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) = Ax −γ, 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 D Weights Resolution Entropy

From time series data, begin with resolution T/2. Record entropy H R. Entropy plots Time D Weights Resolution Entropy

From time series data, begin with resolution T/2. Record entropy H R. Recursively take finer resolutions. Entropy plots Time D Weights Resolution Entropy

Entropy Plots Self-similarity  Linear plot Resolution Entropy ● slope = 5.9

Entropy Plots Self-similarity  Linear plot Resolution Entropy time Uniform: slope=1 slope = 5.9

Entropy Plots Self-similarity  Linear plot Resolution Entropy time Uniform: slope=1Point mass: slope=0 slope = 5.9

13McGlohon, Akoglu, Faloutsos KDD08 Entropy Plots Resolution Entropy Bursty: 0.2 < slope < 0.9 Self-similarity  Linear plot time Uniform: slope=1Point mass: slope=0 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 14

Datasets 15

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 16

Observation 1:λ 1 Power Law (LPL) Q: How does the principal eigenvalue λ 1 change over time A: λ 1 (t) and the number of edges E(t) over time follow a power law with exponent less than 0.5, especially after the ‘gelling point’. 17 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; 18 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: 19 λ 1,w (t) ∝ E(t) β DBLP Author-Conference network Network Traffic

Observation 3: Edge Weights PL Q: How does the weight of an edge relate to “popularity” if its adjacent nodes A: Weight of the link w i,j between two given nodes i and j in a given graph G has a power law relation with the weights w i and w j of the nodes; 20 FEC Committee-to- Candidate network

Outline 1. Motivation 2. Related Work - Patterns - Generators 3. Laws and Observations 4. Proposed graph generator: RTM 5. Sketch of proofs 6. Experiments 7. Conclusion 21

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) 22

Problem Definition cont. 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) 23

One solution: Kronecker Product 24 Intuition : Self-similarity! Communities within communities Recursion yields modular network behavior

One solution: Kronecker Product 25

Recursive Tensor Product(RTM) Use of tensors: 3 rd mode is time Initial tensor I is a realistic graph itself 26 RTM of a (3x3x3) tensor by itself

RTM cont. 27

Outline 1. Motivation 2. Related Work - Patterns - Generators 3. Laws and Observations 4. Proposed graph generator: RTM 5. Sketch of proofs 6. Experiments 7. Conclusion 28

Experimental Results (I) 29 BLOG NETWORK RTM MODEL

Experimental Results (II) 30 RTM MODEL BLOG NETWORK

Conclusion Largest (un)weighted principal eigenvalues are power-law related to the number of edges in real graphs. Weight of an edge is related to the total weights of its incident nodes. Recursive Tensor Multiplication is a recursive method to generate weighted, time-evolving, self-similar, modular networks. 31

Future Directions Largest eigenvalues of the Laplacian matrices Second largest eigenvalue – related to global connectivity – conductance – mixing rate of random walk on graph Probabilistic version of RTM Fitting graphs 32

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