Change-driven model transformations In this paper, we investigate change-driven model transformations, a novel class of transformations, which are directly triggered by complex model changes carried out by arbitrary transactions on the model (e.g. editing operation, transformation, etc). After a classification of relevant change scenarios, we identify challenges for change-driven transformations. As the main technical contribution of the current paper, we define an expressive, high-level language for specifying change-driven transformations as an extension of graph patterns and graph transformation rules. This language generalizes previous results on live model transformations by offering trigger events for arbitrarily complex model changes, and dedicated reactions for specific kinds of changes, making this way the concept of change to be a first-class citizen of the transformation language. We discuss how the underlying transformation engine needs to be adapted in order to use the same language uniformly for different change scenarios. The technicalities of our approach will be discussed on a (1) model synchronization case study with non-materialized target models and (2) a case study on detecting the violation of evolutionary (temporal) constraints in the security requirements engineering domain. The changes of models are a central issue in model-driven software engineering (MDSE). For instance, software engineers continuously change and improve their design models using manual refinement steps or semi-automated model refactoring transformations. However, the change of one model may easily introduce inconsistencies with other models developed by other designers. In case of model-based simulation, the simulator needs to efficiently detect the changes of the underlying model (e.g. to highlight steps which can be executed at this stage).. Unfortunately, the actual notion and representation of change can be very different in practical tools and scenarios. While modern domain-specific modeling environments offer an explicit set of operations, which can be executed on the model by the user in graphical editors, in many other cases, the engineers have no control over the many ways a model may change: any kind of model manipulations are possible in any sequence.. In modern modeling environments (like the Eclipse Modeling Framework, EMF), elementary model changes are reported on-the-fly by some live notification mechanisms to support undo/redo operations. Model versioning frameworks persist the change history of models (history-aware log of model changes, which records causal dependency/ timeliness between such changes) in the form of an external change document. Many complex refactoring operations frequently require the user to preview the requested changes (expressed as change commands) prior to actually executing them to the model. A common concept for all these cases is the change delta, which uniformly captures (the aggregated effect of) a transaction which caused the model to evolve from a previous pre-state to a poststate.. Furthermore, in certain complex tool integration scenarios, we receive absolutely no information about changes, i.e., changes just happen without any trace (notification or change log). Other scenarios include those with observable (but not controllable) changes.. Model transformations, which consume (as input) or produce (as output) change models, in addition to the underlying models themselves, are called change-driven transformations (CDT) [1] . The practically relevant change scenarios impose additional challenges for CDTs. First, change models are typically restricted to contain elementary model changes (i.e. the creation and deletion of certain model elements); change-driven transformations, however, are frequently triggered by complex (aggregated) model changes carried out by a transaction on the model (e.g. user edit, refactoring transformation, etc), which is a key challenge. Moreover, some models can be non-materialized, i.e., when only an external interface is available for query and manipulation, which traditionally requires the development of complex adapters in the modeling environment. In addition, traceability information can also be limited and externalized (i.e. cannot be stored in the host models), which imposes further challenges. Finally, well-formedness restrictions over a trajectory of changing models can express evolutionary constraints.. Despite the large variety of existing model transformation languages and tools, the concept of change is not a first-class citizen in them. In the paper, we argue that in many application scenarios, making the change as part of transformations rules is a promising approach. As a result, appropriate reactions can be based upon observing the current snapshot of the model and also the way how the model evolved.. The execution mechanism of change-driven transformations also differs from traditional batch transformations. First, uncontrolled model changes (which are reported by a model management framework or detected by the CDT engine itself) happen in a transaction. Then, based upon the observed changes, certain change-driven transformation rules may be triggered to manipulate the models in a controlled way. Naturally, model changes induced by these rules are transactions themselves, which may thus trigger further reactions.. In this paper, (1) we propose a high-level change-driven transformation language which allows to unambiguously and succinctly capture the nature of change (as integral part of the language) and to specify the appropriate reactions. (2) Furthermore, implementation techniques of CDTs will be discussed which allow to use a single CDT language uniformly and independently from the actual change scenario and change representation including (a) internal representations (notifications of the modeling environment) versus externalized change models, (b) forward versus backward deltas. The concrete syntax of the language is introduced as an extension to the transformation language of the VIATRA2 framework, and it generalizes previous results on live transformations [2, 3] to the complete set of change scenarios by a precise and generic formalism for change processing.. The technicalities of change-driven transformations will be presented using two motivating case studies. First, in a model synchronization scenario incremental synchronization is carried out on a non-materialized target model (i.e. when only target object identifiers and a model manipulation interface of the target model are known, and the rest of the target model does not exist as an in-memory model within the transformation framework) with weak traceability links. Then, a case study from the security requirements engineering domain will demonstrate how evolutionary (temporal) requirements can be specified and incrementally evaluated using change-driven transformations.. The rest of the paper is structured as follows: Section 2 provides a categorization of changes to be addressed in the paper, and details the main challenges of change-driven transformations. In Sect. 3, a motivating model synchronization case study is introduced as a running example for our paper. Section 4 describes the novel language for change-driven transformations. Section 5 outlines a prototype system architecture for the various cases of change scenarios. Section 6 exemplifies the use of our CDT language in the model synchronization scenario. Specifying and checking evolutionary constraints will be the goal of another case study presented in 7. Section 8 provides a detailed discussion on the advantages and limitations of our approach. Finally, Sect. 9 summarizes related work and Sect. 10 concludes our paper. In the paper, we discussed change-driven transformations, which is a novel class of model transformations aiming to process or derive changes as their input or output. We presented a novel language for specifying change-driven transformations extending a well-established graph transformation language. We also outlined how the same language can be executed in different change scenarios by adapting incremental graph pattern matching engines.. Note that the scenarios discussed in the paper are important but not the only potential application fields of changedriven transformations. In addition to the model synchronization and evolutionary constraints case studies, we have an extended set of applications using CDTs as underlying formalism for the semantic back-annotation of model transformations driven by execution traces [51] .. A primary focus for future work is to elaborate how change-driven transformations can serve as an intermediate language for the efficient execution of existing model transformation used for bidirectional model synchronization languages (like TGGs or QVT relations, and maybe also ATL). Moreover, it is also worth investigating if our change patterns language can be extended to the action part to allow the implicit derivation of change models as output of changedriven rules. Furthermore, we also intend to investigate the correctness and consistency checking of change-driven transformations. Finally, we also aim at using change models for model merging.. the variables and constraints of a specific pattern P, we use V (P) and C(P), respectively. The pattern language [11] of the Viatra2 tool also permits additional constraints such as containment, equality and inequality, or pattern composition, which are not detailed here. Now, we discuss the potential advantages and limitations of change-driven transformation over traditional model transformation techniques. Now an overview is given on various approaches showing similarity to our proposal.. Event-driven techniques Event-driven techniques, which are the technological basis of change-driven model transformations, have been used in many fields. In relational database management systems (RDBMS), even the concept of triggers [30] can be considered as simple operations whose execution is initiated by events. Later, event-condition-action (ECA) rules [12] were introduced for active database systems as a generalization of triggers, and the same idea was adopted in rule engines [31] as well. However, ECA-based approaches lack the support for triggering by complex graph patterns, which is an essential scenario in model-driven development.. High-level transformation specification Event-driven transformation specification can be avoided by using very highlevel transformation specification formalisms. OMG's Query /View/Transformation (QVT) specification [32] , in particular the Relations part, aims at declaratively defining a relation between corresponding source and target models; it is up to an execution platform to exert event-driven behavior in order to maintain this model. While pointing out the advantages of such a solution, [33] highlights issues with the ambiguity of interpretation and implementation of QVT, in the context of bidirectionality.. Inconsistency management Inconsistency management systems aim at ensuring the consistency of multiple views of a software, which is designed by several engineers using tightly or loosely integrated tools. Views can be formulated on different levels of abstraction, and a bidirectional consistency of views is maintained by inconsistency detection and resolution.. Since these systems should typically support informal (e.g., natural language-based) descriptions as views, inconsistency resolution can never be fully automated, and manual user interaction in certain scenarios is unavoidably required, in contrast to our approach, which automatically propagates and transforms change descriptions in a well-defined, rulebased way to the target domain to avoid the appearance of inconsistencies in the target model.. Grundy et al. [34] presents a characteristic representative of inconsistency management systems, which records modification histories in the form of (model-based) change description objects just like our approach. In contrast to our solution, Grundy et al. [34] additionally saves and stores the detected inconsistencies for their possible resolution at a later time. The so-called grouping of inconsistencies in this approach would possibly allow for reaching a goal that is similar to the aim of pattern matching in the current paper; however, in [34] grouping is only used for presentation purposes, i.e., to create change and inconsistency lists for users to interact with.. Olsson and Grundy [35] provide a conceptual architecture and prototype for supporting traceability and inconsistency management between software requirements descriptions, UML-style use case models, and black-box test plans. Relationships between high-level software artefacts are represented by traceability links, which can be defined manually or in a semi-automated way. In contrast to our solution, this approach supports change notifications on a low abstraction level, it can transform only simple modifications automatically, while other changes still need developer intervention.. In graph transformation, [22] presents an approach for consistency management between abstract and concrete syntax representations of visual modeling languages. By their approach, the commands executed through the user interface are explicitly materialized as special command model elements and then processed by triple graph grammar (TGG) rules. This approach is a prime example for the "controlled" change processing scenario, where all possible editing operations are a-priori known; in contrast, our technique primarily targets non-controllable change processing (while we also cover controllable change scenarios through domain-specific command models).. Egyed et al. [16] deals with consistency maintenance in UML models. This paper proposes target incremental techniques to efficiently detect inconsistencies and derive proposed corrections; recommended changes are represented explicitly (such as "DoesExist" and "ShouldExist"). This approach is based on storing very detailed traceability information about rule execution in order to determine when and how a rule should be re-executed for fixing inconsistences; in contrast, our approach is focused on reducing the amount of necessary information persisted in models.. Tratt [15] presents a unidirectional, target incremental batch transformation language for model synchronization. Between two synchronization runs, the user may modify the source as well as target models, and the system will then propagate the changes incrementally, leaving manual target modifications intact. This technique again relies on massive amounts of information cached in traceability models, by copying certain parts of the source model intro traceability models.. Software evolution approaches Software evolution approaches, which focus on the temporal development of system (meta)models, can be considered as a possible application area of our approach, which could generate deltas (for different modeling domains) as inputs for the merging process required in software evolution. However, note that our approach does not further support the merge conflict resolution subtask in any sense.. Mens [6] lays down a wide-range of terminology used in software evolution. According to this framework, snapshot, command, and history scenarios of Sect. 2.1 directly correspond to state-based, forward and backward delta approaches, respectively. Moreover, our solution can be categorized as an operation and intensional change-based approach as model changes are explicitly expressed as transformations, and they are independent from the versions to which they are applied.. The FAMOOS project [36] whose aim was to build a framework to support the evolution and reengineering of object-oriented software systems used languages FAMIX [37] and Hismo [38] for modeling purposes. More specifically, FAMIX is a language-independent model of objectoriented systems, which can be used for exchanging information between reengineering tools. FAMIX can be considered as a simplified metamodel for class diagrams without any support for describing changes. Hismo [38] extends metamodels by adding a time layer on top of the structural information, and it provides a common infrastructure for expressing and combining evolution and structural analyses. The additional time layer enables Hismo to support version control and to calculate changes of models, and in this sense, it could serve as a source of input for our approach, but Hismo has no metamodel for describing changes on a high abstraction level.. Visualization tools in the FAMOOS framework use sideeffect-free OCL-based queries, which can even involve constructs from the time layer, but these queries are imperative from the viewpoint of structural constraint navigation, and they have been used for quantitative structural measurements (e.g., for counting the number of changed methods), in contrast to our approach, which provides declarative graph patterns, which are used to drive and initiate the transformation of change descriptions. Additionally, the Goose tool in FAMOOS uses Prolog rules to search for violations of certain design guidelines. Prolog rules show similarity to our graph patterns in their structure; however, our approach requires no conversion of underlying models, in contrast to Goose, which can operate only on Prolog facts that have to be extracted in advance from FAMIX models.. Narayanan et al. [39] applies graph transformation for metamodel evolution in domain-specific languages. In this approach, GT rules evolve models in a metamodel compliance preserving way. More specifically, they describe the changes themselves inside a single modeling domain, but not the transformation of changes between different domains as in our solution. Moreover, [39] lacks live transformation support.

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Perfect state transfer on quotient graphs We prove new results on perfect state transfer of quantum walks on quotient graphs. Since a graph G has perfect state transfer if and only if its quotient G/π, under any equitable partition π, has perfect state transfer, we exhibit graphs with perfect state transfer between two vertices but which lack automorphism swapping them. This answers a question of Godsil (Discrete Mathematics 312(1):129-147, 2011). We also show that the Cartesian product of quotient graphs k G k /π k is isomorphic to the quotient graph k G k /π, for some equitable partition π. This provides an algebraic description of a construction due to Feder (Physical Review Letters 97, 180502, 2006) which is based on many-boson quantum walk. Perfect state transfer in continuous-time quantum walk on graphs has received considerable attention in quantum information and computation. This is in large part due to its potential applications in quantum information transmission over networks and its role in quantum computation. Recently, continuing ideas developed by Childs [9] , Underwood and Feder [31] used perfect state transfer to show that continuous-time quantum walk is a universal computational model. The notion of perfect state transfer was introduced by Bose [7] in the context of information transfer on linear spin-chains. His original scheme can be generalized to arbitrary graphs (as described by Albanese et al. and Christandl et al. [1, 12, 11] ) which we briefly outline in the following.. Given a weighted graph G on n vertices with adjacency matrix A(G), we imagine a collection of n qubits associated with each vertex of G and arranged so that their interaction is governed by a Hamiltonian H G which depends on the edge structure of G. Here, our collective Hilbert space is u∈V C 2 which is 2 n -dimensional. Suppose an arbitrary one-qubit state |ψ = α 0 |0 + α 1 |1 is located at vertex a of G. Our goal is to move this state to vertex b. For simplicity, we depict a as the leftmost qubit whereas b is the rightmost qubit. The initial configuration has the qubit at vertex a be in state |ψ and the other qubits are in the |0 state, while the final configuration has the qubit at vertex b be in state |ψ with the other qubits being in the |0 state: |start = |ψ a ⊗ |0 ⊗ . . . ⊗ |0 ⊗ |0 b = α 0 |00 . . . 00 + α 1 |10 . . . 00. (1) |final = |0 a ⊗ |0 ⊗ . . . ⊗ |0 ⊗ |ψ b = α 0 |00 . . . 00 + α 1 |00 . . . 01 .. The main goal of perfect state transfer is to achieve, at some time t,. Natural assumptions can be placed on H G which will allow us to view (3) as a continuous-time quantum walk on G.. For example, one typically assumes H G commutes with Z = u∈V Z u , where the latter operator counts the number of qubits in the |1 state 1 . Note that the eigenvalues of Z are given by λ k = −n + 2k, for k = 0, 1, . . . , n. Since |00 . . . 00 belongs to the zero eigenspace of H G , we may focus on the unitary evolution of |10 . . . 00 under e −itH G . The latter state belongs to the eigenspace Λ 1 of Z corresponding to the eigenvalue λ 1 = −n + 2 (since it has exactly one qubit in the |1 state). Thus, e −itH G (α 0 |00 . . . 00 + α 1 |10 . . . 00 ) = α 0 |00 . . . 00 + α 1 e −itH G |10 . . . 00 .. Since H G and Z commute, the eigenspace Λ 1 is H G -invariant; this is because if |z is an eigenvector of Z with eigenvalue λ, then so is H G |z . Thus, the time evolution of exp(−itH G )|10 . . . 00 stays inside the eigenspace Λ 1 . Moreover, we have the following basis states for Λ 1 : |1 = |100 . . . 0 , |2 = |010 . . . 0 , . . . , |n = |000 . . . 1. which forms a natural correspondence with the vertices of G; thus, |a = |1 and |b = |n . Furthermore, suppose H G agrees with A(G) on the subspace Λ 1 where v|H G |u equals the weight ω u,v of the edge (u, v), for all u, v ∈ V . Examples of H G satisfying these assumptions include the XY exchange Hamiltonian H G = 1 2 (u,v)∈E(G) ω u,v (X u X v + Y u Y v ), as well as the Heisenberg exchange (which is related to the Laplacian of G). This shows that the 2 n -dimensional time evolution e −itH G |start can be viewed as a n-dimensional time evolution e −itA(G) |a since the former is confined to the single-excitation subspace Λ 1 . Further background on these connections may be found in [7, 11, 28, 3] .. By the preceding arguments, we may study perfect state tranfer (3) as a continuous-time quantum walk on the graph G (see Farhi and Gutmann [14] ). Thus, without loss of generality, we say a graph G = (V, E) has perfect state transfer (PST) from a to b at time t if. where A(G) is the adjacency matrix of G (thus, focusing on the XY interaction model). This allows us to investigate mathematical properties of the graph G which enable such phenomena to occur. The reduction to quantum walk on graphs was a crucial element in the early works on perfect state transfer (see [7, 1, 12, 11] ). Christandl et al. [12, 11] showed that taking an k-fold Cartesian graph product of either a 2-path or a 3-path (that is, K 2 or P 3 ) with itself yields a high-diameter graph which has perfect state transfer. This follows since both K 2 and P 3 have antipodal perfect state transfer and because the Cartesian product operation preserves perfect state transfer. They also made a crucial connection (see Christandl et al. [11] ). Antipodal PST occurs between vertices marked white.. between hypercubes and weighted paths using the so-called path-collapsing argument 2 . Christandl et al. [11] also observed that the n-path P n , for n ≥ 4, has no antipodal perfect state transfer but a suitably weighted version of it has perfect state transfer. The weighting scheme on P n is derived from a path-collapsed (n − 1)-cube.. In an intriguing work, Feder [15] generalized the construction of the weighted paths with perfect state transfer described by Christandl et al. in [12, 11] . His construction used a many-boson quantum walk on a single primary graph. He showed that this induced a single-boson quantum walk on a secondary graph and that the secondary graph has perfect state transfer if the primary graph does. In this construction, the weighted path of length n is obtained from a (n − 1)-boson quantum walk on K 2 .. In algebraic graph theory, the main question is to find a characterization of graphs which exhibit perfect state transfer. Some progress on specific families of graphs were given by Bernasconi et al. [5] and by Cheung and Godsil [8] for hypercubic graphs and by Bašić and Petković [4] for circulant graphs. Although a general characterization remains beyond the reach of current methods, strong general results towards this goal were recently proved by Godsil [18, 19] . In one of his results, Godsil proved that a necessary condition for G to have perfect state transfer between vertices a and b is that they are cospectral, that is, the vertex-deleted subgraphs G\a and G\b are isomorphic. This intuitively suggests that the neighborhoods around a and b must look similar. In fact, prior to this work, all known examples of graphs with perfect state transfer between vertices a and b admit an automorphism which maps a to b. In [24] , Kay proved that the latter property is necessary for paths, while in [18] , Godsil asked if this necessary condition holds for any graph.. Our goal in the present work is to explore the role of quotient graphs in perfect state transfer. Since quotient graphs naturally arise in the context of equitable partitions, we use this formalization to capture the idea behind path-collapsing arguments [12, 11, 10] . We argue that equitable partitions provide the most natural way to view these arguments since the resulting proofs are more transparent. Moreover, equitable partitions have been studied extensively in algebraic graph theory (see Godsil and McKay [20] and Godsil and Royle [21] ) and have a well-established collection of results which we can build upon.. A main observation we use throughout is the following statement which admits a simple proof: a graph G has perfect state transfer if and only if its quotient graph G/π modulo an equitable partition π has perfect state transfer. Although weaker forms of this statement had appeared in different guises before, we give a simple and direct proof using the machinery of equitable partitions. The necessary condition was used by Christandl et al. [12, 11] to establish that certain weighted paths have perfect state transfer (in contrast to its unweighted variants). Childs et al. [10] used. Figure 2: Feder's weighted lattice PST graphs obtained from k-boson quantum walks on P 3 with k = 2, 3, 4 (see Feder [15] ). Equivalently, these are the quotient graphs P k 3 /π (see Theorem 8) . Antipodal PST occurs between vertices marked white.. the sufficient condition to analyze hitting times of specific graphs related to binary trees. We will use the backward implication of the equivalence to construct new perfect state transfer graphs. In our first application, we use this lifting property to construct a graph with perfect state transfer between two vertices but has no automorphism which maps one vertex to the other. This answers the aforementioned question posed by Godsil [18] .. Using equitable partitions, we also provide an algebraic framework to Feder's construction. We prove that the secondary graph obtained from a k-boson quantum walk on a primary graph G is equivalent to a quotient of the k-fold Cartesian product of G, that is, G k /π, for some equitable partition π. This equivalence is related to works by Audenaart et al. [3] on symmetric powers of graphs and by Osborne [28] on wedge product on graphs. Our work differs from [3] in that we preserve diagonal entries and from [28] in that we work in a symmetric vector space (rather than exterior vector space). A common thread in all these works is the use of algebraic graph theory to provide an explicit connection between many-particle and single-particle quantum walks. Another related work along the same lines was given in [32] . In our algebraic formalism, we employ a model of many-particle quantum walk used by Gamble et al. [16] and by Smith [29] in their works on graph isomorphism.. Finally, we explore Feder's construction when distinct primary graphs with commensurable perfect state transfer (or even periodic) times are used. We prove a composition theorem which shows partial commutativity between the Cartesian product and quotient operators. This mixed construction is akin to perfect state transfer graphs obtained using weak and lexicographic products (see Ge et al. [17] ) and graph joins (see Angeles-Canul et al. [2] ). We found new families of perfect state transfer graphs using cube-like graphs (which were studied by Bernasconi et al. [5] and by Cheung and Godsil [8] ). The graphs derived from these cube-like graphs are different from weighted graphs obtained in Feder's construction.. Our proofs rely on basic ideas from algebraic graph theory and exploit spectral properties of the underlying graphs. In this work, we explored perfect state transfer in quantum walks using equitable partitions. Our main focus is on a strong equivalence of perfect state transfer between a graph and its quotients. Although weaker forms of this equivalence had appeared earlier, we gave a simple and most direct proof which yields a powerful two-way tool (taking lifts and quotients on graphs) to study perfect state transfer.. In lifting, if a perfect state transfer graph is a quotient of another graph, then the parent graph also has perfect state transfer. We used this to construct graphs with perfect state transfer between two vertices but which lack automorphism swapping the vertices; hence, answering a question posed by Godsil in [18] . This question is relevant since, prior to this work, all known graphs with perfect state transfer admit the automorphism property.. In a quotient move, if a graph has perfect state transfer graph, then so does its quotient. These quotient graphs are obtained by forming various equitable partitions of the original graph. We used this to describe Feder's intriguing construction of PST graphs [15] based on many-boson quantum walks. By adopting an explicit model of k-boson quantum walk in [16, 29] , we show that Feder's graphs are quotients of a k-fold Cartesian product of PST graphs. The resulting graphs have perfect state transfer due to the equivalence theorem. This is related to works by Audenaart [3] , by Osborne [28] , and by Wieśniak and Markiewicz [32] which used algebraic graph theory to provide explicit connection between multiple and single excitation subspaces under various coupling schemes on graphs.. It would be interesting to find a property of graphs, for which any graph perfect state transfer graph with this property must admit an automorphism swapping the two perfect state transfer vertices. We leave this as an open question for future work.

B I I O B I O O O O O B I I O O O O B I O O O O O B I I O O O O O O O O O O O O O B I I O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I B I I O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O B I O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O B I I O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O B I O O O O O O O O O O O O B I O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O B I I O B I O O O O O O B I O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O B I O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O B I I O O O O O O O O O O O O O B I I O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O B I I O O O O O O O B I O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O B I I O O O O O O O O O O O


Simulation data structures using SIMULA 67 This paper describes the approach to the teaching of simulation at the University of Montreal. The language used is the present standard of SIMULA, sometimes referred to as SIMULA 67; however, the modelling approach is inspired from GPSS. This combination of an old and a new language has proved very fruitful. Little has so far been published on the new standard for SIMULA and it is hoped that the examples in this paper may serve as a painless introduction to simulation programming in this language. SIMULA i, 2, 3 is at the forefront of language development.. It is a general purpose language with a built-in simulation capability.. The language can best be described as an extension of ALGOL with 3 main features:. "Objects" based on the coroutine concept to allow quasi-parallel processing. An object is a self-contained program with its own local data and actions defined by a generic "class declaration". Examples of objects in simulations would be customers, machines or digital circuits, each of which is capable of independent parallel actions.. Strong list processing capability and dynamic storage allocation under user control. There is automatic garbage collection.. The ability to create hierarchies of concepts which makes it easy to generate special purpose problem oriented languages. More specifically, a class declaration prefixed by the name of another class will have the data and actions of that other class added to its own. To a certain extent, the language may be considered as extendable.. The writing of a simulation program in SIMULA is done in two phases. First, the attributes and actions of the objects that will be used are defined through class declarations.. In a jobshop simulation, for example, useful objects are jobs and machines.. These class declarations are akin to "event routines" in other languages.. Secondly, the required objects are generated and initialised.. In the jobshop example, there will be one class declaration for "job" and another for "machine".. These declarations will in general be rather complex but generation of an object is fairly simple, for example:. " activate new job at time3 ;" and as many jobs and machines as required can be generated in this way. The simulation proper is letting these objects interact over simulated time.. In keeping with the philosophy of extendability, the basic SIMULA language contains a minimum of concepts. However, there is in the compiler a predefined class "simulation" which includes an events chain to implement simulated system time. This class also contains useful scheduling and list processing procedures. These procedures and the events chain, called sequencing set (SQS), are expressed in terms of the basic language. The average simulation user would preface his program block with the class name "simulation" to have access to these procedures, but it is important to note that these are not frozen in the language. A programmer is free to ignore the predefined "simulation" class and redefine the implementation of "simulated system time" if he so wishes.. Similarly, an installation interested in "jobshop" simulation could create a "jobshop" class by adding certain concepts and procedures to "simulation".. A user could then prefix his program with this new class name "jobshop" and treat SIMULA as a specialized jobshop simulation language I. In other words, the first phase of program writing has been eliminated or at least much reduced and the programmer can concentrate on the second phase.. The language therefore allows the user gre~ freedom as to the degree of detail of the modelled objects and the method of passing control between them. This power, however, has attendant disadvantages:. i). The logic of simulation programs is notoriously complex and unless the programmer uses a systematic approach to program writing, debugging is very difficult. With SIMULA, it is very easy to be clever with a resultant loss of clarity. A good programming methodology is essential.. 2). The language does not suggest an approach to modelling.. It was found, when first teaching SIMULA, that students had a tendency to model in too great detail. As a result, they had difficulty in abstracting concepts from one model to adapt them to another. GFSS 4 shows quite a different approach to the design of a simulation language. GPSS forces a "world view" on its user by providing him with predefined objects. GPSS views systems in terms of transactions flowing from one work area to another.. The objects of GPSS are transactions, facilities, storages and logic gates. The transaction is the only object whose behaviour the user can describe.. In fact, the transaction description is the GPSS program.. The other objects are standard and their behaviour predefined. The language is restrictive and it does not have the power of SIMSCRIPT or SIMULA, but the wide use and popularity of GPSS shows that its choice of basic objects is well founded and quite useful for a large class of problems.. GPSS is especially helpful for the beginner on just those two points noted as disadvantages for SIMULA. Model building is'simplified by the presence of the predefined objects which can be used as building blocks. All decisions for scheduling movements through the system are localised in the transaction definition.. The other objects are completely passive in this style of modelling The model logic is therefore easy to debug. Further, GPSS programs for queue-type simulations, for which they are well suited, are very concise.. It is therefore advantageous to combine the two languages.. SIMULA definitions for "facilities" and "storages" will be given as well as the corresponding "entering" and "leaving" procedures. The "transaction" is also defined. By prefixing his program with these definitions, a SIMULA user can program in a way closely resembling GPSS. Essentially, programming is adding to the "transaction" definition.. The concepts of "facilities" and "storages" to represent resources are so fundamental to scheduling or queuing problems that they may be considered.basic simulation data structures. These structures are very close to the concept of "semaphore" introduced by Dijkstra to synchronise loosely connected parallel processes5. The "entering" and "leaving" procedures correspond ~o his "P" and "V" operations.. In this way, the old concepts of GPSS are finding new life in systems programming. Th~ paper will refer to the SIMULA concept of "object" rather than "data structure".. The' two are close although the SIMULA object includes relevant actions as weil as the data structure itself. Conclusions from this work can be drawn on several levels:. i). It has been shown how SIMULA can imitate GPSS. By prefacing his program with the definitions given here, a user can program a suitable model with the same ease and brevity as in GPSS. The use of well defined objects eases model building and later debugging.. The approach has proved quite successful in the introductory teaching of simulation with SIMULA.. 2) SIMULA is not limited to GPSS type models. The power of the language has been shown in the definitions of GPSS objects. The structure of these objects is not frozen as in GPSS and some alternative algorithms for the implementation of "storage" have been suggested.. SIMULA is also suited to other types of models.. The language has been applied to biological epidemic simulations 2 and at Montreal, SIMULA has been used with success to simulate both PERT and digital networks.

B B I O O O O B O O O O O B O B O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O B O O B O O O O O O O B O O O B O O O O O O O O O B O O O O O O B O O O B O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O B B O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O B O O O O B O O O O O B O O O O O O O O O O O O O O O O O O O B O O O O O O O O B O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O B O O O O O O B O O O O O O O O O O O O O B B O O O O O O O O B O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O B O O O O O O O B O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O B B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O B O O B O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O B O B O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O


Dynamic Carrier Allocation Strategies for Mobile Cellular Networks In dynamic carrier allocation (DCA) strategies, carriers are assigned priorities. Based upon how priorities are assigned to carriers, DCA strategies are classified into three classes: static-priority, dynamic-priority, and hybrid-priority strategies. In this paper, we lay out a theoretical foundation for DCA strategies: we first develop the concept of optimal reuse pattern and then propose a model for resource planning as well as three guiding principles for priority assignment in static-and hybrid-priority DCA strategies. These theoretical results will shed light on the strength and weakness of existing DCA strategies and lead us to developing more efficient strategies.Specifically, we propose three new DCA strategies: Column-Wise static-priority (CWSP), Column-Wise hybridpriority (CWHP), and Two-Step dynamic-priority (TSDP). The CWSP and CWHP strategies are obtained from the Geometric strategy by following our guiding principles for priority assignment. The TSDP strategy, on the other hand, was derived from the Nanda-Goodman strategy by applying our notion of optimal reuse patterns. It not only attempts to reuse carriers as compactly as possible (just as other dynamic-priority strategies do), but also goes one step further by bringing the actual carrier reuse pattern closer to the optimal pattern.The performance of the three new DCA strategies are simulated under both uniform and non-uniform traffic distributions. The simulation results show that each one of the three new DCA strategies outperforms existing strategies in its own class, and the TSDP strategy has the best performance among all DCA strategies. We consider a mobile cellular network that employs both TDMA and FDMA multiplexing techniques. The geographical area covered by the network is assumed to be divided into hexagonal cells [3, 8, 10] . Communications to and from mobile hosts (MHs) in each cell are serviced by a base station (BS) located at the center of the cell; the BSs are interconnected via a wired network. The wireless bandwidth available for mobile users is divided into a set of carriers by frequency division, and each carrier is further divided into a number of channels by time division. Each channel can support a call. We assume that carrier is the basic unit of resource allocation, i.e., if a carrier is assigned to a cell, then all channels in that carrier are available for use by mobile users in that cell. A carrier can be used by different cells without intolerable co-channel interference if their geographic distance is no less than a threshold called the minimum reuse distance. This model is the same as that adopted by the GSM pan-European mobile radio network [11] , the Nanda-Goodman DCA strategy [12] , and the Geometric DCA strategy [1] .. The problem of carrier allocation is to assign carriers to cells so as to minimize the overall rate of blocked calls, where a call is said to be blocked if there is no channel available for its use. It is also called a channel allocation problem if time division is not used so that each carrier contains only one channel. Three types of strategies have been proposed for carrier allocations: fixed, flexible, and dynamic strategies [14] . Fixed strategies allocate a fixed set of carriers to each cell, which is allowed to use the allocated carriers and no others. With dynamic strategies [4] , a cell may use any carrier that will not cause co-channel interference. Carriers are not pre-allocated to cells but, rather, assigned on a dynamic and temporary basis. Typically, each carrier is associated with a priority, and when a cell needs a carrier, it picks the one that is available with the highest priority. The carrier is later returned to the system when it is no longer needed by the cell. Flexible strategies [13] combine the aspects of both fixed and dynamic strategies: (1) each cell is allocated a fixed set of permanent carriers and (2) a number of flexible carriers are set aside to be dynamically allocated to cells upon requests.. Of the three types of strategies, dynamic strategies seem most promising and have been the focus of recent research [1, 12, 15] . Various dynamic carrier allocation (DCA) strategies have been proposed; they differ from each other in the way priorities are assigned to carriers. As priorities can be assigned to carriers in a static, dynamic, or hybrid fashion, DCA strategies are further divided into three classes: static-priority, dynamic-priority, and hybridpriority DCA strategies. In a static-priority strategy such as the Geometric strategy [1] , each carrier in each cell is assigned a fixed priority that does not change over time. In a dynamic-priority strategy such as the Nanda-Goodman [12] and Locally-Optimized strategies [15] , a carrier's priority is dynamically computed. A hybrid-priority scheme is something in between: a carrier's priority is calculated as a static base-priority plus a dynamic adaptive-priority.. All the channel borrowing strategies [6, 9, 15] belong to this class.. A DCA strategy is usually evaluated by its failure rate and overhead. The failure rate is the percentage of calls that are blocked during call arrivals or inter-cell handoffs. The overhead of a DCA strategy is measured by the number of messages exchanged among base stations in order to update carrier priorities. Of the three classes of DCA strategies, static-priority strategies obviously have the lowest overhead, and dynamic-priority strategies the highest. As such, one would naturally expect dynamic-priority strategies to outperform static-and hybrid-priority strategies in terms of failure rate. Unfortunately, this is not the case in the literature: the locally-optimized dynamicpriority strategy is inferior to the borrowing with directional channel-locking strategy (which is a hybrid-priority scheme) [15] ; the static-priority-based Geometric strategy was observed in [1] to have a lower failure rate than the dynamic-priority-based Nanda-Goodman strategy. Is dynamic priority inherently not an effective approach? Or efficient dynamic-priority strategies are just yet to be invented?. We wish to answer these questions in this paper. To this end, we lay out a theoretical foundation for DCA strategies: we first develop the concept of optimal reuse pattern and then propose a model for resource planning as well as three guiding principles for priority assignment in static-and hybrid-priority DCA strategies. These theoretical results will shed light on the strength and weakness of existing DCA strategies and lead us to developing more efficient strategies. Specifically, we propose three new DCA strategies: Column-Wise static-priority (CWSP), Column-Wise hybrid-priority (CWHP), and Two-Step dynamic-priority (TSDP). The CWSP and and CWHP schemes are obtained from the Geometric strategy by following our guiding principles for priority assignment. The TSDP strategy, on the other hand, was derived from the Nanda-Goodman strategy by applying our notion of optimal reuse patterns.. It not only attempts to reuse carriers as compactly as possible (just as the Nanda-Goodman and Locally-Optimized dynamic-priority strategies do), but also goes one step further by bringing the actual carrier reuse pattern closer to the optimal pattern.. The proposed strategies are evaluated by simulation against existing schemes. Strategies of the same class (static-, dynamic-, or hybrid-priority) are compared. As strategies of the same class have similar overhead, we use failure rate as the sole criterion. Both uniform and non-uniform traffic distributions are considered. Here is a summary of our simulation results.. Our Column-Wise static-priority strategy has a lower call failure rate than the Geometric strategy, which was the most efficient static-priority strategy known to us. The improvement is up to 30%.. Our Column-Wise hybrid-priority strategy improves by up to 25% over the Borrowing with Directional Channel-Locking strategy, previously the best strategy in the hybrid-priority class.. Among all dynamic-priority strategies, our Two-Step dynamic-priority strategy has the lowest call failure rate.. It reduces the failure rate up to 95% comparing to the Nanda-Goodman strategy.. The TSDP strategy has the lowest call failure rate among all DCA strategies. It reduces the failure rate by 15% to 95% comparing to other DCA strategies considered in this paper.. This paper is organized as follows. Section 2 presents the cellular model and illustrates the basic resource operations and the consistency requirement of priority assignment schemes. Section 3 reviews previous work. Section 4 discusses the resource planning model and three principles for assigning priorities to carriers. Sections 5, 6, and 7 are devoted to discussions of static-, hybrid-, and dynamic-priority DCA strategies, respectively. The simulation models and results are presented in section 8. In this paper, we classified the DCA strategies into three classes: static-priority, hybrid-priority and dynamic-priority strategies. We developed the concept of the optimal carrier reuse pattern which has become the foundation for designing efficient DCA strategies.. We proposed a formal resource planning model and three principles for priority assignment for static-priority and hybrid-priority DCA strategies. Following the principles, we presented the Column-Wise static-priority DCA strategy and the Column-Wise hybrid-priority DCA strategy. Then we proposed the Two-Step Dynamic-Priority DCA strategy which has the best performance among all DCA strategies under both uniform and non-uniform traffic distributions.. All three new DCA strategies proposed in this paper can be implemented in either a centralized or a distributed fashion. The reader is referred to [5] for more detailed on distributed implementations of DCA strategies in mobile cellular networks.. Appendix. We prove Theorem 2 in the following. Proof. Consider m consecutive cells c j;0 ; c j;1 ; : : : ; c j;m?1 (j = 0; 1; 2) in row j (Fig. 9) . Without loss of generality, we assume that n 2 > 0. Let c 0 2;0 ; c 0 2;1 ; : : : ; c 0 2;n 2 ?1 be the cells in row 2 that use x. Let s 2;i (0 i n 2 ? 1) 2 denote 2 s2;n 2 ?1 denotes the set of cells between c 0 2;n 2 ?1 and c 0 2;0 , which is empty if the network is not wrapped-around. Hence, when the cells are wrapped-around, we have m = n 2 + P n 2 ?1 i=0 js 2;i j n 2 + P n 2 ?1 i=0 (3n 0;i + 3n 1;i + 2) = 3n 0 + 3n 1 + 3n 2 ; when the cells are not wrapped-around (s 2;n 2 ?1 is empty), we have m = n 2 + P n 2 ?2 i=0 js 2;i j n 2 + P n 2 ?2 i=0 (3n 0;i + 3n 1;i + 2) = 3n 0 + 3n 1 + 3n 2 ? 2. In both cases, we have n 0 + n 1 + n 2 d m 3 e because n 0 + n 1 + n 2 is an integer.. In a wrapped-around network with three rows, each having m = 3 k 1 cells, the equation holds if and only if js 2;i j = 3 l i + 2 for all i = 0; 1; : : : ; n 2 ? 1. With js 2;i j = 3 l i + 2, at most 3 l i ? 3 cells in s 1;i and at most 3 l i ? 2 cells in s 0;i can use x. According to Lemma 2, the maximum reuse frequency in s 1;i and s 0;i is l i , which is achievable if and only if n 1;i = 0 and n 0;i = l i . It is easy to see that the Geometric strategy follows the first and third principles, but not the second principle. The CWSP strategy differs from the Geometric strategy in that it partially follows the second principle. In the following, we prove it by showing that if the highest-prioritized secondary carrier in a set is available in a cell, those carriers with lower priorities in the same set are also available in the cell with only one exception. . Proof. Because x is available but y is not available in c, we can conclude that y = 2 U(c) otherwise a carrier switch operation between y and the highest-priority available carrier (e.g. x) would have taken place. Hence, y must be an interfered carrier in c, i.e., 9c 0 2 IN(c) s:t: y 2 U(c 0 ). Since y 2 U(c 0 ), we know x = 2 A(c 0 ) otherwise a carrier switch operation between y and the highest-priority available carrier (e.g. x) will take place. Similarly, for any carrier z that has a higher priority than y in c 0 , z 6 2 A(c 0 ).. Also, x = 2 U(c 0 ) because x 2 A(c). Hence, x 2 I(c 0 ).. The only scenario in which Eq. 9 holds is shown in Fig. 4: x is available and y is not available in c because (1) y is in use in c 0 2 IN(c); (2) x is interfered in c 0 ; and (3) any carrier with a higher priority than y is not available in c 0 . Although we cannot describe quantitatively how often this scenario will occur, we can see that it is not a very common situation.. Therefore, if a secondary carrier is available in c, then those carriers in the same set but with lower priorities are likely to be available in c. In other words, the highest-priority available carrier is probably from the richest set, i.e., the CWSP strategy closely follows the second principle. In this section, we briefly review three existing strategies: the Geometric static-priority DCA strategy [1] , the borrowing with directional channel-locking hybrid-priority strategy [15] , and the Nanda-Goodman dynamic-priority DCA strategy [12] .

O B I O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Parsimonious Mahalanobis Kernel for the Classification of High Dimensional Data The classification of high dimensional data with kernel methods is considered in this article. Exploiting the emptiness property of high dimensional spaces, a kernel based on the Mahalanobis distance is proposed. The computation of the Mahalanobis distance requires the inversion of a covariance matrix. In high dimensional spaces, the estimated covariance matrix is ill-conditioned and its inversion is unstable or impossible. Using a parsimonious statistical model, namely the High Dimensional Discriminant Analysis model, the specific signal and noise subspaces are estimated for each considered class making the inverse of the class specific covariance matrix explicit and stable, leading to the definition of a parsimonious Mahalanobis kernel. A SVM based framework is used for selecting the hyperparameters of the parsimonious Mahalanobis kernel by optimizing the so-called radius-margin bound. Experimental results on three high dimensional data sets show that the proposed kernel is suitable for classifying high dimensional data, providing better classification accuracies than the conventional Gaussian kernel. High Dimensional (HD) data sets are commonly available for fully or partially automatic processing: For a relatively low number of samples, n, a huge number of variables, d, is simultaneously accessible. For instance, in hyperspectral imagery, hundreds of spectral wavelengths are recorded for a given pixel; in gene expression analysis, the measure of expression level of thousands of genes is typical; in customer recommendation systems for web services, to each potential client a high number of variables is associated (his past choices, his personal information ...) [1, 2, 3] . For each sample, it is possible to have either numerical or alphabetical variables which can be sparse or with a different signal to noise ratio.. In terms of processing, such data may need to be either classified, clustered, filtered or inversed in a supervised or unsupervised way. Although many algorithms exist in the literature for small or moderate dimensions (from Bayesian methods to Machine Learning techniques) most of them are not well suited to HD data. Actually, HD data pose critical theoretical and practical problems that need to be addressed specifically [2] .. Indeed, HD spaces exhibit non intuitive geometrical and statistical properties when compared to lower dimensional spaces. Most of them do not behave in a similar way as in three dimensional Euclidean spaces (Table 1 summarizes the main properties of HD spaces) [4] . For instance, samples following a uniform law will have a tendency to have a high concentration in the corners [5] . The same property holds for normally distributed data: samples tend to have a high concentration in the tails [6] , making density estimation a difficult task. This problem can be related to the number of parameters t to be estimated to fit a Gaussian than conventional generative methods. However, local kernel methods are sensitive to the size of the dimensionality [24] . A kernel method is said to be local if the decision function value for a new sample depends on the neighbors of that sample in the training set. Since in HD data the neighborhood of a sample is mostly empty, such local methods are negatively impacted by the dimension. For instance, SVM with Gaussian kernel. is such a local kernel method. In this paper, it is proposed to use subspace models to construct a kernel adapted to high dimensional data. The chosen approach for including subspace models in a kernel function is to consider the Mahalanobis distance, d Σc , between two samples for a given class, c, with covariance matrix, Σ c :. Previous works on the Mahalanobis kernel [25, 26, 27, 28] were limited by the effect of dimensionality on the matrix inversion. In [25] , the covariance matrix was computed on the whole training set. The associated implicit model is that the classes share the same covariance matrix, which is not true in practice. Diagonal and full covariance matrices were investigated in [26] for the purpose of classification and in [27] for the purpose of regression. However, in a similar way, the covariance matrix was computed for all the training samples. Computing the covariance matrix for the Mahalanobis distance with all the training samples is equivalent to project the data on all the principal components, scale the variance to one, and then applying the Euclidean distance. By doing so, classes could overlap more than in the original input space and the discrimination between them would be decreased.. In this work, the HDDA model is used for the definition of a class specific covariance matrix adapted for HD data. The specific signal and noise subspaces are estimated for each considered class, ensuring a parsimonious characterization of the classes. Following the HDDA model it is then possible to derive an explicit formulation of the inverse of the covariance matrix, without any regularization or dimension reduction. The parsimonious Mahalanobis kernel is constructed by substituting the Euclidean distance with the Mahalanobis distance computed using the HDDA model. It is proposed in this work to define several hyperparameters in the kernel to control the influence of the signal and noise subspaces in the classification process. These hyperparameters are optimized during the training process by the minimization of the so-called radius margin bound of the SVM classifier. Compared to the previous works on the Mahalanobis kernel for HD data, the proposed method allows the use of a more complex model, a separate covariance matrix per class, with higher efficiency in terms of accuracy. The remainder of the paper is organized as follows. The subspace model and the proposed kernel are discussed in Section 2. The problem of selecting the hyperparameters for classification with SVM is addressed in Section 3. The Section 4 details the estimation of the size of the signal subspace. Results on simulated and real high dimensional data are reported in Section 5. Conclusions and perspectives conclude the paper. In this paper, a novel kernel adapted to high dimensional data has been proposed. The parsimonious Mahalanobis kernel is based on the emptiness property of HD spaces. For each class, the original input space is split into a signal subspace and a noise subspace. Using this assumption, the inversion of the covariance matrix in the Mahalanobis kernel can be accurately computed. The proposed kernel was tested in a SVM framework for the purpose of classification. Experimental results on four data sets demonstrate the potential of the proposed kernel. In each case, the classification accuracy increased compared to the conventional Gaussian kernel and for three cases the proposed kernel showed superior results to simply map the data on the first PCA axes. Consequently, for HD data the HDDA-Mahalanobis kernel should be prefered.. Regarding the computational load, the HDDA-Mahalanobis kernel is more demanding during the training process than the Gaussian kernel since more hyperparameters have to be estimated. Besides that, the HDDA-Mahalanobis kernel is efficient when the dimension of the input space is high. Hence, for moderate or small dimensions, conventional kernels should be preferred.. In the article, only classification is investigated. But other processing could also have been considered, e.g., regression [27] . The critical point is to be able to tune the hyperparameter efficiently, which it is feasible for regression. However, the actual optimization of the hyperparameters is demanding in terms of computations and it is sensitive to local minima. Therefore, a different strategy must be studied.. Perspectives of this work concern the development of new kernels using the HDDA-model. For instance, it is possible to define a new dot product for the conventional polynomial kernel:. Furthermore, a mixture of kernels using the HDDA could be used. From (5), the HDDA-Mahalanobis kernel can be considered as a product of several kernels. In the future, we will investigate the summation of kernels. Finally, a free-parameter alternative to the scree test for estimation of the intrinsic dimension must be addressed. For instance, an maximum likelihood estimator for HDDA exits and must be investigated [47] .

B I I O O O O B I I O O O B I I O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O B O O O O O O O O O O B I I O O O O O O O O O O B I I O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O B I I O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Restricted vertex multicut on permutation graphs Given an undirected graph and pairs of terminals the Restricted Vertex Multicut problem asks for a minimum set of nonterminal vertices whose removal disconnects each pair of terminals. The problem is known to be NP-complete for trees and polynomial-time solvable for interval graphs. In this paper we give a polynomial-time algorithm for the problem on permutation graphs. Furthermore we show that the problem remains NP-complete on split graphs whereas it becomes polynomial-time solvable for the class of co-bipartite graphs. One of the well-studied problems that fall in the area of cut and separation problems is the Multicut problem introduced by Hu in [14] . Given a graph G and a list L of pairs of vertices that are called terminals, the objective for the Multicut problem is to disconnect each terminal pair of the predefined list by removing a minimum set of edges or vertices of G. Problems of this kind arise from areas concerning with the reliability and robustness of network communications [6] . The Multicut problem is NP-complete [7] and several algorithms that approximate a solution on general or restricted graph classes are known [5, 9, 16] , while the parameterized version was proved to be fixedparameter tractable only very recently [3, 17] . This problem includes as a special case the Multiway Cut problem where instead of the list L we are given a set of terminals that need to be pairwise separated from each other; see [7, 11, 15] . Depending on the multicut, that is, the set of vertices or edges whose deletion disconnects each terminal pair, there are three variations of the problem.. The history of multicut problems begins with the edge variation, known as the Edge Multicut, that allows only the removal of edges. If L contains at most two terminal pairs the Edge Multicut problem admits a polynomial-time algorithm [21] whereas for at least three terminal pairs it becomes NP-hard [7] . Furthermore Edge Multicut remains NP-hard even when the input graph is a star (tree of height 1) [10] and therefore excluding any possible polynomial solution on many interesting graph classes. Similar to the Edge Multicut is the Vertex Multicut problem in which one is only allowed to remove vertices of the input graph.. As it was introduced by Cǎlinescu et al. in [5] the Vertex Multicut problem splits into two subproblems depending on whether one is allowed to remove terminal vertices. NP-c means NP-complete, P means polynomial-time algorithm, the asterisk * indicates that the result is obtained here, the pair of parenthesis ( ) means that the complexity is obtained from graph inclusion relationships, and ? stands for unknown complexity.. The Unrestricted Vertex Multicut problem refers to finding any minimum set of vertices of G whose removal separates each terminal pair of L whereas the Restricted Vertex Multicut refers to minimizing a set of nonterminal vertices for the same objective. Unrestricted Vertex Multicut is NP-hard on interval graphs [13] , graphs with bounded treewidth [5] , and planar graphs of bounded degree [5] . From the positive side there is a polynomial-time algorithm for the latter problem on trees [5] . Looking at the line graph of G (that is, the graph representing the adjacencies between the edges of G) an interesting reduction shows that the vertex variant is more general than the edge variant [5] . More precisely, consider an instance of Edge Multicut whose input is a graph G and a set of terminal pairs L, denoted by (G, L). Construct the line graph of G, denoted by L(G) and construct the set of terminal pairs L ′ which contains for each pair (s, t) of L all pairs (e i , f i ) such that e i has s as endpoint and f i has t as endpoint. Then it is known that a solution for the Edge Multicut on (G, L) is a solution for the Unrestricted Vertex Multicut on (L(G), L ′ ) and vice versa [5] . Observe that the line graph of a star graph is a complete graph and, therefore, Unrestricted Vertex Multicut on complete graphs is NP-complete. Thus for many interesting and even constraint graph classes (e.g., cographs, split graphs, and co-bipartite graphs) the Unrestricted Vertex Multicut problem is already NP-hard. Here we focus on the Restricted Vertex Multicut problem. Although many optimization problems that are NP-hard on arbitrary graphs are polynomially solvable on restricted graph classes [4, 12] , not much seems to be known for the restricted variation of multicut on particular graph classes. The problem admits a polynomial-time algorithm for interval graphs [13] , whereas it becomes NP-hard for trees [5, 13] and, thus, for chordal graphs which form a proper superclass of interval graphs. Therefore it is interesting to study the complexity of the Restricted Vertex Multicut problem on graph classes that are characterized without induced trees and are not included in the class of interval graphs.. In this paper we consider the restricted version of the multicut problem on split graphs, permutation graphs, and other related graph classes. Split graphs form a proper subclass of chordal graphs and such graphs are unrelated to interval graphs [12] . We prove that the problem remains NP-hard on split graphs. A natural superclass of interval graphs is the class of co-comparability graphs (complements of comparability graphs) where the complexity of the Restricted Vertex Multicut problem is still unknown. Co-bipartite (complements of bipartite graphs) and permutation graphs are two unrelated subclasses of co-comparability graphs [4, 12] . Interestingly most problems that are hard on co-comparability graphs are already hard on co-bipartite graphs. Here we show that the problem admits a simple and efficient (polynomial-time) solution on co-bipartite graphs and therefore excluding such an approach through a hardness result on co-bipartite graphs. Our main result is a polynomial-time algorithm for the class of permutation graphs. To do so, we take advantage of the notion of scanlines already efficiently applied for other problems such as treewidth and minimum fill-in on permutation graphs [1, 19] . We also give an independent result for cographs that can been seen as a special case of the polynomial-time algorithm on permutation graphs. An overall picture of our results is depicted in Figure 1 . A future research direction is resolving the computational complexity of Restricted Vertex Multicut for larger classes of permutation graphs. It seems that the proposed algorithm for permutation graphs has the potential of being generalizable to larger graph classes, such as trapezoid, or more general, d-trapezoid graphs, or co-comparability graphs of bounded dimension. Note that these graph classes already have an established scanline notion [2] . Moreover permutation graphs are exactly the graphs that are both comparability and co-comparability [4, 12] . For comparability graphs it follows that the problem is NP-complete since every tree is a comparability graph. In the case of co-comparability graphs the complexity status remains open.. Furthermore the characterization given in Corollary 2.4 for P 4 -free graphs implies that the solution consists of a separator set S for each pair of terminals without taking into account its (minimum) cardinality. As an interesting dichotomy result notice that split graphs do not contain P 5 or larger chordless paths and, therefore, Restricted Vertex Multicut remains NP-complete for the class of P k -free graphs for k ≥ 5 whereas due to Corollary 2.4 the problem is polynomial-time solvable for k ≤ 4. For split graphs it is interesting to determine the maximum number k > 2 of terminal pairs for which the problem admits a polynomial solution. We note that both problems Edge Multicut and Restricted Vertex Multicut for general graphs on at least 3 terminal pairs are NP-complete [7, 13] .

O O O O B I O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O B I O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O B I O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O B I B I O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O B I O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Entanglement dynamics of two-particle quantum walks This paper explores the entanglement dynamics generated by interacting two-particle quantum walks on degree-regular and -irregular graphs. We performed spectral analysis of the time-evolution of both the particle probability distribution and the entanglement between the two particles for various interaction strength. While the particle probability distributions are stable and not sensitive to perturbations in the interaction strength, the entanglement dynamics are found to be much more sensitive to system variations. This property may be utilised to probe small differences in the system parameters. Entanglement dynamics has been extensively investigated in the context of quantum chaos and quantum-classical correspondence. Several nonlinear models have been examined for their entanglement dynamics, for example, the N -atom Jaynes-Cummings model [1] [2] [3] , coupled kicked tops [4] [5] [6] [7] , the Dicke model [8] , and Rydberg molecules [9, 10] . Such studies have focused on the connection between the dynamics of classically chaotic systems with the dynamical generation of entanglement in their corresponding quantised systems.. In this work, we study the entanglement dynamics in two-particle discrete-time quantum walks on degree-regular and -irregular graphs. Single-particle quantum walks have already emerged as a useful tool in the development of quantum algorithms [11] [12] [13] [14] [15] [16] [17] [18] . These algorithms depend on interference between the multiple paths that are simultaneously traversed by the quantum walker. In multi-particle quantum walks, the dimension of the state space increases exponentially with the number of particles and there is the additional resource of interaction and entanglement between particles, which may be utilised to develop efficient quantum algorithms to solve practically significant applications.. The theoretical study of discrete-time quantum walks with more than one particle was initiated by Omar et al. [19] , who considered non-interacting two-particle quantum walks on the infinite line. Omar et al. established a role for entanglement in two-particle quantum walks by showing that initial states which are entangled in their coin degrees of freedom can generate two-particle probability distributions in which the positions of the two particles exhibit quantum correlations. Also with non-interacting particles,Štefaňák et al. [20] considered the meeting problem in the discrete-time quantum walk.. In addition to these studies with distinguishable particles, there have also been theoretical investigations of two-photon quantum walks [21, 22] . By increasing the dimension of the coin Hilbert space to incorporate both the direction of photon propagation and polarisation, Pathak and Agarwal [21] introduced a quantum walk in which two photons initially in separable Fock states become entangled through the action of linear optical elements. While the quantum walk studied in [19, 20] requires entangled initial states to generate spatial correlations, the two-photon walk studied in [21] , with its larger coin space, is capable of generating entanglement even from initially separable states. Venegas-Andraca and Bose [23] have also proposed a variant of the two-particle quantum walk in which the particles have a shared coin space. In this system, entanglement is introduced between the spatial degrees of freedom of the particles by performing measurements in the shared coin space.. More recently, Berry and Wang [24] introduced two spatial interaction schemes, which have shown to dynamically generate complex entanglement between the two walking particles.. Rohde et. al.. [25] studied the entanglement dynamics in a single-particle quantum walk between its position states on a finite line with bound and reflecting boundaries. They demonstrated the emergence of quasi-periodicity in the entanglement time series.. In this work, we explore the detailed entanglement dynamics generated by locallyinteracting two-particle quantum walks on degree-regular and -irregular graphs. Our key purpose is to demonstrate that entanglement dynamics arising from two-particle interacting quantum walks on graphs with simple underlying structures can be complex and sensitive to perturbations in the interaction strength parameter. This work provides strong evidence through sufficiently simple examples where the evolution of the system wavefunction is stable and not sensitive to perturbations to the interaction strength, while the entanglement dynamics of the system is much more sensitive to these small changes. Even in cases where the system wavefunction appears to evolve in a more complex manner, the marginal probability distribution is stable compared to the entanglement between the two particles.. Both degree-regular and degree-irregular graphs are studied in this work. Degree-regular graphs are graphs for which every vertex has the same number of neighbours. We studies three simple degree-regular graphs, including the complete K 8 graph (without self-loops), the 3-dimensional hypercube Q 3 , and the joined 2nd generation 3-Cayley tree. The degreeirregular graphs studied are variations of these degree-regular graphs, with several edges removed. This paper is structured as follows. In Section II we describe the mathematical formalism for the interacting two-particle quantum walks and the evaluation of entanglement between the two particles. In Section III, we perform spectral analysis on the time-evolution of probability distribution and entanglement time series of two-particle discrete-time quantum walks for different values of interaction strength. Also presented are the corresponding Feigenbaum diagrams of the frequencies in the time series. Finally, Section IV contains our conclusions. In conclusion, we have demonstrated that while the probability time series for simple degree-regular and degree-irregular graphs such as the K 8 graph and the 2nd generation 3-Cayley tree are generally stable regarding small changes to interaction strength, complex entanglement dynamics can arise from a locally-interacting two-particle quantum walk, and and φ = 0.76π (red, dashed) for a two-particle quantum walk under φ-Grover interaction beginning in an equal superposition initial state.

B I O O O O O O O O B I O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O B I O O O O O O O O O O B I O O O O O O O O O O