TIMES: Temporal Information Maximally Extracted from Structures - - PowerPoint PPT Presentation

TIMES: Temporal Information Maximally Extracted from Structures

Jithin K. Sreedharan

Purdue University

Ananth Grama (Purdue) Abram Magner (UIUC) Wojceich Szpankowski (Purdue)

Jithin K. Sreedharan WWW'18

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The Problem: Recovery of Node Arrival Order

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𝐻 Graph with node label as arrival order Dynamic graph Graph with partial node labels Reverse engineer the dynamic process to infer network trajectories

Jithin K. Sreedharan WWW'18

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Network of biochemical reactions: (protein-protein interaction network)

Cancer proteins tend to be ancient proteins

[Srivastava et al., Nature 2010]

Study of the phylogenetic tree

Social networks:

• nline spam spreading or rumor propagation

Financial transaction networks: flow of capital Spread of infectious diseases:

• rigin and initial carriers

Ebola spread network [Saey, ScienceNews Dec 2015]

Where is arrival order information useful?

Jithin K. Sreedharan WWW'18

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Prior Works

[Bubeck, Devroye, and Lugosi 2016] For preferential and uniform attachment trees, finds the set that contains root node w.h.p. [Frieze and Pegden 2017] For preferential attachment graphs, locates the oldest node by a random walk process. Assumes arrival info is known when a node is sampled. [Shah and Zaman, 2011] Oldest node in Susceptible-Infected epidemic model [Zhu and Ying, 2016] Oldest node in Susceptible-Infected-Recovered epidemic model Several works on "Counting linear extension of partial order sets”

Jithin K. Sreedharan WWW'18

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Formulation

Gn Adversary Estimator {Gt}t=T

t=1

π(GT ) π−1 ≈ σ?

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𝐻 𝜌 𝐻

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𝜌

Graph 𝐻: set of nodes 𝑜 = { 1, … , 𝑜} Youngest node label is 𝑜; Oldest node label is 1

Apply a random permutation 𝜌 from 𝑇,. Observed graph is 𝜌(𝐻) Symmetric group on 𝑜 letters

Graph model and symmetries play a critical role

𝑐 𝑒 𝑓 𝑑 𝑏

Graph model Output 𝜏

Jithin K. Sreedharan WWW'18

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Erdős–Rényi model 𝐻(𝑜, 𝑞) Each pair of nodes receives an edge independently, with probability 𝑞 Preferential Attachment model PA(n, m) § At 𝑢 = 1, 𝐻7 with single vertex (called 1) is created with 𝑛 self loops § At 𝑢 > 1, vertex 𝑢 joins and makes 𝑛 connections to existing nodes in 𝐻:;7. § Each of the 𝑛 connection choice is independent and satisfies Pr[t connects to k|Gt−1] = degt−1(k) 2m(t − 1)

Erdős–Rényi and Preferential Attachment models

Jithin K. Sreedharan WWW'18

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No algorithm can solve the problem with vanishingly small probability of error in the case of preferential attachment and Erdős–Rényi graphs

Total Order: Exact and Approximate Recovery of 𝝆;𝟐

Maximum Likelihood Estimation for Preferential Attachment Graph

MLE gives a large number of equiprobable solutions,

Bad News: Inapproximability results

Minimax risk with worst adversary and best estimator = 1 − 𝑝(1) CML(H) = arg max

σ∈Sn

Pr[π−1 = σ|π(G) = H]

|CML| = en log n−O(n log log n)

Jithin K. Sreedharan WWW'18

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Can we do better?

• Formulation
• Infeasibility results
• Optimization formulation
• A simple approximation algorithm for Preferential Attachment graph
• Experimental results: Facebook Wall post and human brain networks

Jithin K. Sreedharan WWW'18

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1 2 3 4 5 6 7 8 9 10 11 12 Bin 1 Bin 2 Bin 3 Bin 4 Bin 5

Partial Orders and Binning

Look for partial orders instead of total orders Estimator:

Set of all labeled graphs with 𝒐 vertices → Set of all partially ordered set

Jithin K. Sreedharan WWW'18

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How to Measure Efficiency of an Estimator?

Recall

How much are we able to recover from a partial order?

ρ(σ) = E " 1 n

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|{u, v ∈ [n] : u <σ v, π−1(u) < π−1(v)}| # .

Precision

How good are the guessed pairs, irrespective of its number?

# correct pairs in the partial order # pairs in the partial order

# correct pairs in the partial order n

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θ(σ) = E |{u, v ∈ [n] : u <σ v, π−1(u) < π−1(v)}| |{(u, v) : u <σ v}|

• <latexit 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Density

How good are the guessed pairs, irrespective of its number?

# pairs in partial ordeer n

2

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Jithin K. Sreedharan WWW'18

11

Different approach: phrase as an integer program.

max Precision subject to Density ≥ 𝜁 Set of partial orders xu,v : 1{u <φ v} for u, v ∈ [n] max Precision ⇔

Constrained Optimization Problem

Subjected to

For observed graph H = π(G) and ε > 0,

max

φ(H)

P

1u<vn pu,v(H)xu,v

P

1u6=vn xu,v

pu,v(H) := Pr[π−1(u) < π−1(v)|π(G) = H]

θ(σ) = E |{u, v ∈ [n] : u <σ v, π−1(u) < π−1(v)}| |{(u, v) : u <σ v}|

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Recollect

• 1. Antisymmetry: xu,v + xv,u ≤ 1.
• 2. Transitivity: xu,w ≥ xu,v + xv,w − 1 for all u, v, w ∈ [n].
• 3. Minimum density: P

1u6=vn xu,v ≥ ✏

n

2

• .
• 4. Domain restriction: xu,v ∈ {0, 1} for all u, v ∈ [n].

Jithin K. Sreedharan WWW'18

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Estimating 𝒒𝒗,𝒘(𝑰)

1 2 3 4 7 10 11 5 6 8 9 12 2 1 3 4 7 5 10 11 6 8 9 12

We have pu,v(H) = Pr[π−1(u) < π−1(v)|π(G) = H] = |{σ : σ−1∈Γ(H),σ−1(u)<σ−1(v)}|

|Γ(H)|

.

Estimating 𝑞J,K(𝐼) is equivalent to counting linear extensions of a partial order

• - #P-complete in general!

[Karzanov & Khachiyan, Brightwell & Winkler]

• - Approximate counting in

polynomial time (Markov chain algorithm). G DAG of G With the best known technique, 𝑞J,K(𝐼) can be estimated in 𝑃 𝑜N logR 𝑜 w.h.p.

Focus of Preferential Attachment graphs from now on.

Bin 1 Bin 2 Bin 3 Bin 4 Bin 5

Jithin K. Sreedharan WWW'18

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Approximating via Peeling algorithm

1 2 3 4 5 6 7 8 9 10 11 12 G DAG of G

Maximum-Density Precision-1 Estimator

Peeling Estimator Peeling+ Estimator

Outputs only probability-1 pairs given by the DAG

Exact Recovery of the DAG via Peeling

Recursive peeling of minimum degree nodes. Peeled nodes at each step forms a bin DAG recovered + pairs among non-edge nodes in distinct bins Peeling Estimator + pairs among nodes in same bins

The Peeling algorithm recovers all the ordered pairs that hold with probability 1

1 2 3 4 7 10 11 5 6 8 9 12 2 1 3 4 7 5 10 11 6 8 9 12

Jithin K. Sreedharan WWW'18

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0.2 0.4 0.6 0.8 1.0

ε

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

θ

Optimal precision Peeling Peeling+ Perfect-precision

Precision

Upper bound on optimal precision for density ≥ 𝜁

Jithin K. Sreedharan WWW'18

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• No. of nodes, 𝑜 = 5000

How robust is the algorithm?

Uniform Attachment model

PA + UA+addition of edges between existing nodes

Technique Precision Recall Density PA(n, m = 25) 0.958 0.936 0.977 PA(n, M), M ∼ unif{5, 50} 0.691 0.683 0.988 UA(n, m = 25) 0.977 0.967 0.99 UA(n, M), M ∼ unif{5, 50} 0.827 0.823 0.995 Cooper-Frieze (Web graph) model 0.828 0.822 0.993

Experiments: Synthetic Graphs

Result of Peeling estimator

Dataset # Nodes # Edges Genre Precision Recall Density ArXiv High Energy Physics 7.46K 116K Citation 0.708 0.681 0.961 Simple English Wikipedia 100K 1.62M Hyperlink 0.624 0.548 0.878 DBLP CS bibliography 1.13M 5.02M Coauthorship 0.785 0.728 0.927 Facebook Wall post 43.9K 271K Social 0.698 0.657 0.941 SMS network 30.2K 447K Social 0.669 0.610 0.912

Jithin K. Sreedharan WWW'18

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Experiments: Real-World Networks

Result of Peeling estimator Facebook users in New Orleans region. Edge of the form (u, v, t): user u posted on user v’s wall at time t.

Jithin K. Sreedharan WWW'18

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Human Brain Network formation: Cambridge Buckner Dataset

Aim: Recover the evolutionary order among the important regions inside the brain. Conjecture: There exists high correlation between these orderings of species evolved from the same genetic parent.

Experiments: Brain Networks

• The network has 46 nodes, each of which represents a region in the brain
• Data: fMRI resting state images
• Correlation matrix with each element as the Gaussianized version of the Pearson

correlation coefficient.

• Formation of binary adjacency matrix: find a threshold for the correlation matrix

values, above which entries get replaced by 1, and below it by 0.

• The threshold is taken as the largest quantile of elements in the correlation matrix

which just makes the graph formed out of it connected.

Jithin K. Sreedharan WWW'18

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Bin 13 Bin 5 Bin 14 Bin 12 Bin 0 Bin 3

Peeling estimator on Cambridge Buckner Dataset

Jithin K. Sreedharan WWW'18

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Data: Resting state fMRI focusing on the cortex area from 400 healthy young adults. Each network has 300 brain regions.

Human Connectome Data: Histogram of Prominent Regions in Cortex

Jithin K. Sreedharan WWW'18

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Conclusions

§ Formulated the node arrival order problem for temporal networks § Fundamental limits derived for inference of node arrival order § Shown infeasibility of recovering total order § Recovery of partial orders shows promising results § Integer programming framework for general random graph models § Solved the case of standard preferential attachment (PA) model! § Experimental results reveals that our algorithms admit perturbations in PA model § Results on Facebook and human brain networks More details at cs.purdue.edu/homes/jithinks/

(Code and data will be uploaded in a week)

Thank You!