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Political Networks

by: Pierre Huel

Political Networks POL 279

Pierre Huel
GPA 3.56

Zeev Maoz

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Zeev Maoz
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This 24 page Class Notes was uploaded by Pierre Huel on Tuesday September 8, 2015. The Class Notes belongs to POL 279 at University of California - Davis taught by Zeev Maoz in Fall. Since its upload, it has received 54 views. For similar materials see /class/187550/pol-279-university-of-california-davis in Political Science at University of California - Davis.


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Date Created: 09/08/15
Political Networks Methods and Applications Session 5 Cliques Blocks and Blockmodels Topics for Discussion Natural and Derived groupings in networks 0 Cliques Ncliques Derivation of blocks alternative methods 0 Blockmodel analysis of social networks Using cliques and blocks as units of analysis 0 Applications WWFk In many cases and for a wide variety of reasons we may be interested in comparing different subsets ofa given network to each other In some cases these subsets may be determined exogenously for example geographical regions in international networks arties in Congress or parliaments departments in an organization and so forth In such cases we would partition the Sociomatnx into submatrices according to an exogenous criterion However in other cases we may want the structure of relationships in the network to determine these subgroups their sizes number identity of members and relationships between them We focus on partitions ofnetworks that are derived endogenously Two principal types of network partitions are of interest partition of networks into cliquelike structures and partitions into bloc s llques A clique is a fully connected subset of a network Typically SNA deals with cliques of size 3 nodes but for some purposes that we will discuss later on we can actually have smaller cliques dyads isolates A clique can consist of 1 2 N nodes Each member of a clique has direct ties to all other members Note again that most SNA treatments of cliques require that cliques consist of 3 4 N nodes Cliques are based on symmetric nondirectional binary relations Nonsymmetric relations cannot be partitioned into cliques without somehow converting them into symmetric relations 1 There are several ways to symmetrisize asymmetric relations One is to define a symmetric relation as 1 if shagi1 another is to define a relation as symmetric if either 501 or si1 or both Also valued relations must be converted into binary ones in order to allow derivation of cliques Cliques are not mutually exclusive in terms of members However no clique can be a proper subset of another clique Any two cliques derived from a given network must differ with respect to at least one node Consider the following simple network 0 gt muomgt o xo xozp OOm O O OO OO O U ooo xorn 39 This network produces the following clique structure Node Clique 1 1 1 1 0 0 rnUOl39DZD OO O 0 1 1 0 This is the clique membership or clique affiliation matrix 634 This is a typical affiliation matrix with entries cam defined as one if a node i is a member of clique kand zero otherwise Note that cliques I and II overlap with respect to two members A and C but clique I has B in it and II does not whereas II has member D that is not member in clique I We can also generate a clique Membership overlap matrix CMO that takes on the following form C 2 1 2 1 muomgt o xm xmgt 404M4m 044040 OO O39I39I 0 The clique overlap matrix is obtained from CA such that CMOCAgtltCA and its properties are similar to the properties of a sociomatrix obtained from affiliation data ie CMO is of dimension N cmoii the number of clique memberships of node i cmoijcmoji the number of cliques that nodes iand j share in common The cliquebyclique overlap CDC is obtained from COCCA XCA and has the following structure With properties simiIar to cmo yet with the clique as the unit of observation CDC is of dimension k the number of cliques cocii is the no of members in clique i and cocij is the number of nodes that cliques iand jshare in common 4Noo DOOM NO What is the story of cliques Cliques are groupings that are derived from a given network structure on the basis of a definition of complete connectedness every clique has a density of one The most important features of cliques are twofold 1 they are cohesive in the sense that all members within a clique are tied to one another and 2 they are not mutually exclusive so that a node can belong to more than one clique and two cliques can overlap in terms of membership This notion of cohesive groups makes a lot of sense in different political contexts For example alliance memberships trade networks political cliques in congress and parliaments small group interactions in organizations and so forth Clique structure provides another dimension by which we can characterize network structures at various levels We will get back to this point later on Solo unitnon Many RNA quotemuse cllques do noI make for dlscrele parllllons of Ihe nelwork In many tases Ihe overlap ln cllque membershlp ls exlenslve Io Ihe polnI IhaI many cllques are almosI ldenllcal ln Ienns of Ihe ldenIlIy of Ihelr members As noIed Ihe de vallon of cllques ls Ilmlled Io blnaly symmelrltal Ie nondlrecllonal relallons They can by seIIlng any poslllve mlue Io one or by eslabllshlng relallonal Ihresholds However Ihe orlglnal dala In mlued ne or e used Io measure cllque coheslon Agaln more on IhaI laler The resulllng dala does noI provlde us wllh a sense of whelher Ihe Iles wllhln one cllque suggesl a closer relallonshlp among members Ihan Ihe Iles wllhln anoIher cllque Bul Ihls can also be flxed Thls deflnlllon of cllques suggesl IhaI large nelworks tan have a huge number of cllques and Ihelr analysls mlghI be vely dl lcull lack of excluslveness ls problemallc when we wanl Io creale a unlque mapplng such IhaI each node w belong Io slrlclly one subgroup The derhmllon of cllques ls used only on flrsIolder Iles Cllques fonned by lndlrecl Iles are noI lapped ln Ihls appmach Io nelwom parllllonlng NCliques Ncllqlles are dimes Illal reerd dlrecl and Imllrecl rehllons of flrsl second and up In line NM omer They are ohlalned from a readmhlllly mal x of omer 2 n Thus 2dlqlles reflecl fully wnnecled subsels of line nelwom Illal have dlrecl and secondolder Iles 3 Iqlles reflecl fully conneded subsets Illal have dlrecl second mi IIIInIonier Iles and so fonll The analysls of N hues Is sllnllar Io quotml of cllqlles Vel consldered Io be less reslrlcllve and more Incluslve Ilmn cl Iles Democratic Networks Network at Stage Network at Stage II Clique No Clique No State 1 2 3 4 Total State 1 2 3 4 Total 1 1 O O O 1 1 1 O O O 1 1 O O O 1 2 1 O O O 1 3 1 O O 1 2 3 1 O O 1 2 4 1 O O O 1 4 O O O 1 5 1 O 1 O 2 5 1 O 1 O 2 6 1 1 1 1 4 6 1 1 1 1 4 7 O 1 1 O 2 7 O 1 1 O 2 8 O 1 O 1 2 8 O 1 O 1 2 9 O 1 O O 1 9 O 1 O O 1 1 0 O 1 O O 1 1 0 O O O 1 1 1 O 1 O O 1 1 1 O 1 O O 1 Dyads 15 15 3 3 36 Dyads 15 15 3 3 36 DemDem 0 3 0 1 4 De mDem 1 6 0 3 1 0 Prop Dem 0000 0200 0 000 0333 01 1 1 Prop Dem 0067 0400 0000 1 000 0278 Independent Variable Baseline Model All Cliques Nondemocratic Democratic Cliques Cliques AVGREGgt0 AVGREG O Proportion ofMID dyads in Clique No of States in Clique 0144 0144 0096M 0215M 0015 0015 0020 0027 Degree of Clique overlap 0107 0132 70017 0248 with other cliques 0161 0161 0203 0266 Prop Clique Dyads in 70065 70060 0127M 0025 Alliance 0010 0011 0015 0017 Average Capability ratio 70001 70001 70001 70001 across clique dyads 0000 0000 0000 0000 Average Regilne Score of 0001 0 004 Cli ue 0006 0001 Proportion Democracies in 0111 Clique 0026 Constant 0124M 0151M 0284 70121 0032 0035 0054 0124 N 10798 N 10798 N6524 N2374 Cliques200 Wald x216986 Cliques200 Wald x218895 Cliques 188 Wald x210911 Cliques197 Wald x27412 No of States in Clique Degree of Clique overlap with other cliques Prop Clique Dyads in Alliance Average Capability ratio across clique dyads Average Regilne Score of Clique Proportion Democracies in Clique Constant 0008 0003 0372 0033 70010 0002 7538e706 220606 70001 0007 N 10799 Cliques201 Wald x225585 Proportion of War dyads in Clique 009M 0003 0377 0033 70010 0002 7483e706 228606 0021 0005 0005 0007 N 10799 Cliques 201 Wald x227485 0003 0005 0253 0049 700179 0003 700019 Cliques 194 Wald x2 823w 0009 0006 0541 0056 0005 0003 7312606 248e706 0001 0000 70011 0015 N4269 Cliques201 Wald x217537 Blockmodels are models that are based on the partition oia given set of into discrete subgroups Ihese subgroups may be a result of the partitioning rks Ihe placement of units into subgroups deiines their pmitimls Each po contains a 39 39 39 39 o eac m e the pro les of their 39es with other acmrs A key aspect of blockmodelling concerns the identi cation of meaninng roles of actors as members of subgroups and the analysis of the 39 39 a social network 39 J peripheries within the world system we win 39 39 I 39 in the next 39 n we will analyze multirelational networks and multirelational sesslo hlockmo e s Partitioning Networks Again Recall that there are several methods to partition networks into blocks based on similarities between units nodes CONCOR is one approach hierarchical partitioning or multidimensional scaling are other Each approach has advantages and disadvantages but they generally produce similar results Consider the following binary matrix rnUOUJgt o xo xozp o o w o xo xoo OO O U ooo xorn CONCOR partitions this matrix into the following set of blocks w w BLOCK B1 BZ B1 AAAOOJ OAAOOE oooAAfw COCOACW OOOAAEL The same partition is produced by hierarchical scaling and by Multi Dimensional Scaling MDS using the rstorder correlation matrix of structural equivalence Entries in the partition matrix are permuted into blocks Each entry re ects the presence or absence of a tie between units land 1 Note that in this case there are no ties between units that belong to the same block only between units from di erent blocks Instead of examining individual units as elements of a network we can now examine the network in terms of the blocks that were formed by some clustering or scaling method Since in our example we have three blocks we can describe the network in terms of a blockmatrix B of order 3 X 3 where each cell 60 re ects a relationship between block i and block j In our case the blockmatrix is What do the entries mean In our case each entry re ects the density of the block This allows us to de ne the relationship between positions in a blockmodel in a binary fashion For example we can de ne a relationship between positions as zero if the density of the block is lower than the density of the original sociomatrix and one otherwise in our example the density of the original sociomatrix is 05 so that positions B132 B183 are one and so are the symmetrical positions and the rest are zeroes Blockmodels then are de ned by two elements 1 A mapping of units nodes in the original sociomatrices into positions 2 A BxB binary matrix that de nes the relationship between any two blocks Whether a given entry in the B matrix is zero or one is de ned by one of several different criteria discussed in wF We use here the density criterion to define entries Thus the matrix above is given as Consider the following example of an alliancecapability matrix AC Entries in the AC matrix are a product of the row s capabilities and the commitment score entailed in an alliance between row and column In contrast to the previous example this is a valued network matrix Note the density of the sociomatrix is A 0073 A CONCOB procedure yields the following partition The matrix depicts the ties commitment level between states within a given block Note that we defined selfties as 1 and the maximum commitment of 8 i j is 075 Thus the density of the commitment matrgx iBs A 1 3285 0456 2 Z x The density of the blocks is given by i1 11 U B Where Bis the number of units within a block 3 33 1075 B753 25 B 2x U j1 LMw 7 r 39 1000 0214 0000 0214 1000 0000 51 0000 0214 0571 Thus by the density criterion AB 2 AS we have the following blockmodel What can we do once we have blocks and we know the relationship between them Th analys s of blocks LEIH39 mH39HH 111 MLMJ rm y 1 va lz P Advmage ol39 Blockmollelllng Allm mapplng ol39 unltu Into Ilium pmltlons III Is Important ll39we want to lllve l Irreululuwn of I netwan Intn mutually exalullve and exhaustive groups Appllnable to Mill lymmelrlc and nonsymmetrlc networks Appllnable to Mill lllnary and valued dab Flexllrle In terms It method at dellnlng block formation erlterla Ie we can use Illl lerent manure at almllarlly anll Illnlmllurlty tn Genetum blockl Flexllrle In terms It the method el39 blnclunmlelllng e4 coNcon lllerarelllnal austeran MDS etc Flexllrle In terms 0 the milled Il39 generating Image matrices Make for an Interestlng l39oumlltlon l39or meuurlng network properties we will dlwuu IIIII next week IJmitaliona af Bloelanadele 80mm arbitrary parlilionlng of network due to emphasis on discrete Inappan uf Ilnlte Different hincklnedellng methods prndllce different results Different Image matrices hand In different criteria fur determinan fits willlln and helwoen blanks Disordeness Is a disadvantage when we are Interested In overlapping cummilmenti Consider again the commitment matrix in the previous example 1 1 0 0 0 0 Note that two of the cliques are blocks BE CD but two of the cliques are not identical to blocks


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