RTM: Laws and a Recursive Generator for Weighted Time-Evolving Graphs Leman Akoglu, Mary McGlohon, Christos Faloutsos Carnegie Mellon University School of Computer Science 1

Motivation Graphs are popular! Social, communication, network traffic, call graphs… 2 …and interesting surprising common properties for static and un-weighted graphs How about weighted graphs? …and their dynamic properties? How can we model such graphs? for simulation studies, what-if scenarios, future prediction, sampling

Outline 1. Motivation 2. Related Work - Patterns - Generators - Burstiness 3. Datasets 4. Laws and Observations 5. Proposed graph generator: RTM 6. (Sketch of proofs) 7. Experiments 8. Conclusion 3

Graph Patterns (I) Small diameter - 19 for the web [Albert and Barabási, 1999] for the Internet AS topology graph [Faloutsos, Faloutsos, Faloutsos, 1999] Shrinking diameter [Leskovec et al.05] Power Laws 4 y(x) = Axγ, A>0, γ>0 Blog Network time diameter

Graph Patterns (II) 5 DBLP Keyword-to-Conference NetworkInter-domain Internet graph Densification [Leskovec et al.05] and Weight [McGlohon et al.08] Power-laws Eigenvalues Power Law [Faloutsos et al.99] Rank Eigenvalue |E| |W| |srcN| |dstN| Degree Power Law [Richardson and Domingos, 01] In-degree Count Epinions who-trusts-whom graph

Graph Generators Erdős-Rényi (ER) model [Erdős, Rényi 60] Small-world model [Watts, Strogatz 98] Preferential Attachment [Barabási, Albert 99] Edge Copying models [Kumar et al.99], [Kleinberg et al.99], Forest Fire model [Leskovec, Faloutsos 05] Kronecker graphs [Leskovec, Chakrabarti, Kleinberg, Faloutsos 07] Optimization-based models [Carlson,Doyle,00] [Fabrikant et al. 02] 6

Edge and weight additions are bursty, and self- similar. Entropy plots [Wang+02] is a measure of burstiness. Burstiness Time Weights Resolution Entropy Resolution Entropy Bursty: 0.2 < slope < 0.9 slope = 5.9

Outline 1. Motivation 2. Related Work - Patterns - Generators 3. Datasets 4. Laws and Observations 5. Proposed graph generator: RTM 6. Sketch of proofs 7. Experiments 8. Conclusion 8

Datasets 9 Bipartite networks: |N| |E| time 1. AuthorConference 17K, 22K, 25 yr. 2. KeywordConference 10K, 23K, 25 yr. 3. AuthorKeyword 27K, 189K, 25 yr. 4. CampaignOrg 23K, 877K, 28 yr. 1

10 Bipartite networks: |N| |E| time 1. AuthorConference 17K, 22K, 25 yr. 2. KeywordConference 10K, 23K, 25 yr. 3. AuthorKeyword 27K, 189K, 25 yr. 4. CampaignOrg 23K, 877K, 28 yr. 3 Datasets

11 Bipartite networks: |N| |E| time 1. AuthorConference 17K, 22K, 25 yr. 2. KeywordConference 10K, 23K, 25 yr. 3. AuthorKeyword 27K, 189K, 25 yr. 4. CampaignOrg 23K, 877K, 28 yr. Unipartite networks: |N| |E| time 5. BlogNet 60K, 125K, 80 days 6. NetworkTraffic 21K, 2M, 52 months 3 Datasets 20MB

12 Bipartite networks: |N| |E| time 1. AuthorConference 17K, 22K, 25 yr. 2. KeywordConference 10K, 23K, 25 yr. 3. AuthorKeyword 27K, 189K, 25 yr. 4. CampaignOrg 23K, 877K, 28 yr. Unipartite networks: |N| |E| time 5. BlogNet 60K, 125K, 80 days 6. NetworkTraffic 21K, 2M, 52 months 3 Datasets 20MB 5MB 25MB

Outline 1. Motivation 2. Related Work - Patterns - Generators 3. Datasets 4. Laws and Observations 5. Proposed graph generator: RTM 6. Sketch of proofs 7. Experiments 8. Conclusion 13

Observation 1: λ 1 Power Law(LPL) Q1: How does the principal eigenvalue λ 1 of the adjacency matrix change over time? Q2: Why should we care? 14

Observation 1: λ 1 Power Law(LPL) Q1: How does the principal eigenvalue λ 1 of the adjacency matrix change over time? Q2: Why should we care? A2: λ 1 is closely linked to density and maximum degree, also relates to epidemic threshold. A1: 15 λ 1 (t) E(t) α, α 0.5

λ 1 Power Law (LPL) cont. Theorem: For a connected, undirected graph G with N nodes and E edges, without self-loops and multiple edges; λ 1 (G) {2 (1 – 1/N) E} 1/2 For large N, 1/N 0 and λ 1 (G) cE 1/2 16 DBLP Author-Conference network

Observation 2:λ 1,w Power Law (LWPL) Q: How does the weighted principal eigenvalue λ 1,w change over time? A: 17 λ 1,w (t) E(t) β DBLP Author-Conference networkNetwork Traffic

Observation 3: Edge Weights PL(EWPL) Q: How does the weight of an edge relate to popularity if its adjacent nodes? 18 FEC Committee-to- Candidate network w i,j w i * w j Wi,j WiWj j i A:

Outline 1. Motivation 2. Related Work - Patterns - Generators 3. Datasets 4. Laws and Observations 5. Proposed graph generator: RTM 6. Sketch of proofs 7. Experiments 8. Conclusion 19

Problem Definition Generate a sequence of realistic weighted graphs that will obey all the patterns over time. SUGP: static un-weighted graph properties small diameter power law degree distribution SWGP: static weighted graph properties the edge weight power law (EWPL) the snapshot power law (SPL) 20

Problem Definition DUGP: dynamic un-weighted graph properties the densification power law (DPL) shrinking diameter bursty edge additions λ 1 Power Law (LPL) DWGP: dynamic weighted graph properties the weight power law (WPL) bursty weight additions λ 1,w Power Law (LWPL) 21

2D solution: Kronecker Product 22 Idea : Recursion Intuition : Communities within communities Self-similarity Power-laws

2D solution: Kronecker Product 23

3D solution: Recursive Tensor Multiplication(RTM) I X I 1,1,1

3D solution: Recursive Tensor Multiplication(RTM) I X I 1,2,1

3D solution: Recursive Tensor Multiplication(RTM) I X I 1,3,1

3D solution: Recursive Tensor Multiplication(RTM) I X I 1,4,1

3D solution: Recursive Tensor Multiplication(RTM) I X I 2,1,1

3D solution: Recursive Tensor Multiplication(RTM) I X I 3,1,1

3D solution: Recursive Tensor Multiplication(RTM) I

3D solution: Recursive Tensor Multiplication(RTM) I X I 1,1,2

3D solution: Recursive Tensor Multiplication(RTM) I X I 1,2,2

3D solution: Recursive Tensor Multiplication(RTM) I

3D solution: Recursive Tensor Multiplication(RTM) 34 senders recipients t-slices time

3D solution: Recursive Tensor Multiplication(RTM) 35 t1t1 t2t2 t3t3

3D solution: Recursive Tensor Multiplication(RTM) 36 t1t1 t2t2 t3t

Outline 1. Motivation 2. Related Work - Patterns - Generators 3. Datasets 4. Laws and Observations 5. Proposed graph generator: RTM 6. (Sketch of proofs) 7. Experiments 8. Conclusion 37

Experimental Results 38 SUGP: small diameter PL Degree Distribution SWGP: Edge Weights PL Snaphot PL DUGP: Densification PL shrinking diameter bursty edge additions λ 1 PL DWGP: Weight PL bursty weight additions λ 1,w PL Time diameter

Experimental Results 39 SUGP: small diameter PL Degree Distribution SWGP: Edge Weights PL Snaphot PL DUGP: Densification PL shrinking diameter bursty edge additions λ 1 PL DWGP: Weight PL bursty weight additions λ 1,w PL degree count

Experimental Results 40 SUGP: small diameter PL Degree Distribution SWGP: Edge Weights PL Snaphot PL DUGP: Densification PL shrinking diameter bursty edge additions λ 1 PL DWGP: Weight PL bursty weight additions λ 1,w PL |N| |E|

Experimental Results 41 SUGP: small diameter PL Degree Distribution SWGP: Edge Weights PL Snaphot PL DUGP: Densification PL shrinking diameter bursty edge additions λ 1 PL DWGP: Weight PL bursty weight additions λ 1,w PL |E| |W|

Experimental Results 42 SUGP: small diameter PL Degree Distribution SWGP: Edge Weights PL Snaphot PL DUGP: Densification PL shrinking diameter bursty edge additions λ 1 PL DWGP: Weight PL bursty weight additions λ 1,w PL

Experimental Results 43 In-degree In-weight SUGP: small diameter PL Degree Distribution SWGP: Edge Weights PL Snaphot PL DUGP: Densification PL shrinking diameter bursty edge additions λ 1 PL DWGP: Weight PL bursty weight additions λ 1,w PL Out-degree Out-weight

Experimental Results 44 SUGP: small diameter PL Degree Distribution SWGP: Edge Weights PL Snaphot PL DUGP: Densification PL shrinking diameter bursty edge additions λ 1 PL DWGP: Weight PL bursty weight additions λ 1,w PL

Experimental Results 45 SUGP: small diameter PL Degree Distribution SWGP: Edge Weights PL Snaphot PL DUGP: Densification PL shrinking diameter bursty edge additions λ 1 PL DWGP: Weight PL bursty weight additions λ 1,w PL |E| λ1 |E| λ1,w

Conclusion In real graphs, (un)weighted largest eigenvalues are power-law related to number of edges. Weight of an edge is related to the total weights and of its incident nodes. Recursive Tensor Multiplication is a recursive method to generate (1)weighted, (2)time- evolving, (3)self-similar, (4)power-law networks. Future directions: Probabilistic version of RTM Fitting the initial tensor I 46 Wi,j Wi Wj

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