Using a non orthogonal topology and embedded graphs to understand the unified story of the
Using a non orthogonal topology and embedded graphs to understand the unified story of the
Quote from "My Open Letter To You"
Traditional models often rely on orthogonal structures, where elements are categorized along perpendicular, independent dimensions. These systems assume a linear decomposition, where each component exists in isolation, minimizing overlap. This approach simplifies analysis by compartmentalizin
Traditional models often rely on orthogonal structures, where elements are categorized along perpendicular, independent dimensions. These systems assume a linear decomposition, where each component exists in isolation, minimizing overlap. This approach simplifies analysis by compartmentalizing data into distinct, non-interacting categories. While useful for certain applications, it can oversimplify complex systems where elements are inherently interconnected.
A non-orthogonal approach, however, embraces interconnectivity and recursion. Instead of rigid, separable dimensions, elements influence each other dynamically, producing continuous thematic intersections. This means that concepts are not confined to single, isolated bins but rather exist in a fluid state, constantly interacting and shaping one another. This applies in various fields, including theology, narrative analysis, and computational modeling—where meaning and structure cannot be strictly compartmentalized. Understanding these dynamic relationships allows for a more nuanced and accurate representation of complex systems.
Biblical texts frequently exhibit recursive structures, mirroring motifs across books, characters, and historical contexts. Rather than a linear, static progression, themes like bitterness, deliverance, inversion, and testing form non-orthogonal manifolds, where their presence interacts dynamically. This recursive nature suggests that biblical narratives are not merely a sequence of events but a rich tapestry of interwoven themes that gain deeper meaning through their interconnections.
For example:
From an algorithmic perspective, non-orthogonality can be mapped using graph theory and network analysis, providing powerful tools to visualize and understand these complex interconnections.
Using tools like Gephi (for network visualization), Python (for data processing and analysis), Node2Vec (for learning embeddings of nodes in a network), UMAP (Uniform Manifold Approximation and Projection), and t-SNE (t-Distributed Stochastic Neighbor Embedding) (both for dimensionality reduction and visualization of high-dimensional data), we can visualize non-orthogonal clustering, showing that biblical themes evolve relationally, not in strict categorical isolation. These tools allow researchers to uncover hidden patterns and relationships that might not be apparent through traditional linear analysis.
Formalizing a non-orthogonal framework requires a shift in how we conceptualize and model the relationships between elements.
Imagine drawing a complex network, like a social graph or a transportation map, onto a flexible rubber sheet. Now, imagine stretching, twisting, and deforming that sheet in every conceivable way, but without tearing it or gluing any parts together. What happens t
Imagine drawing a complex network, like a social graph or a transportation map, onto a flexible rubber sheet. Now, imagine stretching, twisting, and deforming that sheet in every conceivable way, but without tearing it or gluing any parts together. What happens to the network you drew? Do its connections break? Do new ones form? Or does something fundamental about its structure remain unchanged, even as its appearance wildly transforms?
This intuitive thought experiment leads us to the core question of this exploration: What happens to the fundamental properties of a graph when we continuously deform the topological space in which it is embedded? How do its relationships and structures change, or remain invariant, under such transformations?
The concept of "smearing" embedded graphs—a continuous deformation of the graph within its ambient space—is not just a fascinating mathematical curiosity. It holds broad implications and practical applications across various fields. In network analysis, understanding how graph properties behave under deformation can provide crucial insights into the dynamics of complex systems, from biological networks to communication infrastructures. In data visualization, it can lead to innovative techniques for creating more intuitive, flexible, and interactive representations of complex datasets, allowing for dynamic exploration of relationships. Furthermore, it offers foundational contributions to geometric topology, deepening our understanding of space itself and the structures that can exist within it.
In this post, we will embark on a journey from rigid graph structures to fluid, adaptable representations. We'll define what it means to "smear" a graph, explore the mathematics that governs this process, examine various examples, and finally, discuss its diverse applications and the exciting open questions that remain.
Before diving into the intricacies of smearing, let's establish a common understanding of the foundational concepts.
At its heart, a graph embedding is a way of "drawing" a graph in a specific space. More formally, it's a mapping of a graph G=(V,E) (where V is the set of vertices and E is the set of edges) into a topological space X. This mapping must satisfy two key conditions:
A common example is embedding a planar graph in R2. A planar graph is one that can be drawn on a plane without any edges crossing, except at their shared vertices. Think of drawing a simple house: the vertices are the corners, and the edges are the walls and roof lines. This drawing is a planar embedding. More complex examples include knot-like embeddings of graphs in R3, where edges might twist and intertwine like a knot, but still maintain their fundamental graph structure.
When we embed a graph, certain properties are typically preserved. These include the number of vertices, the number of edges, and the connectivity (i.e., which vertices are connected to which). However, other properties might be lost or altered, such as the specific geometric distances between vertices, the exact lengths of edges, or the angles at which edges meet. These geometric properties are dependent on the specific "drawing" rather than the inherent graph structure.
To understand smearing, we need a brief refresher on topological spaces. A topological space is a set equipped with a structure (a collection of "open sets") that allows us to define concepts like "neighborhood" and "continuity" without needing a notion of distance. This is crucial because smearing is about continuous deformation, not rigid geometric transformations.
The key concept for deformation is a homeomorphism. A homeomorphism is a continuous bijection (a one-to-one and onto mapping) between two topological spaces, whose inverse is also continuous. If two spaces are homeomorphic, they are considered topologically equivalent; one can be continuously deformed into the other without tearing, cutting, or gluing.
This brings us to the intuitive idea of "stretching without tearing." This phrase perfectly encapsulates the essence of topological deformation. Imagine the rubber sheet again: you can stretch it, shrink it, bend it, or twist it, and the network drawn on it will deform accordingly. However, you cannot tear the sheet (which would break connections) or glue two previously separate parts together (which would create new, unintended connections). This allows for significant shape changes while preserving fundamental topological properties like connectivity, the number of connected components, and the presence or absence of "holes" (or cycles) in the graph structure.
In our context, "smearing" refers to a continuous deformation of a graph embedding within its ambient topological space. More precisely, if we have an embedding f:G→X, a smearing operation is a continuous family of embeddings ft:G→X for t∈[0,1], such that f0=f (the initial embedding). This means that as t varies from 0 to 1, the graph's representation in X smoothly transforms from its initial state to a final, "smeared" state.
Visually, imagine the vertices of the graph as tiny beads and the edges as elastic strings connecting them. When you "smear" the graph, these beads move along continuous paths in the space, and the strings stretch and bend to accommodate their new positions. The key is that no string ever breaks, and no string ever passes through another string or bead unless they are already connected.
This concept is closely connected to homotopy and isotopy. A homotopy is a continuous deformation of one continuous map into another. In our case, smearing can be seen as a homotopy of graph embeddings. An isotopy is a special kind of homotopy where all intermediate maps in the deformation are also homeomorphisms onto their images. If a smearing is an isotopy, it means that at every moment during the deformation, the graph remains topologically equivalent to its original embedded form. This distinction is important for understanding which properties are strictly preserved.
Delving deeper, the mathematical framework of smearing allows us to rigorously analyze how graphs behave under continuous deformation.
The core question in smearing is: Which graph properties survive smearing? These are the topological invariants of the embedded graph. Properties like graph connectivity (whether the graph remains a single piece or breaks into multiple components), the number of connected components, and the existence of cycles (loops) are typically preserved. If a graph has a cycle, it will continue to have a cycle after smearing, even if the cycle's shape changes drastically.
It's crucial to differentiate between topological invariants and geometric properties. Topological invariants are properties that are preserved under any homeomorphism. For an embedded graph on a surface, the Euler characteristic of the surface on which it's embedded is a topological invariant. Geometric properties, on the other hand, depend on the specific metric or realization in space. These include edge lengths, angles between edges, and the precise coordinates of vertices. These are generally not preserved under smearing. A straight edge can become curved, and a short edge can become long.
For example, a complete graph Kn (where every vertex is connected to every other vertex) will remain a Kn after any smearing, even if its geometric layout changes. Similarly, a cycle graph Cn (a simple loop of n vertices) will always remain a Cn, no matter how much it's stretched or bent into a different shape. Their essential nature as complete graphs or cycles remains unchanged because their connectivity patterns are topologically invariant.
The smearing process can be mathematically described as a continuous family of embeddings. Let G be a graph and X be a topological space. An embedding of G into X is a continuous map f:G→X such that f is a homeomorphism onto its image f(G). A smearing of f is a continuous map H:G×[0,1]→X such that for each t∈[0,1], the map Ht:G→X defined by Ht(v)=H(v,t) and Ht(e)=H(e,t) is an embedding. Here, [0,1] is our parameter space, often representing "time" or a continuous deformation parameter.
Mathematical formalization involves ensuring that the maps Ht are continuous for all t, and that the overall map H is continuous with respect to both the graph G and the parameter t. This requires tools from point-set topology and potentially differential topology if we are working with smooth manifolds.
Boundary conditions and constraints are essential. For instance, vertices must remain distinct throughout the smearing process. Edges are not allowed to "pass through" other vertices unless they are incident to them. Similarly, the interiors of distinct edge paths must remain disjoint. These constraints ensure that the topological identity of the graph is preserved and that the deformation is "well-behaved."
The study of smeared graphs draws upon significant theorems from topology and graph theory. One fundamental area relates to the existence of embeddings. For instance, Kuratowski's Theorem provides a necessary and sufficient condition for a graph to be planar (i.e., embeddable in R2 without crossings). When considering smearing, questions arise about whether a graph, once embedded, can always be smeared into another desired configuration within the same topological class, or if certain spaces allow for more "flexible" smearings than others.
Existence and uniqueness questions are central. Does a smeared embedding always exist for any continuous deformation? Is the resulting smeared graph unique up to some equivalence (e.g., ambient isotopy of the space)? These questions often lead to deep results in knot theory and geometric topology. For example, the Jordan Curve Theorem (for R2) and its generalizations are foundational for understanding how closed curves (cycles in graphs) divide a space and how these divisions are preserved under continuous deformation.
Classification theorems might exist for specific types of graphs or embeddings. For instance, in knot theory, knots are classified by their topological invariants. Similarly, certain classes of graphs embedded in particular spaces might have classification theorems that describe all possible "smeared" forms they can take, up to topological equivalence. These theorems provide a powerful way to understand the inherent structure of embedded graphs.
To solidify our understanding, let's explore some concrete examples of smearing.
The theoretical concepts of smearing can be brought to life through computation.
# Example: A very basic "smearing" (force-directed layout) in Python using NetworkX
# This is a conceptual example; true continuous smearing is more complex.
import networkx as nx
import matplotlib.pyplot as plt
import numpy as np
def simple_force_layout(graph, iterations=50, learning_rate=0.1, repulsion_strength=0.01, attraction_strength=0.05):
"""
A simplified force-directed layout algorithm to simulate 'smearing'
by iteratively adjusting node positions.
"""
pos = {node: np.random.rand(2) for node in graph.nodes()} # Initial random positions
for i in range(iterations):
forces = {node: np.zeros(2) for node in graph.nodes()}
# Repulsion between all nodes
for u in graph.nodes():
for v in graph.nodes():
if u != v:
diff = pos[u] - pos[v]
dist = np.linalg.norm(diff)
if dist > 0:
forces[u] += (diff / dist) * (repulsion_strength / dist)
# Attraction between connected nodes
for u, v in graph.edges():
diff = pos[v] - pos[u]
dist = np.linalg.norm(diff)
forces[u] += diff * attraction_strength
forces[v] -= diff * attraction_strength
# Update positions
for node in graph.nodes():
pos[node] += forces[node] * learning_rate
return pos
# Create a sample graph (e.g., a cycle graph)
G = nx.cycle_graph(5)
# Perform a simple "smearing" (force-directed layout)
smeared_pos = simple_force_layout(G)
# Plot the smeared graph
plt.figure(figsize=(6, 6))
nx.draw_networkx(G, pos=smeared_pos, with_labels=True, node_color='skyblue', node_size=700, edge_color='gray', font_size=10, font_weight='bold')
plt.title("Smeared Graph (Force-Directed Layout)")
plt.axis('off')
plt.show()
print("Graph nodes:", G.nodes())
print("Graph edges:", G.edges())
print("Final node positions (smeared):", smeared_pos)
The theoretical understanding of smearing embedded graphs translates into powerful applications across various scientific and mathematical domains.
The field of smearing embedded graphs is rich with unsolved problems and exciting avenues for future research.
The concept of smearing embedded graphs in topological spaces offers a powerful lens through which to view and understand complex interconnected systems. By emphasizing the shift from rigid, fixed representations to fluid, adaptable ones, we gain a deeper appreciation for the inherent flexibility and robustness of graph structures under continuous deformation.
This work matters beyond pure mathematics. It has the potential to revolutionize how we visualize, analyze, and understand complex networks in various fields, from uncovering hidden patterns in data to modeling the dynamic behavior of physical and biological systems. The ability to reason about and manipulate the topological essence of graphs, independent of their precise geometric form, opens up new frontiers in research and application.
We encourage you to delve deeper into this fascinating subject. Explore the tools available, experiment with different graph embeddings, and consider how the principles of smearing might apply to problems in your own domain. The journey into the fluid world of smeared graphs is just beginning, and there are countless opportunities for further exploration and groundbreaking research.