Weighted Graphs and Disconnected Components Patterns and a Generator
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Transcript of Weighted Graphs and Disconnected Components Patterns and a Generator
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Weighted Graphs and Disconnected Components
Patterns and a Generator
Mary McGlohon, Leman Akoglu, Christos Faloutsos
Carnegie Mellon University
School of Computer Science
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● In graphs a largest connected component emerges.
● What about the smaller-size components? ● How do they emerge, and join with the large
one?
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“Disconnected” components
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Weighted edges
● Graphs have heavy-tailed degree distribution.● What can we also say about these edges?● How are they repeated, or otherwise weighted?
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Our goals
● Observe “Next-largest connected components”Q1. How does the GCC emerge?
Q2. How do NLCC’s emerge and join with the GCC?
● Find properties that govern edge weightsQ3: How does the total weight of the graph relate to
the number of edges?
Q4: How do the weights of nodes relate to degree?
Q5: Does this relation change with the graph?
● Q6: Can we produce an emergent, generative model
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Outline
● Motivation● Related work● Preliminaries● Data● Observations ● Model● Summary
1 2 3 4 5
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Properties of networks
● Small diameter (“small world” phenomenon)– [Milgram 67] [Leskovec, Horovitz 07]
● Heavy-tailed degree distribution– [Barabasi, Albert 99] [Faloutsos, Faloutsos,
Faloutsos 99]
● Densification– [Leskovec, Kleinberg, Faloutsos 05]
● “Middle region” components as well as GCC and singletons– [Kumar, Novak, Tomkins 06]
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Generative Models
● Erdos-Renyi model [Erdos, Renyi 60]● Preferential Attachment [Barabasi, Albert 99]● Forest Fire model [Leskovec, Kleinberg,
Faloutsos 05]● Kronecker multiplication [Leskovec,
Chakrabarti, Kleinberg, Faloutsos 07]● Edge Copying model [Kumar, Raghavan,
Rajagopalan, Sivakumar, Tomkins, Upfal 00]● “Winners don’t take all” [Pennock, Flake,
Lawrence, Glover, Giles 02]
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Outline
● Motivation● Related work● Preliminaries● Data● Observations ● Model● Summary
1 2 3 4 5 6
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Diameter
● Diameter of a graph is the “longest shortest path”.
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Diameter
● Diameter of a graph is the “longest shortest path”.
diameter=3
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Diameter
● Diameter of a graph is the “longest shortest path”.
● Effective diameter is the distance at which 90% of nodes can be reached.
diameter=3
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Outline
● Motivation● Related work● Preliminaries● Data● Observations ● Model● Summary
1 2 3 4 5
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Unipartite Networks● Postnet: Posts in blogs, hyperlinks
between
● Blognet: Aggregated Postnet, repeated edges
● Patent: Patent citations
● NIPS: Academic citations
● Arxiv: Academic citations
● NetTraffic: Packets, repeated edges
● Autonomous Systems (AS): Packets, repeated edges
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Unipartite Networks● Postnet: Posts in blogs, hyperlinks
between
● Blognet: Aggregated Postnet, repeated edges
● Patent: Patent citations
● NIPS: Academic citations
● Arxiv: Academic citations
● NetTraffic: Packets, repeated edges
● Autonomous Systems (AS): Packets, repeated edges
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(3)
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Unipartite Networks● Postnet: Posts in blogs, hyperlinks
between
● Blognet: Aggregated Postnet, repeated edges
● Patent: Patent citations
● NIPS: Academic citations
● Arxiv: Academic citations
● NetTraffic: Packets, repeated edges
● Autonomous Systems (AS): Packets, repeated edges
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Unipartite Networks
● (Nodes, Edges, Timestamps)● Postnet: 250K, 218K, 80 days
● Blognet: 60K,125K, 80 days
● Patent: 4M, 8M, 17 yrs
● NIPS: 2K, 3K, 13 yrs
● Arxiv: 30K, 60K, 13 yrs
● NetTraffic: 21K, 3M, 52 mo
● AS: 12K, 38K, 6 mo
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Bipartite Networks
● IMDB: Actor-movie network
● Netflix: User-movie ratings
● DBLP: conference- repeated edges
– Author-Keyword
– Keyword-Conference
– Author-Conference
● US Election Donations: $ weights, repeated edges
– Orgs-Candidates
– Individuals-Orgs
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Bipartite Networks
● IMDB: Actor-movie network
● Netflix: User-movie ratings
● DBLP: repeated edges
– Author-Keyword
– Keyword-Conference
– Author-Conference
● US Election Donations: $ weights, repeated edges
– Orgs-Candidates
– Individuals-Orgs
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Bipartite Networks
● IMDB: Actor-movie network
● Netflix: User-movie ratings
● DBLP: repeated edges
– Author-Keyword
– Keyword-Conference
– Author-Conference
● US Election Donations: $ weights, repeated edges
– Orgs-Candidates
– Individuals-Orgs
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Bipartite Networks
● IMDB: 757K, 2M, 114 yr
● Netflix: 125K, 14M, 72 mo
● DBLP: 25 yr
– Author-Keyword: 27K, 189K
– Keyword-Conference: 10K, 23K
– Author-Conference: 17K, 22K
● US Election Donations: 22 yr
– Orgs-Candidates: 23K, 877K
– Individuals-Orgs: 6M, 10M
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Outline
● Motivation● Related work● Preliminaries● Data● Observations● Model● Summary
1 2 3 4 5
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Observation 1: Gelling Point
Q1: How does the GCC emerge?
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Observation 1: Gelling Point
● Most real graphs display a gelling point, or burning off period
● After gelling point, they exhibit typical behavior. This is marked by a spike in diameter.
Time
Diameter
IMDBt=1914
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Observation 2: NLCC behavior
Q2: How do NLCC’s emerge and join with the GCC?
Do they continue to grow in size?Do they shrink?
Stabilize?
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Observation 2: NLCC behavior● After the gelling point, the GCC takes off, but
NLCC’s remain constant or oscillate.
Time
IMDB
CC size
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Outline
● Motivation● Related work● Preliminaries● Data● Observations ● Model● Summary
1 2 3 4 5
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Observation 3
Q3: How does the total weight of the graph relate to the
number of edges?
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Observation 3: Fortification Effect
● $ = # checks ?
|Checks|
Orgs-Candidates
|$|
1980
2004
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Observation 3: Fortification Effect
● Weight additions follow a power law with respect to the number of edges:
– W(t): total weight of graph at t
– E(t): total edges of graph at t
– w is PL exponent
– 1.01 < w < 1.5 = super-linear!
– (more checks, even more $)
|Checks|
Orgs-Candidates
|$|
1980
2004
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Observation 4 and 5
Q4: How do the weights of nodes relate to degree?
Q5: Does this relation change over time?
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Observation 4:Snapshot Power Law
● At any time, total incoming weight of a node is proportional to in degree with PL exponent, iw. 1.01 < iw < 1.26, super-linear
● More donors, even more $
Edges (# donors)
In-weights($)
Orgs-Candidates
e.g. John Kerry, $10M received,from 1K donors
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Observation 5:Snapshot Power Law
● For a given graph, this exponent is constant over time.
Time
exponent
Orgs-Candidates
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Outline
● Motivation● Related work● Preliminaries● Data● Observations ● Q6: Is there a generative, “emergent”
model?● Summary
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Goals of model
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● a) Emergent, intuitive behavior● b) Shrinking diameter● c) Constant NLCC’s● d) Densification power law● e) Power-law degree distribution
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Goals of model
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● a) Emergent, intuitive behavior● b) Shrinking diameter● c) Constant NLCC’s● d) Densification power law● e) Power-law degree distribution
= “Butterfly” Model
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Butterfly model in action
● A node joins a network, with own parameter.
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pstep
“Curiosity”
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Butterfly model in action
● A node joins a network, with own parameter.
● With (global) phost, chooses a random host
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phost “Cross-disciplinarity”
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Butterfly model in action
● A node joins a network, with own parameters.
● With (global) phost, chooses a random host – With (global) plink, creates link
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plink“Friendliness”
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Butterfly model in action
● A node joins a network, with own parameters.
● With (global) phost, chooses a random host – With (global) plink, creates link
– With pstep travels to random neighbor
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pstep
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Butterfly model in action
● A node joins a network, with own parameters.
● With (global) phost, chooses a random host – With (global) plink, creates link
– With pstep travels to random neighbor. Repeat.
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plink
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Butterfly model in action
● A node joins a network, with own parameters.
● With (global) phost, chooses a random host – With (global) plink, creates link
– With pstep travels to random neighbor. Repeat.
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pstep
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Butterfly model in action
● Once there are no more “steps”, repeat “host” procedure:– With phost, choose new host, possibly link, etc.
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Butterfly model in action
● Once there are no more “steps”, repeat “host” procedure:– With phost, choose new host, possibly link, etc.
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Butterfly model in action
● Once there are no more “steps”, repeat “host” procedure:– With phost, choose new host, possibly link, etc.
– Until no more steps, and no more hosts.
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plink
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Butterfly model in action
● Once there are no more “steps”, repeat “host” procedure:– With phost, choose new host, possibly link, etc.
– Until no more steps, and no more hosts.
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pstep
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a) Emergent, intuitive behavior
Novelties of model:● Nodes link with probability
– May choose host, but not link (start new component)
● Incoming nodes are “social butterflies”– May have several hosts (merges components)
● Some nodes are friendlier than others– pstep different for each node
– This creates power-law degree distribution (theorem)
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Validation of Butterfly
● Chose following parameters:– phost= 0.3
– plink = 0.5
– pstep ~ U(0,1)
● Ran 10 simulations● 100,000 nodes per simulation
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b) Shrinking diameter
● Shrinking diameter– In model, gelling usually occurred around N=20,000
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Nodes
Diam-eter
N=20,000
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● Constant / oscillating NLCC’s
Nodes
NLCCsize
c) Oscillating NLCC’s
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N=20,000
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d) Densification power law
● Densification:– Our datasets had a=(1.03, 1.7)
– In [Leskovec+05-KDD], a= (1.1, 1.7)
– Simulation produced a = (1.1,1.2)
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Nodes
EdgesN=20,000
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e) Power-law degree distribution
● Power-law degree distribution– Exponents approx -2
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Degree
Count
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53McGlohon, Akoglu, Faloutsos KDD08
Summary
● Studied several diverse public graphs– Measured at many timestamps
– Unipartite and bipartite
– Blogs, citations, real-world, network traffic
– Largest was 6 million nodes, 10 million edges
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54McGlohon, Akoglu, Faloutsos KDD08
Summary
● Observations on unweighted graphs:A1: The GCC emerges at the “gelling point”
A2: NLCC’s are of constant / oscillating size
● Observations on weighted graphs:A3: Total weight increases super-linearly with edges
A4: Node’s weights increase super-linearly with degree, power law exponent iw
A5: iw remains constant over time
● A6: Intuitive, emergent generative “butterfly” model, that matches properties
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55McGlohon, Akoglu, Faloutsos KDD08
References[Barabasi+99] Barabasi, A. L. & Albert, R. (1999), 'Emergence of scaling in random networks',
Science 286(5439), 509--512.
[Erdos+60] Erdos, P. & Renyi, A. (1960), 'On the evolution of random graphs', Publ. Math. Inst. Hungary. Acad. Sci. 5, 17-61.
[Faloutsos*99] Faloutsos, M.; Faloutsos, P. & Faloutsos, C. (1999), 'On Power-law Relationships of the Internet Topology', SIGCOMM, 251-262.
[Kumar+99]. R. Kumar, P. Raghavan, S. Rajagopalan, D. Sivakumar, A. Tomkins, and Eli Upfal. Stochastic models for the Web graph. Proceedings of the 41th FOCS. 2000, pp. 57-65
[Kumar+06] Kumar, R.; Novak, J. & Tomkins, A. (2006), Structure and evolution of online social networks, in 'KDD '06: Proceedings of the 12th ACM SIGKDD International Conference on Knowedge Discover and Data Mining', pp. 611—617.
[Leskovec+05KDD] Leskovec, J.; Kleinberg, J. & Faloutsos, C. (2005), Graphs over time: densification laws, shrinking diameters and possible explanations, in 'KDD '05.
[Leskovec+07] Leskovec, J.; Faloutsos, C. Scalable modeling of real graphs using Kronecker Multiplication. ICML 2007.
[Milgram+67] Milgram, S. (1967), 'The small-world problem', Psychology Today 2, 60—67.
[Pennock+02] Winners don’t take all: Characterizing the competition for links on the web PNAS 2002
[Wang+2002] Wang, M.; Madhyastha, T.; Chang, N. H.; Papadimitriou, S. & Faloutsos, C. (2002), 'Data Mining Meets Performance Evaluation: Fast Algorithms for Modeling Bursty Traffic', ICDE.
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56McGlohon, Akoglu, Faloutsos KDD08
Contact usLeman Akoglu
www.andrew.cmu.edu/~lakoglu
Christos Faloutsos
www.cs.cmu.edu/~christos
Mary McGlohon
www.cs.cmu.edu/~mmcgloho
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● From time series data, begin with resolution r= T/2.
● Record entropy HR
57McGlohon, Akoglu, Faloutsos KDD08
Entropy plots [Wang+2002]
Time
Weights
Resolution
Entropy
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● From time series data, begin with resolution r= T/2.
● Record entropy HR`
58McGlohon, Akoglu, Faloutsos KDD08
Entropy plots
Time
Weights
Resolution
Entropy
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● From time series data, begin with resolution r= T/2.
● Record entropy HR
● Recursively take finer resolutions.
59McGlohon, Akoglu, Faloutsos KDD08
Entropy plots
Time
Weights
Resolution
Entropy
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● From time series data, begin with resolution r= T/2.
● Record entropy HR
● Recursively take finer resolutions.
60McGlohon, Akoglu, Faloutsos KDD08
Entropy plots
Time
Weights
Resolution
Entropy
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61McGlohon, Akoglu, Faloutsos KDD08
Entropy Plots
● Self-similarity Linear plot
Resolution
En
trop
y s= 0.59
● Self-similarity Linear plot●
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62McGlohon, Akoglu, Faloutsos KDD08
Entropy Plots
● Self-similarity Linear plot
Resolution
En
trop
y s= 0.59
● Self-similarity Linear plot● Uniform: slope of plot s=1.
time
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63McGlohon, Akoglu, Faloutsos KDD08
Entropy Plots
● Self-similarity Linear plot
Resolution
En
trop
y s= 0.59
● Self-similarity Linear plot● Uniform: slope of plot s=1. Point mass: s=0
time time
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64McGlohon, Akoglu, Faloutsos KDD08
Entropy Plots
● Self-similarity Linear plot
Resolution
En
trop
y s= 0.59
● Self-similarity Linear plot● Uniform: slope of plot s=1. Point mass: s=0
time time
Bursty: 0.2 < s < 0.9