arXiv subjects

Dean Kraizberg

Publications and source records attributed to Dean Kraizberg.

4 recordsLinked to original sources

arXiv2026

Dynamic Averaging on Regular Graphs

We study a dynamic averaging process on finite regular graphs with bounded, time-varying load arrivals. At each discrete time $t$, an edge is chosen uniformly at random, a load $0\le w_t \le 1$ is introduced, and the total load of its two endpoints together with $w_t$ is divided equally between them. Starting from the flat configuration, we obtain a pairwise concentration bound governed by the effective resistance between vertices and use generic chaining to derive a general upper bound on the expected gap between the largest and smallest loads. As a consequence, we show that every $d$-regular graph has expected gap $O_d(\sqrt n)$, uniformly in time and over all deterministic arrival sequences. Applying our general bound to the discrete two-dimensional torus yields the sharp $O(\log n)$ upper bound, improving the best previously known bound. For the cycle, whenever the arriving loads are bounded away from zero, we prove that the expected gap is $Ω(\sqrt n)$ for all sufficiently large times. Together with our upper bound, this confirms the conjecture of Alistarh, Nadiradze, and Sabour that the expected gap on the cycle is of order $\sqrt n$.

math.PR↗

arXiv2026

Entropy Bounds for Local Coordination and Graph Amenability

We study local pure coordination games on finite graphs. In these games, each vertex must choose one of two symmetric actions using only local information, and the cost is the average disagreement across edges. Hutchcroft, Rospuskova, and Tamuz showed that if such local coordination can be achieved with low cost, then the underlying graph must be amenable, or hyperfinite, but their quantitative bound has a square root loss. We improve this loss in the unbiased binary setting. The main idea is to associate with each player's local output a probability measure that records, along an ordered list of information sources, the mutual information with that output. For binary outputs, two players who usually agree have nearby associated measures, with a bound given by the binary entropy of their disagreement probability. Combining this estimate with a grand coupling theorem yields an improved amenability bound of order $\varepsilon\log(1/\varepsilon)$, where $\varepsilon$ is the average disagreement. We also show that the square root loss in the earlier general theorem is essentially unavoidable for non-binary coordination profiles. Thus the binary assumption is not merely technical: it is what makes the improved entropy bound possible.

cs.GT↗

arXiv2026

A Weak Structural Form of Commutative Equivalence in Finite Codes

We investigate the structural relationship between prefix-free codes over the binary alphabet and a class of unlabeled rooted trees, which we call \emph{symmetric} trees. We establish a canonical correspondence between prefix-free codes and symmetric trees, preserving not only the lengths of codewords but also some additional commutative structure. Using this correspondence, we provide a result related to the commutative equivalence conjecture. We show that for every code, there exists a prefix-free code such that, for each fixed word length, the sums of powers of two determined by the occurrences of a distinguished symbol are equal.

cs.IT↗

arXiv2026

Winning Criteria for Open Games: A Game-Theoretic Approach to Prefix Codes

We study two-player games with alternating moves played on infinite trees. Our main focus is on the case where the trees are full (regular) and the winning set is open (with respect to the product topology on the tree). Gale and Stewart showed that in this setting one of the players always has a winning strategy, though it is not known in advance which player. We present simple necessary conditions for the first player to have a winning strategy, and establish an equivalence between winning sets that guarantee a win for the first player and maximal prefix codes. Using this equivalence, we derive a necessary algebraic condition for winning, and exhibit a family of games for which this algebraic condition is in fact equivalent to winning. We introduce the concept of coverings, and show that by covering the tree of the game with an infinite labeled tree corresponding to the free group, we can use "game-theoretic tools" to derive a simple trait of maximal prefix codes.

math.OC↗