arXiv subjects

Marcel Blattner

Publications and source records attributed to Marcel Blattner.

10 recordsLinked to original sources

arXiv2026

Quot-Stack Moduli and Transverse Deformations of Graded Metabelian Lie Algebras

We identify the graded metabelian locus inside the deformation theory of positively graded Lie algebras. Let $M(U)$ be the free metabelian Lie algebra on $U$ and $B(U)=M(U)'$. For every finite graded rank vector $h$, we prove that the moduli stack of such algebras is equivalent to the quotient stack $[\mathrm{Quot}^{\mathrm{gr}}h(B(U))/\mathrm{GL}(U)]$. At a quotient $B(U)\to C$ with kernel $N$, its tangent complex is the two-term complex from $\mathrm{End}(U)$ to $\mathrm{Hom}{\mathrm{Sym}(U)}(N,C)_0$. For every algebra $\mathfrak{g}$ in this stack, restriction to $\Lambda^2\mathfrak{g}'$ induces a defect map on $H^2_0(\mathfrak{g};\mathfrak{g})$, and we prove that its kernel is $H^0$ of the Quot-stack tangent complex. Thus a first-order deformation is tangent to the metabelian locus exactly when its derived--derived restriction vanishes. We also recover the inverse-system module degree by degree from intrinsic lower-central tensors; on the level locus its terminal tensor suffices. For the $14$-dimensional algebra attached to a regular pencil of binary quartics, exact computation gives $\dim H^2_0=11$ and $H^3_0=0$. Its effective miniversal graded deformation germ is formally smooth of dimension $11$, while its metabelian subgerm is formally smooth of dimension $3$. Terminal restriction identifies the $8$-dimensional normal space with derived--derived brackets. Using the pencil's classical $V_4$-symmetry, we determine its induced representation on the graded tangent space and show that four negative-weight primary obstruction maps are surjective.

math.RA↗

arXiv2025

Tangential Action Spaces: Geometry, Memory and Cost in Holonomic and Nonholonomic Agents

Living systems balance energetic efficiency with the capacity for path-dependent effects. We introduce Tangential Action Spaces (TAS), a geometric framework that models embodied agents as hierarchies of manifolds linked by projections from physical states to cognitive representations and onward to intentions. Lifts from intentions back to actions may follow multiple routes that differ in energy cost and in whether they leave memory-like traces. Under explicit assumptions, we prove: (i) if the physical-to-cognitive map is locally invertible, there is a unique lift that minimises instantaneous energy and yields no path-dependent memory; any memory requires strictly positive excess energy. (ii) If multiple physical states map to a cognitive state (a fibration), the energy-minimising lift is the metric-weighted pseudoinverse of the projection. (iii) In systems with holonomy, excess energy grows quadratically with the size of the induced memory for sufficiently small loops, establishing a local cost-memory law. These results motivate a classification of embodied systems by the origin of path dependence: intrinsically conservative, conditionally conservative, geometrically nonconservative, and dynamically nonconservative. Numerical examples illustrate each case. We also present a reflective extension (rTAS) in which perception depends on a learnable model state; a block metric formalises an effort-learning trade-off, and cross-curvature terms couple physical and model holonomy. Simulations of single- and two-agent settings show role asymmetries and sensitivity to coupling. TAS provides a geometric language linking embodiment, memory, and energetic cost, yielding testable predictions and design guidelines for biological and robotic systems.

eess.SY↗

arXiv2016

Continuous-Time Quantum Walks on Directed Bipartite Graphs

This paper investigates continuous-time quantum walks on directed bipartite graphs based on a graph's adjacency matrix. We prove that on bipartite graphs, probability transport between the two node partitions can be completely suppressed by tuning a model parameter $α$. We provide analytic solutions to the quantum walks for the star and circulant graph classes that are valid for an arbitrary value of the number of nodes $N$, time $t$ and the model parameter $α$. We discuss quantitative and qualitative aspects of quantum walks based on directed graphs and their undirected counterparts. Numerical simulations of quantum walks on circulant graphs show complex interference phenomena and how complete suppression of transport is achieved near $α=π/2$. By proving two mirror symmetries around $α=0$ and $π/2$ we show that these quantum walks have a period of $π$ in $α$. We show that undirected edges lose their effect on the quantum walk at $α=π/2$ and present non-bipartite graphs that exhibit suppression of transport. Finally, we analytically compute the Hamiltonians of quantum walks on the directed ring graph.

quant-ph↗

arXiv2013

The Role of Trends in Evolving Networks

Modeling complex networks has been the focus of much research for over a decade. Preferential attachment (PA) is considered a common explanation to the self organization of evolving networks, suggesting that new nodes prefer to attach to more popular nodes. The PA model results in broad degree distributions, found in many networks, but cannot explain other common properties such as: The growth of nodes arriving late and Clustering (community structure). Here we show that when the tendency of networks to adhere to trends is incorporated into the PA model, it can produce networks with such properties. Namely, in trending networks, newly arriving nodes may become central at random, forming new clusters. In particular, we show that when the network is young it is more susceptible to trends, but even older networks may have trendy new nodes that become central in their structure. Alternatively, networks can be seen as composed of two parts: static, governed by a power law degree distribution, and a dynamic part governed by trends, as we show on Wiki pages. Our results also show that the arrival of trending new nodes not only creates new clusters, but also has an effect on the relative importance and centrality of all other nodes in the network. This can explain a variety of real world networks in economics, social and online networks, and cultural networks. Products popularity, formed by the network of people's opinions, exhibit these properties. Some lines of products are increasingly susceptible to trends and hence to shifts in popularity, while others are less trendy and hence more stable. We believe that our findings have a big impact on our understanding of real networks.

physics.soc-ph↗

arXiv2013

Preferential Attachment in Online Networks: Measurement and Explanations

We perform an empirical study of the preferential attachment phenomenon in temporal networks and show that on the Web, networks follow a nonlinear preferential attachment model in which the exponent depends on the type of network considered. The classical preferential attachment model for networks by Barabási and Albert (1999) assumes a linear relationship between the number of neighbors of a node in a network and the probability of attachment. Although this assumption is widely made in Web Science and related fields, the underlying linearity is rarely measured. To fill this gap, this paper performs an empirical longitudinal (time-based) study on forty-seven diverse Web network datasets from seven network categories and including directed, undirected and bipartite networks. We show that contrary to the usual assumption, preferential attachment is nonlinear in the networks under consideration. Furthermore, we observe that the deviation from linearity is dependent on the type of network, giving sublinear attachment in certain types of networks, and superlinear attachment in others. Thus, we introduce the preferential attachment exponent $β$ as a novel numerical network measure that can be used to discriminate different types of networks. We propose explanations for the behavior of that network measure, based on the mechanisms that underly the growth of the network in question.

physics.soc-ph↗

arXiv2012

Recommendation systems in the scope of opinion formation: a model

Aggregated data in real world recommender applications often feature fat-tailed distributions of the number of times individual items have been rated or favored. We propose a model to simulate such data. The model is mainly based on social interactions and opinion formation taking place on a complex network with a given topology. A threshold mechanism is used to govern the decision making process that determines whether a user is or is not interested in an item. We demonstrate the validity of the model by fitting attendance distributions from different real data sets. The model is mathematically analyzed by investigating its master equation. Our approach provides an attempt to understand recommender system's data as a social process. The model can serve as a starting point to generate artificial data sets useful for testing and evaluating recommender systems.

physics.soc-ph↗

arXiv2012

B-Rank: A top N Recommendation Algorithm

In this paper B-Rank, an efficient ranking algorithm for recommender systems, is proposed. B-Rank is based on a random walk model on hypergraphs. Depending on the setup, B-Rank outperforms other state of the art algorithms in terms of precision, recall (19% - 50%), and inter list diversity (20% - 60%). B-Rank captures well the difference between popular and niche objects. The proposed algorithm produces very promising results for sparse and dense voting matrices. Furthermore, a recommendation list update algorithm is introduced,to cope with new votes. This technique significantly reduces computational complexity. The implementation of the algorithm is simple, since B-Rank needs no parameter tuning.

physics.data-an↗

arXiv2008

Heat Conduction Process on Community Networks as a Recommendation Model

Using heat conduction mechanism on a social network we develop a systematic method to predict missing values as recommendations. This method can treat very large matrices that are typical of internet communities. In particular, with an innovative, exact formulation that accommodates arbitrary boundary condition, our method is easy to use in real applications. The performance is assessed by comparing with traditional recommendation methods using real data.

physics.soc-ph↗

arXiv2007

When are recommender systems useful?

Recommender systems are crucial tools to overcome the information overload brought about by the Internet. Rigorous tests are needed to establish to what extent sophisticated methods can improve the quality of the predictions. Here we analyse a refined correlation-based collaborative filtering algorithm and compare it with a novel spectral method for recommending. We test them on two databases that bear different statistical properties (MovieLens and Jester) without filtering out the less active users and ordering the opinions in time, whenever possible. We find that, when the distribution of user-user correlations is narrow, simple averages work nearly as well as advanced methods. Recommender systems can, on the other hand, exploit a great deal of additional information in systems where external influence is negligible and peoples' tastes emerge entirely. These findings are validated by simulations with artificially generated data.

cs.IR↗

arXiv2006

Exploring an opinion network for taste prediction: an empirical study

We develop a simple statistical method to find affinity relations in a large opinion network which is represented by a very sparse matrix. These relations allow us to predict missing matrix elements. We test our method on the Eachmovie data of thousands of movies and viewers. We found that significant prediction precision can be achieved and it is rather stable. There is an intrinsic limit to further improve the prediction precision by collecting more data, implying perfect prediction can never obtain via statistical means.

physics.soc-ph↗