15-382 C OLLECTIVE I NTELLIGENCE – S19 L ECTURE 34: N ETWORKS 1 T EACHER : G IANNI A. D I C ARO
N ETWORK S CIENCE § Barabasi, “Network Science” § Easley & Kleinberg, “Networks, Crowds, and Markets: Reasoning about a Highly Connected World” § Newman, “Networks” 15781 Fall 2016: Lecture 22 2
C OMPLEX SYSTEMS AS NETWORKS Many complex systems can be represented as networks Any complex system has an associated network of communication / interaction among the components § Nodes = components of the complex system § Links = interactions between them This and fol ollow owing g slides are adapted from om Kri Kristina Le Lerman’s sl slides s
D IRECTED VS . U NDIRECTED N ETWORKS Undirected Directed § Undirected links § Directed links o Interactions flow both ways o interaction flows one way § Examples § Examples o Social networks: people and o WWW: web pages and hyperlinks friendship o Twitter follower graph o Atoms in a crystal o Animal relations, prey-predator o Countries in geographic maps
H OW DO WE CHARACTERIZE NETWORKS ? § Size o Number of nodes o Number of links § Degree o Average degree o Degree distribution § Diameter § Clustering coefficient § …
N ODE DEGREE 2 2 3 4 4 3 1 1 Directed networks Undirected networks § Indegree: § Node degree: number of links to other nodes /3 = 1 𝑙 ' /3 = 2, 𝑙 ) /3 = 0, 𝑙 * /3 = 1] [𝑙 # [𝑙 # = 2, 𝑙 ' = 3, 𝑙 ) = 2, 𝑙 * = 1] § Outdegree: § Number of links 567 = 1 𝑙 ' 567 = 1, 𝑙 ) 567 = 2, 𝑙 * 567 = 0] [𝑙 # 1 𝑀 = 1 2 . 𝑙 / § Total degree = in + out /0# § Number of links § Average degree 1 1 /3 = . 567 𝑀 = . 𝑙 / 𝑙 / 1 𝑙 = 1 𝑙 / = 2𝑀 /0# /0# 𝑂 . 𝑂 Average degree: 𝑀/𝑂 § /0#
D EGREE DISTRIBUTION § Degree distribution 𝑞 : is the probability that a randomly selected node has degree 𝑙 𝑞 : = 𝑂 : /𝑂 o Where 𝑂 : is number of nodes of degree 𝑙 clique (fully connected graph) regular lattice 5 karate club friendship network regular lattice 4
D EGREE DISTRIBUTION IN REAL NETWORKS Degree distribution of real-world networks is highly heterogeneous, i.e., it can vary significantly hubs
R EAL NETWORKS ARE SPARSE § Real network § Complete graph 𝑀 ≪ 𝑂(𝑂 − 1)/2
M ATHEMATICAL REPRESENTATION OF DIRECTED GRAPHS § Adjacency list 1 o List of links 4 [(1,2), (2,4), (3,1), (3,2)] 2 3 j § Adjacency matrix i 0 1 0 0 𝑂×𝑂 matrix 𝑩 such that 0 0 0 1 𝐵 /B = 1 1 0 0 o 𝐵 /B = 1 if link (𝑗, 𝑘) exists 0 0 0 0 o 𝐵 /B = 0 if there is no link
U NDIRECTED VS . DIRECTED GRAPHS 1 1 4 4 3 2 3 2 0 1 0 0 0 1 1 0 0 0 0 1 1 0 1 1 𝐵 /B = 𝐵 /B = 1 1 0 0 1 1 0 0 0 0 0 0 0 1 0 0 Symmetric
P ATHS AND DISTANCES IN NETWORKS § Path: sequence of links (or nodes) from one node to another § Walk: a Path of length 𝑜 from one node to another, that can include repeated nodes / links (e.g., [1-2-1]) § Shortest Path: path with the shortest distance between two nodes § Diameter: Shortest paths between most distant nodes
C OMPUTING PATHS / DISTANCES Number of walks 𝑂 /B between nodes i and j The minimum 𝑚 such that § can be calculated using the adjacency matrix 𝐵 G /B > 0 gives the distance § 𝐵 /B gives paths of length 𝑒 = 1 (in hops) between 𝑗 and 𝑘 𝐵 ' /B gives #walks of length 𝑒 = 2 § 𝐵 G /B gives #walks of length 𝑒 = 𝑚 § 0 1 1 0 1 1 0 1 1 4 𝐵 /B = 2 1 1 1 1 1 0 0 3 2 1 3 1 0 0 1 0 0 𝐵 ' /B = 1 1 2 1 𝐵 /B = 𝑏 /B § 1 0 1 1 𝐵 ' )* = 𝑏 )# 𝑏 #* + 𝑏 )' 𝑏 '* + 𝑏 )) 𝑏 )* + 𝑏 )* 𝑏 ** + 𝑏 )L 𝑏 L* + 𝑏 )M 𝑏 M* § 𝑏 )* 𝑏 ** is the # of walks from 3 to 1 multiplied by the # of § 2 4 3 1 walks from 1 to 4 à # of walks from 3 to 4 through 1 4 2 4 3 𝑏 ): 𝑏 :* is the # of walks from 3 to 𝑙 multiplied by the # of § 𝐵 ) /B = walks from 𝑙 to 4 à # of walks from 3 to 4 through 𝑙 3 4 2 1 1 3 1 0 Sum of all two-steps walks between 3 and 4 §
A VERAGE DISTANCE IN NETWORKS clique: 𝑒 = 1 regular lattice (ring): 𝑒 = 𝑃(𝑂) karate club friendship network: 𝑒 = 2.44 regular lattice (square): 𝑒 = 𝑃( 𝑂)
C LUSTERING Clustering g coe oefficient captures the probability of neighbors of a given § node 𝑗 to be linked oefficient of a vertex 𝑗 in a graph quantifies how close § Loc ocal clustering g coe its neighbors are to being a clique
P ROPERTIES OF REAL WORLD NETWORKS § Real networks are fundamentally different from what we’d expect o Degree distribution • Real networks are scale-free o Average distance between nodes • Real networks are small world o Clustering • Real networks are locally dense § What do we expect? o Create a model of a network. Useful for calculating network properties and thinking about networks.
R ANDOM NETWORK MODEL § Networks do not have a regular structure § Given N nodes, how can we link them in a way that reproduces the observed complexity of real networks? § Let connect nodes at random! § Erdos-Renyi model of a random network o Given N isolated nodes o Select a pair of nodes. Pick a random number between 0 and 1. If the number > 𝑞 , create a link o Repeat previous step for each remaining node pair o Average degree: 𝑙 = 𝑞(𝑂 − 1) § Easy to compute properties of random networks
R ANDOM NETWORKS ARE TRULY RANDOM N=12, p=1/6 N=100, p=1/6 Average degree: 𝑙 = 𝑞(𝑂 − 1)
D EGREE DISTRIBUTION IN RANDOM NETWORK § Follows a binomial distribution § For sparse networks, <k> << N, Poisson distribution. o Depends only on <k>, not network size N
R EAL NETWORKS DO NOT HAVE P OISSON DEGREE DISTRIBUTION degree (followers) distribution activity (num posts) distribution
S CALE FREE PROPERTY WWW hyperlinks distribution on: 𝒒 𝒍 ~𝒍 T𝜹 Pow ower-law distribution § Networks whose degree distribution follows a power-law free networks distribution are called sc scale fr § Real network have hubs
R ANDOM VS SCALE - FREE NETWORKS Random networks and scale-free networks are very different. Differences are apparent when degree distribution is plotted on log scale. 0 1 2 3 10 10 10 10 0 10 - = 0 . 5 f ( x ) cx -1 10 -2 10 = - x f ( x ) c -3 10 - = 1 f ( x ) cx loglog -4 10
T HE M ILGRAM EXPERIMENT § In 1960’s, Stanley Milgram asked 160 randomly selected people in Kansas and Nebraska to deliver a letter to a stock broker in Boston. o Rule: can only forward the letter to a friend who is more likely to know the target person § How many steps would it take?
T HE M ILGRAM EXPERIMENT § Within a few days the first letter arrived, passing through only two links. § Eventually 42 of the 160 letters made it to the target, some requiring close to a dozen intermediates. § The median number of steps in completed chains was 5.5 à “ six degr grees of of separation on ”
F ACEBOOK IS A VERY SMALL WORLD § Ugander et al. directly measured distances between nodes in the Facebook social graph (May 2011) o 721 million active users o 68 billion symmetric friendship links o the average distance between the users was 4.74
S MALL WORLD PROPERTY § Distance between any two nodes in a network is surprisingly short o “six degrees of separation”: you can reach any other individual in the world through a short sequence of intermediaries § What is small? o Consider a random network with average degree 𝑙 o Expected number of nodes a distance d is 𝑂(𝑒)~ 𝑙 V o Diameter 𝑒 WXY ~ log 𝑂 / log 𝑙 o Random networks are small
W HAT IS IT SURPRISING ? o not ha § Regu gular lattices (e.g. g., physical ge geogr ography) do have e the the small wor orld prop operty o Distances grow polynomially with system size o In networks, distances grow logarithmically with network size
S MALL WORLD EFFECT IN RANDOM NETWORKS Wa Watts-Stroga ogatz mode odel § Start with a regular lattice, e.g., a ring where each node is connected to immediate and next neighbors. o Local clustering is 𝐷 = 3/4 § With probability 𝑞 , rewire link to a randomly chosen node o For small 𝑞 , clustering remains high, but diameter shrinks o For large 𝑞 , becomes random network
S MALL WORLD NETWORKS § Small wor orld networ orks constructed using Watts-Strogatz model have small average distance and high clustering, just like real networks § Long-distance links, joining distant local clusters regular lattice random network Clustering Average distance p
S OCIAL NETWORKS ARE SEARCHABLE § Milgram experiments showed that o Short chains exist! o People can find them! • Using only local knowledge (who their friends are, their location and profession) • How are short chains discovered with this limited information? • Hint: geographic information? [Milgram]
K LEINBERG MODEL OF GEOGRAPHIC LINKS § Incorporate geographic distance in the distribution of links Distance between nodes Link to all nodes within distance r , then add is d q long range links with probability d -a
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