Inferring the roles of individuals in collective systems using information-theoretic measures of influence

Abstract

In collective systems, influence of individuals can permeate an entire group through indirect interactionscom-plicating any scheme to understand individual roles from observations. A typical approach to understand an individuals influence on another involves consideration of confounding factors, for example, by conditioning on other individuals outside of the pair. This becomes unfeasible in many cases as the number of individuals increases. In this article, we review some of the unforeseen problems that arise in understanding individual influence in a collective such as single cells, as well as some of the recent works which address these issues using tools from information theory.

Keywords: causal inference; data-driven approach; information theory; single cells collectives; transfer entropy.

Figure 1

Graph representations for models for different interaction types. (A) Type A. (B) Type B. (C) Type C. (D) Type D. [Reprinted with permission from Ref. [15]. Copyright ©2023, American Association for the Advancement of Science.]

Figure 2

Particle interaction depends on position. (A) the solid particle’s motion is influenced by the dotted particle within its interaction domain, but not vice versa. (B) both particles influence each other as they are within each other’s interaction domains. (C) particles move independently since they are outside each other’s interaction domains. [Reprinted with permission from Ref. [23]. Copyright ©2023 AIP Publishing LLC.]

Figure 3

Derivative of average TE with respective of the cutoff distance λ, $\frac{⟨\mathit{T}\mathit{E}{⟩}_{\mathit{\lambda }}}{\mathit{d}\mathit{\lambda }}$ as a function of λ. Inset: Average TE, ⟨TE⟩_λ as a function of cutoff distance λ. [Reprinted with permission from Ref. [23]. Copyright ©2023 AIP Publishing LLC.]

Figure 4

The derivative of the average inward TE for particle 1 with respect to the cutoff distance λ is depicted as a function of λ. In the inset, the average inward TE of particle 1 is illustrated as a function of the cutoff distance λ. [Reprinted with permission from Ref. [23]. Copyright ©2023 AIP Publishing LLC.]

Figure 5

S as a function of noise level η₀ (in units of π radians) for a model with different numbers of leaders and followers. (A) S_L→F, and (B) S_F→L as a function of η₀ (in units of π radians) for four agents with one leader and three followers (blue) and two leaders and two followers (red). (C) S_L→F and (D) S_F→L for eight agents with one leader and seven followers (blue), two leaders and six followers (red), three leaders and five followers (yellow), and four leaders and four followers (purple). (E) S_L→F and (F) S_F→L with three followers and one leader (blue), two leaders (red), and three leaders (yellow). (G) Graph representation of model A, where there is one leader and three followers. (H) Graph representation of model A, where there are two leaders and two followers. [Reprinted with permission from Ref. [15]. Copyright ©2023, American Association for the Advancement of Science.]

Figure 6

T as a function of noise level η₀ (in units of π radians) for model A with one leader and different numbers of followers. Here, N_F=1 (blue), 3 (red), 7 (yellow), and 15 (purple), where the number of leader is alwaysone. [Reprinted with permission from Ref. [15]. Copyright ©2023, American Association for the Advancement of Science.]

Figure 7

Schematic illustrates particle groups with diverse alignment scores (A^R), with the thick shaded arrow representing a PIV vector defined over a grid, and the ovals with arrows representing the moving particles along with their respective directions of movement. (a) A^R close to –1 due to opposing PIV vector; (b) A^R close to 0 for randomly moving particles; (c) A^R close to 1 when PIV vector aligns with particle movement. [Reprinted with permission from Ref. [25]. Copyright ©2023 Springer Nature Limited.]

Figure 8

The alignment score A^R is plotted against time t for various radius R values under different noise conditions: (a) ${\mathit{\eta }}_0=\frac{\mathit{\pi }}{6}$, (b) ${\mathit{\eta }}_0=\frac{3\mathit{\pi }}{6}$, (c) η₀=π, and (d) ${\mathit{\eta }}_0=\frac{11\mathit{\pi }}{6}$. [Reprinted with permission from Ref. [25]. Copyright ©2023 Springer Nature Limited.]

Figure 9

The landscape of A^R at ${\mathit{\eta }}_0=\frac{11\mathit{\pi }}{6}$ (high noise) is depicted in relation to the normalized PIV grid size, $\frac{\mathit{\gamma }}{{\mathit{\gamma }}_0\left(\mathit{N}\right)}$, and normalized radius, $\frac{\mathit{R}}{{\mathit{R}}_0\left(\mathit{N}\right)}$, for a particle count of (a) N=100 and (b) N=300. [Reprinted with permission from Ref. [25]. Copyright ©2023 Springer Nature Limited.]