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Vertex Centrality Reconstruction in an Inverse Problem for Information Diffusion

Yixian Gao, Songshuo Li, Yang Yang

Abstract

We consider an inverse problem in information diffusion modeled by random walks on combinatorial graphs. The problem concerns reconstruction of vertex centrality from the distribution of the first passage times observed on a subset of vertices. We adapt the boundary control method to obtain a direct algorithm that computes the unobserved vertex centrality. The algorithm is numerically implemented and validated on small graphs.

Vertex Centrality Reconstruction in an Inverse Problem for Information Diffusion

Abstract

We consider an inverse problem in information diffusion modeled by random walks on combinatorial graphs. The problem concerns reconstruction of vertex centrality from the distribution of the first passage times observed on a subset of vertices. We adapt the boundary control method to obtain a direct algorithm that computes the unobserved vertex centrality. The algorithm is numerically implemented and validated on small graphs.
Paper Structure (15 sections, 7 theorems, 80 equations, 6 figures, 1 algorithm)

This paper contains 15 sections, 7 theorems, 80 equations, 6 figures, 1 algorithm.

Table of Contents

  1. introduction
  2. Background
  3. Problem Formulation
  4. Literature Review
  5. The Contribution
  6. Blagovescenskii-type Identity
  7. Representation of Graph Heat Solutions on
  8. Blagovescenskii-type Identity
  9. Reconstruction Procedure
  10. Construction of controls
  11. Reconstruction of the vertex centrality on .
  12. Reconstruction algorithm
  13. Numerical experiment
  14. Experiment 1: a graph with eight vertices
  15. Experiment 2: the graph with nine vertices

Key Result

Lemma 2.1

MR4598377 The solution $U^f|_{\mathbb{Z}_{2T}\times B}$ with $f\in \ell^2(\mathbb{Z}_T\times B)$ can be written in terms of $f$ and $r(t, x, y)|_{\mathbb{Z}_{2T}\setminus\{0\} \times B \times B}$ as follows: for $(t,x)\in \mathbb Z_{2T}\times B$.

Figures (6)

  • Figure 1: $X$ and $B$ in Experiment 1.
  • Figure 2: The singular values of $[W^{*}W]$ and $\mathbf{H}$ in Experiment 1. The minimum singular values are $1.6083*10^{-5}$ and $1.5844*10^{-17}$, respectively.
  • Figure 3: The ground-truth $\mu_x$, the reconstructed $\mu_x'$ and the absolute errors in Experiment 1.
  • Figure 4: $X$ and $B$ in Experiment 2.
  • Figure 5: The singular values of $[W^{*}W]$ and $\mathbf{H}$ in Experiment 2. The minimum singular values are $1.0922*10^{-4}$ and $0.0023$, respectively.
  • ...and 1 more figures

Theorems & Definitions (15)

  • Lemma 2.1
  • proof
  • Remark 2.2
  • Lemma 2.3
  • Proposition 2.4
  • proof
  • Definition 2.5
  • Remark 2.6
  • Proposition 2.7
  • proof
  • ...and 5 more