Computational Social Science, Lecture 06: Networks, Part II

Networks Part II Sharad Goel Columbia University Computational Social Science: Lecture 6 March 1, 2013

Corporate E-mail Communication [ Adamic & Adar, 2004 ] via Easley & Kleinberg

Networks/Graphs Nodes/vertices people, organizations, webpages, computers Edges represent connections between pairs of nodes

Distance Length of the shortest path between two nodes

Breadth-first Search iteratively explore nodes one layer at a time

# initialize distances dist = {} for u in G: dist[u] = NA dist [u0] = 0 d=0 periphery = { u0 } while len(periphery) > 0: # find nodes one step away from the periphery next_level = {} for u in periphery: next_level += { w for w in neighbors[u] if dist[w] == NA } # update distances d += 1 for u in next_level: dist[u] = d # update periphery periphery = next_level

BFS @ scale undirected network Input edge list, starting node u0 Output Distance to all nodes from u0

BFS @ scale undirected network Input: edge list, distances (u, d) 1. join distances with edge list 2. foreach (u, d, w) output (w, d+1) [ also output (u0, 0) ] 3. group by w, and output min d

Connected Components undirected network Input Edge list Output List of nodes for each component

Connected Graph There is a path between every pair of nodes

Connected Component A connected subset of nodes that is not contained in any larger connected subset

Connected Components undirected network 1. Select a node u0 that has not yet been assigned 2. BFS starting from u0 3. Record nodes reached by BFS

Consider the global human social network, with an edge between every pair of friends Is this network connected?

Consider the global human social network, with an edge between every pair of friends Is this network connected? No – there are people with no (living) friends, who are hence isolated from the rest of the network

Consider the global human social network, with an edge between every pair of friends Is there a “giant” connected component?

Consider the global human social network, with an edge between every pair of friends Is there more than one “giant” component?

Consider the global human social network, with an edge between every pair of friends Is there more than one “giant” component? No – unlikely to have two large disconnected sets of people

Consider the global human social network, with an edge between every pair of friends Is there more than one “giant” component? No – unlikely to have two large disconnected sets of people Historically it was more likely e.g., pre-Columbian America & Eurasia

Consider the global human social network, with an edge between every pair of friends On average, how far are people from one another?

The Small-world Experiment Stanley Milgram, 1967 296 people were randomly selected in Omaha and Wichita Packages sent to the selected individuals with instructions to forward to a particular stock broker in Boston through a chain of people they knew on a first-name basis.

The Small-world Experiment Stanley Milgram, 1967 Of the 296 packages, 232 did not reach target Of the 64 that did arrive, average path length was 6 “Six degrees of separation”

Small-world phenomenon Is “six degrees” big or small?

Small-world phenomenon navigational vs. topological

The Anatomy of the Facebook Social Graph J. Ugander, B. Karrer, L. Backstrom, C. Marlow 721 million users, 69 billion edges 5 degrees of separation

Edge list  degree distribution undirected network Input Edge list Output Degree distribution

3 1 2 5 4 7 6 Degree of node u # of edges incident on u

Edge list  degree distribution undirected network Map input: (u, w) output: (u, w), key := u output: (w, u), key := w Reduce input: u, {w1, …, wk} output: u, k

Edge list  degree distribution undirected network Map input: u, k identity, key := k Reduce input: k, {u1, …, um} output: k, m

An email network of 130M users Edges indicate reciprocated communication

An email network of 130M users Edges indicate reciprocated communication (log-log plot)

Triadic closure 1. Opportunity 2. Incentive 3. Commonality

Counting Triangles undirected network Input adjacency list Output Number of triangles incident on each node

Counting Triangles In memory for u in nodes: triangles[u] = 0 for w in neighbors[u]: triangles[u] += len(neighbors[w] & neighbors[u]) triangles[u] = triangles[u] / 2

Counting Triangles @ scale Every node needs to know to which nodes it is connected and to which nodes its neighbors are connected

Counting Triangles @ scale Map input: u {w1, …, wk} foreach wi: output wi u {w1, …, wk} Reduce In memory triangle count

Homophily the tendency of individuals to associate with similar others “birds of a feather flock together”

Birds of a Feather: Homophily in Social Networks McPherson, Smith-Lovin, Cook race, sex, age, religion, education, occupation, social class, behaviors, attitudes, abilities, aspirations

Homophily 1. Preference 2. Influence 3. Opportunity

Computing Homophily Input Edge list, race of each individual Output Distribution of race among friends White Black Latino Asian White Black Latino Asian

Computing Homophily 1. Join edges (u, v) by u, demographics (w, race) by w 2. Join edges (u, v, urace) by v, demographics (w, race) by w

Computing Homophily 1. Join edges (u, v) by u, demographics (w, race) by w 2. Join edges (u, v, urace) by v, demographics (w, race) by w 3. Group edges (u, v, urace, vrace) by sorted([urace, vrace])

Computing Homophily 1. Join edges (u, v) by u, demographics (w, race) by w 2. Join edges (u, v, urace) by v, demographics (w, race) by w 3. Group edges (u, v, urace, vrace) by sorted([urace, vrace]) 4. Count edges in each group 5. Normalize the table

How do ideas and products spread through society?

The structure of diffusion 93% 5% 1% 0.3% 0.3%

Computing the structure of diffusion Input Twitter network Time-stamped “adoptions” of 1B URLs Output Distribution of cascade structures

Computing the structure of diffusion We assume v influenced u to adopt link t if v is the last of u’s contacts to adopt before u 2 3 5 9

Computing the structure of diffusion Draw a labeled edge from v to u 3 t 5

Computing the structure of diffusion Group edges by their labels (URL)

Computing the structure of diffusion Compute the connected components for each forest corresponding to a URL

Computing the structure of diffusion Definition. Two (rooted) trees are isomorphic if they are identical under a relabeling of the vertices.

x (x) (x, x) ((x)) (x, (x)) Basis. The canonical name c(T) for the one-node tree T is x. Induction. If T has more than one node, let T1, . . . ,Tk denote the subtrees of the root indexed such that c(T1) ≤ c(T2) ≤ · · · ≤ c(Tk) under the lexicographic order. Then the canonical name for T is (c(T1), . . . ,c(Tk)). Aho et al. [1974]

Computing the structure of diffusion Compute the canonical name for each tree in the URL forests

Computing the structure of diffusion Count the number of trees of each type

Computing the structure of diffusion 1. Join adoptions (link, u, uts) by u, edges (u, w) by u 2. Join (link, u, uts, w) by (link, w), adoptions (link, w, wts) by (link, w)

Computing the structure of diffusion 1. Join adoptions (link, u, uts) by u, edges (u, w) by u 2. Join (link, u, uts, w) by (link, w), adoptions (link, w, wts) by (link, w) 3. Group (link, u, uts, w, wts) by (link, u)

Computing the structure of diffusion 1. Join adoptions (link, u, uts) by u, edges (u, w) by u 2. Join (link, u, uts, w) by (link, w), adoptions (link, w, wts) by (link, w) 3. Group (link, u, uts, w, wts) by (link, u) 4. Output unique “parent” edge (link, u, uts, w, wts) for each group

Computational Social Science, Lecture 06: Networks, Part II

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Computational Social Science, Lecture 06: Networks, Part II