Papers

Let $G$ be a simple graph. A dissociation set of $G$ proposed by Yannakakis in $1981$ is defined as a set of vertices that induces a subgraph in which every vertex has a degree of at most $1$. A dissociation set is maximal if it is not contained as a proper subset in any other dissociation set. In $2025$, Wang et al.\cite{ZiyuanWang} established that for any tree $T$ of order $n\geq 4$, the number of maximal dissociation sets in $T$ is at most $3^{\frac{n-1}{3}}+\frac{n-1}{3}$ and characterized the extremal trees attaining the upper bound. They also proposed a conjecture about the upper bound of the maximal dissociation set. In this paper, we consider this conjecture and show that the maximum number of maximal dissociation sets in a tree of order $n(n\geq 3)$ is $g(n)$, where \[ g(n) = \begin{cases} n, & n=3,4,5,6,\\ 3^{\frac{n-1}{3}}+\frac{n-1}{3}, & n \equiv 1 \pmod{3},~n\geq7,\\ 4\cdot 3^{\frac{n-5}{3}}+n-5, & n \equiv 2 \pmod{3},~n\geq8, \\ 16\cdot 3^{\frac{n-9}{3}}+3n-25, & n \equiv 0 \pmod{3},~n\geq12~\text{and }~n\neq21, \\ 19, & n=9, \\ 1349, & n=21. \end{cases} \] We also characterize the extremal trees with the maximum number of maximal dissociation sets.

We give a geometric description of nilpotent representations of equioriented cyclic quivers over a field via an explicit bijection. Over a finite field, this yields formulas for the number of nilpotent representations and for the probability that a representation is nilpotent. We also give a refinement by rank. Over the Boolean semiring, we identify nilpotent semirepresentations with directed acyclic graphs, derive a recursive formula for their number, and determine the asymptotic decay rate of the probability of nilpotence.

Fluid intelligent metasurfaces (FIMs) reshape antenna geometry to improve angular sensitivity. This paper studies their role in cell-free integrated sensing and communication, where distributed access points provide distinct observation directions. Under calibrated path gains, we derive the conditional angular Fisher information matrix, retain cross-AP coupling, and map it to a position-domain Cramér–Rao bound (CRB). A diagonal-information specialization exposes the dependence on angular diversity, while a multi-target extension quantifies the loss from jointly estimating other target positions. We use CRB-driven coordinate search to design transmit and receive shapes and evaluate sensing performance, equal-resource AP scaling, and MRT-based communication–sensing allocation. In the considered four-Tx/four-Rx configuration, shape design reduces the position root CRB from 0.300 to 0.227 m at 10 dB reference SNR. Two-target results further illustrate how cross-target coupling increases the joint localization bound at small separations.

The purpose of this paper is to study the $U$-module structure of the embedded contact homology of a class of prequantization circle bundles over closed symplectic surfaces. We assume that the Euler number of each circle bundle is strictly less than the Euler characteristic of the corresponding surface. We give an explicit description of the $U$-map with respect to a basis of filtered ECH defined via cobordism maps. We use the formula for the $U$-map to compute the ECH spectrum and, as an application, the ECH capacities of certain complements of symplectic divisors. Furthermore, we show that the ECH capacities of closed integral symplectic manifolds are infinite.

For a graph $G$, Schur positivity of $X_G$ implies that $G$ is strongly nice, and every strongly nice graph is nice. We construct three infinite families separating these properties. We first give a connected family $F_t$, $t\ge6$, that is strongly nice but not Schur positive. We then prove that homogeneous strongly nice symmetric functions with nonnegative monomial coefficients are closed under multiplication, and hence that strongly nice graphs are closed under disjoint union. As an application, for $H=K_{3,3}-e$, the graphs \[ M_t=H\sqcup K_t,\qquad t\ge3, \] form a disconnected family that is strongly nice but not Schur positive. Finally, we define \[ N_r=K_r\vee(K_2\sqcup2K_1),\qquad r\ge2, \] and prove that every $N_r$ is connected and nice but not strongly nice. We also introduce the level-$k$ nice property and show that the level depth of $N_r$ is $4r!$.

The Karush–Kuhn–Tucker (KKT) residual is a fundamental measure of first-order optimality and, under an error bound condition, is comparable to the distance to the KKT solution set up to constant factors. Despite the $O(K^{-1})$ ergodic rates known for objective error and feasibility violations, we show that the KKT residual of classical ADMM cannot, in general, satisfy a uniform $O(K^{-1})$ bound. Specifically, we construct a fixed-dimensional, horizon-dependent family of two-block convex optimization problems for which the KKT residual is $\Omega(K^{-1/2})$ at both the last iterate and the equal-weight ergodic average at the prescribed horizon $K$. Consequently, a uniform $O(K^{-1})$ KKT residual bound is impossible for either output.

We introduce a general framework for composite inclusion problems with affine constraints, covering both monotone and semimonotone regimes. The central idea is a new interpretation of the constrained inclusion through an operator-vector pair that separates the implicit inclusion from the affine constraint: the former is handled through possibly preconditioned resolvent evaluations, while the latter is handled through an explicit forward step in an auxiliary variable. This yields a single abstract iteration applicable to multioperator inclusions, linearly coupled inclusions, and block-separable inclusions with affine constraints. The freedom in choosing the operator-vector pair enables the systematic construction of problem-adapted splitting algorithms, including new schemes for several important problem classes. Exploiting the orthogonal decomposition induced by the constraint subspace, we develop a unified and streamlined convergence analysis and establish weak and strong convergence guarantees under semimonotonicity assumptions. When specialized to multioperator inclusions, the framework permits general bounded linear operator coefficients, rather than only scalar coefficients, and therefore accommodates preconditioned resolvents. The resulting schemes recover several existing methods while extending them to previously uncovered regimes, and in several important cases, require weaker assumptions and admit provably larger admissible parameter ranges.

Let \(p\colon \mathcal{A}\to G\) be a Fell bundle over a locally compact Hausdorff second countable étale groupoid \(G\) and let \(\mathrm{C}^*(G; \mathcal{A})\) be the associated Fell bundle \(\mathrm{C}^*\)-algebra. Suppose \(\sigma^c\) is the real dynamics on \(\mathrm{C}^*(G;\mathcal{A})\) induced by a real-valued \(1\)-cocycle \(c\). We establish an affine homeomorphism between the \(\sigma^c\)-ground state space of \(\mathrm{C}^*(G;\mathcal{A})\) and the state space of \(\mathrm{C}^*(G(Z); \mathcal{A}|_{G(Z)})\), where \(Z\) is the boundary set of \(c\) and \(G(Z)\) is the boundary groupoid. In particular, \(\mathrm{C}^*(G; \mathcal{A})\) admits \(\sigma^c\)-ground states if and only if \(Z\neq \emptyset\). We further investigate the relation between ground states and KMS\(_{\infty}\) states and give a crystallization interpretation of the boundary \(\mathrm{C}^*\)-algebra \(\mathrm{C}^*(G(Z); \mathcal{A}|_{G(Z)})\) when the cocycle is locally constant. Finally, we apply our results to several classes of examples, including twisted \(\mathrm{C}^*\)-algebras of Deaconu–Renault groupoids and twisted higher-rank graph \(\mathrm{C}^*\)-algebras, obtaining explicit descriptions of their ground states.

For complex \(m\)-homogeneous polynomials, let \(D_m\) denote the optimal dimension-free constant in the polynomial Bohnenblust–Hille inequality. We prove that, for every \(B>1/2\), there exists \(K_B>0\) such that \[ D_m\le K_B m^B,\qquad m\ge1. \] More precisely, we obtain the square-root-scale estimate \[ D_m\le \sqrt m\,\exp\!\bigl(C\sqrt{\log m}\,\log\log m\bigr). \] The proof combines coefficient-preserving degree reduction with a weighted bootstrap, separating balanced and dominant degree profiles. We also determine the critical linear-dimensional asymptotics: \[ D_{m,n_m}\longrightarrow 2 \qquad\text{whenever}\qquad \frac{n_m}{m}\longrightarrow1, \] and consequently \(\liminf_{m\to\infty}D_m\ge2\). Further applications give two-sided bounds for homogeneous Sidon constants and a quantitative logarithmic remainder in the multidimensional Bohr-radius asymptotic.

We prove that for every ergodic transformation $T$ on an infinite standard probability space both the automorphism group (i.e. centralizer) $\mathrm{Aut}(T)$ and its reversing automorphism group are realizable as symmetry groups of graphings. This is an analogue of Sabidussi's realization of arbitrary graph-automorphism groups, and provides numerous examples of graphing automorphism groups carrying no compatible compact topology, answering a question of Lovasz'. Another consequence of discussion and ensuing constructions is the existence of large mutually locally-globally equivalent graphing families with highly variable symmetry.

Let $L$ be a finite-dimensional Lie algebra over a field $F$. The comaximal graph $\Gamma(L)$ has as vertices the proper nonzero subalgebras of $L$, two of them adjacent whenever they generate $L$; its structure was previously classified for Lie algebras of dimension at most 3 over finite fields. Here we extend that work in two directions. First, we obtain explicit formulas for the number of triangles $t(\Gamma(L))$ for every three-dimensional Lie algebra over $\F_q$. Second, we extend the classification to several four-dimensional families over $\mathbb{F}_q$, the abelian, Heisenberg, and filiform algebras, and $\mathfrak{gl}_2(\F_q)$. We also relate graph-theoretic properties of $\Gamma(L)$, such as completeness and the role of the Frattini subalgebra, to structural properties of $L$, including supersolvability. These results yield new combinatorial invariants for finite-dimensional Lie algebras over finite fields.

Let $k\in\mathbb N$ satisfy $1<k<n/2$, and let $\Sigma^p\subset\Sn^n$ be a closed smooth embedded submanifold. We prove that a complete conformal metric $g=v^{-2}g_{\Sn^n}$ on $\Sn^n\setminus\Sigma$ satisfying $\lambda(g^{-1}A_g)\in\Gk$ and $\sigma_k(g^{-1}A_g)=\kappa>0$ must obey \[ p\leq p_k(n), \] where $p_k$ is the model threshold determined by $\Hh^{p+1}\times\Sn^{n-p-1}$. When $k=2$ and $n=m^2$, we construct a smooth complete equality example on $\Sn^n\setminus\Sn^{(m^2-m-2)/2}$. We also prove that the strict inequality $p<p_k(n)$ holds under a finite positive linear-contact hypothesis.

This cumulative habilitation thesis studies probabilistic circuits (PCs) as a powerful and tractable framework for reasoning and learning under uncertainty in artificial intelligence (AI). It first advocates for probability as a core language for AI, emphasizing its connections to logic and information theory; the conceptual simplicity of probabilistic reasoning—based primarily on the sum and product rules; the parallels between probabilistic inference and human cognition; and the role of probability in optimal decision making. However, probability also faces significant computational challenges, as probabilistic inference is NP-hard in almost all probabilistic models. PCs address these challenges through structural constraints that ensure exact computation of a wide range of inference queries in polynomial time, such as marginals, conditionals, most probable explanations, expectations, and more advanced inference tasks. This thesis synthesizes a decade of research across foundations, algorithmic developments, and empirical validation of PCs. Key contributions highlighted in this work are foundational theory of PCs, Bayesian approaches for learning PCs, scalable implementations and integration with deep learning, hybrid models that combine PCs with intractable models, and connections with symbolic machine learning paradigms. This is the first part of my Habilitation Thesis. The second part is omitted, as it comprises the cumulative part of the thesis and has been published at various venues (see Chapter 5).

In this manuscript, we establish sharp bounds of the third-order Toeplitz determinant and a particular case of the generalized Zalcman functional for a class of holomorphic mappings defined on the unit ball in a complex Banach space. The obtained estimates yield corresponding bounds for several subclasses of starlike mappings as special cases and also provide higher-dimensional extensions of certain known results from the classical one-dimensional theory.

We present a systematic study on the role of invariant connections in geometric mechanics. We first develop a comprehensive theory of reduction under left- and right-invariant connections on Lie groups, showing that the Euler–Poincaré and Lie–Poisson equations are independent of the connection. We then introduce a novel connection-dependent variational principle, where the Lagrangian depends on the velocity parallel-transported back to the initial point of the curve. For Cartan–Schouten connections, this leads to an integro-differential Euler–Poincaré equation that exhibits two distinct sources of nonlocality in time: a path-dependent term encoded in the parallel transport and a future-dependent term arising from a curvature integral. We reformulate this equation as a two-point boundary value problem and present a two-level numerical scheme. The general theory is illustrated on the Heisenberg group, where the equations simplify due to nilpotency, and on the rotation group, where the full integro-differential structure is retained.

We define a family of Turaev-Viro type invariants for hyperbolic $3$-manifolds with cusp ends, extending the invariants introduced in \cite{LMSWY} for hyperbolic $3$-manifolds with totally geodesic boundary. These invariants are constructed from what we call the ideal $\mathrm{U}_{q\tilde q}\mathfrak{sl}(2;\mathbb R)$-$6j$ symbols, which are variants of the $6j$-symbols associated with the positive representations of the modular double of $\mathrm{U}_q\mathfrak{sl}(2;\mathbb R)$. We also prove that these invariants decay exponentially, with the exponential decay rate determined by the hyperbolic volume of the manifold.

We prove a measurable realization theorem for hyperuniform Delone point processes. In dimensions \(d\geq2\), for every prescribed \(q\geq1\), every essentially free ergodic p.m.p.\ action of \(\mathbb R^d\) admits, at every sufficiently large prescribed intensity, a generating Delone realization whose return-time point process \(\eta\) is measurably isomorphic to the original action and whose Bartlett spectrum \(\sigma_\eta\) satisfies \[ \sigma_\eta(B_\varepsilon)=o(\varepsilon^{2q}) \qquad(\varepsilon\downarrow0). \] Thus arbitrarily high finite-order low-frequency suppression can be imposed without changing the prescribed measurable dynamics. The same realizations can be chosen with surface-order ball variance and linear rigidity to any prescribed finite order, while also being maximally rigid and almost surely bounded-displacement equivalent to a lattice. For essentially free Euclidean-motion actions whose translation subaction is ergodic, the construction can be made isotropic and \(V\)-ergodic, and hence \(V\)-weakly mixing. In dimension one, every essentially free ergodic flow admits generating Delone realizations with logarithmic interval discrepancy, maximal rigidity, and near-quadratic decay of the Bartlett spectrum at the origin.

We identify a non-Hermitian counterpart of dynamical tunnelling induced by a localised absorbing region in a symmetric confining potential. The absorbing region, described by an imaginary potential barrier, induces an effective double-well structure with a pair of long-lived states that have little population in the barrier. In contrast to the periodic tunnel oscillations through a real barrier, the system slowly evolves towards equal population of the two sides, while the absorbing region remains only weakly populated. We demonstrate this generic mechanism in two continuous potentials, and capture its essence analytically in a minimal $3\times 3$ matrix model. We propose a realistic optical waveguide setup for the experimental observation of the effect.

In classical information theory, a source is modeled by a single, precisely known probability distribution. However, in the increasingly complex communication networks full of unanticipated, nonstationary, and heterogeneous random events, the assumption of precise and well-defined probability distributions to describe random variables appears somewhat idealized. Therefore, it is important to characterize the uncertainty of distributions of source messages, subject to relaxing the assumption of deterministic probability models for analyzing information sources in information theory. Based on the nonlinear expectation theory, a novel axiomatical system that extends classical probability theory, this paper investigates the information sources whose distributions themselves are uncertain, and refers to them as uncertain-distribution sources. We generalize the fundamental concept information entropy to nonlinear information entropy, which describes the measurement of the amount of information contained in a uncertain-distribution source. By using the strong law of large numbers under sublinear expectation, we establish a nonlinear source coding theorem, which not only shows that the nonlinear information entropy is the upper bound for the infimum of achievable coding rate of uncertain-distribution sources under the maximum error probability criterion, but also determines a cluster point of the coding rate of uncertain-distribution sources under the minimum error probability criterion. Our findings reveal that the introduction of nonlinear expectation theory allows for a more comprehensive understanding of information sources.

We consider single-facility median location problems on graphs where two conflicting cost values are associated with the edges. As an example, suppose that a decision maker wants to locate one new facility, e. g., a pizza delivery place, which uses bicycles for delivery. The two objective functions could then be the total traveling time and the total number of left turns, as the latter are very risky. Path choices then depend on the preferences of the decision maker, and in general, there may not exist a unique optimal path between a customer and a new facility location. In this paper, we consider the location decision and the routing decision in a coupled problem, i. e., we search for an optimal location and for consistent delivery paths simultaneously. We introduce the concept of consistent paths, where we assume that the choice of a path from the facility to a demand node implies certain preferences. All paths of a solution are consistent if the preferences of all paths do not contradict each other. We present an algorithm that computes a minimum complete set of efficient solutions with consistent path choices and illustrate the results at example instances in the city of Wuppertal in Germany.

It is an open problem to determine whether surface groups can embed in Thompson's group $V$. We prove that any finitely generated subgroup of Thompson's group $T$ without abelian groups of arbitrary high rank is either virtually abelian or virtually free. In particular, surface groups don't embed in $T$, answering a question of Belk and Moore.

In Vilenkin's group we present an algorithm for constructing a step scaling function with a given support and a constant on given cosets.

We study the dynamic behavior of the free boundary of the Hele-Shaw flow with a nonnegative source, in which the pressure satisfies $-\Delta p=c$ in $\{p>0\}$ and the free boundary moves with normal velocity $V=|\nabla p|$. For initial domains satisfying an interior ball condition, we prove that the patch $\{p>0\}$ expands at a locally positive rate, so that its hitting time is locally Lipschitz; this is optimal, since distinct portions of the patch may collide. We also show that the free boundary is locally either regular, or of collision type, or undergoing parabolic-scale hole closings of various dimensions: such a space-time characterization of singularities is new in Hele-Shaw flow. We further show that, near any time at which no collision occurs, the space-time free boundary is a $C^1$ hypersurface. This implies that at such times every free boundary point has a space-time normal: the space-time unit normal is parallel to the time direction precisely where the spatial free boundary has singular geometry. The results apply in particular to the tumor growth model with nutrients introduced by Perthame, Quirós, and Vázquez, but the analysis of singularities is new even for the classical injection problem.

A system with two periodic directions carries two commuting holonomies \(a=(u,v)\in\R^2/\Z^2\). We determine their distinguished values while separating lattice, arithmetic, and observable effects. Maximizing the lowest twisted eigenvalue places \(a\) at a deep hole of the momentum lattice. For every rectangular torus the maximizer is antiperiodic, so complex multiplication is sufficient for torsion optima but not necessary. Let \(G_\tau^{-1}\) be the dual metric and \(D_\tau(a)\) the normalized zeta determinant of the twisted Laplacian. At the rotation-fixed deep holes of the square and hexagonal lattices, symmetry gives the exact determinant response \(-\operatorname{Hess}_a\log D_\tau=2\pi(\Im\tau)G_\tau^{-1}\). With spectral wavenumber \(\kappa=2\pi\), the lowest manifolds are fourfold and threefold, with gaps \(2\kappa^2\) and \(4\kappa^2/3\); phase errors split them linearly while their centroids remain stationary. We then give a finite-device realization: an \(8\times8\) microring lattice closed by two phase-controlled seams. At a reported coupling scale of \(16\) GHz, its exact square-lattice spectrum has a \(17.32\) GHz shell gap and a \(1.92\) GHz doublet separation for a \(0.1\) holonomy error; a triangular-link configuration gives a threefold qutrit manifold with an \(18.11\) GHz gap. This is a quantitative spectroscopy proposal, not a claim of topological protection or a completed device.

Spectral characterization of graphs for various graph matrices constitutes a central topic in spectral graph theory. Let $G$ be a graph with adjacency matrix $A(G)$, diagonal degree matrix $\Deg(G)$, distance matrix $D(G)$, and transmission matrix \(\Trs(G)\), respectively. Recently, Alfaro and Zapata (2024) introduced the degree-distance matrices \(\Ddegp(G)=\Deg(G)+D(G)\) and \(\Ddeg(G)=\Deg(G)-D(G)\), together with the transmission-adjacency matrices \(\Atrsp(G)=\Trs(G)+A(G)\) and \(\Atrs(G)=\Trs(G)-A(G)\). Based on computational evidence for trees on at most \(20\) vertices, they conjectured that all trees are determined by the spectra of \(\Ddegp\) as well as \(\Ddeg\). In this paper, we disprove these conjectures by constructing an infinite family of pairs of non-isomorphic trees. More precisely, for each integer \(r\ge 3\), we construct a pair of trees on \(17r-15\) vertices which are simultaneously cospectral with respect to the following six matrices \[ A,\quad L,\quad Q,\quad D,\quad \Ddegp,\quad \Ddeg . \] The construction is based on an \(r\)-regularized leaf extension and an equitable-partition reduction. We also record a simple sign-switching observation for transmission-adjacency matrices: if \(G\) is bipartite, then \(\Atrs(G)\) and \(\Atrsp(G)\) are similar via a diagonal \(\{\pm1\}\)-matrix and have the same Smith normal form. Consequently, for trees, the spectral and Smith normal form problems for \(\Atrs\) and \(\Atrsp\) are equivalent.

Let $G$ be the affine Kac–Moody group of type $\widetilde A_{n-1}$, $B$ be an Iwahori subgroup in $G$, $\mathcal{F}=G/B$ be the flag variety, and $W$ be the Weyl group of $G$. Given distinct involutions $w_1$, $w_2\in W$, we prove that the tangent cones $C_{w_1}$, $C_{w_2}$ to the corresponding Schubert subvarieties $X_{w_1}$ and $X_{w_2}$ of $\mathcal{F}$ at the point $p=e\mod B$ do not coincide as subvarieties of the tangent space to $\mathcal{F}$ at the point $p$. This generalizes similar results in the finite-dimensional setting. The main technical tools we used are combinatorics of the embeddings of the Weyl groups of different ranks and coadjoint orbits for the unipotent radical of the group $B$.

We construct $U(1)$-symmetric $SU(2)$ instantons on the Taub-NUT space with generic holonomy, and instanton number and magnetic charge $(k,m)=(2,0)$, $(1,1)$, and $(2,-1)$. We construct these instantons via the bow construction, a Nahm-like non-linear transformation which identifies them with a set of bow data solving an integrable ODE and a series of algebraic relations on an interval. We show how the bow data recovers the data for all corresponding $U(1)$-symmetric instantons on flat $\mathbb{R}^4$ and calorons on $\mathbb{R}^3\times S^1$ in various limits, and we demonstrate the existence of Taub-NUT instantons that do not limit to $\mathbb{R}^4$ instantons nor calorons.

We introduce Cayley-tree POTP, obtained by imposing the pseudo-orbit equations of a finitely generated group action only along a spanning tree of a Cayley graph. For zero-dimensional actions, we characterize this property by equicontinuity along normalized replacement paths; for subshifts, the criterion is expressed in the right-coset space of the common left-period subgroup. These criteria characterize virtual freeness and, for commensurated subgroup pairs, identifies Cayley-tree POTP of the coset full shift with relative quasi-tree geometry and a finite Bass–Serre decomposition. For infinite-index VFP pairs, Cayley-tree POTP of the coset full shift is equivalent to virtual cohomological codimension one, although ordinary POTP holds for every such shift. Finally, we prove that the strong topological Rokhlin property passes to finite-index overgroups. Consequently every finitely generated virtually free group has this property, answering the virtually cyclic case posed by Doucha, and Cayley-tree POTP is generic for its Cantor actions.

The Porous Medium Equation (PME), given by $u_t = \Delta(u^m)$ for $m > 1$, is a degenerate nonlinear parabolic partial differential equation that arises in various physical applications such as fluid flow in porous media, heat transfer in plasmas, and population dynamics. It is known for its nonlinear diffusion and finite propagation speed. In this paper, we study numerical solutions of the one-dimensional direct and inverse PME using Physics-Informed Neural Networks (PINNs), and compare them with classical numerical methods and available analytical and manufactured solutions. While PINNs provide a flexible framework for solving both forward and inverse problems, we show that the standard inverse formulation suffers from a strong sensitivity to the initial guess, leading to only local convergence. To address this issue, we propose a novel two-stage PINN training framework for the inverse problem, which significantly improves convergence stability and allows reliable recovery of the unknown parameter even for poor initial guesses. Overall, the proposed approach demonstrates that PINNs are a flexible and accurate alternative to classical methods for the 1D PME, and the introduced two-stage training strategy substantially improves their robustness in inverse problems, providing a solid basis for extensions to more complex geometries and higher-dimensional cases.

We propose and analyze a time-staggered numerical scheme for mean field game (MFG) systems with individual noise on finite graphs. Numerically solving such coupled forward–backward systems is delicate because the density evolves in the open probability simplex and the coefficients may degenerate at its boundary. The scheme preserves mass and satisfies a discrete fundamental identity compatible with the Lasry–Lions monotonicity argument, leading to uniqueness of the numerical solution. By establishing a timestep-uniform positive lower bound for the density and uniform bounds for the value variable, we prove first-order convergence for every interior discrete solution. For potential MFGs, we establish a variational characterization by identifying the scheme with the KKT system of a convex discrete action, yielding existence of the discrete solution and an optimization-based realization. The resulting optimization problem is solved by a feasible primal–dual Newton method in mass-preserving coordinates. Numerical experiments confirm the predicted convergence rate and illustrate topology-dependent transport and congestion-driven route choice.