arXiv subjects

Bernard Mans

Publications and source records attributed to Bernard Mans.

At least 19 recordsLinked to original sources

arXiv2026

Online Treasure Hunt in Vertex-Permuted Dynamic Rings

We study the problem of treasure hunt by a group of $k \geq 1$ agents in vertex-permuted dynamic rings (VP). In this model, the $n$ vertices remain on a ring but are permuted at each time step. We first show that treasure hunt is impossible for any $k \leq n-3$ agents, if there are no restrictions on the sequence of permutations used in the dynamic ring. We then study the $VP(\delta)$ setting, in which for every pair $i, j$ of vertices, the edge $(i, j)$ is guaranteed to appear within $\delta$ steps. We show that the class $VP(\delta)$ is feasible only for $\delta \geq \left\lceil \frac{n-1}{2}\right\rceil$. For the one-agent case, we show a tight bound of $\Theta(\delta n)$ on the worst-case search time as well as competitive ratio of any online algorithm for treasure hunt, provided $\delta \geq 2n$. We then give an optimal algorithm for $k$ agents, thereby showing that $k$ agents can obtain a speedup of $k$ on the worst-case search time. Finally, in the R-VP setting, in which in every step, the vertices are arranged as a ring according to a random permutation, we show that treasure hunt takes expected $\Theta(n)$ steps against an oblivious adversary and $\Theta(n \log n)$ steps against an adaptive adversary.

cs.DC↗

arXiv2026

Asymptotics of Parking Search in Hyperfractal Networks

We study the asymptotic behaviour of the distance to the first available parking slot in a recursive Manhattan street network endowed with a hyperfractal intensity structure, where slot-release events occur according to Poisson processes along the streets. We establish, by analysing the associated self-similar harmonic sums via Mellin-transform asymptotics, a power-law decay of the expected distance as the total intensity grows, with exponent equal to the inverse of the hyperfractal dimension. In particular, the scaling exponent depends only on the large-scale geometry of the network. We further prove that this exponent is robust under random multiplicative modulations of the street intensities: mild stochastic heterogeneity affects only the multiplicative constant. Similar scaling behaviour holds for the variance, the number of turns before parking, and for a jump-over variant of the search strategy.

math.PR↗

arXiv2023

Functional graphs of families of quadratic polynomials

We study functional graphs generated by several quadratic polynomials, acting simultaneously on a finite field of odd characteristic. We obtain several results about the number of leaves in such graphs. In particular, in the case of graphs generated by three polynomials, we relate the distribution of leaves to the Sato-Tate distribution of Frobenius traces of elliptic curves. We also present extensive numerical results which we hope may shed some light on the distribution of leaves for larger families of polynomials.

math.NT↗

arXiv2022

Balanced Allocation on Hypergraphs

We consider a variation of balls-into-bins which randomly allocates $m$ balls into $n$ bins. Following Godfrey's model (SODA, 2008), we assume that each ball $t$, $1\le t\le m$, comes with a hypergraph $\mathcal{H}^{(t)}=\{B_1,B_2,\ldots,B_{s_t}\}$, and each edge $B\in\mathcal{H}^{(t)}$ contains at least a logarithmic number of bins. Given $d\ge 2$, our $d$-choice algorithm chooses an edge $B\in \mathcal{H}^{(t)}$, uniformly at random, and then chooses a set $D$ of $d$ random bins from the selected edge $B$. The ball is allocated to a least-loaded bin from $D$, with ties are broken randomly. We prove that if the hypergraphs $\mathcal{H}^{(1)},\ldots, \mathcal{H}^{(m)}$ satisfy a \emph{balancedness} condition and have low \emph{pair visibility}, then after allocating $m=Θ(n)$ balls, the maximum number of balls at any bin, called the \emph{maximum load}, is at most $\log_d\log n+O(1)$, with high probability. The balancedness condition enforces that bins appear almost uniformly within the hyperedges of $\mathcal{H}^{(t)}$, $1\le t\le m$, while the pair visibility condition measures how frequently a pair of bins is chosen during the allocation of balls. Moreover, we establish a lower bound for the maximum load attained by the balanced allocation for a sequence of hypergraphs in terms of pair visibility, showing the relevance of the visibility parameter to the maximum load. In Godfrey's model, each ball is forced to probe all bins in a randomly selected hyperedge and the ball is then allocated in a least-loaded bin. Godfrey showed that if each $\mathcal{H}^{(t)}$, $1\le t\le m$, is balanced and $m=O(n)$, then the maximum load is at most one, with high probability. However, we apply the power of $d$ choices paradigm, and only query the load information of $d$ random bins per ball, while achieving very slow growth in the maximum load.

cs.DS↗

arXiv2021

COVID-19 vaccination strategies on dynamic networks

Coronavirus disease (COVID-19), which was caused by SARS-CoV-2, has become a global public health concern. A great proportion of the world needs to be vaccinated in order to stop the rapid spread of the disease. In addition to prioritising vulnerable sections of the population to receive the vaccine, an ideal degree-based vaccination strategy uses fine-grained contact networks to prioritise vaccine recipients. This strategy is costly and impractical due to the enormous amount of specific contact information needed. It also does not capture indirect famine or aerosol-based transmission. We recently proposed a new vaccination strategy called Individual's Movement-based Vaccination (IMV), which takes into account both direct and indirect transmission and is based on the types of places people visit. IMV was shown to be cost-efficient in the case of influenza-like diseases. This paper studies the application of IMV to COVID-19 using its documented transmission parameters. We conduct large scale computer simulations based on a city-wide empirical mobility dataset to evaluate the performance and practicability of the strategy. Results show that the proposed strategy achieves nearly five times the efficiency of random vaccination and performs comparably to the degree-based strategy, while significantly reducing the data collection requirements.

cs.SI↗

arXiv2021

Connecting flying backhauls of UAVs to enhance vehicular networks with fixed 5G NR infrastructure

This paper investigates moving networks of Unmanned Aerial Vehicles (UAVs), such as drones, as one of the innovative opportunities brought by the 5G. With a main purpose to extend connectivity and guarantee data rates, the drones require hovering locations due to limitations such as flight time and coverage surface. We provide analytic bounds on the requirements in terms of connectivity extension for vehicular networks served by fixed Enhanced Mobile BroadBand (eMBB) infrastructure, where both vehicular networks and infrastructures are modeled using stochastic and fractal geometry as a model for urban environment. We prove that assuming $n$ mobile nodes (distributed according to a hyperfractal distribution of dimension $d_F$) and an average of $ρ$ Next Generation NodeB (gNBs), distributed like an hyperfractal of dimension $d_r$ if $ρ=n^θ$ with $θ>d_r/4$ and letting $n$ tending to infinity (to reflect megalopolis cities), then the average fraction of mobile nodes not covered by a gNB tends to zero like $O\left(n^{-\frac{(d_F-2)}{d_r}(2θ-\frac{d_r}{2})}\right)$. Interestingly, we then prove that the average number of drones, needed to connect each mobile node not covered by gNBs is comparable to the number of isolated mobile nodes. We complete the characterisation by proving that when $θ<d_r/4$ the proportion of covered mobile nodes tends to zero. We provide insights on the intelligent placement of the "garage of drones", the home location of these nomadic infrastructure nodes, such as to minimize what we call the "flight-to-coverage time". We provide a fast procedure to select the relays that will be garages (and store drones) in order to minimize the number of garages and minimize the delay. Finally we confirm our analytical results using simulations carried out in Matlab.

cs.DC↗

arXiv2021

Characterizing the Energy Trade-Offs of End-to-End Vehicular Communications using an Hyperfractal Urban Modelling

We characterize trade-offs between the end-to-end communication delay and the energy in urban vehicular communications with infrastructure assistance. Our study exploits the self-similarity of the location of communication entities in cities by modeling them with an innovative model called "hyperfractal". We show that the hyperfractal model can be extended to incorporate road-side infrastructure and provide stochastic geometry tools to allow a rigorous analysis. We compute theoretical bounds for the end-to-end communication hop count considering two different energy-minimizing goals: either total accumulated energy or maximum energy per node. We prove that the hop count for an end-to-end transmission is bounded by $O(n^{1-α/(d_F-1)})$ where $α<1$ and $d_F>2$ is the fractal dimension of the mobile nodes process. This proves that for both constraints the energy decreases as we allow choosing routing paths of higher length. The asymptotic limit of the energy becomes significantly small when the number of nodes becomes asymptotically large. A lower bound on the network throughput capacity with constraints on path energy is also given. We show that our model fits real deployments where open data sets are available. The results are confirmed through simulations using different fractal dimensions in a Matlab simulator.

cs.DC↗

arXiv2020

Tight Analysis of Asynchronous Rumor Spreading in Dynamic Networks

The asynchronous rumor algorithm spreading propagates a piece of information, the so-called rumor, in a network. Starting with a single informed node, each node is associated with an exponential time clock with rate $1$ and calls a random neighbor in order to possibly exchange the rumor. Spread time is the first time when all nodes of a network are informed with high probability. We consider spread time of the algorithm in any dynamic evolving network, $\mathcal{G}=\{G^{(t)}\}_{t=0}^{\infty}$, which is a sequence of graphs exposed at discrete time step $t=0,1\ldots$. We observe that besides the expansion profile of a dynamic network, the degree distribution of nodes over time effect the spread time. We establish upper bounds for the spread time in terms of graph conductance and diligence. For a given connected simple graph $G=(V,E)$, the diligence of cut set $E(S, \overline{S})$ is defined as $ρ(S)=\min_{\{u,v\}\in E(S,\overline{S})}\max\{\bar{d}/d_u, \bar{d}/d_v\}$ where $d_u$ is the degree of $u$ and $\bar{d}$ is the average degree of nodes in the one side of the cut with smaller volume (i.e., ${\mathtt{vol}}{(S)}=\sum_{u\in S}d_u$). The diligence of $G$ is also defined as $ρ(G)=\min_{ \emptyset\neq S\subset V}ρ(S)$. We show that the spread time of the algorithm in $\mathcal{G}$ is bounded by $T$, where $T$ is the first time that $\sum_{t=0}^TΦ(G^{(t)})\cdotρ(G^{(t)})$ exceeds $C\log n$, where $Φ(G^{(t)})$ denotes the conductance of $G^{(t)}$ and $C$ is a specified constant. We also define the absolute diligence as $\overlineρ(G)=\min_{\{u,v\}\in E}\max\{1/d_u,1/d_v\}$ and establish upper bound $T$ for the spread time in terms of absolute diligence, which is the first time when $\sum_{t=0}^T\lceilΦ(G^{(t)})\rceil\cdot \overlineρ(G^{(t)})\ge 2n$. We present dynamic networks where the given upper bounds are almost tight.

cs.DS↗

arXiv2020

Vaccination strategies on dynamic networks with indirect transmission links and limited contact information

Infectious diseases are still a major global burden for modern society causing 13 million deaths annually. One way to reduce the morbidity and mortality rates from infectious diseases is through preventative or targeted vaccinations. Current vaccination strategies, however, rely on the highly specific individual contact information that is difficult and costly to obtain, in order to identify influential spreading individuals. Current approaches also focus only on direct contacts between individuals for spreading, and disregard indirect transmission where a pathogen can spread between one infected individual and one susceptible individual that visit the same location within a short time-frame without meeting. This paper presents a novel vaccination strategy that relies on coarse-grained contact information, both direct and indirect, that can be easily and efficiently collected. Rather than tracking exact contact degrees of individuals, our strategy uses the types of places people visit to estimate a range of contact degrees for individuals, considering both direct and indirect contacts. We conduct extensive simulations to evaluate the performance of our strategy in comparison to the state of the art's vaccination strategies. Results show that our strategy achieves comparable performance to the oracle approach and outperforms all existing strategies when considering indirect links.

cs.SI↗

arXiv2020

On the equational graphs over finite fields

In this paper, we generalize the notion of functional graph. Specifically, given an equation $E(X,Y) = 0$ with variables $X$ and $Y$ over a finite field $\mathbb{F}_q$ of odd characteristic, we define a digraph by choosing the elements in $\mathbb{F}_q$ as vertices and drawing an edge from $x$ to $y$ if and only if $E(x,y)=0$. We call this graph as equational graph. In this paper, we study the equational graphs when choosing $E(X,Y) = (Y^2 - f(X))(λY^2 - f(X))$ with $f(X)$ a polynomial over $\mathbb{F}_q$ and $λ$ a non-square element in $\mathbb{F}_q$. We show that if $f$ is a permutation polynomial over $\mathbb{F}_q$, then every connected component of the graph has a Hamiltonian cycle. Moreover, these Hamiltonian cycles can be used to construct balancing binary sequences. By making computations for permutation polynomials $f$ of low degree, it appears that almost all these graphs are strongly connected, and there are many Hamiltonian cycles in such a graph if it is connected.

math.CO↗

arXiv2019

Generating dynamic contact graphs with indirect links

Graph models are widely used to study diffusion processes in contact networks. Recent data-driven research has highlighted the significance of indirect links, where interactions are possible when two nodes visit the same place at different times (SPDT), in determining network structure and diffusion dynamics. However, how to generate dynamic graphs with indirect links for modeling diffusion remains an unsolved challenge. Here, we present a dynamic contact graph model for generating contact networks with direct and indirect links. Our model introduces the concept of multiple concurrently active copies of a node for capturing indirect transmission links. The SPDT graph model builds on activity driven time-varying network modelling for generating dynamic contact networks using simple statistical distributions. This model is fitted with a large city-scale empirical dataset using maximum likelihood estimation methods. Finally, the performance of the model is evaluated by analysing the capability of capturing the network properties observed in empirical graphs constructed using the location updates of a social networking app and simulating SPDT diffusion processes. Our results show that, in comparison to current graph models that only include direct links, our graph model with indirect links match empirical network properties and diffusion dynamics much more closely.

cs.SI↗

arXiv2019

Blockchain moderated by empty blocks to reduce the energetic impact of crypto-moneys

While cryptocurrencies and blockchain applications continue to gain popularity, their energy cost is evidently becoming unsustainable. In most instances, the main cost comes from the required amount of energy for the Proof-of-Work, and this cost is inherent to the design. In addition, useless costs from discarded work (e.g., the so-called Forks) and lack of scalability (in number of users and in rapid transactions) limit their practical effectiveness. In this paper, we present an innovative scheme which eliminates the nonce and thus the burden of the Proof-of-Work which is the main cause of the energy waste in cryptocurrencies such as Bitcoin. We prove that our scheme guarantees a tunable and bounded average number of simultaneous mining whatever the size of the population in competition, thus by making the use of nonce-based techniques unnecessary, achieves scalability without the cost of consuming a large volume of energy. The technique used in the proof of our scheme is based on the analogy of the analysis of a green leader election. The additional difference with Proof-of-Work schemes (beyond the suppression of the nonce field that is triggering most of the waste), is the introduction of (what we denote as) "empty blocks" which aim are to call regular blocks following a staircase set of values. Our scheme reduces the risk of Forks and provides tunable scalability for the number of users and the speed of block generation. We also prove using game theoretical analysis that our scheme is resilient to unfair competitive investments (e.g., "51 percent" attack) and block nursing.

cs.CR↗

arXiv2019

Information Dissemination Speed in Delay Tolerant Urban Vehicular Networks in a Hyperfractal Setting

This paper studies the fundamental communication properties of urban vehicle networks by exploiting the self-similarity and hierarchical organization of modern cities. We use an innovative model called "hyperfractal" that captures the self-similarities of both the traffic and vehicle locations but avoids the extremes of regularity and randomness. We use analytical tools to derive theoretical upper and lower bounds for the information propagation speed in an urban delay tolerant network (i.e., a network that is disconnected at all time, and thus uses a store-carry-and-forward routing model). We prove that the average broadcast time behaves as $n^{1-δ}$ times a slowly varying function, where $δ$ depends on the precise fractal dimension. Furthermore, we show that the broadcast speedup is due in part to an interesting self-similar phenomenon, that we denote as {\em information teleportation}. This phenomenon arises as a consequence of the topology of the vehicle traffic, and triggers an acceleration of the broadcast time. We show that our model fits real cities where open traffic data sets are available. We present simulations confirming the validity of the bounds in multiple realistic settings, including scenarios with variable speed, using both QualNet and a discrete-event simulator in Matlab.

cs.NI↗

arXiv2019

Indirect interactions influence contact network structure and diffusion dynamics

Interaction patterns at the individual level influence the behaviour of diffusion over contact networks. Most of the current diffusion models only consider direct interactions among individuals to build underlying infectious items transmission networks. However, delayed indirect interactions, where a susceptible individual interacts with infectious items after the infected individual has left the interaction space, can also cause transmission events. We define a diffusion model called the same place different time transmission (SPDT) based diffusion that considers transmission links for these indirect interactions. Our SPDT model changes the network dynamics where the connectivity among individuals varies with the decay rates of link infectivity. We investigate SPDT diffusion behaviours by simulating airborne disease spreading on data-driven contact networks. The SPDT model significantly increases diffusion dynamics (particularly for networks with low link densities where indirect interactions create new infection pathways) and is capable of producing realistic disease reproduction number. Our results show that the SPDT model is significantly more likely to lead to outbreaks compared to current diffusion models with direct interactions. We find that the diffusion dynamics with including indirect links are not reproducible by the current models, highlighting the importance of the indirect links for predicting outbreaks.

cs.SI↗

arXiv2018

A Graph Model with Indirect Co-location Links

Graph models are widely used to analyse diffusion processes embedded in social contacts and to develop applications. A range of graph models are available to replicate the underlying social structures and dynamics realistically. However, most of the current graph models can only consider concurrent interactions among individuals in the co-located interaction networks. However, they do not account for indirect interactions that can transmit spreading items to individuals who visit the same locations at different times but within a certain time limit. The diffusion phenomena occurring through direct and indirect interactions is called same place different time (SPDT) diffusion. This paper introduces a model to synthesize co-located interaction graphs capturing both direct interactions, where individuals meet at a location, and indirect interactions, where individuals visit the same location at different times within a set timeframe. We analyze 60 million location updates made by 2 million users from a social networking application to characterize the graph properties, including the space-time correlations and its time evolving characteristics, such as bursty or ongoing behaviors. The generated synthetic graph reproduces diffusion dynamics of a realistic contact graph, and reduces the prediction error by up to 82% when compare to other contact graph models demonstrating its potential for forecasting epidemic spread.

cs.SI↗

arXiv2018

Impact of Indirect Contacts in Emerging Infectious Disease on Social Networks

Interaction patterns among individuals play vital roles in spreading infectious diseases. Understanding these patterns and integrating their impact in modeling diffusion dynamics of infectious diseases are important for epidemiological studies. Current network-based diffusion models assume that diseases transmit through interactions where both infected and susceptible individuals are co-located at the same time. However, there are several infectious diseases that can transmit when a susceptible individual visits a location after an infected individual has left. Recently, we introduced a diffusion model called same place different time (SPDT) transmission to capture the indirect transmissions that happen when an infected individual leaves before a susceptible individual's arrival along with direct transmissions. In this paper, we demonstrate how these indirect transmission links significantly enhance the emergence of infectious diseases simulating airborne disease spreading on a synthetic social contact network. We denote individuals having indirect links but no direct links during their infectious periods as hidden spreaders. Our simulation shows that indirect links play similar roles of direct links and a single hidden spreader can cause large outbreak in the SPDT model which causes no infection in the current model based on direct link. Our work opens new direction in modeling infectious diseases.

cs.SI↗

arXiv2017

On Functional Graphs of Quadratic Polynomials

We study functional graphs generated by quadratic polynomials over prime fields. We introduce efficient algorithms for methodical computations and provide the values of various direct and cumulative statistical parameters of interest. These include: the number of connected functional graphs, the number of graphs having a maximal cycle, the number of cycles of fixed size, the number of components of fixed size, as well as the shape of trees extracted from functional graphs. We particularly focus on connected functional graphs, that is, the graphs which contain only one component (and thus only one cycle). Based on the results of our computations, we formulate several conjectures highlighting the similarities and differences between these functional graphs and random mappings.

math.NT↗

arXiv2015

Functional Graphs of Polynomials over Finite Fields

Given a function $f$ in a finite field ${\mathbb F}_q$ of $q$ elements, we define the functional graph of $f$ as a directed graph on $q$ nodes labelled by the elements of ${\mathbb F}_q$ where there is an edge from $u$ to $v$ if and only if $f(u) = v$. We obtain some theoretic estimates on the number of non-isomorphic graphs generated by all polynomials of a given degree. We then develop a simple and practical algorithm to test the isomorphism of quadratic polynomials that has linear memory and time complexities. Furthermore, we extend this isomorphism testing algorithm to the general case of functional graphs, and prove that, while its time complexity increases only slightly, its memory complexity remains linear. We exploit this algorithm to provide an upper bound on the number of functional graphs corresponding to polynomials of degree $d$ over ${\mathbb F}_q$. Finally, we present some numerical results and compare function graphs of quadratic polynomials with those generated by random maps and pose interesting new problems.

math.NT↗