arXiv subjects

Piet Van Mieghem

Publications and source records attributed to Piet Van Mieghem.

At least 19 recordsLinked to original sources

arXiv2026

Flow Subgraphs and Flow Network Design under End-to-End Power Dissipation Constraints

We investigate how the underlying graph of a network supports a flow between a source node and a destination node and propose to compute the expected number of nodes and links that contribute to transferring items in random graphs. Since the transportation is associated with a \quotes{cost} or \quotes{power dissipation}, we further address how to construct a graph given predetermined end-to-end power dissipation, which can be reduced to the \quotes{inverse effective resistance problem} that asks for a weighted graph in which the effective resistance matrix equals a predetermined demand matrix. We propose a heuristic algorithm, \quotes{Resistor Gap Pruning} (RGP), which provides sparse graphs closely approximating the demand effective resistance and which shows stable performance across different demand scenarios.

math-ph↗

arXiv2026

Geometric Organization and Inference of Shortest Path Nodes in Soft Random Geometric Graphs

The shortest path problem is related to many dynamic processes on networks, ranging from routing in communication networks to signaling in molecular interaction networks. When the network is fully known, the shortest path problem can be solved precisely and in polynomial time. If, however, the network of interest is only partially observable, the shortest path problem is no longer straightforward. Inspired by the shortest path problem in partially observable networks, we investigate the geometric properties of shortest paths in {\it Euclidean} Soft Random Geometric Graphs (SRGGs). We find that shortest paths are aligned along geodesic curves connecting shortest path endpoints. The strength of the shortest path alignment, as quantified by the average distance to geodesic from shortest path nodes and the average path stretch, is higher for larger SRGGs with short-range connections. In addition, we find that the strength of the shortest path alignment is non-monotonic with respect to the average degree of the SRGG. Based on these observations, we establish the conditions under which the alignment of shortest paths may be sufficiently strong to allow the identification of shortest path nodes based on their proximity to geodesic curves. We show that in partially observable networks with uncertain node positions, our geometric approach can outperform network-based shortest-path algorithms. In practical settings, our findings may have applications to navigation, wireless routing, and flow characterization in infrastructure networks.

physics.soc-ph↗

arXiv2025

Corrigendum to "Degree-Based Approximations for Network Reliability Polynomials". Comment on J. Complex Networks 2025, 13, cnaf001

Our original paper \cite{VanMieghem2025} described the stochastic approximation $\overline{rel}_G(p)=\bigl[1-ϕ_D(1-p)\bigr]^{N}$ in \cite[eq. (2.2)]{VanMieghem2025} and the first-order approximation $(R_1)_G(p)=\prod_{i=1}^{N}\!\bigl[1-(1-p)^{d_i}\bigr]$ in \cite[eq. (4.1)]{VanMieghem2025} as upper bounds for the all-terminal reliability polynomial $rel_G(p)$. The present corrigendum clarifies that the unique upper bound is $\Pr[\hat D_{\min}\geq 1]$, which is difficult to compute exactly, because we must account for correlated node-isolation events. Both the stochastic approximation $\overline{rel}_G$ and the first-order approximation $(R_1)_G$ ignore those correlations, assume independence and, consequently, do not always upperbound $rel_G(p)$ as stated previously. The complete graph $K_{3}$ is a counterexample, where both approximations lie below the exact reliability polynomial $rel_{K_3}(p)$, illustrating that they are not upper bounds. Moreover, as claimed in \cite{VanMieghem2025}, the first-order approximation $(R_1)_G$ is not always more accurate than the stochastic approximation $\overline{rel}_G$. We show by an example that the relative accuracy of the stochastic approximation $\overline{rel}_G$ and the first-order approximation $(R_1)_G$ varies with the graph $G$ and the link operational probability $p$. }{network robustness, node failure, probabilistic graph, reliability polynomial

physics.soc-ph↗

arXiv2025

Node Reliability: Approximation, Upper Bounds, and Applications to Network Robustness

This paper discusses the reliability of a graph in which the links are perfectly reliable but the nodes may fail with certain probability p. Calculating graph node reliability is an NP-Hard problem. We introduce an efficient and accurate Monte Carlo method and a stochastic approximation for the node reliability polynomial based solely on the degree distribution. We provide the formulas for the node reliability polynomial of both Erdos-Renyi graphs and Random Geometric graphs. The phase transition in the node reliability of Erdos-Renyi graphs is also discussed. Additionally, we propose two increasingly accurate upper bounds for the node reliability polynomial solely based on the graph's degree distributions. The advantages and disadvantages of these two upper bounds are thoroughly compared. Beyond the computation of node reliability polynomials, we also estimate the number of cut sets and present a solution to the reliability-based network enhancement problem.

eess.SY↗

arXiv2025

The Laplacian matrix of weighted threshold graphs

Threshold graphs are generated from one node by repeatedly adding a node that links to all existing nodes or adding a node without links. In the weighted threshold graph, we add a new node in step $i$, which is linked to all existing nodes by a link of weight $w_i$. In this work, we consider the set ${\cal A}_N$ that contains all Laplacian matrices of weighted threshold graphs of order $N$. We show that ${\cal A}_N$ forms a commutative algebra. Using this, we find a common basis of eigenvectors for the matrices in ${\cal A}_N$. It follows that the eigenvalues of each matrix in ${\cal A}_N$ can be represented as a linear transformation of the link weights. In addition, we prove that, if there are just three or fewer different weights, two weighted threshold graphs with the same Laplacian spectrum must be isomorphic.

math.CO↗

arXiv2025

Some Laplacian eigenvalues can be computed by matrix perturbation

Based on matrix perturbation theory, closed-form analytic expansions are studied for a Laplacian eigenvalue of an undirected, possibly weighted graph, which is close to a unique degree in that graph. An approximation is presented to provide an analytic estimate of a Laplacian eigenvalue and complements bounds on Laplacian eigenvalues in spectral graph theory. Then, we apply the Euler summation and numerically analyze how the structure of graphs influences the convergence of the corresponding Euler series. Moreover, we obtain the explicit form of the perturbation Taylor series and its Euler series of almost regular graphs in which only one node has a unique degree and all remaining nodes have the same degree. We find that the Euler series possesses a superior convergence range than the perturbation Taylor series for almost regular graphs.

math.SP↗

arXiv2025

The spectral degree exponent of a graph

We propose the spectral degree exponent as a novel graph metric. Although Hofmeister \cite{HofmeisterThesis} has studied the same metric, we generalise Hofmeister's work to weighted graphs. We provide efficient iterative formulas and bounds for the spectral degree exponent and provide highly accurate asymptotic expansions for the spectral degree exponent for several families of graphs. Furthermore, we uncover a close relation between the spectral degree exponent and the well-known degree assortativity, by showing high correlations between the two metrics in all small graphs, several random graph models and many real-world graphs.

math.CO↗

arXiv2024

Generating Temporal Contact Graphs Using Random Walkers

We study human mobility networks through timeseries of contacts between individuals. Our proposed Random Walkers Induced temporal Graph (RWIG) model generates temporal graph sequences based on independent random walkers that traverse an underlying graph in discrete time steps. Co-location of walkers at a given node and time defines an individual-level contact. RWIG is shown to be a realistic model for temporal human contact graphs, which may place RWIG on a same footing as the Erdos-Renyi (ER) and Barabasi-Albert (BA) models for fixed graphs. Moreover, RWIG is analytically feasible: we derive closed form solutions for the probability distribution of contact graphs.

cs.SI↗

arXiv2024

Transition from time-variant to static networks: timescale separation in NIMFA SIS epidemics

We extend the N-Intertwined Mean-Field Approximation (NIMFA) for the Susceptible-Infectious-Susceptible (SIS) epidemiological process to time-varying networks. Processes on time-varying networks are often analysed under the assumption that the process and network evolution happen on different timescales. This approximation is called timescale separation. We investigate timescale separation between disease spreading and topology updates of the network. We introduce the transition times $\mathrm{\underline{T}}(r)$ and $\mathrm{\overline{T}}(r)$ as the boundaries between the intermediate regime and the annealed (fast changing network) and quenched (static network) regimes, respectively, for a fixed accuracy tolerance $r$. By analysing the convergence of static NIMFA processes, we analytically derive upper and lower bounds for $\mathrm{\overline{T}}(r)$. Our results provide insights/bounds on the time of convergence to the steady state of the static NIMFA SIS process. We show that, under our assumptions, the upper-transition time $\mathrm{\overline{T}}(r)$ is almost entirely determined by the basic reproduction number $R_0$ of the network. The value of the upper-transition time $\mathrm{\overline{T}}(r)$ around the epidemic threshold is large, which agrees with the current understanding that some real-world epidemics cannot be approximated with the aforementioned timescale separation.

math.DS↗

arXiv2023

Analytic solution of Markovian epidemics without re-infections on heterogeneous networks

Most epidemic processes on networks can be modelled by a compartmental model, that specifies the spread of a disease in a population. The corresponding compartmental graph describes how the viral state of the nodes (individuals) changes from one compartment to another. If the compartmental graph does not contain directed cycles (e.g. the famous SIR model satisfies this property), then we provide an analytic, closed-form solution of the continuous-time Markovian compartmental model on heterogeneous networks. The eigenvalues of the Markovian process are related to cut sets in the contact graph between nodes with different viral states. We illustrate our finding by analytically solving the continuous-time Markovian SI and SIR processes on heterogeneous networks. We show that analytic extensions to e.g. non-Markovian dynamics, temporal networks, simplicial contagion and more advanced compartmental models are possible. Our exact and explicit formula contains sums over all paths between two states in the SIR Markov graph, which prevents the computation of the exact solution for arbitrary large graphs.

physics.soc-ph↗

arXiv2023

Time Dynamics of the Dutch Municipality Network

Based on data sets provided by Statistics Netherlands and the International Institute of Social History, we investigate the Dutch municipality merging process and the survivability of municipalities over the period 1830-2019. We examine the dynamics of the population and area per municipality and how their distributions evolved during the researched period. We apply a Network Science approach, where each node represents a municipality and the links represent the geographical interconnections between adjacent municipalities via roads, railways, bridges or tunnels which were available in each specific yearly network instance. Over the researched period, we find that the distributions of the logarithm of both the population and area size closely follow a normal and a logistic distribution respectively. The tails of the population distributions follow a power-law distribution, a phenomenon observed in community structures of many real-world networks. The dynamics of the area distribution are mainly determined by the merging process, while the population distribution is also driven by the natural population growth and migration across the municipality network. Finally, we propose a model of the Dutch Municipality Network that captures population increase, population migration between municipalities and the process of municipality merging. Our model allows for predictions of the population and area distributions over time.

physics.soc-ph↗

arXiv2023

Topological properties and organizing principles of semantic networks

Interpreting natural language is an increasingly important task in computer algorithms due to the growing availability of unstructured textual data. Natural Language Processing (NLP) applications rely on semantic networks for structured knowledge representation. The fundamental properties of semantic networks must be taken into account when designing NLP algorithms, yet they remain to be structurally investigated. We study the properties of semantic networks from ConceptNet, defined by 7 semantic relations from 11 different languages. We find that semantic networks have universal basic properties: they are sparse, highly clustered, and many exhibit power-law degree distributions. Our findings show that the majority of the considered networks are scale-free. Some networks exhibit language-specific properties determined by grammatical rules, for example networks from highly inflected languages, such as e.g. Latin, German, French and Spanish, show peaks in the degree distribution that deviate from a power law. We find that depending on the semantic relation type and the language, the link formation in semantic networks is guided by different principles. In some networks the connections are similarity-based, while in others the connections are more complementarity-based. Finally, we demonstrate how knowledge of similarity and complementarity in semantic networks can improve NLP algorithms in missing link inference.

cs.CL↗

arXiv2023

Reporting delays: a widely neglected impact factor in COVID-19 forecasts

Epidemic forecasts are only as good as the accuracy of epidemic measurements. Is epidemic data, particularly COVID-19 epidemic data, clean and devoid of noise? Common sense implies the negative answer. While we cannot evaluate the cleanliness of the COVID-19 epidemic data in a holistic fashion, we can assess the data for the presence of reporting delays. In our work, through the analysis of the first COVID-19 wave, we find substantial reporting delays in the published epidemic data. Motivated by the desire to enhance epidemic forecasts, we develop a statistical framework to detect, uncover, and remove reporting delays in the infectious, recovered, and deceased epidemic time series. Our framework can uncover and analyze reporting delays in 8 regions significantly affected by the first COVID-19 wave. Further, we demonstrate that removing reporting delays from epidemic data using our statistical framework may decrease the error in epidemic forecasts. While our statistical framework can be used in combination with any epidemic forecast method that intakes infectious, recovered, and deceased data, to make a basic assessment, we employed the classical SIRD epidemic model. Our results indicate that the removal of reporting delays from the epidemic data may decrease the forecast error by up to 50. We anticipate that our framework will be indispensable in the analysis of novel COVID-19 strains and other existing or novel infectious diseases.

q-bio.PE↗

arXiv2022

Linear Clustering Process on Networks

We propose a linear clustering process on a network consisting of two opposite forces: attraction and repulsion between adjacent nodes. Each node is mapped to a position on a one-dimensional line. The attraction and repulsion forces move the nodal position on the line, depending on how similar or different the neighbourhoods of two adjacent nodes are. Based on each node position, the number of clusters in a network, together with each node's cluster membership, is estimated. The performance of the proposed linear clustering process is benchmarked on synthetic networks against widely accepted clustering algorithms such as modularity, the Louvain method and the non-back tracking matrix. The proposed linear clustering process outperforms the most popular modularity-based methods, such as the Louvain method, while possessing a comparable computational complexity.

cs.SI↗

arXiv2022

Number of paths in a graph

The $k$-th power of the adjacency matrix of a simple undirected graph represents the number of walks with length $k$ between pairs of nodes. As a walk where no node repeats, a path is a walk where each node is only visited once. The set of paths constitutes a relatively small subset of all possible walks. We introduce three types of walks, representing subsets of all possible walks. Considered types of walks allow for deriving an analytic solution for the number of paths of a certain length between node pairs in a matrix form. Depending on the path length, different approaches possess the lowest computational complexity. We also propose a recursive algorithm for determining all paths in a graph, which can be generalised to directed (un)weighted networks.

math.CO↗

arXiv2021

The Mittag-Leffler function

We review the function theoretical properties of the Mittag-Leffler function $E_{a,b}\left( z\right) $ in a self-contained manner, but also add new results; more than half is new!

math.FA↗

arXiv2021

Two-population SIR model and strategies to reduce mortality in pandemics

Despite many studies on the transmission mechanism of the Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), it remains still challenging to efficiently reduce mortality. In this work, we apply a two-population Susceptible-Infected-Removed (SIR) model to investigate the COVID-19 spreading when contacts between elderly and non-elderly individuals are reduced due to the high mortality risk of elderly people. We discover that the reduction of connections between two populations can delay the death curve but cannot well reduce the final mortality. We propose a merged SIR model, which advises elderly individuals to interact less with their non-elderly connections at the initial stage but interact more with their non-elderly relationships later, to reduce the final mortality. Finally, immunizing elderly hub individuals can also significantly decrease mortality.

physics.soc-ph↗

arXiv2021

Clustering for epidemics on networks: a geometric approach

Infectious diseases typically spread over a contact network with millions of individuals, whose sheer size is a tremendous challenge to analysing and controlling an epidemic outbreak. For some contact networks, it is possible to group individuals into clusters. A high-level description of the epidemic between a few clusters is considerably simpler than on an individual level. However, to cluster individuals, most studies rely on equitable partitions, a rather restrictive structural property of the contact network. In this work, we focus on Susceptible-Infected-Susceptible (SIS) epidemics, and our contribution is threefold. First, we propose a geometric approach to specify all networks for which an epidemic outbreak simplifies to the interaction of only a few clusters. Second, for the complete graph and any initial viral state vectors, we derive the closed-form solution of the nonlinear differential equations of the N-Intertwined Mean-Field Approximation (NIMFA) of the SIS process. Third, by relaxing the notion of equitable partitions, we derive low-complexity approximations and bounds for epidemics on arbitrary contact networks. Our results are an important step towards understanding and controlling epidemics on large networks.

physics.soc-ph↗