Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An adaptation of InfoMap to absorbing random walks using absorption-scaled graphs

Esteban Vargas Bernal, Mason A. Porter, Joseph H. Tien

TL;DR

The paper presents a principled adaptation of InfoMap to absorbing random walks by using absorption-scaled graphs and Markov time sweeping, enabling detection of communities shaped by node-specific absorption rates. It introduces a map function L^{(a)} for absorbing Markov chains, proves its convergence to the standard map function as absorption vanishes, and connects absorption-scaled graphs to fundamental and absorption inverses. Through toy examples and ring-lattice SIR experiments, it shows that heterogeneous absorption can produce effective communities that differ markedly from structure-based partitions and can significantly influence epidemic dynamics. The work provides theoretical and computational tools to study dynamics-driven communities and demonstrates that absorption-aware methods can yield insights into disease duration, final size, and peak in structured populations. Code and examples illustrate practical applicability, with broad implications for mobility, epidemiology, and information spread on networks.

Abstract

InfoMap is a popular approach to detect densely connected "communities" of nodes in networks. To detect such communities, InfoMap uses random walks and ideas from information theory. Motivated by the dynamics of disease spread on networks, whose nodes can have heterogeneous disease-removal rates, we adapt InfoMap to absorbing random walks. To do this, we use absorption-scaled graphs (in which edge weights are scaled according to absorption rates) and Markov time sweeping. One of our adaptations of InfoMap converges to the standard version of InfoMap in the limit in which the node-absorption rates approach $0$. We demonstrate that the community structure that one obtains using our adaptations of InfoMap can differ markedly from the community structure that one detects using methods that do not account for node-absorption rates. We also illustrate that the community structure that is induced by heterogeneous absorption rates can have important implications for susceptible-infected-recovered (SIR) dynamics on ring-lattice networks. For example, in some situations, the outbreak duration is maximized when a moderate number of nodes have large node-absorption rates.

An adaptation of InfoMap to absorbing random walks using absorption-scaled graphs

TL;DR

The paper presents a principled adaptation of InfoMap to absorbing random walks by using absorption-scaled graphs and Markov time sweeping, enabling detection of communities shaped by node-specific absorption rates. It introduces a map function L^{(a)} for absorbing Markov chains, proves its convergence to the standard map function as absorption vanishes, and connects absorption-scaled graphs to fundamental and absorption inverses. Through toy examples and ring-lattice SIR experiments, it shows that heterogeneous absorption can produce effective communities that differ markedly from structure-based partitions and can significantly influence epidemic dynamics. The work provides theoretical and computational tools to study dynamics-driven communities and demonstrates that absorption-aware methods can yield insights into disease duration, final size, and peak in structured populations. Code and examples illustrate practical applicability, with broad implications for mobility, epidemiology, and information spread on networks.

Abstract

InfoMap is a popular approach to detect densely connected "communities" of nodes in networks. To detect such communities, InfoMap uses random walks and ideas from information theory. Motivated by the dynamics of disease spread on networks, whose nodes can have heterogeneous disease-removal rates, we adapt InfoMap to absorbing random walks. To do this, we use absorption-scaled graphs (in which edge weights are scaled according to absorption rates) and Markov time sweeping. One of our adaptations of InfoMap converges to the standard version of InfoMap in the limit in which the node-absorption rates approach . We demonstrate that the community structure that one obtains using our adaptations of InfoMap can differ markedly from the community structure that one detects using methods that do not account for node-absorption rates. We also illustrate that the community structure that is induced by heterogeneous absorption rates can have important implications for susceptible-infected-recovered (SIR) dynamics on ring-lattice networks. For example, in some situations, the outbreak duration is maximized when a moderate number of nodes have large node-absorption rates.
Paper Structure (24 sections, 12 theorems, 69 equations, 12 figures, 2 tables, 4 algorithms)

This paper contains 24 sections, 12 theorems, 69 equations, 12 figures, 2 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Background on InfoMap
  3. The map function and the standard InfoMap algorithm
  4. Markov time sweeping
  5. Markov time sweeping and adaptations of InfoMap to absorbing random walks
  6. A map function $L^{(a)}$ for Markov chains with an absorbing state
  7. A map function for an absorbing random walk
  8. Construction of a map function for an absorbing random walk
  9. The distribution $\pi_\delta^{(a)}$
  10. The transition-probability matrix $P_\delta$
  11. Main results for the map function $L^{(a)}$
  12. Relating $\tilde{G}(D_\delta, I)$ to $\tilde{G}(D_\delta, \mathbf{0})$
  13. Examples
  14. A three-node network
  15. A four-clique network
  16. ...and 9 more sections

Key Result

Proposition 1

Suppose that the Markov chain with transition-probability matrix $AW^{-1}$ is regular. Let $\vec{\delta}$ be a node-absorption-rate vector in which all entries are strictly positive, and let $D_\delta := \operatorname{diag}\{\vec{\delta}\}$. Let $N = (I - A(W+D_\delta)^{-1})^{-1}$ be the fundamental

Figures (12)

  • Figure 1: Consider an absorbing random walk on the depicted four-node network, and suppose that the absorption rate of node 2 is much larger than the absorption rates of the other nodes. Detecting communities via modularity maximization or the standard InfoMap algorithm produces a partition of the network into a single community that includes all nodes. However, the flow of an absorbing random walk is trapped in either the set $\{1\}$ (in dark blue) or the set $\{3,4\}$ (in light blue). Consequently, a partition that separates node 1 from nodes 3 and 4 better captures the dynamics of an absorbing random walk than a partition of the network into a single community.
  • Figure 2: An absorption-scaled graph. (a) A graph $G$ with absorption-rate vector $\vec{d}$. The pink node has a large absorption rate. (b) The associated absorption-scaled graph $\tilde{G}$, where the arrow length is proportional to the corresponding edge weight.
  • Figure 3: The map function $L^{(a)}(M,A,\vec{\delta},\pi_0)$ for all five possible partitions $M$ of the three-node network with adjacency matrix (\ref{['eqn:A_3nodes']}). The node-absorption-rate vector is $\vec{\delta} = (\delta_1,\delta_2,\delta_3)^{\operatorname{\,T}}$, with $\delta_1 = \delta_3 = 0.1$ and $0.1 \leq \delta_2 \leq 10$. The initial distribution is $\pi_0 = (1/3,1/3,1/3)^{\operatorname{\,T}}$.
  • Figure 4: An example three-node network. The node-absorption rate of node 2 is greater than or equal to the node-absorption rates of nodes 1 and 3.
  • Figure 5: The values of $L(M, P_l(D_\delta, \mathbf{0}, 1/20))$ for all five possible partitions $M$ of the three-node network in Figure \ref{['ex1']} with $\delta_1 = \delta_3 = 0.1$ and Markov time $t = 1/20$.
  • ...and 7 more figures

Theorems & Definitions (29)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Definition 6
  • ...and 19 more