Alright, future engineers!
A **Tree** is an undirected graph where any two vertices are connected by exactly one path (no cycles). Ex: A graph with N vertices & N-1 edges (no cycles) is a tree. Pro-Tip: Perfect for modeling hierarchical structures like file systems!
New paper, with Kirill Kovalenko, Gonzalo Contreras, Stefano Boccaletti and Rubén Sánchez.
People have noticed that, in higher-order networks, synchronization is often explosive, and that cluster synchronization happens very rarely, if ever. We explain why, by showing that symultaneous dynamical equitability across layers or interaction orders is necessary and sufficient for cluster synchronization, except if the coupling functions depend linearly on each other. Since the probability of randomly satisfying this condition is exceedingly low, cluster synchronization is precluded in such networks.
https://www.nature.com/articles/s42005-026-02543-5
#mathematics #physics #science #networks #complexity #HigherOrderNetworks #MultiplexNetworks #synchronization #dynamicalsystems #graphs #graphtheory #equitability #ClusterSynchronization #ExplosiveSynchronization
This image is of a simulation of a simple directed #hypergraph , but using an n-ary line rather than a set for the #hyperedge
I have written a working #graphTheory editor for binary edges, where nodes are extended to sets (a #Higraph) , and am contemplating the complexity of n-ary edges with increasing apprehension. It requires refactoring just about the entire edge drawing codebase - 100's of changes across ~2000 lines of #Python.
Is it worth it? Please comment/ vote in the poll below
Vertex Degree: The number of edges connected to a vertex in a graph. Ex: In a social network, your degree is how many friends you have. Pro-Tip: Sum of all degrees is always 2 * (num of edges) – the Handshaking Lemma!
https://arxiv.org/abs/2603.12358
Here is the *third* manuscript coming out of the "Topics in Ramsey theory" online-only problem-solving session (https://sparse-graphs.mimuw.edu.pl/doku.php?id=sessions:2025sessions:2025session1) of the Sparse (Graphs) Coalition, which took place less than a year ago.
It is still surprising to realise what one can make of such events, if they are set up well.
#combinatorics #remoteconferences #graphtheory #extremalcombinatorics #openscience
An ordered graph is a graph whose vertex set is equipped with a total order. The ordered complete graph $K_N^<$ is the complete graph with vertex set $[N]$ equipped with the natural ordering of the integers. Given an ordered graph $H$, the ordered Ramsey number $R_<(H)$ is the smallest integer $N$ such that every red/blue edge-colouring of $K_N^<$ contains a monochromatic copy of $H$ with vertices appearing in the same relative order as in $H$. Balko, Cibulka, Král, and Kyn\v cl asked whether, among all ordered paths on $n$ vertices, the ordered Ramsey number is minimised by the alternating path $\mathrm{AP}_n$ -- the ordered path with vertex set $[n]$ such that the vertices encountered along the path are $1, n, 2, n - 1,3, n-2,\dots$. Motivated by this problem, we make progress on establishing the value of $R_<(\mathrm{AP}_n)$ by proving that \[ R_{<}(\mathrm{AP}_n)\leq \left(2+\frac{\sqrt{2}}{2}+o(1)\right)n. \] We then use similar methods to determine the exact ordered Turán number of $\mathrm{AP}_n$, and study the ordered Ramsey and Turán numbers of several related ordered paths.
A #Higraph milestone: Blobs (nodes as sets) now work! Grab a 'parent' blob, and all the children move. Edges connect anywhere on the blob, and default to be orthogonal to the point of contact. graphML read and write working.
Now on to proper hyperedges!
Differential Logic • 18
If we follow the classical line which singles out linear functions as ideals of simplicity then we may complete the analytic series of the proposition in the following way.
The next venn diagram shows the differential proposition we get by extracting the linear approximation to the difference map at each cell or point of the universe What results is the logical analogue of what would ordinarily be called the differential of but since the adjective differential is being attached to just about everything in sight the alternative name tangent map is commonly used for whenever it’s necessary to single it out.
To be clear about what’s being indicated here, it’s a visual way of summarizing the following data.
To understand the extended interpretations, that is, the conjunctions of basic and differential features which are being indicated here, it may help to note the following equivalences.
Capping the analysis of the proposition in terms of succeeding orders of linear propositions, the final venn diagram of the series shows the remainder map which happens to be linear in pairs of variables.
Reading the arrows off the map produces the following data.
In short, is a constant field, having the value at each cell.
Resources
cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon (1) (2)
cc: Research Gate • Structural Modeling • Systems Science • Syscoi
I think this notation is starting to get somewhere now.
kipriyanovich
Charo del Genio
Grant_H
Ross Kang
Linotype Pilgrim
Peter Cock
Inquiry Into Inquiry
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