class: middle, left, inverse, title-slide .title[ # UT-Tyler SNA Bookclub ] .subtitle[ ## Analyzing social networks using R
Chapter12 Equivalence ] .author[ ### Chad (Chungil Chae) ] .institute[ ### Wenzhou-Kean University ] .institute[ ###
] .date[ ### November 07, 2023, Draft v0.1 ] --- <style> .center2 { margin: 0; position: absolute; top: 50%; left: 50%; -ms-transform: translate(-50%, -50%); transform: translate(-50%, -50%); } </style> <!-- xaringanExtra 셋팅 -->
<!-- 환경설정 --> <!-- 슬라이드 내에 돌릴 코드 --> <!-- Comment --> # Introduction .pull-left[  ] .pull-right[ - [Chungil Chae (chad)](https://scholar.google.com/citations?hl=en&user=c4lRBrkAAAAJ) - Wenzhou-Kean University - College of Business and Public Management, Business Analytics - Assistant Professor - Reserach Interest - Social Network Analysis - Social Sequence Analysis - Human Behavior - Knowledge Sharing - Learning Experience / Leanring Path - Cognitive Architecture - Path - Management -> Engineering(HRD in Tech Mgs) -> Education(WFED/CIED) -> Engineering(Cognitive Science) -> Management(Business Analysis)] --- # Outline - Structural Equivalence - Profile Similarity - Blockmodels - Optimization - Regular Equivanence - The REGE algorithm - Core-Periphery Models - Summary - Problem and Exercises --- # Chapter12: Equivalence .boxgreen[ .boxtitle[ **EQUIVALENCE** ] - In social network analysis, equivalence refers to the concept of nodes (or actors) in a network having **similar structural roles or positions** (Sailer, 1978). - Nodes that are deemed "equivalent" share comparable patterns of ties or relationships with other nodes in the network (Borgatti at al., 2018). ]  ??? As highlighted in the previous summary, there are different notions of equivalence. For instance, Lorrain and White's (1971) structural equivalence suggests that nodes are equivalent if they have identical relational ties to the same other nodes. On the other hand, White and Reitz's (1983) regular equivalence broadens this view, allowing nodes to be equivalent if they relate in similar ways to equivalent nodes, not necessarily the same ones. This notion of equivalence, particularly relational equivalence, underscores the intricate interdependencies between roles in networks, such as the teacher-student dynamic where their roles are defined in relation to each other. In essence, equivalence in social network analysis provides a framework to identify and classify nodes based on the similarity of their structural positions or roles within the broader network. Social role concepts have been pivotal for social theorists since the mid-20th century. While some focus on the rights and obligations that come with these roles, others, like Nadel (1957), consider them from a relational standpoint, emphasizing the interdependencies between roles, such as the teacher-student relationship. Social networks are ideal settings for understanding roles relationally, as they involve various relations among a set of participants. An essential aspect of social roles is their definition across multiple relations, highlighting the importance of considering various ties simultaneously. In network analysis, nodes with similar structural roles are termed 'relationally equivalent.' This concept stands in contrast to the relational or community detection methods discussed in Chapter 1. Positional approaches identify nodes with similar structures, whereas cohesive subgroup methods identify closely connected node clusters. Definitions accommodating both directed and undirected networks have emerged, and they've evolved to analyze imperfect data. Over time, network researchers have crafted several structure-based equivalence concepts, each having its distinct value. This chapter specifically delves into two primary ideas: Lorrain and White's (1971) structural equivalence and White and Reitz's (1983) regular equivalence, with the latter encompassing the former. The chapter also sheds light on core-periphery structures, anchored on these two concepts, starting with structural equivalence. --- # Structural Equivalence .font150[**Structurally equivalent actors share other similarities:**] ## What this mean? - Similarity in attitudes (Erickson, 1988); behaviors (Burt, 1980) - The idea that persons adapt to their social environments, and therefore actors with similar social environments will tend to have certain similarities. - Substitutability: Nodes that are structurally equivalent are structurally indistinguishable and therefore substitutable. - Data reduction device because it enables us - to build a simplified model - high-level description of the relations within the network. ??? - Structurally equivalent actors share other similarities: - attitudes (Erickson, 1988) - behaviors (Burt, 1980). In other words, actors in the same structural equivalence class tend ot show a certain amount of homogeneity, very much like actors in the same cohesive subgroup. - Thus, one mechanism underlying the relationship between struc- tural equivalence and homogeneity si the idea that persons adapt ot their social environments, and therefore actors with similar social environments will tend to have certain similarities. - Another way to think about structural equivalence is in terms of substitutability. Nodes that are structurally equivalent are structurally indistinguishable and therefore substitutable. - A final reason for paying attention to structural equivalence si that we can use it as a kind of data reduction device because ti enables us to build a simplified model of the network without sacrificing essential features. Hence, ti provides a high-level description of the relations within the network. --- - Two actors are structuraly equivalent if: - they both send ties to the same third parties. - and receive ties from the same third parties. <img src="ch12_equivalence_en_files/figure-html/unnamed-chunk-3-1.png" width="100%" /> ??? - Two actors i and j are structurally equivalent if for every actor k diferent from i and ,j whenever i is connected to k, then j is also connected to k, and if i is not connected to k then neither is j. - If i connected ot itself then so is ,j and is not connected ot itself then neither is j -os that both actors share the same relationship with themselves. - Structural Equivalence: If two people, let's call them i and j, have the same connections to everyone else in a group, they're considered "structurally equivalent." So, if i knows someone, j does too, and if i doesn’t, then j doesn’t either. - Self-connection: If i is connected to themselves (like being self-reliant or self-focused), j is the same. If i isn't, then j isn't either. This means both i and j have the same relationship with themselves. - Multirelational Example: Imagine a diagram with 5 people (a1, a2, a3, a4, and a5) and three types of relationships (R1, R2, R3). In one relationship type (R1), a1 and a2 have the same connections, and so do a3, a4, and a5. But in the other two types (R2 and R3), only a1 & a2 and a3 & a4 share the same connections. a5 doesn't match with anyone in these two relationship types. - Visual Representation: If you look at a diagram and erase the names of two people who have the same connections to everyone else, you wouldn't know who was who. They're indistinguishable because they have the same relationship patterns. If they're structurally equivalent, everything about their relationships in the network is the same - their popularity, importance, number of mutual friends, etc. - Valued Data: Sometimes, relationships have values (like how close they are). For two people to be structurally equivalent, not only should they know the same people, but their closeness values with those people should also match. --- # Profile Similarity ## What it is? - Profile similarity refers to the measure of how alike the network connections or "profiles" of two actors are within a network. - By comparing the rows and columns of an adjacency matrix **(profiles)** - Different measures, such as matches, correlation, and Euclidean distance - These matrices can then be subjected to standard classification techniques like hierarchical clustering and multidimensional scaling. ## Why? - Structural equivalence is a theoretical mathematical model used to measure the degree of similarity in the network connections between actors. - In real-world data, exact structural equivalence is rare, - but the concept serves as a foundational principle for understanding how actors in a network may have similarities. ??? ??? Profile similarity refers to the measure of how alike the network connections or "profiles" of two actors are within a network. In the context of structural equivalence, it involves comparing the rows and columns of an adjacency matrix, which represent the connections of actors, to determine the extent of their structural equivalence. Different measures, such as matches, correlation, and Euclidean distance, can be used to compare these profiles and quantify their similarity. This similarity then informs the construction of a structural equivalence matrix, which showcases the degree of likeness between actors in the network. Structural equivalence is a theoretical mathematical model used to measure the degree of similarity in the network connections between actors. In real-world data, exact structural equivalence is rare, but the concept serves as a foundational principle for understanding how actors in a network may have similarities. To determine if two actors are structurally equivalent, we only need to know their direct connections, making ego-network data enough for calculation. The adjacency matrix provides the necessary information, where each actor's connections are represented as rows and columns, known as profiles. These profiles can be compared using various measures like matches, correlation, and Euclidean distance to determine the structural equivalence of actors. For instance, when comparing profiles of two actors in a directed network, their corresponding rows and columns in the adjacency matrix are matched and checked for similarity. This process creates a structural equivalence matrix (S), which showcases the similarity between actors. This matrix is square and symmetric, with its size determined by the number of actors. Depending on the measure used, the matrix can represent either similarity (high values indicating more similarity) or distance (high values indicating more differences). These matrices can then be subjected to standard classification techniques like hierarchical clustering and multidimensional scaling. --- ## For Example - The study of relationships among 18 monks in a monastery. - Hierarchical clustering of the matrix revealed specific groups of monks that had similarities in their connections, somewhat aligning with observations made during the study. - This method of determining structural equivalence provided a more detailed partition of the monk groups than the original observations, demonstrating the value and precision of the approach. <img src="resource/img/pe1.png" width="1500" height="300" /> ??? One real-world example is the study of relationships among 18 monks in a monastery. Using data about the esteem and disesteem relations among the monks, a structural equivalence matrix was created. While no two monks were found to be perfectly equivalent, certain pairs had high similarity. Hierarchical clustering of the matrix revealed specific groups of monks that had similarities in their connections, somewhat aligning with observations made during the study. This method of determining structural equivalence provided a more detailed partition of the monk groups than the original observations, demonstrating the value and precision of the approach (Sampson, 1969). --- - Sampson defined four sorts fo relation: affect, esteem, influence and sanction. - The esteem relation, which we split into two relations of esteem and disesteem. - Each novitiate ranked the other monks, giving their top three choices (esteem) and - his bottom three choices (disesteem). - A value of 3 represents the first (highest) choice and 1 the third choice. - However, not all monks rated exactly three others, and some used equal scores, e.g., assigning two 3s. <img src="resource/img/pe1.png" width="1500" height="300" /> --- - Shortly after these data were collected, **Gregory, Basil, Elias and Simplicius** were expelled Almost immediately, **John-Bosco** departed voluntarily. - A few days later **Hugh, Boniface, Mark and Albert left**, and within a week **Victor and Amand** departed. - One month later **Romuald** also left. <img src="resource/img/pe1.png" width="1500" height="300" /> --- Sampson grouped the monks as follows: - The Young Turks - Winfrid, John-Bosco, Gregory, Hugh, Boniface, Mark, Albert - The Loyal Opposition - Bonaventure, Ambrose, Berthold, Peter, Louis - The Outcasts - Basil, Elias, Simplicius - Indeterminate - Romuald, Victor, Amand <img src="resource/img/pe1.png" width="1500" height="300" /> --- .code16[ ``` ## ROMUALD BONAVENTURE AMBROSE BERTHOLD PETER ## ROMUALD 0 0 0 0 0 ## BONAVENTURE 0 0 1 0 3 ## AMBROSE 0 0 0 0 3 ## BERTHOLD 0 0 0 0 3 ## PETER 2 0 0 1 0 ``` ] .code16[ ``` ## Note that "Euclidean" and "AbsDiff" result in a matrix where high values represent differences. "MatchesN", "PosMatchesN", "Product", "Pearson", "Kendall" and "Spearman" result in a matrix where high values represent similarities./n ``` ``` ## ROMUALD BONAVENTURE AMBROSE BERTHOLD PETER ## ROMUALD 0.00 7.00 6.56 8.25 10.82 ## BONAVENTURE 7.00 0.00 5.10 8.31 11.92 ## AMBROSE 6.56 5.10 0.00 8.89 12.17 ## BERTHOLD 8.25 8.31 8.89 0.00 10.34 ## PETER 10.82 11.92 12.17 10.34 0.00 ``` ] ??? --- ``` ## Note that "Euclidean" and "AbsDiff" result in a matrix where high values represent differences. "MatchesN", "PosMatchesN", "Product", "Pearson", "Kendall" and "Spearman" result in a matrix where high values represent similarities./n ``` <img src="ch12_equivalence_en_files/figure-html/unnamed-chunk-10-1.png" width="100%" /> - Berthold, Victor, Bonaventure, Ambrose - Mark, Romuald, Winfrid, Hugh, Boniface, Albert - Basil, Elias, Simplicius - John-Bosco, Gregory - Peter, Louis, Amand ??? - To obtain a clustering based on structural equivalence - the esteem and disesteem matrices were submitted to the StructuralEquivalence() function. - The function constructs a profile for each actor by appending its row in the disesteem matrix to its row in the esteem matrix and calculates the level to which each pair of actors are structurally equivalent. - Because the data are not symmetric, it also appends to these profiles the columns of each data matrix, yielding profiles of the order of 4n cells in length. - Euclidean distance was selected as the choice for measuring the amount of structural equivalence between the profiles. The results are shown in Matrix 12.3, where the distances have been rounded up to the first decimal place. - Since we used Euclidean distance, a value of zero would indicate perfect structural equivalence. The only zero values are on the diagonal (since actors are of course structurally equivalent of themselves), so no two actors are perfectly structurally equivalent. - The smallest value is 5.1, achieved by Bonaventure and Ambrose, and separately, Albert and Boniface. The least similar pair of actors is Peter and Mark, who have a score of 13.38. - To obtain some form of structurally equivalent groups, we can submit this matrix to either amultidimensional scaling routine or a clustering method. Fgiure 12. shows the clustering dendrogram of the hierarchical clustering for the structural equivalence matrix given ni Matrix 123.. Usnig xUCNIET this is done with the function HierarchicalClustering(). - Berthold, Victor, Bonaventure, AmbroseMark, Romuald, Winfrid, Hugh, Boniface, AlbertBasil, Elias, SimpliciusJohn-Bosco, GregoryPeter, Louis, AmandGroups 2 and 4 together make up Sampson's Young Turks, while groups 1 and 5 form Sampson's Loyal Opposition. Group 3 is exactly Sampson's Outcasts. - The structural equivalence partition is more refined than Sampson's in that it splits apart some of his groups, but does not contradict it aside from splitting the 'indeterminate' (Romuald, Victor, Amand) across 3 groups. --- # Blockmodels - .font150[**Blockmodeling** is a process used in network analysis to simplify and understand the structure of a network. ] - Blockmodeling is a method used to understand and represent the structure of a network by grouping structurally equivalent actors (or nodes) together. - .font150[**Image matrix** is a simplified or reduced matrix by structural equivalence] - This matrix provides a concise representation of the network, highlighting the primary relationships and connections among the groups of nodes. - To visually represent these connections, the matrix can be displayed using white and black squares, where - black indicates a connection (value of 1) and - white indicates the absence of a connection (value of 0). - For networks with valued data (not just binary connections), - the matrix might represent average tie strengths or other values, - giving insight into the strength or nature of relationships. ??? Everyday Analogy for Blockmodeling: Imagine you're a student at a large school, and this school has numerous clubs and groups. Each club has its activities, and various students participate in them. When you first enter the school, you hardly know anything about all these clubs and groups. Nevertheless, through conversations with friends, you start to get an overview of what clubs exist, which clubs have similar activities, and which ones are in competition with each other. At this point, your brain tries to process and simplify all this complex information. For instance, you might categorize music-related clubs as the 'Music Group', sports-related clubs as the 'Sports Group', and science-related clubs as the 'Science Group'. By doing this classification, you can more easily understand the relationships within each group and between groups, making your school life smoother. In this sense, the way you cognitively simplify the relationships of clubs and groups in school works similarly to Blockmodeling. Blockmodeling is the process of simplifying relationships within a complex network for easier understanding, and it operates on principles similar to how you perceive the relationships of clubs and groups in school. Blockmodeling is a method used to understand and represent the structure of a network by grouping structurally equivalent actors (or nodes) together. Once nodes are grouped based on their structural equivalence, a simplified or reduced matrix, known as an 'image matrix', is produced. This matrix provides a concise representation of the network, highlighting the primary relationships and connections among the groups of nodes. In its pure form, the blocks in the matrix are either entirely filled with 1s (indicating connections) or 0s (indicating no connections), termed as '1-blocks' and '0-blocks'. However, in real-world data, these blocks may not be perfect, and the extent of their perfection indicates how well the data has been partitioned into structurally equivalent blocks. To visually represent these connections, the matrix can be displayed using white and black squares, where black indicates a connection (value of 1) and white indicates the absence of a connection (value of 0). This visual representation aids in understanding the underlying patterns in the data, such as identifying core groups or communities and understanding interactions between them. For networks with valued data (not just binary connections), the blocks in the matrix might represent average tie strengths or other values, giving insight into the strength or nature of relationships. When applied to real-world scenarios, like understanding relationships among monks based on esteem and disesteem, blockmodeling can reveal deeper insights, uncovering patterns and structural properties that might not be immediately apparent from the raw data. For instance, it can highlight groups that are cohesive or isolated, or provide insights into the nature and strength of interactions between different groups. --- .pull-left[ ``` ## a1 a2 a3 a4 a5 ## a1 1 1 1 1 1 ## a2 1 1 1 1 1 ## a3 1 1 0 0 0 ## a4 1 1 0 0 0 ## a5 1 1 0 0 0 ``` ] .pull-right[ ``` ## {a1,a2} {a3,a4,a5} ## {a1,a2} 1 1 ## {a3,a4,a5} 1 0 ``` ] <img src="ch12_equivalence_en_files/figure-html/unnamed-chunk-13-1.png" width="100%" /> --- ## Esteem network .pull-left[ <img src="ch12_equivalence_en_files/figure-html/unnamed-chunk-14-1.png" width="100%" /> ] .pull-right[ ``` ## ## . ------------------------------------------------------------------ ## . Number of valid cells: 306 ## . which corresponds to: 100 % of considered cells. ## . ------------------------------------------------------------------ ``` ``` ## 1 2 3 4 ## 1 0.357 0.020 0.000 0 ## 2 0.082 0.476 0.000 0 ## 3 0.286 0.143 NaN 0 ## 4 0.000 0.143 0.667 1 ``` ] --- ## Disesteem network .pull-left[ <img src="ch12_equivalence_en_files/figure-html/unnamed-chunk-16-1.png" width="100%" /> ] .pull-right[ ``` ## ## . ------------------------------------------------------------------ ## . Number of valid cells: 306 ## . which corresponds to: 100 % of considered cells. ## . ------------------------------------------------------------------ ``` ``` ## 1 2 3 4 ## 1 0.024 0.163 0.143 0.476 ## 2 0.184 0.048 0.286 0.571 ## 3 0.143 0.143 NaN 0.333 ## 4 0.429 0.048 0.000 0.000 ``` ] --- # Optimization
1. Performing a profile analysis 2. Partition our adjacency matrix. - to use an optimization method to find a good partition of the data --- - Assessing how close a partition is to an ideal blockmodel by examining the entries of each block. - To compare two different partitions on the same data, we just need to count the number of changes that are required to make the blockmodel fit (by minimizing the number of errors) the ideal structure of Os and 1s. - We call this the 'error', and we can now try to optimize the fit over all the possible assignments of actors to different groups. --- ## Four-block solution of optimization (Sampson 'liking' data) .pull-left[ <img src="ch12_equivalence_en_files/figure-html/unnamed-chunk-20-1.png" width="100%" /> ] .pull-right[ ``` ## ## . ------------------------------------------------------------------ ## . Number of valid cells: 306 ## . which corresponds to: 100 % of considered cells. ## . ------------------------------------------------------------------ ``` ``` ## 1 2 3 4 ## 1 1.000 0.042 0.000 0.250 ## 2 0.042 0.143 0.292 0.281 ## 3 0.000 0.167 0.833 0.000 ## 4 0.083 0.062 0.000 0.750 ``` ] ??? We have constructed our blockmodels by first performing a profile analysis and then using this to partition our adjacency matrix. An alternative si to use an optimization method to find a good partition of the data (Panning, 1982; Batagelj et al., 1992b). As already mentioned, we are able to assess how close a partition is to an ideal blockmodel by examining the entries of each block. To compare two different partitions on the same data, we just need to count the number of changes that are required to make the blockmodel fit (by minimizing the number of errors) the ideal structure of Os and 1s. We call this the 'error', and we can now try to optimize the fit over all the possible assignments of actors to different groups. For valued data, it is not possible to simply count the number of errors. In this situation, more sophis- ticated fit functions are used, but the principle remains the same. There are some specialist software packages that take this approach, and Pajek (Batagelj and Mrvar, 1998) ni particular has sophisticated implementations and extensions which we will briefly discuss later in this chapter. Implementations in Rare limited, and so we will present just one simple example. We demonstrate the optimization method on the dichotomized Sampson data for the l'ik- ing' relation at time point 3. Figure 12.6 is a four-block optimization of the data in which we look for 0-blocks and complete blocks using the function xBlockOptimize(). In this exam- ple, since the monks were asked only to rank their top three choices, it si not possible for the larger blocks to have al 1s. This means that the larger blocks cannot be made to fit wel, with the consequence that the technique struggles to find good solutions. The researcher needs to be aware that, regardless of the inherent structure, the method will produce an answer. This answer may not be particularly good - it may simply be the best of a bad set. It is always good practice to examine the results carefully to see fi they fit the model well. A second issue is that there may be a number of solutions that have the same fi.t The number of errors in Figure 12.6 is 40, but there are many other solutions that have 40 errors. Many of the other solutions are very similar - it is easy to see, for example, that swapping Peter and Louis would not change the error count. --- # Regular Equivanence .font150[Regular equivalence refers to a concept wherein two actors may not necessarily be connected to the same other actors, but they are connected in structurally similar ways.] (White and Reitz, 1983). - Regular equivalence relaxes the stringent condition on the 1-blocks in blockmodeling. - It is not essential for every block to be composed of 1s; rather, there should be at least one 1 in each row and column. ??? ??? Explanation: In structural equivalence, for two actors to be equivalent, they must be connected to exactly the same others. For instance, if two teachers teach the same set of students, they are considered structurally equivalent. However, in regular equivalence, the teachers need to teach at least one student each, and likewise, all students should have at least one teacher. As an example, in Matrix 12.5, there are three structural roles or groups that are regularly equivalent. In the first group, nodes (a1, a2, a3, a4) are equivalent and share the characteristic that each is tied to someone (not necessarily the same individual) within their own group and group 3, but not tied to anyone in group 2. This is termed the 'structural signature' of this set of nodes. Methods similar to structural equivalence can be employed to identify regular equivalence. The optimization method merely requires a minor adjustment to the fit measure, but in all other aspects, it is identical to that of structural equivalence. This concept of regular equivalence aids in simplifying and understanding complex network structures through the grouping of nodes based on their positions and connection patterns within the network. --- ## Regular blockmodel of Schwimmer's gift-giving network .pull-left[ <img src="ch12_equivalence_en_files/figure-html/unnamed-chunk-23-1.png" width="100%" /> ] .pull-right[ ``` ## ## . ------------------------------------------------------------------ ## . Number of valid cells: 462 ## . which corresponds to: 100 % of considered cells. ## . ------------------------------------------------------------------ ``` ``` ## 1 2 3 ## 1 0.167 0.204 0.206 ## 2 0.204 0.000 0.214 ## 3 0.206 0.214 0.000 ``` ] --- # The REGE algorithm .font150[The REGE (Regular Equivalence) algorithm is a method designed to address the extension of profile similarity to regular equivalence.] - Unlike structural equivalence, which focuses on the direct connections between actors, the REGE algorithm considers profiles of actor equivalences. - This method iteratively progresses toward an equivalence matrix, utilizing interim scores as equivalence measures between iterations. --- - Complexity: - Transitioning from profile similarity to regular equivalence is not straightforward. For structural equivalence, the measure is localized, based on direct connections to other actors. - However, in regular equivalence, profiles of actor equivalences must be compared, making the process more intricate. The equivalences aren't predetermined and become evident only once the process concludes. - Iterative Process: - The REGE algorithm functions through an iterative procedure. - It continually moves toward a final equivalence matrix, using interim scores as temporary measures of equivalence between iterations. --- **Important Features:** - Directed Data: - The algorithm is exclusively tailored for directed data - Interpretation of Values : - The similarity scores generated - Conversion Option: - If the data is binary, the adjacency matrix can be converted to a geodesic distance matrix and then applied to the REGE algorithm - Unique Solution: - The REGE algorithm aims to converge to the maximum equivalence - Multiple Relations: - The REGE algorithm is versatile and can be applied to datasets containing multiple relations. ??? - Directed Data: The algorithm is exclusively tailored for directed data containing at least one actor with either zero outdegree (termed a sink) or zero indegree (referred to as a source). - Interpretation of Values: The similarity scores generated can be challenging to understand, except for scores of 100, which indicate absolute equivalence. These scores are generally employed to partition data, akin to the method used in profile structural equivalence. - Conversion Option: If the data is binary, the adjacency matrix can be converted to a geodesic distance matrix and then applied to the REGE algorithm. This gives the algorithm a more intricate dataset to work with, reducing the risk of obtaining oversimplified outputs. - Unique Solution: The REGE algorithm aims to converge to the maximum equivalence (i.e., the one with the minimum equivalence classes). This ensures a unique solution, eliminating the potential challenges of multiple solutions presented by the direct method. - Multiple Relations: The REGE algorithm is versatile and can be applied to datasets containing multiple relations. --- Practical Application: - Using the Sampson esteem and disesteem data as an example, the algorithm classified various monks into clusters. - This grouping helped identify core group members, outliers, and leaders within the dataset. - It's worth noting that structurally equivalent actors, barring isolates, are more likely to be part of cohesive subgroups. - However, this limitation doesn't apply to regular equivalence, allowing it to represent roles independent of group membership. --- ``` ## ROMUALD BONAVENTURE AMBROSE BERTHOLD PETER ## ROMUALD 100.00000 30.13558 18.71628 30.25135 31.17075 ## BONAVENTURE 30.13558 100.00000 85.29479 57.25563 47.21449 ## AMBROSE 18.71628 85.29479 100.00000 55.46088 47.36068 ## BERTHOLD 30.25135 57.25563 55.46088 100.00000 74.25654 ## PETER 31.17075 47.21449 47.36068 74.25654 100.00000 ``` <img src="ch12_equivalence_en_files/figure-html/unnamed-chunk-27-1.png" width="100%" /> ``` ## CL_9 ## ROMUALD 1 ## BONAVENTURE 2 ## AMBROSE 2 ## BERTHOLD 3 ## PETER 4 ## LOUIS 5 ## VICTOR 3 ## WINFRID 6 ## JOHN_BOSCO 7 ## GREGORY 4 ## HUGH 2 ## BONIFACE 2 ## MARK 2 ## ALBERT 2 ## AMAND 8 ## BASIL 9 ## ELIAS 9 ## SIMPLICIUS 9 ``` --- ``` ## CL_9 ## ROMUALD 1 ## BONAVENTURE 2 ## AMBROSE 2 ## BERTHOLD 3 ## PETER 4 ## LOUIS 5 ## VICTOR 3 ## WINFRID 6 ## JOHN_BOSCO 7 ## GREGORY 4 ## HUGH 2 ## BONIFACE 2 ## MARK 2 ## ALBERT 2 ## AMAND 8 ## BASIL 9 ## ELIAS 9 ## SIMPLICIUS 9 ``` --- <img src="ch12_equivalence_en_files/figure-html/unnamed-chunk-29-1.png" width="100%" /> --- ``` ## ## . ------------------------------------------------------------------ ## . Number of valid cells: 306 ## . which corresponds to: 100 % of considered cells. ## . ------------------------------------------------------------------ ``` ``` ## 1 2 3 4 5 6 7 8 9 ## 1 NaN 0.000 0.000 0.000 0.000 0.0 0.000 0 0.000 ## 2 0 0.400 0.167 1.417 0.333 0.5 0.833 0 0.000 ## 3 0 0.083 1.000 1.500 1.000 0.0 0.000 0 0.000 ## 4 1 0.083 0.750 0.000 1.500 0.0 1.500 0 0.000 ## 5 0 0.833 0.000 0.000 NaN 0.0 0.000 0 0.000 ## 6 0 0.167 0.000 1.000 0.000 NaN 3.000 0 0.000 ## 7 0 0.167 1.500 0.000 0.000 2.0 NaN 0 0.000 ## 8 0 0.833 0.000 0.000 1.000 0.0 0.000 NaN 0.000 ## 9 0 0.000 0.000 0.500 0.000 0.0 1.000 1 1.833 ``` --- # Core-Periphery Models .font150[**A core-periphery structure** is a prevalent equivalence pattern observed in social networks and other fields]. - This structure divides the nodes of a network into two distinct groups: the core and the periphery. --- ## Components of the Structure: Core: These are the central nodes that are highly connected to each other. They play a dominant role in the network. Periphery: These nodes are less connected or not connected to other peripheral nodes. They have fewer connections compared to core nodes and are generally linked to the core rather than to other periphery nodes. ## Interactions within the Structure: - There are four types of interactions: - Core-to-core interactions. - Periphery-to-periphery interactions. - Core-to-periphery interactions. - Periphery-to-core interactions. - In an undirected network, the core-to-periphery and the periphery-to-core interactions are mirror images of each other. --- ## Characteristics of an Ideal Core-Periphery Structure: - The core block would be densely packed with connections (represented as a 1-block). - The peripheral block would have no connections within itself (represented as a 0-block). - The structure is such that every peripheral member is connected to every core member. ## Modifications to the Ideal Structure: - Real-world data might not align perfectly with the idealized core-periphery pattern. As such, conditions are sometimes relaxed. - One approach overlooks the off-diagonal blocks and focuses only on the core and periphery blocks. However, it's crucial to ensure some core-periphery interaction in the network. - The 'discrete core-periphery method' measures how well nodes fit the ideal core model. This method is efficient, especially for non-valued data. --- ## Issues and Alternatives: - The discrete model might be too simplistic as it strictly places nodes either in the core or in the periphery. Some researchers suggest introducing a semi-periphery, which adds complexity. - An alternative method, the continuous core-periphery model, uses a matrix to assign a coreness value to each node, indicating its significance in the network. ## Visualization and Application: - The continuous model's Z matrix can aid in visualizing the data, with core nodes placed centrally and peripheral nodes around the edges. - For example, analyzing citation data among top social work journals revealed a strong core-periphery structure, where core journals cite other core journals frequently, while peripheral journals don't cite each other often. --- > A core-periphery structure in networks identifies central nodes (core) that are heavily connected, and peripheral nodes that have fewer connections and are often tied to the core. This structure helps in understanding the dynamics, hierarchy, and flow of information in networks. --- ``` ## GROUP_1_E10 ## AMH 2 ## ASW 2 ## BJSW 2 ## CAN 2 ## CCQ 2 ## CSWJ 2 ## CW 1 ## CYSR 2 ## FR 2 ## IJSW 2 ## JGSW 2 ## JSP 2 ## JSWE 1 ## PW 2 ## SCW 1 ## SSR 1 ## SW 1 ## SWG 2 ## SWHC 2 ## SWRA 1 ``` --- <img src="ch12_equivalence_en_files/figure-html/unnamed-chunk-33-1.png" width="100%" /> --- ``` ## ## . ------------------------------------------------------------------ ## . Number of valid cells: 400 ## . which corresponds to: 100 % of considered cells. ## . ------------------------------------------------------------------ ``` ``` ## 1 2 ## 1 0.9444444 0.2142857 ## 2 0.3928571 0.1122449 ``` --- <img src="ch12_equivalence_en_files/figure-html/unnamed-chunk-36-1.png" width="100%" /> --- <img src="ch12_equivalence_en_files/figure-html/unnamed-chunk-37-1.png" width="100%" /> --- # Summary - Structural Positions in Multirelational Networks: - Actors in similar positions often exhibit similar behavioral patterns. - Positions are captured using equivalences. - Types of Equivalences: - Structural Equivalence: - Actors connected to the same others. - Profile methods measure similarity in adjacency matrix rows/columns. - Result: proximity matrix showing equivalence extent between actor pairs. - Regular Equivalence: - Actors connected to equivalent others. - REGE algorithm measures profile similarity with fuzzy sets iteration. - Relaxation: 1-blocks need at least one 1 in every row/column. --- - Application on Real Data: - Concepts must be relaxed for real data application. - Proximity matrices from both types can be clustered to find equivalent groups. - Equivalence amount can be gauged by examining induced groups. - Structural equivalence: blocks of all 0s or all 1s. - Regular equivalence: at least one 1 in each row/column. - Optimization and Blockmodels: - Equivalence measures enable optimization for best fit. - Partitioned matrices, or blockmodels, provide data summary. - Core-Periphery Model: - Core actors interact among themselves. - Peripheral actors interact with core, but not other peripherals. - Common in social network data; crucial for understanding group dynamics. --- Network Analysis. Lecture 7. Structural Equivalence and Assortative Mixing https://www.youtube.com/watch?v=d5RUMz1gBeo <center> <iframe width="560" height="315" src="https://www.youtube.com/embed/d5RUMz1gBeo?si=4mVkkI98THQqAx1z" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" allowfullscreen></iframe> </center> --- # References <p><cite>Borgatti, S. P., M. G. Everett, and J. C. Johnson (2022). <em>Analyzing social networks using R</em>. Sage.</cite></p> <p><cite>Everett, M. G. (1985). “Role similarity and complexity in social networks”. In: <em>Social Networks</em> 7.4, pp. 353–359.</cite></p> <p><cite>Sailer, L. D. (1978). “Structural equivalence: Meaning and definition, computation and application”. In: <em>Social networks</em> 1.1, pp. 73–90.</cite></p> <p><cite>Sampson, S. (1969). “Crisis in a cloister (Unpublished doctoral dissertation)”. In: <em>Ithaca, NY: Cornell University</em>.</cite></p> <!-- 메뉴얼 및 주요 1. 제목과 '#' 내용 사이에 한줄 공백 2. --- 슬라이드 나누기 앞 뒤로 한줄씩 공백 3. 코멘트는 HTML 코멘트 4. 2 컬럼 .pull-left[내용 왼쪽] .pull-right[내용 오른쪽] 5. 프리젠테이션 코멘트 ??? 다음에 내용 6. 가능한 Theme들 names(xaringan:::list_css()) 7. 헤더 코드 깨질떄 줄바꿨다가 다시 해주면 됨 8. 시작하고 프리젠테이션 모드가 안될때는 좀 기다리면 됨 9. tachyons은 metropolitan 테마에선 적용안됨 레벨1 글씨가 깨짐(색상, 위치) 10 내용중에 레퍼런스 리스트를 본문에 삽입하고 싶을때 #print(bib[key = "textbook"], # .opts = list(check.entries = FALSE, # style = "html", # bib.style = "authoryear")) --> <!-- Comment -->