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Learning the Influence Graph of a Markov Process that Randomly Resets to the Past

Sudharsan Senthil, Avhishek Chatterjee

TL;DR

The paper tackles learning the directed influence graph of high-dimensional stochastic processes when distant past states can retake influence, breaking standard Markov assumptions. It introduces the Past Influence Model (PIM), which incorporates random resets to the past, and develops PIMRecGreedy, a memoryless entropy-based greedy method for recovering neighborhoods with finite-sample guarantees. A key contribution is a formal directed conditional-entropy framework and a rigorous sample-size bound showing high-probability recovery with complexity on the order of O(d^2 + d log |V|). The work enables reliable structure learning in social networks and financial risk contexts where past events re-emerge, and it provides a foundation for extending learning under non-Markov dynamics.

Abstract

Learning the influence graph G of a high-dimensional Markov process is central to many application domains, including social networks, neuroscience, and financial risk analysis. However, in many of these applications, future states of the process are occasionally and unpredictably influenced by a distant past state, thus destroying the Markovianity. To study this practical issue, we propose the past influence model (PIM), which captures the occasional "random resets to past" by modifying the Markovian dynamics in [1], which, in turn, is a non-linear generalization of the dynamics studied in [2], [3]. The recursive greedy algorithm proposed in this paper recovers any bounded degree $G$ when the number of ``jumps back in time" is order-wise smaller than the total number of samples, and the algorithm does not require memory.

Learning the Influence Graph of a Markov Process that Randomly Resets to the Past

TL;DR

The paper tackles learning the directed influence graph of high-dimensional stochastic processes when distant past states can retake influence, breaking standard Markov assumptions. It introduces the Past Influence Model (PIM), which incorporates random resets to the past, and develops PIMRecGreedy, a memoryless entropy-based greedy method for recovering neighborhoods with finite-sample guarantees. A key contribution is a formal directed conditional-entropy framework and a rigorous sample-size bound showing high-probability recovery with complexity on the order of O(d^2 + d log |V|). The work enables reliable structure learning in social networks and financial risk contexts where past events re-emerge, and it provides a foundation for extending learning under non-Markov dynamics.

Abstract

Learning the influence graph G of a high-dimensional Markov process is central to many application domains, including social networks, neuroscience, and financial risk analysis. However, in many of these applications, future states of the process are occasionally and unpredictably influenced by a distant past state, thus destroying the Markovianity. To study this practical issue, we propose the past influence model (PIM), which captures the occasional "random resets to past" by modifying the Markovian dynamics in [1], which, in turn, is a non-linear generalization of the dynamics studied in [2], [3]. The recursive greedy algorithm proposed in this paper recovers any bounded degree when the number of ``jumps back in time" is order-wise smaller than the total number of samples, and the algorithm does not require memory.

Paper Structure

This paper contains 9 sections, 6 theorems, 37 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. The Social Network Model
  3. Modeling
  4. Some Mathematical Structures
  5. Probabilistically teleporting to the past
  6. The Learning Problem
  7. Algorithm to Recover the Influence Graph
  8. Simulation results
  9. Conclusion

Key Result

Theorem 1

Let $\tilde{A}$ be the matrix whose $ij^{th}$ element is $\alpha_i \alpha_{ij}$, and $\rho(\tilde{A})$ is the largest eigenvalue of $\tilde{A}$. Also, let $\bar{\mu} := \max_{v} \sup_{x \in [0,1]} \mu_v(x).$ The PIMRecGreedy($\kappa$) algorithm recovers the true influence graph with probability at l Where, $c>0, |\xi|:=|\chi|^{(2+|P|_{max})}$, $|\chi| \leq (\frac{\bar{M}(\bar{M}+1)}{2}+2)$, $\beta

Figures (5)

  • Figure 1: $|V| = 5, d = 5$, and $\overline{M} = 1$
  • Figure 2: $|V| = 10, d = 10$, and $\overline{M} = 2$
  • Figure 3: A) $K_3$ with a pendant, B) $S_3$, and C) Tree $|V| = 5$.
  • Figure 4: $d = 10$ and $\overline{M} = 1$
  • Figure 5: $|V| = 10, d = 10$, $T = 3500$, and $\overline{M} = 2$

Theorems & Definitions (10)

  • Theorem 1
  • Lemma 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof