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**Network Reconstruction under Compressive Sensing**

By: Motahareh Eslami Mehdiabadi Sharif University of Technology Authors: Payam Siyari, Hamid R. Rabiee Mostafa Salehi, Motahareh EslamiMehdiabadi

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**Outline Introduction Related Work Network Reconstruction**

Compressive Sensing Problem Formulation Proposed Framework: CS-NetRec Experimental Evaluation Conclusion

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**Introduction Large Scale Many Systems Modeled as Networks Partial**

Unknown Structure Many Systems Modeled as Networks Partial Observations

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**Introduction (cont’d)**

Network Reconstruction Problem: Given a network with missing edges Assumptions: Certain observable quantities on the network Can have partial observations Process-> Node values Goal: Uncover network structure

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**Introduction (cont’d)**

Network Reconstruction Problem: (a) An example network Figure 1: An example of the network reconstruction problem.

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**Introduction (cont’d)**

Network Reconstruction Problem: (b) The example network with no Information about the edges Figure 1: An example of the network reconstruction problem.

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**Introduction (cont’d)**

Network Reconstruction Problem: (c) A partial observation from the network structure. The process output is f(v1,v2,v3,v5,v6). Figure 1: An example of the network reconstruction problem.

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**Introduction (cont’d)**

Network Reconstruction Problem: (c) Another partial observation from the network structure. The process output is f(v2,v3,v4,v5,v6). Figure 1: An example of the network reconstruction problem.

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**Introduction (cont’d)**

Network Reconstruction Problem: Encountered in many real-world applications: Inaccuracies in uncovering the Protein interaction data [1]. Network Reconstruction Problem arises in many fileds. For example, in analysis of protein interaction data,there are several inaccuracies in reports about uncovering of the network Figure 2: Protein interaction network in yeast image from

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**Introduction (cont’d)**

Network Reconstruction Problem: Encountered in many real-world applications: In the social networks analysis, particularly online social networks (OSNs), there is missing data due to several reasons: Security User privacy Data aggregation overhead, etc. In recommender systems, especially in OSNs

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**Introduction (cont’d)**

Contributions Reconstructing the underlying network without any knowledge about the topological features of the underlying network. Proposing a novel and general framework based on the rich mathematical framework of Compressive Sensing (CS) for the first time

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**Network Reconstruction**

Related Work Network Reconstruction Compressive Sensing Two distinct fields

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**Network Reconstruction**

Problems Network Completion: Model-based approaches [8,9] Matrix Completion: Completing a low-rank data matrix[13] Link prediction: Predicting future edges [14,15] Network Inference: Diffusion network extraction [16,17] Solutions proposed Network Completion: However, in this model - > Kronecker graphs model Matrix Completion: predict its missing elements – challenges : several properties of real networks (heavy-tailed degree distribution) are not considered in the formulation of … Classic link prediction problem- predict based on the past snapshots of the networks Network Inference: only observe node infection times but not the sources of infection which means who infect whom...in the upcoming slides

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Compressive Sensing Efficient sparse signal recovery [20, 21] Basic idea The under-sampled data of a sparse signal have all the information needed about that signal. Combining l1-minimization and random matrices The development of compressive sensing began with the work of Donoho & Candes. [2006]

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**Compressive Sensing Underdetermined linear systems**

N: umber of equations < M: number of unknowns =>It is obvious we have infinite number of solutions.

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Compressive Sensing Typically, the solution of a linear system can be obtained by the least square minimization.

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**Compressive Sensing Looking for sparse solutions**

But when we are looking for for sparse solutions, you are Facing such optimization problem. Looking for sparse solutions Combinatorial, NP-Hard

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Compressive Sensing The option that we have here is to use other l-p norms to see if they can help. The reason is that minimizing objective norms such as l1 and l2 norms can be done efficiently using many algorithms. It is proven, For sufficiently sparse solutions, l1 can lead to the same solution as l-zero solution.

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**Compressive Sensing LASSO [22,23]**

One step ahead, we Combine the l-1 minimzantion and least squares to have the following problem namely lasso which is a well-known problem in sparse signal recovery and machine learning. LASSO [22,23]

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**CS in Network Analysis Compressive Sensing**

Mainly studied in signal & image processing [26-28] CS in Network Analysis Mostly in the field of Wireless Sensor Networks [29–31] Used for Network Tomography: Inference based on end-to-end measurements [32] Also in network traffic monitoring [33,34] and P2P networks [35] Network tomography:Link delays

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Problem Formulation 21

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**Can be run several times**

Preliminaries External Process Can be run several times Measures a value for each node as the output Diffusion of information, e.g. news headlines, virus, rumor, etc.

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**(a) A news blogs network**

Problem Formulation Information Diffusion & The Cascading Behavior (a) A news blogs network Figure 3: An example of information diffusion on a news blogs network.

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**Problem Formulation Information Diffusion & The Cascading Behavior:**

Consequence of time stamps : cascading behaviour (b) Example cascade Figure 3: An example of information diffusion on a news blogs network.

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**Problem Formulation Information Diffusion & The Cascading Behavior:**

Cascade Hit times = <(A, tA), (B, tB), (D, tD), (G, tG), (E, tE), > (b) Example cascade Figure 3: An example of information diffusion on a news blogs network.

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Problem Formulation Conditional probability of observing cascade c spreading from u to v [16] To transform this problem to a compressive sensing framework, first, we should understand the nature of information propagation.

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**β: The probability that a cascade**

Problem Formulation Conditional probability of observing cascade c spreading from u to v [16] The likelihood of a cascade Spreading in a given tree pattern T [16] β: The probability that a cascade will continue

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**Computationally expensive!**

Problem Formulation Conditional probability of observing cascade c spreading from u to v [16] The likelihood of a cascade Spreading in a given tree pattern T [16] As we are not aware of the true tree of cascade, we have to consider all possible trees… The probability that a cascade c can occur in the graph G [16] Computationally expensive!

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Problem Formulation Conditional probability of observing cascade c spreading from u to v [16] The likelihood of a cascade Spreading in a given tree pattern T [16] The probability that a cascade c can occur in the graph G [16] The approximated tree and its corresponding probability

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**Problem Formulation (b) Example cascade**

Figure 4: An example of information diffusion on a news blogs network.

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**(c) Most likely cascade tree shown by dotted links**

Problem Formulation The most probable cascade tree for the news cascade It can be seen that it is not necessary that the most likely cascade be the true cascade. However, it is a reasonable approximation. (c) Most likely cascade tree shown by dotted links Figure 4: An example of information diffusion on a news blogs network.

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**Proposed Framework Defining cascade probabilities as an inner product:**

Where:

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**Proposed Framework(CS-NetRec)**

Each inner product as an equation in linear system Each equation = a cascade.

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Proposed Framework

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**Experimental Evaluation**

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**Dataset * Synthetic * Real Erdos-Reyni (ER) Small-World**

Barabasi-Albert(BA) Core-Priphery (Kronecker) * Real Network Node # Edge# Football 115 615 Neural (C.elegans) 306 2345 Airport (USTop500) 500 2980 36

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**Precision Recall F-measure Evaluation**

The number of correctly inferred diffusion links divided by the total number of inferred links. Precision The number of correctly diffusion links divided by the total number of links in the network. Recall A trade-off between Precision and Recall F-measure 37

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**Cascade Dependency Synthetic Networks Real Networks**

For more cascades, we traverse more edges on the graph=> each edge is repeated in more equations of the system more measurements=> more accuracy Synthetic Networks Real Networks

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**Performed in ER network**

The Effect of Sparsity Performed in ER network

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**Performance Comparison**

(with NetInf) BA ER Small World Core Kron.

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**Performance Comparison**

(with NetInf) C.elegans Football US Top500

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**Conclusion A Novel and general framework for Network Reconstruction**

Based on the rich mathematical framework of Compressive Sensing “for the first time” As a special case : information networks Reconstruct the underlying network without any knowledge about the topological features.

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Future Work Utilize other processes and features/similarities in the graphs, e.g. delay, distance, etc. Use incidence matrix or Laplacian matrix. Dimensionality reduction

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Q&A

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