class: center, middle, inverse, title-slide .title[ # Social Networks Theories and Methods ] .subtitle[ ## Centrality, or How to stand out ] .author[ ###
James Hollway
] --- class: center, middle .pull-1[.circleon[![](https://globalquiz.org/media/pic/400/8340.jpg)]] .pull-1[.circleon[![](https://cdn.imgbin.com/14/23/6/imgbin-people-connected-tXALjkJraJY0PjsUKek18tXrH.jpg)]] .pull-1[.circleon[![](https://i.pinimg.com/originals/8a/08/f1/8a08f1d673c8b64ddb067de8d8d879d8.jpg)]] --- class: center, middle .pull-1[.circleon[![](https://encrypted-tbn0.gstatic.com/images?q=tbn:ANd9GcTRYdGQnVNT9XMKmui2UXPFtMEa7Gx1Ao7cEevLYvQ-hxTvO6huAUIUAmLRWRr9i2mC5So&usqp=CAU)]] .pull-1[.circleon[![](https://static.thenounproject.com/png/1653609-200.png)]] .pull-1[.circleon[![](https://static.thenounproject.com/png/3120664-200.png)]] --- class: center, middle # Centrality .pull-1[.circleon[![](https://encrypted-tbn0.gstatic.com/images?q=tbn:ANd9GcTRYdGQnVNT9XMKmui2UXPFtMEa7Gx1Ao7cEevLYvQ-hxTvO6huAUIUAmLRWRr9i2mC5So&usqp=CAU)]] .pull-1[.circleoff[![](https://static.thenounproject.com/png/1653609-200.png)]] .pull-1[.circleoff[![](https://static.thenounproject.com/png/3120664-200.png)]] --- ## What is ‘centrality’? Being central in a network is a central concept in networks research... .center[![](https://encrypted-tbn0.gstatic.com/images?q=tbn:ANd9GcTRYdGQnVNT9XMKmui2UXPFtMEa7Gx1Ao7cEevLYvQ-hxTvO6huAUIUAmLRWRr9i2mC5So&usqp=CAU)] Why could “being central” be important... - in your private life? - in your professional life? - in a social network from your field of interest? --- ## Centrality questions - Does a node have access to lots of resources? - Does a node have access to different resources? - Does a node connect different parts of the network? - Does a node control the interaction between other nodes? - Would the failure of a node cause a system to collapse? - Is a node considered popular by others? - Which nodes’ esteem matters? --- background-image: url(https://www.epfl.ch/research/domains/transportation-center/wp-content/uploads/2019/03/iStock-warning-1024x576.jpg) background-size: contain .pull-right-1[ ## Warning Not _one_ kind of centrality Many, with several in regular use: - Degree - Closeness - Betweenness - Eigenvector - ... They need to be deployed with care, *theoretically*... ] --- background-image: url(https://i1.wp.com/beinspired.no/wp-content/uploads/2018/09/Periodic-table-Centrality.gif?resize=997%2C540) --- ## Degree centrality .pull-left[ .red[Q]: How many edges does a node have? .red[T]: About e.g. activity, popularity, attention .red[M]: Outdegree, indegree, or the sum of both `$$c_{D+}(i) = \sum_{j\in V} x_{ij}$$` `$$c_{D-}(i) = \sum_{j\in V} x_{ji}$$` `$$c_{D}(i) = c_{D+}(i) + c_{D-}(i)$$` .footnote[See also strength and weighted centrality extensions, e.g. Opsahl et al. (2010)] ] .pull-right[ <img src="ISON_L3_Centrality_files/figure-html/degree-1.png" width="504" /> ] --- .center[![:scale 55%](https://divisbyzero.files.wordpress.com/2008/09/konigsberggraph.jpg) ] -- .pull-left[ <img src="ISON_L3_Centrality_files/figure-html/bernie-1.png" width="504" /> ] .pull-right[.pull-down[ *Do you know more or fewer people in the class than others in the class on average?* ]] --- ### Do you have more or fewer friends than your friends? -- Option 1: Data collection - go on Facebook, count how many “friends” you have - click on every one of those “friends” and see how many “friends” they have - get a late dinner... -- Option 2: Graph theory <!-- $$ \frac{\sum d_v}{n} $$ --> <!-- $$ \frac{\sum d_v^2}{\sum d_v} $$ --> $$ \frac{\sum d_v^2}{\sum d_v} ≥ \frac{\sum d_v}{n} $$ q.e.d. --- ### Friendship paradox .pull-left-2[ <img src="ISON_L3_Centrality_files/figure-html/adol-1.png" width="504" /> ] .pull-right-1[ - 8 people, 10 ties = 20 friends - First number is number of friends (ties) - 20/8 people = 2.5 friends on average - Second number (in parentheses) is average number of friends of friends - 23.92/8 = 2.99 friends of friends on average That is, you are not likely to be most popular `\((1/n)\)`, but you are likely to be connected to most popular `\((d^*/n)\)` ] .footnote[Feld (1991) “Why your friends have more friends than you do” *American Journal of Sociology* 96(6): 1464-1477.] --- class: middle ### Task - EASY: _Can you think of a network in which nobody has fewer friends than their friends?_ - MEDIUM: _Can you think of a network in which almost everybody has fewer friends than their friends?_ - HARD: _Can you think of a network where most have more friends than their friends?_ --- .pull-left[ ### Similar situations - Class paradox - larger class than average - [Lovers paradox](https://www.psychologytoday.com/intl/blog/the-scientific-fundamentalist/200911/why-your-friends-have-more-friends-you-do) - more lovers than you - Mothers paradox - more children than average - Disease paradox - friends get flu first ] .pull-right[![](http://www.umasocialmedia.com/socialnetworks/wp-content/uploads/2012/09/classsizeparadox.png)] --- ### **Big** lesson of the day - Networks can fool people. - Just like how the earth seems flat because we're on it, people may get incorrect ideas about society because they're _in_ it. --- ## Betweenness centrality .pull-left[ AKA .red[medial] centrality .red[Q]: How many shortest paths go through a certain node? .red[T]: About e.g. brokering, transmission, innovation .red[M]: `$$c_{B}(i) = \sum_{j, k \in V \not i} \delta(j,k|i)$$` - `\(\delta(j,k|i)\)`: fraction of shortest paths from `\(j\)` to `\(k\)` where `\(i\)` is an intermediate node - naïve calculation complex ] .pull-right[ <img src="ISON_L3_Centrality_files/figure-html/between-1.png" width="504" /> ] --- background-image: url(https://d3i71xaburhd42.cloudfront.net/409ccf2c137b26d02af120c38a349bcd61302319/2-Figure1-1.png) --- background-image: url(https://d3i71xaburhd42.cloudfront.net/409ccf2c137b26d02af120c38a349bcd61302319/2-Figure2-1.png) --- background-image: url(https://i.pinimg.com/originals/53/33/5d/53335d1c81b0493b4a0cdfdc28fd6b39.jpg) background-size: contain --- ## Closeness centrality .pull-left[ AKA .red[radial] centrality .red[Q]: How short are the distances to all other nodes? .red[T]: About capacity to e.g. communicate, diffuse, draw on all networked resources .red[M]: Inverse of the sum over all dyadic distances, `\(d(i,j)\)` `$$c_{C}(i) = \frac{1}{\sum_{j \in V} d(i,j)}$$` ] .pull-right[ <img src="ISON_L3_Centrality_files/figure-html/close-1.png" width="504" /> ] --- ### How many people do we need to be close to to be happy? Dunbar's number (150) can be more differentiated: - 3-5 closest BFFs in your intimate circle - 9-15 friends and family who are your go-to comrades in your friendship circle - 30-45 allies and colleagues who are in your co-participation circle - 90-135 wider acquaintances who are in your exchange circle See [also](http://www.smbc-comics.com/comics/1427381663-20150326.png) --- ## Eigenvector centrality .pull-left[ AKA .red[feedback] centrality (remarkably old method: Leontief 1941, Seeley 1949) .red[Q]: How connected is the node to important (central) neighbours? .red[T]: About e.g. power, influence, support .red[M]: [Leading eigenvector](https://setosa.io/ev/eigenvectors-and-eigenvalues/) of sociomatrix `$$c_{E}(i) = \alpha \sum_{j \in N(i)} d(i,j)$$` - with `\(\partial = 1/\alpha\)` being an eigenvalue of the adjacency matrix and `\(N(i)\)` the neighbour nodes of `\(i\)` - In undirected networks, `\(C_E(i)\)` with `\(\alpha = 1\)` equals the degree centrality (stationary distribution of a random walk) ] .pull-right[ <img src="ISON_L3_Centrality_files/figure-html/eigen-1.png" width="504" /> ] --- ### Assortativity - Assortative mixing is a bias in favour of ties between similar others... - Where based on attributes, called .red[homophily] - Where based on structures, called .red[assortativity] - Usually just Pearson correlation coefficient between nodes' degrees --- ### From undirected to directed .pull-left[HITS ![](https://fmsasg.com/SocialNetworkAnalysis/SocialNetworkAnalysis_HubAuthority-sm.png)] .pull-right[ .red[Hubs] point to many authorities - eigenvector centrality of matrix `\(M_{hub} = AA^T\)` .red[Authorities] pointed to by many hubs - eigenvector centrality of matrix `\(M_{auth} = A^TA\)` ] --- background-image: url("figure-html/centrality-test.pdf") --- ## Which centrality should I use? -- - Option 1: just try all and see? -- - Option 2: create an index of all of them? -- - **No**, option 3: derive correct centrality measure based on theoretical assumptions (see also Zweig 2016) It all depends on research question and meaning of the edges: - support? - flow? - power? - access? -- Ultimately ordinal, not interval, scale: there is no "unit of centrality" - makes comparing across measures difficult - makes comparing across networks difficult (even after normalisation) .footnote[See also Brandes (2016) and Schoch & Brandes (2016)] --- class: center, middle # Centralisation .pull-1[.circleoff[![](https://encrypted-tbn0.gstatic.com/images?q=tbn:ANd9GcTRYdGQnVNT9XMKmui2UXPFtMEa7Gx1Ao7cEevLYvQ-hxTvO6huAUIUAmLRWRr9i2mC5So&usqp=CAU)]] .pull-1[.circleon[![](https://static.thenounproject.com/png/1653609-200.png)]] .pull-1[.circleoff[![](https://static.thenounproject.com/png/3120664-200.png)]] --- background-image: url(https://www.researchgate.net/profile/Upul-Jayasinghe/publication/316042146/figure/fig1/AS:614002224799778@1523400946963/Centralized-vs-Decentralized-vs-Distributed-Networks.png) background-size: contain --- ## Centralisation - .red[Q]: How central is a network’s most central node compared to all others? - .red[M]: equation needs to: - work out the largest centrality - calculate the sum in differences in centrality between each node and all other nodes - choose the maximum of this - divide this quantity by the theoretically largest such sum of differences in any network of the same size Therefore, every centrality measure has its own centralisation measure: `$$\max \sum_{i=1}^N C_x(p^*) - C_x(p_i)$$` --- ## Degree distributions .center[![:scale 70%](https://www.researchgate.net/profile/Guilherme-Ferraz-De-Arruda/publication/324745118/figure/fig5/AS:619148941987843@1524628019643/In-a-we-present-the-degree-distribution-of-an-Erdoes-and-Renyi-network-with-N-10-5-and.png)] --- background-image: url(https://www.mdpi.com/entropy/entropy-22-00052/article_deploy/html/images/entropy-22-00052-g001.png) background-size: contain ??? Answer key: - a = betweenness - b = closeness - c = eigenvector - d = degree - e = harmonic - f = Katz --- class: center, middle # Visualization .pull-1[.circleoff[![](https://encrypted-tbn0.gstatic.com/images?q=tbn:ANd9GcTRYdGQnVNT9XMKmui2UXPFtMEa7Gx1Ao7cEevLYvQ-hxTvO6huAUIUAmLRWRr9i2mC5So&usqp=CAU)]] .pull-1[.circleoff[![](https://static.thenounproject.com/png/1653609-200.png)]] .pull-1[.circleon[![](https://static.thenounproject.com/png/3120664-200.png)]] --- background-image: url(https://i0.wp.com/www.mattdancho.com/assets/network_visualization_goals.png) background-size: contain --- class: center, middle # Summary .pull-1[.circleon[![](https://encrypted-tbn0.gstatic.com/images?q=tbn:ANd9GcTRYdGQnVNT9XMKmui2UXPFtMEa7Gx1Ao7cEevLYvQ-hxTvO6huAUIUAmLRWRr9i2mC5So&usqp=CAU)]] .pull-1[.circleon[![](https://static.thenounproject.com/png/1653609-200.png)]] .pull-1[.circleon[![](https://static.thenounproject.com/png/3120664-200.png)]] -- .pull-left[.pull-down[What questions do you have for me?]] --- background-image: url(https://devhumor.com/content/uploads/images/February2017/cat-computer-beep.jpg) background-size: contain