strong and weak ties - graz university of...
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Strong and Weak TiesWeb Science (VU) (706.716)
Elisabeth Lex & Denis Helic
KTI, TU Graz
May 29, 2017
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Outline
1 Strong and Weak Ties
2 General principles about structural properties of social networks
3 The Strength of Weak Ties
4 Graph Partitioning and Betweenness Measures
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Strong and Weak Ties
Strong and Weak Ties
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Strong and Weak Ties
Motivation: Mark Granovetter
Sociologist at Stanford UniversityBest known for his theory “Thestrength of weak ties”Theory focuses on spread ofinformation in social networksStrongly influenced by the Milgramexperiment on social networks
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Strong and Weak Ties
Ties and Tie Strength
(Interpersonal) Ties: Information carrying connections between peoplein a network, relationships between the nodes3 types: strong, weak or absentAccording to Granovetter, strength of ties is a combination of
amount of timeemotional intensityintimacy (mutual confiding)reciprocal services that characterize the tie
For a friendship network, link strength could represent frequency ofcontact, proximity, length of friendship, or subjective assessment ofclosenessStrong ties are those with a larger weight (e.g. the friends you seemore often or have known for longer)
Can you give examples of strong/weak ties?Elisabeth Lex & Denis Helic (KTI, TU Graz) Networks May 29, 2017 5 / 58
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Strong and Weak Ties
Example Application for Strong and Weak Ties
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Strong and Weak Ties
Granovetter’s Experiment
Granovetter launched study on job seeking in the 1960ies and 1970ies toinvestigate usefulness of different types of ties in certain situations:
How people find out about new jobs?People find the information through personal contactsBut: contacts were often acquaintances (weak ties) rather than closefriends (strong ties)!Why acquaintances are most helpful?
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General principles about structural properties of social networks
General principles about structuralproperties of social networks
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General principles about structural properties of social networks
Strong Triadic Closure
Property among three nodes A, B, C, such that if strong tie existsbetween A-B and A-C, there is a weak or strong tie between B-CToo extreme for real-world networks, but good simplificationUseful to understand networks and predict network evolutionE.g. Trust networks: If A trusts B and A trusts C, B will probablytrust C as well.
Figure: Formation of E(B,C), since B and C they have common neighbor A.
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General principles about structural properties of social networks
Triadic Closure
Occurs in social networks due to increased opportunity for two nodeswith common neighbor to meet and to create at least a weak tieReasons for triadic closure:
Opportunity: B and C have a common friend A so there is anincreased chance they will end up knowing each other.Trust: B and C are friends with A. Gives them a basis for trusting eachother that an arbitrary pair of unconnected people might lack.Incentive: Node B has incentive to bring A and C together to decreaseeffort it takes to maintain two separate relationships
Networks that fullfil triadic closure are highly connected - thus highclustering coefficient
E.g. Bearman and Moody (2004): low clustering coefficiant amongteenagers implies higher probability of suicideImplication: low clustering coefficient implies few good friends.
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General principles about structural properties of social networks
Applications for Triadic Closure Principle
Can be used to model how networks will evolveCan predict development of ties in networkShows progression of connectivity
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General principles about structural properties of social networks
Measuring the Extent of Triadic Closure
Repetition: CC of node A is defined as the probability that tworandomly selected friends of A are friends with each other.CC for a node is a measure of the connectedness of its neighbourhoodLocal CC of a node is a measure how transitive connections are in anetworkI.e. a friend of a friend is also a friend
CC of A = Number of connections between friends of APossible number of connections between friends of A
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General principles about structural properties of social networks
Repetition: Example for Clustering coefficient (CC)
Figure: Clustering Coefficiant computation for node A
CC of A = Number of connections between friends of APossible number of connections between friends of A = 1
6
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The Strength of Weak Ties
The Strength of Weak Ties
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The Strength of Weak Ties
Usefulness of certain types of ties in certain situations
Remember: Weak ties more helpful to find out about new jobsGranovetter: Weak ties can act as a “bridge” that spans parts of asocial networkConnect otherwise disconnected social groups
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The Strength of Weak Ties
Bridges and Local Bridges
DefinitionA bridge between node A and node B exists if deleting the bridging edgewould cause node A and node B to lie in two different components
Figure: Example of a bridge between the nodes A and B
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The Strength of Weak Ties
Local Bridges (1/2)
Bridges rare in real-world social networks (Small World Property)Local bridge occurrs when node acts as gatekeeper between 2neighbors who are otherwise not connectedLocal bridge: if nodes A and B have NO friends in common, i.e. ifdeleting the edge between A and B increases distance between A andB to value strictly more than 2
Figure: Edge E(A,B) is a local bridge since A and B have no common friends
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The Strength of Weak Ties
Local Bridges (2/2)
Figure: Edge E(A,B) is a local bridge since A and B have no common friends
Is there another local bridge in this example apart from E(A,B)?
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The Strength of Weak Ties
Local Bridges (2/2)
Figure: Edge E(A,B) is a local bridge since A and B have no common friends
Is there another local bridge in this example apart from E(A,B)?No - Local bridges never form the side of any triangle in the network!
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The Strength of Weak Ties
Span of Local Bridges
Span of a local bridge is the distance btw its endpoints if the edge isdeletedLength of the shortest path between two nodesLocal bridges with large span play similar role as bridges since theirendpoints provide access to parts of network that would otherwise befar away
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The Strength of Weak Ties
Span of Local Bridges
What is the span of the local bridge E(A,B)?
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The Strength of Weak Ties
Span of Local Bridges
What is the span of the local bridge E(A,B)? 4
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The Strength of Weak Ties
Theory of distance contraction (Small World Property)
We define a range 𝑟 of a link (𝑖, 𝑗) as the distance ℓ𝑖𝑗 between 𝑖 and𝑗 when the link has been deletedWe define a link as a shortcut if its range 𝑟 > 2Local bridges are shortcutsSpan of a local bridge is the range 𝑟 of the shortcutSociology vs. mathematics terminology
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The Strength of Weak Ties
Remember Granovetter’s experiment
Why are acquaintances more important?
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The Strength of Weak Ties
Remember Granovetter’s experiment
Why are acquaintances more important?
A, C, D, and E will all tend to be exposed to similar sources of info, whileA’s link to B offers access to things A otherwise wouldn’t necessarily hearabout!
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The Strength of Weak Ties
Local bridges and weak ties
The relationship between local bridges and weak ties through the strongtriadic closure is
if a node A satisfies strong triadic closure and is involved in atleast two strong ties then any local bridge adjacent to A must be aweak tie
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The Strength of Weak Ties
Local bridges and weak ties
Proof by contradiction:Node A satisfies strong triadic closure and is involved in at least 2strong tiesSuppose E(A,B) is a local bridgeStrong Triadic Closure says: E(B,C) must existBut since E(A,B) is a bridge, E(B,C) must NOT exist since endpointsof a bridge have no friends in common
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The Strength of Weak Ties
Summary: The Strength of Weak Ties
Weak ties (acquaintances) are social ties that connect us to newsources of informationWeak ties link together tightly knit communities, each containing alarge number of strong tiesLocal bridge property directly related to their weakness as social tiesDual role of being weak connections but also valuable links to parts ofthe network that are harder to accessLocal Bridges = Shortcuts ≈ Weak Ties (Small World)Put it all together: weak ties are very strong indeed!
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The Strength of Weak Ties
Example: Weak Ties
Example: Spread of information/rumors in social networksStudies have shown that people rarely act on mass-media informationunless it is also transmitted through personal ties [Granovetter 2003,p 1274]Information/rumors moving through strong ties is much more likely tobe limited to a few cliques than that going via weak ones, bridges willnot be crossed
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The Strength of Weak Ties
Summary
Bridges: edges when removed will find their endpoints in differentconnected componentsLocal bridges: edges whose endpoints are not in triangles2 types of edges: weak and strong tiesStrong triadic closure: Two strong ties imply a third strong / weak tieLocal bridges are weak ties
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The Strength of Weak Ties
Real world validation: Onnela et al., 2007 (1/5)
Investigated a large cellphone network with 4 millions usersEdge between two users if they called each other within theobservation time period of 18 months84% of nodes in a giant component, i.e. a single connectedcomponent containing most of the individuals in the networkTie strength: time spent in a phone conversation
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The Strength of Weak Ties
Real world validation: Onnela et al., 2007 (2/5)
Extended definition of local bridge to neighbourhood overlap todetect “almost” local bridgesSince only very small fraction of edges in the cell-phone data werelocal bridges
Neighbourhood Overlap = # nodes who are neighbours of both A and B# nodes who are neighbours of at least A or B
Neighbourhood Overlap is 0 if E(A,B) is a local bridge
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The Strength of Weak Ties
Real world validation: Onnela et al., 2007 (3/5)
Neighbourhood Overlap = # nodes who are neighbours of both A and F# nodes who are neighbours of at least A or F
E.g. Neighborhood Overlap of nodes A-F:Denominator determined by B, C, D, E, G, JNominator: only C is a neighbor of both A and FResult: Neighborhood overlap: 1/6
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The Strength of Weak Ties
Real world validation: Onnela et al., 2007 (4/5)
Figure: y-axis: neighborhood overlap 𝑂, x-axis: cummulative tie strength𝑃𝑐𝑢𝑚(𝑤), i.e. fraction of links with tie strength smaller than 𝑤 (percentile).
Plot shows relation between tie strength and neighborhood overlapOverlap increases as a function of cummulative tie strengthWhen tie strength decreases, overlap decreases, i.e. weak ties becomealmost local bridges
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The Strength of Weak Ties
Real world validation: Onnela et al., 2007 (5/5)
To support the hypothesis that weak ties tend to link together moretightly knit communities, Onnela et al. perform two simulations:
Removing edges in decreasing order of tie strength, the giantcomponent shrank graduallyRemoving edges in increasing order of tie strength, the giantcomponent shrank more rapidly and at some point then startedfragmenting into several components
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The Strength of Weak Ties
Web Science: research methodology
In this example we saw another research methodology in Web scienceSo far: we observed a phenomenon and empirically analyzed itWe then designed a model that captures the phenomenon and try tovalidate itIn this study: we started with a (social) theory and searched for theempirical evidenceWe needed to come up with means of quantifiying theoreticalmeasures on complex and large scale networks
Be careful when investigating networks only in respect to structuralproperties - because one knows relatively little about the meaning orsignificance of any particular node or edge!
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The Strength of Weak Ties
Strong and weak ties in large social networks: Facebook(1/4)
Marlow et al. (2009) investigated 4 types of Facebook networks (usageover a 1-month period): “To what extent is each social link actually usedfor social interaction beyond being listed?”
All friendsMaintained relationships (passive, e.g. following a user)One way communication of a user (e.g. liking content the other userposted)Reciprocal (mutual) communication (e.g. chatting on the wall)
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The Strength of Weak Ties
Strong and weak ties in large social networks: Facebook(3/4)
Figure: 4 types of networks on Facebook and their graphsElisabeth Lex & Denis Helic (KTI, TU Graz) Networks May 29, 2017 38 / 58
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The Strength of Weak Ties
Strong and weak ties in large social networks: Facebook(2/4)
They found thatAll networks thin out when links represent stronger ties.As the number of total friends increases, the number of reciprocalcommunication links levels out at slightly more than 10
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The Strength of Weak Ties
Strong and weak ties in large social networks: Facebook(4/4)
Figure: Active Friendships in Facebook: Users report large number of friends, upto 500. Mutual communication(strong ties) between 10 and 20, maintainedcommunication (weak ties) under 50.
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The Strength of Weak Ties
Passive Engagement
The power of media like Facebook:Maintained relationships (weak ties) enable passive engagementKeeping up with others by reading news about them also in absenceof communicationWeak ties are the middle ground between strong ties (mutualcommunication) and inactive ties (friends only listed)If only mutual communication allowed - small list of friendsWeak ties maintain the social network highly connected: everyone ispassively engaged with each other and events/news propagate veryquickly
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The Strength of Weak Ties
Strong and weak ties in large social networks: Twitter(1/2)
Huberman et al. (2009) investigated weak ties on Twitter:Dataset: Followers graph (directed)Strong ties of user A: users to whom A directly communicatesthrough tweetsWeak ties of A: users that A follows without direct communication
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The Strength of Weak Ties
Strong and weak ties in large social networks: Twitter(2/2)
Figure: Total number of user’s strong ties as function of number of followees onTwitter
Below 50 strong ties even for over 1000 followees (weak ties)Elisabeth Lex & Denis Helic (KTI, TU Graz) Networks May 29, 2017 43 / 58
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Graph Partitioning and Betweenness Measures
Graph Partitioning and BetweennessMeasures
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Graph Partitioning and Betweenness Measures
Objectives
Define densely connected regions of a network - Graph partitioningalgorithm to identify densely connected regionsbreaks a network into a set of nodes densely connected with eachother with edgessparser interconnections between the regions
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Graph Partitioning and Betweenness Measures
Example
Figure: Co-authorship network of physicists and applied mathematicians.
Weak ties seem to link together groups of tightly-knit nodes, eachgroup containing a large number of strong tiesHow can we find these groups, aka communities?
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Graph Partitioning and Betweenness Measures
Why are communities interesting?
For example to:Detect clusters of customers with similar interests and use them topersonalize recommendation systemsStudy relationships among nodes to detect nodes that connectcommunities and thus act as information brokersIdentification of clusters in networks can be exploited e.g. to improvePageRank…
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Graph Partitioning and Betweenness Measures
Another Example
One carate club splits into twoafter a conflict, networkstructure shows that both arestill heavily interconnectedAre there edges between theclubs that occur at lower densitythan edges within the clubs?
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Graph Partitioning and Betweenness Measures
Dividing graph into nested regions
Larger regions containing severalsmaller onesDivisive methods: Repeatedlyidentify and remove edgesconnecting densely connectedregions.Example: break first at the 7-8edge, and then the nodes intonodes 7 and 8Agglomerative methods:Repeatedly identify and mergenodes that likely belong in thesame regionMerge 4 triangles and then pairsof triangles (via nodes 7 and 8)
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Graph Partitioning and Betweenness Measures
Divisive removal of Bridges
Idea: remove bridges and localbridgesBoth form part of the shortestpath between pairs of nodes indifferent parts of the networkHowever: which when several?Which if none exist?
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Graph Partitioning and Betweenness Measures
Generalize the Role of Bridges and Local Bridges
Look for the edges that carry the most of “traffic” in a networkWithout the edge, paths between many pairs of nodes may have to be“re-routed” a longer wayThese edges tend to link different densely-connected regionsGood candidates for removal in a divisive methodSo, for each edge, we compute the total amount of “flow” it carriesto find the ones with max traffic
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Graph Partitioning and Betweenness Measures
Traffic in a Network: Edge Betweenness
For nodes A and B connected by a path assume 1 unit of “flow”Divide flow evenly along all possible shortest paths from A to Bif k shortest paths from A and B, 1/k units of flow pass along eachE.g. 7 total units flow over 7-8 to get from 1 to nodes 8-147 x 7 = 49 total units flow over 7-8 from nodes 1-7 to 8-14Edge betweenness of E(7-8) = 49
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Graph Partitioning and Betweenness Measures
Exploiting Edge Betweeness For Community Detection
Divisive: Remove edges with high betweenness
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Graph Partitioning and Betweenness Measures
Girvan-Newman Method
Based on successively deleting edges of high betweenness as they are themost important edges for connecting networks
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Graph Partitioning and Betweenness Measures
Example
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Graph Partitioning and Betweenness Measures
Girvan-Newman Method
Algorithm is divisive and recursiveTries to find an optimal way of cutting the graph into two pieces, andthen it does the same on the pieces.Each time it finds a way to split a graph into two optimal pieces, itcalls itself on the pieces.Terminates when it is given a graph with only one node.Returns list of all the components it has found
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Graph Partitioning and Betweenness Measures
Challenges when computing betweenness
According to definition: consider all the shortest paths between allpairs of nodes (“sum of the fraction of all-pairs shortest paths thatpass through E(A,B)”)Computationally intensive!One solution: Method based on Breadth-first Search (BFS)Use approximative algorithms
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Graph Partitioning and Betweenness Measures
Thanks for your attention - Questions?
Slides use figures from Chapter 3 of Networks, Crowds and Markets byEasley and Kleinberg (2010)http://www.cs.cornell.edu/home/kleinberg/networks-book/Elisabeth Lex & Denis Helic (KTI, TU Graz) Networks May 29, 2017 58 / 58