matrix Y, which is possible only because the number of columns in X equals the number of rows in Y. (Why is Y multiplied by X not possible?) The result is 5x2 matrix Z: 0 1 1 0 0 1 0 1 1 0 0 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 2 2 0 2 2 2 0 0 2 = Z XY 4 2 4 4 2 2 2 2 0 0 (0*0 + 1*2 + 1*2 + 0*2 + 0*0 = 4) (0*2 + 1*0 + 1*2 + 0*0 + 0*2 = 2) (1*0 + 0*2 + 1*2 + 1*2 + 0*0 = 4) (1*2 + 0*0 + 1*2 + 1*0 + 0*2 = 4) (0*0 + 1*2 + 0*2 + 0*2 + 1*0 = 2) (0*2 + 1*0 + 0*2 + 0*0 + 1*2 = 2) (0*0 + 0*2 + 1*2 + 0*2 + 0*0 = 2) (0*2 + 0*0 + 1*2 + 0*0 + 0*2 = 2) (0*0 + 0*2 + 0*2 + 0*2 + 0*0 = 0) (0*2 + 0*0 + 0*2 + 0*0 + 0*2 = 0) =

  9. To multiply dual network matrices in UCINET, use Data/Transpose to transpose matrix A into matrix A’, then use Tools/Matrix Algebra to multiply this pair of matrices in a specific order. Post-multiplying the two-mode 1998 GIS matrix A by its transpose, using the UCINET Algebra command: OxO=prod(MAT_A,MAT_ATR), creates a 10x10 orgs-by-orgs matrix (OxO). 1 2 3 4 5 6 7 8 9 10 AP CO DE HP IB IN MI NE OR SU -- -- -- -- -- -- -- -- -- -- 1 APPLE 3 0 0 1 0 1 1 0 0 0 2 COMPAQ 0 19 4 3 7 7 14 1 2 0 3 DELL 0 4 5 2 4 1 2 0 1 0 4 HP 1 3 2 16 4 0 9 0 3 2 5 IBM 0 7 4 4 17 4 6 1 2 3 6 INTEL 1 7 1 0 4 16 9 1 0 0 7 MICROSOFT 1 14 2 9 6 9 31 1 2 1 8 NETSCAPE 0 1 0 0 1 1 1 5 3 1 9 ORACLE 0 2 1 3 2 0 2 3 7 1 10 SUN 0 0 0 2 3 0 1 1 1 8 The main diagonal shows that Microsoft was the most active in forming partnerships (31) and Apple the least active (3). The off-diagonal entries reveal that Microsoft and Compaq partnered most often (14 times), but 13 pairs formed no partnerships that year

  10. Command ExE=prod(MAT_ATR,MAT_A) creates a 54x54 event-by-event matrix (ExE). Its main diagonal shows the number of organizations forming the strategic alliance (e.g., 4 partners in event #1) and the off-diagonal shows the number of actors common to two alliances (e.g., 3 organizations participated in both alliance #1 and alliance #34). To save space, I display only the first five rows of the event-by-event matrix: 111111111122222222223333333333444444444455555 123456789012345678901234567890123456789012345678901234 ------------------------------------------------------ 1 410022201211100211111020000120011321131100102212312000 2 121111100011111111111110101121122111111100002111211000 3 012111100011111111110100111111111001101100011110110101 4 011211110011021101111201202122211101001100011110210110 5 211133301222011302220110101221111112012200102311221000 UCINET’s NetDraw Visualization Software can display ties among both sets in an affiliation network. Click File/Open/Ucinet dataset/2-Mode network and create the following diagram.

  11. What evidence that a Wintel coalition (Microsoft & Intel) was opposed by the NOIS (Netscape, Oracle, IBM, Sun) in the 1998 GIS strategic alliance spider web?

  12. Each one-mode matrix can be analyzed using conventional network methods to reveal properties of the inter-actor or inter-event indirect ties • Descriptive values for the OxO dual network are: • Density: • Mean alliances = 2.31 per dyad (s.d. = 2.94) • Network Centralization: • Degree = 24.0%; Microsoft = 45, Compaq = 38, Apple = 3 • Closeness = 49.7%; Microsoft = 9, IBM = 10, Apple =15 • Betweenness = 8.6%; Microsoft = 4.1, HP = 2.5, Apple & Sun = 0.2 • Clique Analysis (using only binary ties): 1: Compaq Dell HP IBM Microsoft Oracle 2: Compaq Dell IBM Intel Microsoft 3: Compaq IBM Intel Microsoft Netscape 4: Compaq IBM Microsoft Netscape Oracle 5: IBM Microsoft Netscape Oracle Sun 6: HP IBM Microsoft Oracle Sun 7: Apple HP Microsoft 8: Apple Intel Microsoft

  13. Multidimensional scaling (stress=0.14) with hierarchical clusters (complete link)

  14. 2-Mode Data Exercise “US National Labor Policy Domain Event Interests Network UCINET matrix input.txt” is a data file of 117 organizations’ interests in 36 policy events in 1988 (Knoke et al. 1996). • Run UCINETData/Import/DLto create and save this large file as a 2-mode orgs-by-events affiliation matrix. • Data/Extractto reduce the large file by keeping no more than 25 orgs and 10 events (read codebooks and examine matrix to decide which ones). • Display with UCINET NetDraw’s File/Open/Ucinet dataset/2-Mode network (try various display options). Interpret structural relations of orgs & events. • Data/Transposeto transpose the reduced network into a 2-mode events-by-orgs affiliation matrix and save. • Tools/Matrix Algebra to create both 1-mode OxO and ExE matrices. By visual inspection, which orgs/pairs were the most/least active? Which policy events/pairs attracted the biggest/smallest audiences? • Analyze various network properties of both 1-mode matrices (e.g., density, centrality, multidimensional scaling) and interpret your findings.

  15. Relational Algebra Relational algebra, also known as role algebra, uncovers the structure of social roles by investigating indirect connections across multiple relations. Role structures describe how social roles are associated in the networks, independent of individual actors occupying those roles. Scott Boorman and Harrison White (1976) proposed using role algebra methods as an extension of blockmodel analysis. But, relational algebra can be learned and applied independently of blockmodeling. • RELATIONAL ALGEBRA– a formal structure consisting of two elements - sets of relations and operations to manipulate those relations: • Dichotomous primitive relations (generator relations) among actors, represented by capital letters; e.g., relations F and E {for Friend & Enemy} • A composition operation () that combines two or more primitive relations. Compound relation FE (Enemy of my Friend) results from composition where tie i(F  E)j occurs if there exists some third actor k such that iFk and kEj

  16. High Tech Managers’ Roles Consider these graphs and matrices representing the advice-giver (A) and friendship (F) images for a three-position blockmodel of David Krackhardt’s 21 high-tech managers (Wasserman & Faust 1994:439-442). The circular arrows show that high densities of ties occur among the individual managers within three jointly-occupied blocks: ADVICE (A) FRIENDSHIP (F) a ● a ● c ● c ● b ● b ● Aabc a 0 1 1 b 0 1 1 c 0 0 0 Fabc a 1 0 1 b 0 0 0 c 1 1 1 What role structure is revealed by the four compound relations?

  17. Forming Compound Relations A compound relation is formed by the Boolean multiplication of two or sociomatrices. It resembles ordinary matrix multiplication, except that any cell entry greater than “0” is replaced by a “1”. A nonzero entry means that a compound relation exist between a pair of blocks/actors. Use Tools/Matrix Algebra in UCINET to open a window in which to write the matrix multiplication commands. (Consult the Help Manual entries under “Algebra Package” and “Algebra, Binary Operation” for proper syntax.) For Krackhardt’s three manager blocks, the Boolean matrix product command forAFis: AF=bprod(Advice,Friendship) AFabc a 1 1 1 b 1 1 1 c 0 0 0 Each 1 entry in AF identifies an advice-giving block connected via an intermediary position to a friendship block (i.e., “friends of advisors”): In the original advice network A, block a gives advice to block c (aAc), while in the friendship network F, block c cites block b as its friend (cFb). Hence, the compound relation AF reveals indirect connection from block a to block b: (aAc)(cFb) = (aAFb). The nonzero diagonal blocks in the original image matrices are always used when composing a relational algebraic compound. Thus, block b’s compound tie to block c involves the latter’s within-block friendships: (bAc)(cFc) = (bAFc).

  18. FA, “advisors of friends,” shows how one block might use their friends to contact an advisory block. For example, block c cites block a as friend and block a advises block b. Thus, the compound matrix reveals: (cFa)(aAb) = (cFAb). FAabc a 0 1 1 b 0 0 0 c 0 1 1 The FF product identifies that classic compound relation, “a friend of a friend.” Because block b cites no direct friends, it also can’t reach any indirect friends. But, the other blocks have friends who are connected to other friends: (aFc)(cFb) = (aFFb). FFabc a 0 1 1 b 0 0 0 c 0 1 1 Finally, the AA composition yields the “advisors of advisors” (evidently the experts’ mavens). But this compound matrix is identical to the original advice-giving image! The composition AA=A reveals that this advising network is transitive; for example, the compound (aAb)(bAc)=(aAAc) which is already a direct tie (aAc). Hence, we don’t need AA because A includes all those compound relations within its direct advising ties. AAabc a 0 1 1 b 0 1 1 c 0 0 0

  19. Words & Equivalence Composition can involve sequences longer than two compounded primitive relations; e.g., AFFAAFA. A string of letters is a word, whose length is the number of primitive relations in it. Role algebraists inductively generate a “dictionary” of the unique words (matrices/images), with the fewest letters, required for a complete description of a multiple-network system’s social role structure. When a researcher generates a longer new word, she compares its sociomatrix to see whether any simpler word already in the dictionary also has that longer word’s sociomatrix. Words with identical matrices or images are equivalent, and the set of all words with identical images comprise an equivalence class. For Krackhardt’s high-tech manager data, the five shortest unique words in its dictionary areA, F, AF, FA, andFF(but notAA). See W&F (1994:440) for some examples of longer words in those equivalence classes.

  20. Role Table • In a multiplication table, or role table, each row and column entry corresponds to a unique primitive or compound relation. Instead of displaying network images (as W&F show in Fig. 11.2), each equivalence class in the table is labeled by the graph’s word. The cell entries in the table contain the smallest word resulting from multiplying the row matrix by the column matrix. • Matrix multiplication is associative: the order of performing successive multiplications does not affect the result: ABC=(AB)C=A(BC) • Matrix multiplication is not commutative: the result of multiplying two matrices may differ by the sequential order: AB≠BA • In mathematical theory, a semigroup is defined as a set elements with an associative binary operator on it. Thus, a social network semigroup is the set of images/matrices formed by a set of relations and the composition operation (Boyd 1990; Pattison 1993). If all compositions of the primitive relations are also members of the set, then a semigroup is closed under associative matrix multiplication. A role table contains “all possible images that can result from the operation of composition on the primitive relations” (W&F 1994:437).

  21. The Dictionary The role table for the Krackhardt managers’ advice and friendship networks shows that composing any pairs of the five unique words in the dictionary yields four of these words (see W&F Fig. 11.5). For example, multiplying (AF)(FA) = (AFFA). But, the first three terms on the rightside can be factored: (AFF)=(AF)(F) and we find in the table that (AF)(F)=(AF). Hence, by substitution (AFFA)=(AFF)(A)=(AF)(A). Finally, the role table shows that (AF)(A)=(A), so the multiplication (AF)(FA)=(AFFA) reduces to just (A), as displayed in row 3 column 4. A F AF FA FF A A AF AF A FF F FA FF FF FA FF AF A AF AF A AF FA FA FF FF FA FF FF FA FF FF FA FF Rows 1 & 3 are identical, as are rows 2, 4, & 5, as well as two column pairs, 1 & 4 and 2 & 3. These identities suggest opportunity to simplify the social role structure of Krackhardt’s high-tech managers (next).

  22. Simplifying a Role Table Role table simplification involves reducing the number of network images or words while preserving important structural properties. Each image in the initial set is mapped onto a smaller number of images in the simplified set. “The reduction of the role table is a partition of the distinct images, S, into a smaller collection of classes, Q.” (W&F 1994:443). Unfortunately, a unique or “best” reduction may not be possible for some networks. • Image simplification strategies include: • Substantive approachesthat combine images with the identical meaning or similar operation; • Sociometric approachesthat equate images with similar ties which may differ substantively. Sociometric similarity could be assessed using correlation to measure association, or finding images that are contained within (subset of) another image. See W&F (1994:444-445) for an application of the latter technique to the advice and friendship role table.

  23. Homomorphic Reduction A homomorphic reduction of an original role table involves a mapping that preserves the composition operation. More than one image may exist. • One homomorphic image for the A&F role table permutes and partitions the 5x5 table into two groups that produce “nearly identical results”: {1,3} and {2,4,5} which have the word equivalences {A,AF} and {F,FA,FF}. The reduced matrix expresses a first letter law that “any two elements always result in an element that is in the same class as the first element of the composition“ (447). • Another homomorphic image groups {1,4} and {2,3,5}, with word equivalences {A,FA} and {F,AF,FF}. This reduction satisfies a last letter law that “the composition of any two elements results in an element that is in the same class as the second element of the composition” (448). If the same or comparable relations are measured for two or more network systems, their role tables can be compared on formal similarities and/or differences. Boorman and White’ (1976) joint homomorphic reduction of two role structures summarizes common features. It involves two mappings that preserve the composition operation, resulting in a new multiplication role table that is a reduction of both original tables. As the union of two roles structures, the role table contains all the word equations appearing in one or both systems. For example, Breiger and Pattison (1978) interpreted the joint homomorphic reduction of three political elite networks in a German town and a U.S. city as an instance of Granovetter’s strong-weak tie hypothesis.

  24. Role Algebra Exercise A blockmodel of the “send policy information” and “give support” networks in the U.S. labor domain yields two 4-block images, whose main members are: (A) unions; (B) public interest groups; (C) businesses; (D) federal government. InfoABC D A 1 0 0 0 B 1 1 0 1 C 1 0 1 1 D 1 0 0 1 SupportABC D A 1 1 0 1 B 1 1 0 0 C 0 0 0 1 D 0 0 0 1 • Create text files for both image matrices and add DL commands that enable you to import and save these networks as two UCINET files. • Use matrix algebra to compute four possible Boolean matrix products corresponding to the compound relations among Information and Support (e.g., II, IS, etc.). What are your substantive observations? • Produce a role table containing the complete dictionary of all the unique words that can be formulated by compounding Information & Support. Provide a substantive interpretation of the patterns you observe in this national policy domain.

  25. References Boyd, John P. 1990. Social Semigroups: A Unified Theory of Scaling and Blockmodeling as Applied to Social Networks. Fairfax, VA: George Mason University Press. Boorman Scott A. and Harrison White C. 1976. “Social Structure from Multiple Networks, II: Role Structures.” American Journal of Sociology 81:1384-1446. Breiger, Ronald L. 1974. “The Duality of Persons and Groups.” Social Forces 53:181-190. Breiger, Ronald L. and Philippa E. Pattison. 1978. “The Joint Role Structure of Two Communities’ Elites.” Sociological Methods & Research 7:213-226. Knoke, David, Franz Urban Pappi, Jeffrey Broadbent and Yutaka Tsujinaka. 1996. Comparing Policy Networks: Labor Politics in the U.S., Germany, and Japan. New York: Cambridge University Press. Krackhardt, David. 1999. “Ties That Torture: Simmelian Tie Analysis in Organizations.” Research in the Sociology of Organizations 16:183-210. Moreno, Jacob L. 1934. Who Shall Survive? Washington, DC: Nervous and Mental Disease Publishing Co. Pattison, Philippa E. 1993. Algebraic Models for Social Networks. New York: Cambridge University Press. Wasserman, Stanley and Katherine Faust. 1994. Social Network Analysis: Methods and Applications. New York: Cambridge University Press.

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