Lattice structures play a central role in spectral graph theory, offering analytical insight into diffusion, synchronization, and transport processes on regular discrete spaces. While their spectral properties are completely characterized in the classical graph setting, an extension to hypergraphs, where interactions involve more than two nodes, remains largely unexplored in the matrix-based formulation. In this work, we generalize the notion of a lattice to the hypergraph framework and study its Laplacian spectra under two alternative definitions: the clique Laplacian, obtained through pairwise projection, and the hyperedge-based Laplacian, defined via normalized hyperedge incidences. For both definitions, we derive the corresponding Laplacian matrices, analyze their eigenvalue spectra, and discuss how they reflect the underlying topological and dynamical structure of the hyperlattice. Our main result is a theorem giving a full spectral characterization in the periodic case, together with a reflective open-boundary analogue whose spectrum retains a separable trigonometric structure. The obtained eigenvalues are expressed explicitly in terms of the hyperedge size, the number of directional families, and the lattice side length, thereby capturing how the geometry of higher-order interactions shapes the spectral structure.

Improving both safety and efficiency in traffic systems remains a central challenge, especially in the context of increasingly mixed traffic with autonomous and human-driven vehicles. Although many indicators assess either safety or efficiency, few capture both dimensions simultaneously at a microscopic level. This paper introduces two new traffic indicators: Efficiency Index (EI), which measures local speed and spacing regularity, and Safety and Efficiency Index (SEI), which combines EI with a time-to-collision safety component. A tunable version, SEMI, allows greater sensitivity to risk by penalizing safety-critical interactions. Using real-world traffic flow data and simulation via SUMO, we tested these indicators in varying penetration rates of autonomous vehicles. The results show that while AVs improve efficiency across the board, the safety gains become especially pronounced in dense traffic. These findings offer a flexible and interpretable tool for researchers and practitioners in traffic engineering, vehicle automation, and public policy. The proposed indicators can inform the design of AV control models, traffic management strategies, and infrastructure planning where safety-efficiency trade-offs must be explicitly addressed.

In this paper we introduce a spectrum-preserving relation between graphs with loops and graphs without loops. Our approach generalizes the spectral results obtained on (m,k)- stars to a wider class of graphs, namely (m,k,s)- stars with or without loops. The proposed equivalence of the two classes of graphs allows to study pseudographs as simple graphs, by extending the techniques developed for simple graphs to pseudographs, without losing information, and it could be relevant for applications of graph theory to complex systems physics and neural networks. Finally, in order to make the demonstrated results easily applicable, we have provided a public Github repository where Python code that allows straightforward implementations of the outcomes is made available.

This work focuses on the potential impacts of the autonomous vehicles in a mixed traffic condition represented in traffic simulator Simulation of Urban MObility (SUMO) with real traffic flow. Specifically, real traffic flow and speed data collected in 2002 and 2019 in Gothenburg were used to simulate daily flow variation in SUMO. In order to predict the most likely drawbacks during the transition from a traffic consisting only manually driven vehicles to a traffic consisting only fully-autonomous vehicles, this study focuses on mixed traffic with different percentages of autonomous and manually driven vehicles. To realize this aim, several parameters of the car following and lane change models of autonomous vehicles are investigated in this paper. Along with the fundamental diagram, the number of lane changes and the number of conflicts are analyzed and studied as measures for improving road safety and efficiency. The study highlights that the autonomous vehicles' features that improve safety and efficiency in 100% autonomous and mixed traffic are different, and the ability of autonomous vehicles to switch between mixed and autonomous driving styles, and vice versa depending on the scenario, is necessary.

The spectral theory of the normalized Laplacian for chemical hypergraphs is further investigated. The signless normalized Laplacian is introduced and it is shown that its spectrum for classical hypergraphs coincides with the spectrum of the normalized Laplacian for bipartite chemical hypergraphs. Furthermore, the spectra of special families of hypergraphs are established.

The safety of heterogeneous traffic is a vital topic in the oncoming era of autonomous vehicles (AVs). The cooperative vehicle infrastructure system (CVIS) is considered to improve heterogeneous traffic safety by connecting and controlling AVs cooperatively, and the connected AVs are so-called connected and automated vehicles (CAVs). However, the safety impact of cooperative control strategy on the heterogeneous traffic with CAVs and human-driving vehicles (HVs) has not been well investigated. In this paper, based on the traffic simulator SUMO, we designed a typical highway scenario of on-ramp merging and adopted a cooperative control method for CAVs. We then compared the safety performance for two different heterogeneous traffic systems, i.e. AV and HV, CAV and HV, respectively, to illustrate the safety benefits of the cooperative control strategy. We found that the safety performance of the CAV and HV traffic system does not always outperform that of AV and HV. With random departSpeed and higher arrival rate, the proposed cooperative control method would decrease the conflicts significantly whereas the penetration rate is over 80%. We further investigated the conflicts in terms of the leading and following vehicle types, and found that the risk of a AV/CAV followed by a HV is twice that of a HV followed by another HV. We also considered the safety effect of communication failure, and found that there is no significant impact until the packet loss probability is greater than 30%, while communication delay’s impact on safety can be ignored according to our experiments.

The purpose of this paper is to introduce a model to study structures which are widely present in public transportation networks. We show that, through hypergraphs, one can describe these structures and investigate the relation between their spectra. To this aim, we extend the structure of (m, k)-stars on graphs to hypergraphs: the (m, k)-hyperstars on hypergraphs. Also, by giving suitable conditions on the hyperedge weights, we prove the existence of matrix eigenvalues of computable values and multiplicities, where the matrices considered are Laplacian, adjacency and transition matrices. By considering separately the case of generic hypergraphs and uniform hypergraphs, we prove that two kinds of vertex set reductions on hypergraphs with (m, k)-hyperstars are feasible, keeping the same eigenvalues with reduced multiplicity. Finally, some useful eigenvector properties are derived up to a product with a suitable matrix, and we relate these results to Fiedler spectral partitioning on the hypergraph.

In this work we propose a new cooperation strategy between connected autonomous vehicles in on-ramps merging scenarios and we implement the cut-in risk indicator (CRI) to investigate the safety effect of the proposed strategy. The new cooperation strategy considers a pair of vehicles approaching an on-ramp. The strategy then makes decisions on the target speeds/accelerations of both vehicles, possible lane changing, and a dynamic decision-making approach in order to reduce the risk during the cut-in manoeuvre. In this work, the CRI was first used to assess the risk during the merging manoeuvre. For this purpose, scenarios with penetration rates of autonomous vehicles from 20% to 100%, with step of 10%, both connected and non-connected autonomous vehicles were evaluated. As a result, on average a 35% reduction of the cut-in risk manoeuvres in connected autonomous vehicles compared to non-connected autonomous vehicles is obtained. It is shown through the analysis of probability density functions characterising the CRI distribution that the reduction is not homogeneous across all indicator values, but depends on the penetration rate and the severity of the manoeuvre.

In this work we focus on future projected impacts of the autonomous vehicles in a realistic condition representing mixed traffic. By using real flow and speed data collected in 2002 and 2019 in the city of Gothenburg, we replicated and simulated the daily flow variation in SUMO. The expansion of the city in recent years was reflected in an increase in road users, and it is reasonable to expect it will increase further. Through simulations, it was possible to project this increase and to predict how this will impact the traffic in future. Furthermore, the composition of vehicle types in the future traffic can be expected to change through the introduction of autonomous vehicles. In order to predict the most likely drawbacks during the transition from a traffic consisting only manually driven vehicles to a traffic consisting only fully-autonomous vehicles, we focus on mixed traffic with different percentages of autonomous and manually driven vehicles. To realize this aim, several parameters of the car following and lane change models of autonomous vehicles are investigated in this paper. Along with the fundamental diagram, the number of lane changes and the number of conflicts are analyzed and studied as measures for improving road safety and efficiency.

As an indicator of the stability of spectral clustering of an undirected weighted graph into k clusters, the kth spectral gap of the graph Laplacian is often considered. The kth spectral gap is characterized in this paper as an unstructured distance to ambiguity, namely as the minimal distance of the Laplacian to arbitrary symmetric matrices with vanishing kth spectral gap. As a conceptually more appropriate measure of stability, the structured distance to ambiguity of the k-clustering is introduced as the minimal distance of the Laplacian to Laplacians of graphs with the same vertices and edges but with weights that are perturbed such that the kth spectral gap vanishes. To compute a solution to this matrix nearness problem, a two-level iterative algorithm is proposed that uses a constrained gradient system of matrix differential equations in the inner iteration and a one-dimensional optimization of the perturbation size in the outer iteration. The structured and unstructured distances to ambiguity are compared on some example graphs. The numerical experiments show, in particular, that selecting the number k of clusters according to the criterion of maximal stability can lead to different results for the structured and unstructured stability indicators.

The aim of this paper is to highlight and investigate the effects of increasing presence rate of autonomous vehicles (AVs) in terms of traffic safety and traffic flow characteristics. For this purpose, using existing driver models in traffic simulator SUMO we identify and analyze those parameters that characterize and distinguish AVs' driving from manual driving in a heterogeneous traffic context. While it is essential to identify the parameters for traffic flow characteristics of heterogeneous fleets compared to homogeneous ones comprising manually driven vehicles (MV) only (i.e. current status), the safety aspects must be also accounted for. In order to combine these two fundamental aspects of heterogeneous traffic, we used a complete description of a highway driving scenario. The scenario integrates the perceptions of different type of vehicles (i.e. AV and MV) involved and the reaction times of human drivers and decision-making units of autonomous vehicles, to explore the impact of both the rate of AV presence and the perturbation in perception capabilities in highway scenarios.

This paper is an attempt to analyze the notion of ecological niche as a community of different species and of ecosystem as a set of niches in order to formulate a dynamical model for an ecosystem. Our assumption is that the concept of fitness landscape allows to model the phenotype dynamics of an ensemble of species as a stochastic process. To take into account the interaction structure of different communities in the niches and the environment we introduce an ecological fitness potential to formulate a Lotka-Volterra system which describes the evolution of a mutual ecosystem in presence of finite resources. To explicitly consider the effect of fluctuations in the numerousness of the species, we associate a master equation to the average Lotka-Volterra system and we study the conditions of existence of a detailed balance equilibrium (i.e. a thermodynamic equilibrium) for the ecosystem. The explicit solution for the equilibrium probability distribution is a multinomial negative distribution and we discuss the relation between the detailed balance condition and relative species abundance distribution in the framework of Hubbell's neutral theory. Moreover the theoretical distribution implies the existence of a correlation among the relative species distribution associated to the different communities. We use numerical simulations to illustrate the results on simple models.

The purpose of this paper is to introduce a model to study structures which are widely present in public transportation networks. We show that, through hypergraphs, one can describe these structures and investigate the relation between their spectra. To this aim, we extend the structure of $(m,k)$-stars on graphs to hypergraphs: the $(m,k)$-hyperstars on hypergraphs. Also, by giving suitable conditions on the hyperedge weights we prove the existence of matrix eigenvalues of computable values and multiplicities, where the matrices considered are Laplacian, adjacency and transition matrices. By considering separately the case of generic hypergraphs and uniform hypergraphs, we prove that two kinds of vertex set reductions on hypergraphs with $(m,k)$-hyperstar are feasible, keeping the same eigenvalues with reduced multiplicity. Finally, some useful eigenvectors properties are derived up to a product with a suitable matrix, and we relate these results to Fiedler spectral partitioning on the hypergraph.

This material is a supplement to our paper entitled Mathematical Definitions of Scene and Scenario for Analysis of Automated Driving Systems in MixedTraffic Simulations. In this case study we focus on a complex one-minute scenario taken from a real driving data set, specifically from driver D1 of UAH-DriveSet [1], i.e. 20151110175712-16km-D1-NORMAL1-SECONDARY. Here the ego-vehicle, initially in a car following condition, attempts to overtake a truck but is blocked by the oncoming traffic in the opposite lane, and finally executes the overtaking maneuver when there is enough window of time and appropriate set of conditions.

This paper introduces a unified mathematical definition for describing commonly used terms encountered in systematical analysis of automated driving systems in mixed-traffic simulations. The most significant contribution of this work is in translating the terms that are clarified previously in literature into a mathematical set and function based format. Our work can be seen as an incremental step towards further formalisation of Domain-Specific-Language (DSL) for scenario representation. We also extended the previous work in the literature to allow more complex scenarios by expanding the model-incompliant information using set-theory to represent the perception capacity of the road-user agents. With this dynamic perception definition, we also support interactive scenarios and are not limited to reactive and pre-defined agent behavior. Our main focus is to give a framework to represent realistic road-user behavior to be used in simulation or computational tool to examine interaction patterns in mixed-traffic conditions. We believe that, by formalising the verbose definitions and extending theprevious work in DSL, we can support automatic scenario generation and dynamic/evolving agent behavior models for simulating mixed traffic situations and scenarios. In addition, we can obtain scenarios that are realistic but also can represent rare-conditions that are difficult to extract from field-tests and real driving data repositories.

A novel algorithmic approach is presented for the problem of partitioning a connected undirected weighted graph under constraintssuch as cardinality or membership requirements or must-link and cannot-link constraints. Such constrained problems can be NP-hardcombinatorial optimization problems. They are restated as matrix nearness problems for the weight matrix of the graph, where theobjective is to minimize the distance between the given weight matrix and perturbed weight matrices for which a functional of aneigenvalue and eigenvector of the graph Laplacian takes its minimal value. A key element in the numerical solution of these matrixnearness problems is the use of a gradient system of matrix differential equations for the functional.

In this paper we study two classes of graphs, the (m,k)-stars and l-dependent graphs, investigating the relation between spectrum characteristics and graph structure: conditions on the topology and edge weights are given in order to get values and multiplicities of Laplacian matrix eigenvalues. We prove that a vertex set reduction on graphs with (m,k)-star subgraphs is feasible, keeping the same eigenvalues with reduced multiplicity. Moreover, some useful eigenvectors properties are derived up to a product with a suitable matrix. Finally, we relate these results with Fiedler spectral partitioning of the graph. The physical relevance of the results is shortly discussed.

In this paper we introduce a spectra preserving relation between graphs with loops and graphs without loops. This relation is achieved in two steps. First, by generalizing spectra results got on (m, k)-stars to a wider class of graphs, the (m, k, s)-stars with or without loops. Second, by defining a covering space of graphs with loops that allows to remove the presence of loops by increasing the graph dimension. The equivalence of the two class of graphs allows to study graph with loops as simple graph without loosing information.

Given a connected undirected weighted graph, we are concerned with problems related to partitioning the graph. First of all we look for the closest disconnected graph (the minimum cut problem), here with respect to the Euclidean norm. We are interested in the case of constrained minimum cut problems, where constraints include cardinality or membership requirements, which leads to NP-hard combinatorial optimization problems. Furthermore, we are interested in ambiguity issues, that is in the robustness of clustering algorithms that are based on Fiedler spectral partitioning. The above-mentioned problems are restated as matrix nearness problems for the weight matrix of the graph. A key element in the solution of these matrix nearness problems is the use of a constrained gradient system of matrix differential equations.

Statistical mechanics points out as fluctuations have a relevant role for systems near critical points. We study the effect of traffic fluctuations and the transition to congested states for a stochastic dynamical model of traffic on a road network. The model simulates a finite population that moves from one road to another according to random transition probabilities. In such a way, we mimic the traffic fluctuations due to the granular feature of traffic and the dynamics at the crossing points. Then the amplitude of traffic flow fluctuations is proportional to the average flow as suggested by empirical observations. Assuming a parabolic shaped flow-density relation, there exists an unstable critical point for the road dynamics and the system can perform a phase transition to a congested state, where some roads reach their maximal capacity. We apply a statistical physics approach to study the onset congestion and we characterize analytically the relation between the fluctuations amplitude and the appearance of congested nodes. We verify the results by means of numerical simulations on a Manhattan-like road network. Moreover we point out the existence of oscillating regimes, where traffic oscillations back propagate on the road network, whose onset depend sensitively from the traffic fluctuations and that have a strong influence on the hysteresis cycles of the systems when the traffic load is modulated. The comparison between the numerical simulations and the empirical traffic data recorded by an inductive-loop traffic detector system (MTS system) on the county roads of the Emilia Romagna region in Italy is shortly discussed.