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arxiv: 1907.10667 · v1 · pith:UXN6RHDSnew · submitted 2019-07-23 · ⚛️ physics.soc-ph · cs.SI

Influence and Betweenness in Flow Models of Complex Network Systems

Olexandr Polishchuk This is my paper

Pith reviewed 2026-05-24 17:16 UTC · model grok-4.3

classification ⚛️ physics.soc-ph cs.SI
keywords complex network systemsflow adjacency matricesinfluence measuresbetweenness measuresnetwork robustnesstargeted attacksfunctional importanceflow models
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The pith

Flow adjacency matrices define influence and betweenness measures that quantify the contribution of nodes and edges to flows in complex network systems and the losses from their blockage.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper develops a flow-based approach to analyzing complex network systems by introducing flow adjacency matrices. These matrices support definitions of influence measures for nodes, such as strength, power, domain, and diameter, along with betweenness measures for nodes and edges. The resulting indicators show how much each element contributes to flow movement and estimate the system losses if that element is blocked or attacked. Similar concepts apply to subsystems. Readers might care because this offers a method to pinpoint vulnerabilities in real-world networks like transportation or information systems.

Core claim

The paper establishes that flow adjacency matrices provide the foundation for defining the strength, power, domain and diameter of influence for nodes, and the measure, power, domain and diameter of betweenness for nodes and edges. These quantities directly quantify the contribution of each element to the motion of flows through the system and allow prediction of the losses that would result from blocking the element. The same notions extend to subsystems and the entire network system, with examples illustrating their use in practical investigations of real complex networks.

What carries the argument

Flow adjacency matrices, which model flow interactions between nodes and serve as the basis for deriving influence and betweenness indicators of nodes, edges, and subsystems.

If this is right

  • Nodes with high influence measures contribute more to overall flow motion, so their removal causes greater disruption.
  • Betweenness measures identify nodes and edges that lie on many flow paths, making them critical for maintaining connectivity.
  • Expected losses from targeted attacks can be calculated in advance using these indicators.
  • Subsystems can be ranked by their influence and betweenness to assess their role in the whole system.
  • These metrics support analysis of network robustness against failures or blockages.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • These measures might differ from standard graph centrality in networks where flow directions or capacities matter.
  • Testing the measures on simulated networks with known flow dynamics could validate their predictive power for losses.
  • Extending the approach to time-varying flows could address dynamic network behaviors not covered here.

Load-bearing premise

That the introduced flow adjacency matrices correctly capture the relevant flow dynamics and functional relationships in the network.

What would settle it

Observing a complex network system where removing a node with high calculated betweenness does not produce the predicted level of flow loss or disruption.

Figures

Figures reproduced from arXiv: 1907.10667 by Olexandr Polishchuk.

Figure 4
Figure 4. Figure 4: Domains of output ( out RS,ext – vertical lines) and input influence ( in GS,ext – horizontal lines) of NS subsystem (subsystem nodes are reflected by squares) [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
read the original abstract

This paper provides the analysis for functional approaches of complex network systems research. In order to study the behavior of these systems the flow adjacency matrices were introduced. The concepts of strength, power, domain and diameter of influence of complex network nodes are analyzed for the purpose of determining their importance in the systems structure. The notions of measure, power, domain and diameter of betweenness of network nodes and edges are introduced to identify their significance in the operation process of network systems. These indicators quantitatively express the contribution of the corresponding element for the motion of flows in the system and determine the losses that are expected in the case of blocking this node or edge or targeted attack on it. Similar notions of influence and betweenness are introduced to determine the functional importance of separate subsystems of network system and the system as a whole. Examples of practical use of the obtained results during investigation of real complex network systems are given.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces flow adjacency matrices as a basis for analyzing complex network systems and defines new measures of node/edge influence (strength, power, domain, diameter) and betweenness (measure, power, domain, diameter), extending these to subsystems. These quantities are asserted to quantify each element's contribution to flow motion and to determine expected losses from blocking, removal, or targeted attack, with examples of application to real systems provided.

Significance. If the proposed measures were shown through derivation or validation to predict flow contributions and removal losses, they would represent a functional extension of centrality concepts tailored to flow networks, with potential utility in infrastructure and transportation systems analysis. The subsystem-level generalizations are a constructive addition. However, the manuscript presents these primarily as definitions without anchoring derivations or empirical mappings, limiting the assessed significance.

major comments (2)
  1. [Abstract, §3] Abstract and the definitions in §3: the central claim that the influence and betweenness indicators 'quantitatively express the contribution ... and determine the losses that are expected' is asserted without a derivation, conservation relation, or explicit mapping from the flow adjacency matrix entries to flow volumes or removal impacts; this link is load-bearing for the stated purpose.
  2. [§5] §5 (examples): the practical-use cases illustrate the new quantities but supply no quantitative comparison to simulated flow losses, empirical removal data, or baseline centrality measures, so the predictive content of the claims remains untested.
minor comments (2)
  1. Notation for 'power' and 'domain' of influence/betweenness should be distinguished more clearly from existing network-science terminology to avoid reader confusion.
  2. [§2] The manuscript would benefit from an explicit statement of the axioms or assumptions underlying the flow adjacency matrix construction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments. We address each major point below, clarifying the conceptual basis of the definitions while agreeing to strengthen the manuscript where the mapping or validation can be made more explicit.

read point-by-point responses

  1. Referee: [Abstract, §3] Abstract and the definitions in §3: the central claim that the influence and betweenness indicators 'quantitatively express the contribution ... and determine the losses that are expected' is asserted without a derivation, conservation relation, or explicit mapping from the flow adjacency matrix entries to flow volumes or removal impacts; this link is load-bearing for the stated purpose.

    Authors: The flow adjacency matrix is defined so that each entry directly represents a flow volume or capacity between nodes. The influence and betweenness quantities are then obtained by summation and maximization operations over these entries; the contribution of a node or edge is therefore expressed by construction as the aggregate flow it participates in. Expected losses upon removal follow immediately by subtracting the relevant submatrix contributions. We acknowledge that an explicit one-paragraph derivation linking matrix entries to loss expressions is not present in §3 and will insert it in the revision to make the mapping fully transparent. revision: partial

  2. Referee: [§5] §5 (examples): the practical-use cases illustrate the new quantities but supply no quantitative comparison to simulated flow losses, empirical removal data, or baseline centrality measures, so the predictive content of the claims remains untested.

    Authors: Section 5 is written to show how the new measures are computed and interpreted on concrete infrastructure networks. We agree that direct numerical comparisons against simulated removal losses or against classical centrality indices would strengthen the demonstration of predictive utility. Because the original examples do not contain such side-by-side evaluations, we will add a short quantitative comparison (using standard betweenness as baseline) to one of the real-system cases in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No circularity; new measures introduced by definition with no reduction to inputs shown

full rationale

The provided abstract and structure introduce flow adjacency matrices and define influence/betweenness notions (strength, power, domain, diameter, etc.) as new indicators. These are then asserted to 'quantitatively express the contribution... and determine the losses' under blocking. No equations, fitting procedures, or self-citations are visible that would reduce the claimed predictive power back to the definitions by construction. The central claims remain definitional assertions rather than derivations that collapse to their inputs. No load-bearing self-citation chains or uniqueness theorems from prior author work are referenced. This is the common case of a paper that is self-contained against external benchmarks at the level of definitions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities; the new measures are presented as definitions without stated foundations or fitting procedures.

pith-pipeline@v0.9.0 · 5675 in / 1069 out tokens · 41231 ms · 2026-05-24T17:16:41.874484+00:00 · methodology

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