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Jure Leskovec, CMU Kevin Lang, Anirban Dasgupta and Michael Mahoney Yahoo! Research

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Communities: Sets of nodes with lots of connections inside and few to outside (the rest of the network) Assumption: Networks are (hierarchically) composed of communities (modules) Communities, clusters, groups, modules

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How community like is a set of nodes? Want a measure that corresponds to intuition Conductance (normalized cut): Φ(S) = # edges cut / # edges inside Small Φ(S) corresponds to more community-like sets of nodes

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Score: Φ(S) = # edges cut / # edges inside

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Bad community Φ=5/7 = 0.7

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Score: Φ(S) = # edges cut / # edges inside Better community Φ=5/7 = 0.7 Bad community Φ=2/5 = 0.4

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Score: Φ(S) = # edges cut / # edges inside Better community Φ=5/7 = 0.7 Bad community Φ=2/5 = 0.4 Best community Φ=2/8 = 0.25

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We define: Network community profile (NCP) plot Plot the score of best community of size k Search over all subsets of size k and find best: Φ(k=5) = 0.25 NCP plot is intractable to compute Use approximation algorithm

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Dolphin social network Two communities of dolphins NCP plot Network

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Zachary’s university karate club social network During the study club split into 2 The split (squares vs. circles) corresponds to cut B NCP plotNetwork

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Collaborations between scientists in Networks NCP plotNetwork

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Hierarchical network Geometric (grid-like) network – Small social networks – Geometric and – Hierarchical network have downward NCP plot

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Previously researchers examined community structure of small networks (~100 nodes) We examined more than 70 different large social and information networks Large real-world networks look completely different!

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Typical example: General relativity collaboration network (4,158 nodes, 13,422 edges)

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Better and better communities Worse and worse communities Best community has 100 nodes Community score Community size

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Whiskers are responsible for downward slope of NCP plot Whisker is a set of nodes connected to the network by a single edge NCP plot Largest whisker

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Each new edge inside the community costs more NCP plot Φ=2/1 = 2 Φ=8/3 = 2.6 Φ=64/11 = 5.8 Each node has twice as many children

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Network structure: Core-periphery, jellyfish, octopus Whiskers are responsible for good communities Denser and denser core of the network Core contains 60% node and 80% edges

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What if we allow cuts that give disconnected communities? Cut all whiskers and compose communities out of them

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Community score Community size We get better community scores when composing disconnected sets of whiskers Connected communities Bag of whiskers

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Rewired network: random network with same degree distribution

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What is a good model that explains such network structure? None of the existing models work Pref. attachment Small World Geometric Pref. Attachment Flat Down and Flat Flat and Down

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Forest Fire: connections spread like a fire New node joins the network Selects a seed node Connects to some of its neighbors Continue recursively As community grows it blends into the core of the network

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rewired network Bag of whiskers

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Whiskers: Largest whisker has ~100 nodes Independent of network size Dunbar number: a person can maintain social relationship to 150 people Bond vs. identity communites Core: Newman et al. analyzed 400k node product network ▪ Largest community has 50% nodes ▪ Community was labeled “miscelaneous”

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NCP plot is a way to analyze network community structure Our results agree with previous work on small networks But large networks are fundamentally different Large networks have core-periphery structure Small well isolated communities blend into the core of the networks as they grow

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