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Source Detection in Networks using the Stationary Distribution of a Markov Chain

Yael Sabato, Amos Azaria, Noam Hazon

TL;DR

The paper tackles identifying the diffusion source in networks under the Independent Cascade model using maximum likelihood estimation, noting that exact computation is intractable. It introduces a principled Markov-chain based method that leverages the stationary distribution, via the Markov chain tree theorem, to aggregate over all possible diffusion trees. Two network-to-Markov-chain conversion schemes, Self-Loops and No-Loops, enable exact or near-exact recovery of the source likelihoods from a single stationary distribution, with the No-Loops variant showing favorable performance under sampling. Experiments on 14 random graph families and 9 real networks demonstrate that the proposed approach outperforms existing baselines, offering a scalable ML-based solution for IC diffusion source detection. This method has practical impact for rapid and principled source tracing in social networks and related diffusion settings.

Abstract

Nowadays, the diffusion of information through social networks is a powerful phenomenon. One common way to model diffusions in social networks is the Independent Cascade (IC) model. Given a set of infected nodes according to the IC model, a natural problem is the source detection problem, in which the goal is to identify the unique node that has started the diffusion. Maximum Likelihood Estimation (MLE) is a common approach for tackling the source detection problem, but it is computationally hard. In this work, we propose an efficient method for the source detection problem under the MLE approach, which is based on computing the stationary distribution of a Markov chain. Using simulations, we demonstrate the effectiveness of our method compared to other state-of-the-art methods from the literature, both on random and real-world networks.

Source Detection in Networks using the Stationary Distribution of a Markov Chain

TL;DR

The paper tackles identifying the diffusion source in networks under the Independent Cascade model using maximum likelihood estimation, noting that exact computation is intractable. It introduces a principled Markov-chain based method that leverages the stationary distribution, via the Markov chain tree theorem, to aggregate over all possible diffusion trees. Two network-to-Markov-chain conversion schemes, Self-Loops and No-Loops, enable exact or near-exact recovery of the source likelihoods from a single stationary distribution, with the No-Loops variant showing favorable performance under sampling. Experiments on 14 random graph families and 9 real networks demonstrate that the proposed approach outperforms existing baselines, offering a scalable ML-based solution for IC diffusion source detection. This method has practical impact for rapid and principled source tracing in social networks and related diffusion settings.

Abstract

Nowadays, the diffusion of information through social networks is a powerful phenomenon. One common way to model diffusions in social networks is the Independent Cascade (IC) model. Given a set of infected nodes according to the IC model, a natural problem is the source detection problem, in which the goal is to identify the unique node that has started the diffusion. Maximum Likelihood Estimation (MLE) is a common approach for tackling the source detection problem, but it is computationally hard. In this work, we propose an efficient method for the source detection problem under the MLE approach, which is based on computing the stationary distribution of a Markov chain. Using simulations, we demonstrate the effectiveness of our method compared to other state-of-the-art methods from the literature, both on random and real-world networks.
Paper Structure (16 sections, 3 theorems, 16 equations, 5 figures, 5 tables)

This paper contains 16 sections, 3 theorems, 16 equations, 5 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminaries
  4. Directed Rooted Trees
  5. The Diffusion Model
  6. Markov Chain
  7. Irreducibility
  8. Stationary Distribution
  9. Vector Notations
  10. Problem Statement
  11. The Markov Chain Approach
  12. Conversion Methods
  13. The Self-Loops Method
  14. The no-loops Method
  15. Experiments
  16. ...and 1 more sections

Key Result

Theorem 1

(Markov chain tree theorem) Given a finite state irreducible Markov chain, For every $s_i\in S_M$, the unique Stationary Distribution $\Pi_i$ equals to $\hat{\Psi}_i$. namely:

Figures (5)

  • Figure 1: Graph examples.
  • Figure 2: The Markov chains that are obtained by applying the self loops method on the graphs of Figure \ref{['fig:two graphs']}.
  • Figure 3: The Markov chain that is obtained by stage (1) of the no-loops method for the social network graph in \ref{['fig:complex_ex']}.
  • Figure 4: The average number of times in which the no-loops and the self-loops methods, using direct calculation and using random walks with various number of steps, find the correct source node on the random graphs.
  • Figure 5: The average number of times in which the no-loops and the self-loops methods, using direct calculation and using random walks, find the correct source node on the real-world networks.

Theorems & Definitions (7)

  • Definition 1: Source Detection
  • Definition 2: ML-Source
  • Theorem 1
  • Theorem 2
  • proof
  • Theorem 3
  • proof