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Balance with Memory in Signed Networks via Mittag-Leffler Matrix Functions

Yu Tian, Ernesto Estrada

TL;DR

It is shown that the ML balance index can be obtained from first principles on the basis of a nonconservative diffusion dynamic, and that it accounts for the memory of the system about the past, by diminishing the penalization that long cycles typically receive in other matrix functions.

Abstract

Structural balance is an important characteristic of graphs/networks where edges can be positive or negative, with direct impact on the study of real-world complex systems. When a network is not structurally balanced, it is important to know how much balance still exists in it. Although several measures have been proposed to characterize the degree of balance, the use of matrix functions of the signed adjacency matrix emerges as a very promising area of research. Here, we take a step forward to using Mittag-Leffler (ML) matrix functions to quantify the notion of balance of signed networks. We show that the ML balance index can be obtained from first principles on the basis of a nonconservative diffusion dynamic, and that it accounts for the memory of the system about the past, by diminishing the penalization that long cycles typically receive in other matrix functions. Finally, we demonstrate the important information in the ML balance index with both artificial signed networks and real-world networks in various contexts, ranging from biological and ecological to social ones.

Balance with Memory in Signed Networks via Mittag-Leffler Matrix Functions

TL;DR

It is shown that the ML balance index can be obtained from first principles on the basis of a nonconservative diffusion dynamic, and that it accounts for the memory of the system about the past, by diminishing the penalization that long cycles typically receive in other matrix functions.

Abstract

Structural balance is an important characteristic of graphs/networks where edges can be positive or negative, with direct impact on the study of real-world complex systems. When a network is not structurally balanced, it is important to know how much balance still exists in it. Although several measures have been proposed to characterize the degree of balance, the use of matrix functions of the signed adjacency matrix emerges as a very promising area of research. Here, we take a step forward to using Mittag-Leffler (ML) matrix functions to quantify the notion of balance of signed networks. We show that the ML balance index can be obtained from first principles on the basis of a nonconservative diffusion dynamic, and that it accounts for the memory of the system about the past, by diminishing the penalization that long cycles typically receive in other matrix functions. Finally, we demonstrate the important information in the ML balance index with both artificial signed networks and real-world networks in various contexts, ranging from biological and ecological to social ones.
Paper Structure (16 sections, 7 theorems, 45 equations, 8 figures, 4 tables)

This paper contains 16 sections, 7 theorems, 45 equations, 8 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Structural balance
  4. Laplacians and Mittag-Leffler matrix function
  5. Motivation
  6. Main results
  7. Nonconservative fractional diffusion and balance
  8. How global balance is accounted for
  9. How memory is accounted for
  10. Properties of the balance index with memory
  11. Examples of applications
  12. Gene regulatory networks
  13. Spatial ecological networks
  14. Social networks in rural villages
  15. A useful approximation
  16. ...and 1 more sections

Key Result

Theorem 2.1

A signed graph $G$ is structurally balanced if and only if there is a bipartition of the node set into $V=V_1\cup V_2$ with $V_1$ and $V_2$ being mutually disjoint and one of them being nonempty, s.t. any edge between the two is negative while any edge within each node subset is positive.

Figures (8)

  • Figure 1: The five switching isomorphism types of signed Petersen graph excluding the unsigned one, where solid lines represent positive edges and dashed lines represent negative ones.
  • Figure 2: Change of the difference of $Tr\left(A^{k}\right)/k!$ between graphs c) and d) of Fig. \ref{['Petersen graphs']}.
  • Figure 3: (left) Contour plot of the values of $K_{\alpha}\left(C_{n}\right)$ for unbalanced cycles with $n$ vertices for values of $0.1\leq\alpha\leq1.$ (right) Difference of the balance index $K_\alpha$ for graph c) versus that for graph d) in Fig. \ref{['Petersen graphs']} for values of $0.2\leq \alpha \leq 1$.
  • Figure 4: (left) Plot of the difference in balance $K_{\alpha}$ as a function of $\alpha$ for the gene regulatory networks of yeast and B. subtilis. (middle) Plot of the percentages of negative cycles $C_{n}^{-}$ for $3\leq n\leq11$ for the gene regulatory networks of yeast and B. subtilis. The percentages are calculated as $100\cdot C_{n}^{-}/\left(C_{n}^{+}+C_{n}^{-}\right).$ (right) Plot of the nonmonotonic change of $K_{\alpha}$ as a function of $\alpha$ for the gene regulatory network of yeast.
  • Figure 5: (left) Change of the average balance $K_{\alpha}$ for $0.4\leq\alpha\leq1$ of the signed ecological networks in the four major Spanish locations studied. (right) Plot of the change of the Pearson correlation coefficient $r^{2}$ of the balance indices with memory $K_{\alpha}$ for $0.4\leq\alpha\leq1$ versus the amount of precipitation in mm on the four major locations studied in this work.
  • ...and 3 more figures

Theorems & Definitions (14)

  • Theorem 2.1: structure theorem for balance harary_1953_balance
  • Definition 2.2
  • Proposition 3.1
  • proof
  • Proposition 4.1
  • proof
  • Corollary 4.2
  • Definition 4.3: martin2023fractional
  • Remark 4.4
  • Lemma 4.5: martin2023fractional
  • ...and 4 more