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Related Concept Videos

Graphs of Functions01:30

Graphs of Functions

Graphs of functions provide a visual representation of how output values change in response to varying inputs. Each point on the graph corresponds to an ordered pair, where the x-coordinate (independent variable) determines the horizontal position and the y-coordinate (dependent variable) determines the vertical position. Linear functions like y = x give a straight line, indicating a constant rate of change.Nonlinear functions display more complex behaviors. Even power functions generate...
Graphs of Two-Variable Functions01:27

Graphs of Two-Variable Functions

A weather map provides a practical example of a function of two variables. Across a wide region such as the United States, temperatures vary from one location to another. Each location can be identified by two geographic coordinates: longitude and latitude. Since a single temperature value is assigned to each coordinate pair, the situation can be represented mathematically as a function with two inputs and one output.In mathematical notation, longitude and latitude can be labeled as x and y,...
Graphs of Equations in Two Variables01:30

Graphs of Equations in Two Variables

An equation with two variables, typically written in the form y = f(x) or Ax + By = C, describes a relationship between quantities represented by x and y. Each solution to such an equation is an ordered pair (x, y) that satisfies the equation when substituted. These pairs can be represented graphically to understand the variables' relationship visually.A common technique for constructing the graph of a two-variable equation is to create a value table. Begin by choosing several values for the...
Graphs of Polar Equations01:17

Graphs of Polar Equations

The polar coordinate system represents points using a distance from a central point (the pole) and an angle from a reference direction (the polar axis). Unlike rectangular coordinates, polar coordinates are ideal for graphing curves with radial symmetry or periodic behavior.Some general forms of graphs in polar coordinates include the following:Equation of a Circle (Centered at the Pole):A graph where the radius remains constant for all angles traces a circle centered at the pole:Equation of a...
Second Derivatives and the Shape of a Graph01:29

Second Derivatives and the Shape of a Graph

The second derivative of a function provides essential information about a graph's curvature and how it changes over an interval. It helps determine whether a function is concave upward or concave downward and identifies points where the curvature changes. These properties are fundamental in analyzing real-world scenarios, such as changes in road elevation, population growth, and economic trends.A function f(x) is considered concave upward on an interval if its graph lies above all its tangent...
Graphical Representation of Inequalities01:28

Graphical Representation of Inequalities

The graph of the equation where y equals x squared forms a curve known as a parabola. This curve acts as a boundary in the coordinate plane, dividing it into distinct regions based on the relative position of points.When the equality sign in the equation is replaced with an inequality—such as greater than, less than, greater than or equal to, or less than or equal to—the graphical representation changes from a single curve into a broader shaded area that signifies the set of all points...

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Related Experiment Video

Updated: Jul 19, 2026

Inherent Dynamics Visualizer, an Interactive Application for Evaluating and Visualizing Outputs from a Gene Regulatory Network Inference Pipeline
10:44

Inherent Dynamics Visualizer, an Interactive Application for Evaluating and Visualizing Outputs from a Gene Regulatory Network Inference Pipeline

Published on: December 7, 2021

Degree landscapes in scale-free networks.

Jacob Bock Axelsen1, Sebastian Bernhardsson, Martin Rosvall

  • 1NBI, Blegdamsvej 17, Dk 2100, Copenhagen, Denmark. bock@nbi.dk

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|October 10, 2006
PubMed
Summary

We introduce a landscape analogy to understand real-world network structures. This approach visualizes network topology by mapping node degrees to mountains and valleys, aiding in network analysis.

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Modeling the Functional Network for Spatial Navigation in the Human Brain
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Modeling the Functional Network for Spatial Navigation in the Human Brain

Published on: October 13, 2023

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Last Updated: Jul 19, 2026

Inherent Dynamics Visualizer, an Interactive Application for Evaluating and Visualizing Outputs from a Gene Regulatory Network Inference Pipeline
10:44

Inherent Dynamics Visualizer, an Interactive Application for Evaluating and Visualizing Outputs from a Gene Regulatory Network Inference Pipeline

Published on: December 7, 2021

Modeling the Functional Network for Spatial Navigation in the Human Brain
05:55

Modeling the Functional Network for Spatial Navigation in the Human Brain

Published on: October 13, 2023

Area of Science:

  • Network Science
  • Complex Systems
  • Data Analysis

Background:

  • Real-world networks exhibit broad degree distributions.
  • Understanding network topology is crucial for various applications.
  • Existing models may not fully capture the complexity of these distributions.

Purpose of the Study:

  • To generalize the degree-organizational view of real-world networks.
  • To introduce a landscape analogy for visualizing network topology.
  • To quantify network structures using landscape features.

Main Methods:

  • Generalizing the degree-organizational view of networks.
  • Utilizing a landscape analog with mountains (high-degree nodes) and valleys (low-degree nodes).
  • Measuring mountain widths and separations to quantify topology.
  • Generating ridge landscapes to model spatially constrained networks.

Main Results:

  • Correlated degrees map to smooth landscapes (e.g., social networks).
  • Hierarchical networks correspond to one-mountain landscapes (e.g., the Internet).
  • Degree-disassortative networks without hierarchy resemble rough landscapes with multiple mountains.

Conclusions:

  • The landscape analogy provides a novel framework for network analysis.
  • Topological features can be quantified by landscape metrics.
  • Ridge landscapes model networks with spatial or functional constraints.