Social Networks Slides by : I. Koutsopoulos (AUEB), Source:L. Adamic, SN Analysis, Coursera course
Introduction
Political blogs
Organizations
Facebook networks
Ingredient networks
SN representation
Networks are sets of nodes connected by edges. What are networks? node Network Graph edge points lines vertices edges, arcs math nodes links computer science sites bonds physics actors ties, relations sociology
Network elements: edges Directed (also called arcs, links) A -> B A likes B, A gave a gift to B, A is B s child Undirected A <-> B or A B A and B like each other A and B are siblings A and B are co-authors
Edge attributes Examples weight (e.g. frequency of communication) ranking (best friend, second best friend ) type (friend, relative, co-worker) properties depending on the structure of the rest of the graph: e.g. betweenness
Directed networks girls school dormitory dining-table partners, 1 st and 2 nd choices (Moreno, The sociometry reader, 1960) Ada Louise Lena Marion Adele Jane Cora Eva Frances Maxine Mary Robin Martha Anna Edna Betty Ruth Jean Alice Laura Helen Ellen Hazel Hilda Ella Irene
Data representation adjacency matrix edgelist adjacency list
Adjacency matrices Representing edges (who is adjacent to whom) as a matrix A ij = 1 if node i has an edge to node j = 0 if node i does not have an edge to j A ii = 0 unless the network has self-loops A ij = A ji if the network is undirected, or if i and j share a reciprocated edge
Example adjacency matrix 1 2 3 A = 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 5 4 0 0 0 0 1 1 1 0 0 0
Edge list Edge list 2, 3 2, 4 3, 2 3, 4 4, 5 5, 2 5, 1 1 5 2 4 3
Adjacency lists Adjacency list is easier to work with if network is large sparse quickly retrieve all neighbors for a node 1: 2: 3 4 3: 2 4 4: 5 5: 1 2 1 5 2 4 3
Computing metrics degree & degree distribution connected components
Degree: which node has the most edges????
Nodes Node network properties from immediate connections indegree how many directed edges (arcs) are incident on a node outdegree how many directed edges (arcs) originate at a node degree (in or out) number of edges incident on a node indegree=3 outdegree=2 degree=5 from the entire graph centrality (betweenness, closeness)
Is everything connected?
Connected components Strongly connected components Each node within the component can be reached from every other node in the component by following directed links Strongly connected components B C D E A G H F Weakly connected components: every node can be reached from every other node by following links in either direction A B E D C F H G Weakly connected components A B C D E G H F A B C F G In undirected networks one talks simply about connected components E D H
Giant component if the largest component encompasses a significant fraction of the graph, it is called the giant component
Erdös and Rényi
Erdös-Renyi: simplest network model Assumptions nodes connect at random network is undirected Key parameter (besides number of nodes N) : p or M p = probability that any two nodes share and edge M = total number of edges in the graph
what they look like after spring layout
Degree distribution (N,p)-model: For each potential edge we flip a biased coin with probability p we add the edge with probability (1-p) we don t
Emergence of the giant component http://ccl.northwestern.edu/netlogo/models/giantcomponent
Percolation on a 2D lattice http://www.ladamic.com/netlearn/netlogo501/latticepercolation.html
size of giant component Percolation threshold Percolation threshold: how many edges need to be added before the giant component appears? As the average degree increases to z = 1, a giant component suddenly appears average degree av deg = 0.99 av deg = 1.18 av deg = 3.96
Giant component another angle How many other friends besides you does each of your friends have? By property of degree distribution the average degree of your friends, you excluded, is z so at z = 1, each of your friends is expected to have another friend, who in turn have another friend, etc. the giant component emerges
Average shortest path How many hops on average between each pair of nodes? again, each of your friends has z = avg. degree friends besides you ignoring loops, the number of people you have at distance l is z l
friends at distance l N l =z l scaling: average shortest path l av l av ~ log N log z
Between-ness
is counting the edges enough?
Stanford Social Web (ca. 1999) network of personal homepages at Stanford
different notions of centrality In each of the following networks, X has higher centrality than Y according to a particular measure Y X Y X Y X X Y indegree outdegree betweenness closeness
what does degree not capture? In what ways does degree fail to capture centrality in the following graphs?
Brokerage not captured by degree Y X
betweenness: capturing brokerage intuition: how many pairs of individuals would have to go through you in order to reach one another in the minimum number of hops? X Y
betweenness: definition C ( i) g ( i) / B j k jk g jk Where g jk = the number of shortest paths connecting jk g jk (i) = the number that actor i is on. Usually normalized by: C B ' (i) = C B (i )/[(n -1)(n- 2)/2] number of pairs of vertices excluding the vertex itself
betweenness on toy networks non-normalized version: A B C D E A lies between no two other vertices B lies between A and 3 other vertices: C, D, and E C lies between 4 pairs of vertices (A,D),(A,E),(B,D),(B,E) note that there are no alternate paths for these pairs to take, so C gets full credit
betweenness on toy networks non-normalized version:
betweenness on toy networks non-normalized version: A B C E why do C and D each have betweenness 1? They are both on shortest paths for pairs (A,E), and (B,E), and so must share credit: ½+½ = 1 D
closeness What if it s not so important to have many direct friends? Or be between others But one still wants to be in the middle of things, not too far from the center
need not be in a brokerage position Y X X X X Y Y Y
Closeness is based on the length of the average shortest path between a node and all other nodes in the network Closeness Centrality: closeness: definition C c ( i) N j 1 d( i, j) 1 Normalized Closeness Centrality C C ' (i) = (C C (i))/(n -1)
Eigenvector centrality How central you are depends on how central your neighbors are
Eigenvector centrality in directed networks PageRank brings order to the Web: it's not just the pages that point to you, but how many pages point to those pages, etc. more difficult to artificially inflate centrality with a recursive definition an important page, e.g. slashdot if a web page is slashdotted, it gains attention Many webpages scattered across the web
Ranking pages by tracking a drunk A random walker following edges in a network for a very long time will spend a proportion of time at each node which can be used as a measure of importance
Trapping a drunk Problem with pure random walk metric: Drunk can be trapped and end up going in circles
Ingenuity of the PageRank algorithm Allow drunk to teleport with some probability e.g. random websurfer follows links for a while, but with some probability teleports to a random page (bookmarked page or uses a search engine to start anew)
PageRank PageRank example: probable location of random walker after 1 step 20% teleportation probability 1 6 8 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 t=0 2 7 0.2 0.1 0 1 2 3 4 5 6 7 8 3 4 5 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 t=1 1 2 3 4 5 6 7 8 slide adapted from: Dragomir Radev
Coordination game and clustering coefficient
networked coordination game choice between two things, A and B (e.g. basketball and soccer) if friends choose A, they get payoff a if friends choose B, they get payoff b if one chooses A while the other chooses B, their payoff is 0
coordinating with one s friends Let A = basketball, B = soccer. Which one should you learn to play? fraction p = 3/5 play basketball fraction p = 2/5 play soccer
which choice has higher payoff? d neighbors p fraction play basketball (A) (1-p) fraction play soccer (B) if choose A, get payoff p * d *a if choose B, get payoff (1-p) * d * b so should choose A if p d a (1-p) d b or p b / (a + b)
two equilibria everyone adopts A everyone adopts B
what happens in between? What if two nodes switch at random? Will a cascade occur? example: a = 3, b = 2 payoff for nodes interaction using behavior A is 3/2 as large as what they get if they both choose B nodes will switch from B to A if at least q = 2/(3+2) = 2/5 of their neighbors are using A
how does a cascade occur suppose 2 nodes start playing basketball due to external factors (e.g. they are bribed with a free pair of shoes by some devious corporation)
you pick the initial 2 nodes A larger example (Easley/Kleinberg Ch. 19) does the cascade spread throughout the network? http://www.ladamic.com/netlearn/netlogo412/cascademodel.html
implications for viral marketing if you could pay a small number of individuals to use your product, which individuals would you pick?
Clustering Global clustering coefficient 3 x number of triangles in the graph number of connected triples of vertices C = 3 x number of triangles in the graph number of connected triples
Local clustering coefficient (Watts&Strogatz 1998) For a vertex i The fraction pairs of neighbors of the node that are themselves connected Let n i be the number of neighbors of vertex i C i = # of connections between i s neighbors max # of possible connections between i s neighbors Ci directed = # directed connections between i s neighbors n i * (n i -1) Ci undirected = # undirected connections between i s neighbors n i * (n i -1)/2
Local clustering coefficient (Watts&Strogatz 1998) Average over all n vertices C 1 n i C i i n i = 4 max number of connections: 4*3/2 = 6 3 connections present C i = 3/6 = 0.5 link present link absent
Explanation n i = 3 there are 2 connections present out of max of 3 possible C i = 2/3 i