🖍️Mark Needham - Collected Quotes

"A random walk, in general, is sometimes described as being similar to how a drunk person traverses a city. They know what direction or end point they want to reach but may take a very circuitous route to get there. The algorithm starts at one node and somewhat randomly follows one of the relationships forward or backward to a neighbor node. It then does the same from that node and so on, until it reaches the set path length. ('We say somewhat randomly because the number of relationships a node has, and its neighbors have, influences the probability a node will be walked through.)'" (Mark Needham & Amy E Hodler, "Graph Algorithms: Practical Examples in Apache Spark and Neo4j", 2019)

"Adding graph features and context improves predictions, especially in situations where connections matter. [...] Unfortunately, many machine learning approaches today miss a lot of rich contextual information. This stems from ML’s reliance on input data built from tuples, leaving out a lot of predictive relationships and network data. Furthermore, contextual information is not always readily available or is too difficult to access and process. Even finding connections that are four or more hops away can be a challenge at scale for traditional methods. Using graphs, we can more easily reach and incorporate connected data." (Mark Needham & Amy E Hodler, "Graph Algorithms: Practical Examples in Apache Spark and Neo4j", 2019)

"At the most abstract level, graph analytics is applied to forecast behavior and prescribe action for dynamic groups. Doing this requires understanding the relationships and structure within the group. Graph algorithms accomplish this by examining the overall nature of networks through their connections. With this approach, you can understand the topology of connected systems and model their processes." (Mark Needham & Amy E Hodler, "Graph Algorithms: Practical Examples in Apache Spark and Neo4j", 2019)

"Betweenness Centrality makes the assumption that all communication between nodes happens along the shortest path and with the same frequency, which isn’t always the case in real life. Therefore, it doesn’t give us a perfect view of the most influential nodes in a graph, but rather a good representation." (Mark Needham & Amy E Hodler, "Graph Algorithms: Practical Examples in Apache Spark and Neo4j", 2019)

"Centrality algorithms are used to understand the roles of particular nodes in a graph and their impact on that network. They’re useful because they identify the most important nodes and help us understand group dynamics such as credibility, accessibility, the speed at which things spread, and bridges between groups." (Mark Needham & Amy E Hodler, "Graph Algorithms: Practical Examples in Apache Spark and Neo4j", 2019)

"Community formation is common in all types of networks, and identifying them is essential for evaluating group behavior and emergent phenomena. The general prin‐ ciple in finding communities is that its members will have more relationships within the group than with nodes outside their group. Identifying these related sets reveals clusters of nodes, isolated groups, and network structure. This information helps infer similar behavior or preferences of peer groups, estimate resiliency, find nested relationships, and prepare data for other analyses. Community detection algorithms are also commonly used to produce network visualization for general inspection." (Mark Needham & Amy E Hodler, "Graph Algorithms: Practical Examples in Apache Spark and Neo4j", 2019)

"Feature extraction is a way to distill large volumes of data and attributes down to a set of representative descriptive attributes. The process derives numerical values (fea‐ tures) for distinctive characteristics or patterns in input data so that we can differenti‐ ate categories in other data. It’s used when data is difficult for a model to analyze directly - perhaps because of size, format, or the need for incidental comparisons. Feature selection is the process of determining the subset of extracted features that are most important or influential to a target goal. It’s used to surface predictive importance as well as for efficiency." (Mark Needham & Amy E Hodler, "Graph Algorithms: Practical Examples in Apache Spark and Neo4j", 2019)

"Graph algorithms provide one of the most potent approaches to analyzing connected data because their mathematical calculations are specifically built to operate on relationships. They describe steps to be taken to process a graph to discover its general qualities or specific quantities. Based on the mathematics of graph theory, graph algo‐ rithms use the relationships between nodes to infer the organization and dynamics of complex systems. Network scientists use these algorithms to uncover hidden infomation, test hypotheses, and make predictions about behavior." (Mark Needham & Amy E Hodler, "Graph Algorithms: Practical Examples in Apache Spark and Neo4j", 2019)

"Graph embedding is the representation of the nodes and relationships in a graph asfeature vectors. [...] Graph embedding uses graph data slightly differently than in connected feature extraction. It enables us to represent entire graphs, or subsets of graph data, in a numerical format ready for machine learning tasks. This is especially useful for unsu‐pervised learning, where the data is not categorized because it pulls in more contextual information through relationships. Graph embedding is also useful for data exploration, computing similarity between entities, and reducing dimensionality to aid in statistical analysis." (Mark Needham & Amy E Hodler, "Graph Algorithms: Practical Examples in Apache Spark and Neo4j", 2019)

"In classic graph theory, an acyclic graph that is undirected is called a tree. In computer science, trees can also be directed. A more inclusive definition would be a graph where any two nodes are connected by only one path. Trees are significant for understanding graph structures and many algorithms. They play a key role in designing networks, data structures, and search optimizations to improve categorization or organizational hierarchies." (Mark Needham & Amy E Hodler, "Graph Algorithms: Practical Examples in Apache Spark and Neo4j", 2019)

"Keep in mind that centrality measures represent the importance of a node in comparison to other nodes. Centrality is a ranking of the potential impact of nodes, not a measure of actual impact. For example, you might identify the two people with the highest cen‐ trality in a network, but perhaps policies or cultural norms are in play that actually shift influence to others. Quantifying actual impact is an active research area to develop additional influence metrics." (Mark Needham & Amy E Hodler, "Graph Algorithms: Practical Examples in Apache Spark and Neo4j", 2019)

"Machine learning is not artificial intelligence (AI), but a method for achieving AI. ML uses algorithms to train software through specific examples and progressive improvements based on expected outcome - without explicit programming of how to accomplish these better results. Training involves providing a lot of data to a model and enabling it to learn how to process and incorporate that information." (Mark Needham & Amy E Hodler, "Graph Algorithms: Practical Examples in Apache Spark and Neo4j", 2019)

"Make it a habit to run Connected Components to test whether a graph is connected as a preparatory step for general graph analysis. Performing this quick test can avoid accidentally running algorithms on only one disconnected component of a graph and getting incorrect results." (Mark Needham & Amy E Hodler, "Graph Algorithms: Practical Examples in Apache Spark and Neo4j", 2019)

"Sometimes the most important cog in the system is not the one with the most overt power or the highest status. Sometimes it’s the middlemen that connect groups or the brokers who the most control over resources or the flow of information. Betweenness Centrality is a way of detecting the amount of influence a node has over the flow of information or resources in a graph. It is typically used to find nodes that serve as a bridge from one part of a graph to another." (Mark Needham & Amy E Hodler, "Graph Algorithms: Practical Examples in Apache Spark and Neo4j", 2019)

"Putting together the right mix of features can increase accuracy because it fundamentally influences how our models learn. Because even modest improvements can make a significant difference, our focus in this chapter is on connected features. Connected features are features extracted from the structure of the data. These features can be derived from graph-local queries based on parts of the graph surrounding a node, or graph-global queries that use graph algorithms to identify predictive elements within data based on relationships for connected feature extraction." (Mark Needham & Amy E Hodler, "Graph Algorithms: Practical Examples in Apache Spark and Neo4j", 2019)

"Use Degree Centrality if you’re attempting to analyze influence by looking at the number of incoming and outgoing relationships, or find the “popularity” of individual nodes. It works well when you’re concerned with immediate connectedness or near-term probabilities. However, Degree Centrality is also applied to global analysis when you want to evaluate the minimum degree, maximum degree, mean degree, and standard deviation across the entire graph." (Mark Needham & Amy E Hodler, "Graph Algorithms: Practical Examples in Apache Spark and Neo4j", 2019)

"When using community detection algorithms, be conscious of the density of the relationships. If the graph is very dense, you may end up with all nodes congregating in one or just a few clusters. You can counteract this by filtering by degree, relationship weights, or similarity metrics. On the other hand, if the graph is too sparse with few connected nodes, you may end up with each node in its own cluster. In this case, try to incorporate additional relationship types that carry more relevant information." (Mark Needham & Amy E Hodler, "Graph Algorithms: Practical Examples in Apache Spark and Neo4j", 2019)

"Without peripheral and related information, solutions that attempt to predict behav‐ ior or make recommendations for varying circumstances require more exhaustive training and prescriptive rules. This is partly why AI is good at specific, well-defined tasks, but struggles with ambiguity. Graph-enhanced ML can help fill in that missing contextual information that is so important for better decisions." (Mark Needham & Amy E Hodler, "Graph Algorithms: Practical Examples in Apache Spark and Neo4j", 2019)

📊Graphical Representation: Graphics We Live By (Part VIII: List of Items in Power BI)

Graphical Representation Series

Introduction

There are situations in which one needs to visualize only the rating, other values, or ranking of a list of items (e.g. shopping cart, survey items) on a scale (e.g. 1 to 100, 1 to 10) for a given dimension (e.g. country, department). Besides tables, in Power BI there are 3 main visuals that can be used for this purpose: the clustered bar chart, the line chart (aka line graph), respectively the slopegraph:

Main Display Methods

Main Display Methods

For a small list of items and dimension values probably the best choice would be to use a clustered bar chart (see A). If the chart is big enough, one can display also the values as above. However, the more items in the list, respectively values in the dimension, the more space is needed. One can maybe focus then only on a subset of items from the list (e.g. by grouping several items under a category), respectively choose which dimension values to consider. Another important downside of this method is that one needs to remember the color encodings. 

This downside applies also to the next method - the use of a line chart (see B) with categorical data, however applying labels to each line simplifies its navigation and decoding. With line charts the audience can directly see the order of the items, the local and general trends. Moreover, a line chart can better scale with the number of items and dimension values.

The third option (see C), the slopegraph, looks like a line chart though it focuses only on two dimension values (points) and categorizes the line as "down" (downward slope), "neutral" (no change) and "up" (upward slope). For this purpose, one can use parameters fields with measures. Unfortunately, the slopegraph implementation is pretty basic and the labels overlap which makes the graph more difficult to read. Probably, with the new set of changes planned by Microsoft, the use of conditional formatting of lines would allow to implement slope graphs with line charts, creating thus a mix between (B) and (C).

This is one of the cases in which the Y-axis (see B and C) could be broken and start with the meaningful values. 

Table Based Displays

Especially when combined with color encodings (see C & G) to create heatmap-like displays or sparklines (see E), tables can provide an alternative navigation of the same data. The color encodings allow to identify the areas of focus (low, average, or high values), while the sparklines allow to show inline the trends. Ideally, it should be possible to combine the two displays.  

Table Displays and the Aster Plot

One can vary the use of tables. For example, one can display only the deviations from one of the data series (see F), where the values for the other countries are based on AUS. In (G), with the help of visual calculations one can also display values' ranking. 

Pie Charts

Pie charts and their variations appear nowadays almost everywhere. The Aster plot is a variation of the pie charts in which the values are encoded in the height of the pieces. This method was considered because the data used above were encoded in 4 similar plots. Unfortunately, the settings available in Power BI are quite basic - it's not possible to use gradient colors or link the labels as below:

Source Data as Aster Plots

Sankey Diagram

A Sankey diagram is a data visualization method that emphasizes the flow or change from one state (the source) to another (the destination). In theory it could be used to map the items to the dimensions and encode the values in the width of the lines (see I). Unfortunately, the diagram becomes challenging to read because all the lines and most of the labels intersect. Probably this could be solved with more flexible formatting and a rework of the algorithm used for the display of the labels (e.g. align the labels for AUS to the left, while the ones for CAN to the right).

Sankey Diagram

Data Preparation

A variation of the above image with the Aster Plots which contains only the plots was used in ChatGPT to generate the basis data as a table via the following prompts:

• retrieve the labels from the four charts by country and value in a table
• consolidate the values in a matrix table by label country and value

The first step generated 4 tables, which were consolidated in a matrix table in the second step. Frankly, the data generated in the first step should have been enough because using the matrix table required an additional step in DAX.

Here is the data imported in Power BI as the Industries query:

let
    Source = #table({"Label","Australia","Canada","U.S.","Japan"}
, {
 {"Credit card","67","64","66","68"}
, {"Online retail","55","57","48","53"}
, {"Banking","58","53","57","48"}
, {"Mobile phone","62","55","44","48"}
, {"Social media","74","72","62","47"}
, {"Search engine","66","64","56","42"}
, {"Government","52","52","58","39"}
, {"Health insurance","44","48","50","36"}
, {"Media","52","50","39","23"}
, {"Retail store","44","40","33","23"}
, {"Car manufacturing","29","29","26","20"}
, {"Airline/hotel","35","37","29","16"}
, {"Branded manufacturing","36","33","25","16"}
, {"Loyalty program","45","41","32","12"}
, {"Cable","40","39","29","9"}
}
),
    #"Changed Types" = Table.TransformColumnTypes(Source,{{"Australia", Int64.Type}, {"Canada", Int64.Type}, {"U.S.", Number.Type}, {"Japan", Number.Type}})
in
    #"Changed Types"

Transforming (unpivoting) the matrix to a table with the values by country:

IndustriesT = UNION (
    SUMMARIZECOLUMNS(
     Industries[Label]
     , Industries[Australia]
     , "Country", "Australia"
    )
    , SUMMARIZECOLUMNS(
     Industries[Label]
     , Industries[Canada]
     , "Country", "Canada"
    )
    , SUMMARIZECOLUMNS(
     Industries[Label]
     , Industries[U.S.]
     , "Country", "U.S."
    )
    ,  SUMMARIZECOLUMNS(
     Industries[Label]
     , Industries[Japan]
     , "Country", "Japan"
    )
)

Notes:

The slopechart from MAQ Software requires several R language libraries to be installed (see how to install the R language and optionally the RStudio). Run the following scripts, then reopen Power BI Desktop and enable running visual's scripts.

install.packages("XML")
install.packages("htmlwidgets")
install.packages("ggplot2")
install.packages("plotly")

Happy (de)coding!

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🐍Python: PySpark and GraphFrames (Test Drive)

Besides the challenges met during configuring the PySpark & GraphFrames environment, also running my first example in Spyder IDE proved to be a bit more challenging than expected. Starting from an example provided by the DataBricks documentation on GraphFrames, I had to add 3 more lines to establish the connection of the Spark cluster, respectively to deactivate the context (only one SparkContext can be active per Java VM).

The following code displays the vertices and edges, respectively the in and out degrees for a basic graph. 

from graphframes import *
from pyspark.context import SparkContext
from pyspark.sql.session import SparkSession

#establishing a connection to the Spark cluster (code added)
sc = SparkContext('local').getOrCreate()
spark = SparkSession(sc)

# Create a Vertex DataFrame with unique ID column "id"
v = spark.createDataFrame([
  ("a", "Alice", 34),
  ("b", "Bob", 36),
  ("c", "Charlie", 30),
  ("d", "David", 29),
  ("e", "Esther", 32),
  ("f", "Fanny", 36),
  ("g", "Gabby", 60)
], ["id", "name", "age"])
# Create an Edge DataFrame with "src" and "dst" columns
e = spark.createDataFrame([
  ("a", "b", "friend"),
  ("b", "c", "follow"),
  ("c", "b", "follow"),
  ("f", "c", "follow"),
  ("e", "f", "follow"),
  ("e", "d", "friend"),
  ("d", "a", "friend"),
  ("a", "e", "friend")
], ["src", "dst", "relationship"])

# Create a GraphFrame
g = GraphFrame(v, e)

g.vertices.show()
g.edges.show()

g.inDegrees.show()
g.outDegrees.show()

#stopping the active context (code added)
sc.stop()

Output:

id name age a Alice 34 b Bob 36 c Charlie 30 d David 29 e Esther 32 f Fanny 36 g Gabby 60

src dst relationship a b friend b c follow c b follow f c follow e f follow e d friend d a friend a e friend

id inDegree f 1 e 1 d 1 c 2 b 2 a 1

id outDegree f 1 e 2 d 1 c 1 b 1 a 2

Notes:
Without the last line, running a second time the code will halt with the following error: 
ValueError: Cannot run multiple SparkContexts at once; existing SparkContext(app=pyspark-shell, master=local) created by __init__ at D:\Work\Python\untitled0.py:4

Loading the same data from a csv file involves a small overhead as the schema needs to be defined explicitly. The same output from above should be provided by the following code:

from graphframes import *
from pyspark.context import SparkContext
from pyspark.sql.session import SparkSession
from pyspark.sql.types import * 

#establishing a connection to the Spark cluster (code added)
sc = SparkContext('local').getOrCreate()
spark = SparkSession(sc)

nodes = [
    StructField("id", StringType(), True),
    StructField("name", StringType(), True),
    StructField("age", IntegerType(), True)
]
edges = [
    StructField("src", StringType(), True),
    StructField("dst", StringType(), True),
    StructField("relationship", StringType(), True)
    ]

v = spark.read.csv(r"D:\data\nodes.csv", header=True, schema=StructType(nodes))

e = spark.read.csv(r"D:\data\edges.csv", header=True, schema=StructType(edges))

# Create a GraphFrame
g = GraphFrame(v, e)

g.vertices.show()
g.edges.show()

g.inDegrees.show()
g.outDegrees.show()

#stopping the active context (code added)
sc.stop()

The 'nodes.csv' file has the following content:
id,name,age
"a","Alice",34
"b","Bob",36
"c","Charlie",30
"d","David",29
"e","Esther",32
"f","Fanny",36
"g","Gabby",60

The 'edges.csv' file has the following content:
src,dst,relationship
"a","b","friend"
"b","c","follow"
"c","b","follow"
"f","c","follow"
"e","f","follow"
"e","d","friend"
"d","a","friend"
"a","e","friend"

Note:
There should be no spaces between values (e.g. "a", "b"), otherwise the results might deviate from expectations. 

Now, one can go and test further operations on the graph thus created:

#filtering edges 
gl = g.edges.filter("relationship = 'follow'").sort("src")
gl.show()
print("number edges: ", gl.count())

#filtering vertices
#gl = g.vertices.filter("age >= 30 and age<40").sort("id")
#gl.show()
#print("number vertices: ", gl.count())

# relationships involving edges and vertices
#motifs = g.find("(a)-[e]->(b); (b)-[e2]->(a)")
#motifs.show()

Happy coding!

🐍Python: Installing PySpark and GraphFrames on a Windows 10 Machine

One of the To-Dos for this week was to set up the environment so I can start learning PySpark and GraphFrames based on the examples from Needham & Hodler’s free book on Graph Algorithms. Therefore, I downloaded and installed the Java SDK 8 from the Oracle website (requires an Oracle account) and the latest stable version of Python (Python 3.9.2), downloaded and unzipped the Apache Spark package locally on a Windows 10 machine, respectively the Winutils tool as described here.

The setup requires several environment variables that need to be created, respectively the Path variable needs to be extended with further values (delimited by ";"). In the end I added the following values:

Variable	Value
HADOOP_HOME	D:\Programs\spark-3.0.2-bin-hadoop2.7
SPARK_HOME	D:\Programs\spark-3.0.2-bin-hadoop2.7
JAVA_HOME	D:\Programs\Java\jdk1.8.0_281
PYTHONPATH	D:\Programs\Python\Python39\
PYTHONPATH	;%SPARK_HOME%\python
PYTHONPATH	%SPARK_HOME%\python\lib\py4j-0.10.9-src.zip
PATH	%HADOOP_HOME%\bin
PATH	%SPARK_HOME%\bin
PATH	%PYTHONPATH%
PATH	%PYTHONPATH%\DLLs
PATH	%PYTHONPATH%\Lib
PATH	%JAVA_HOME%\bin

I tried then running the first example from Chapter 3 using the Spyder IDE, though the environment didn’t seem to recognize the 'graphframes' library. As long it's not already available, the graphframes .jar file (e.g. graphframes-0.8.1-spark3.0-s_2.12.jar) corresponding to the installed Spark version must be downloaded and copied in the Spark folder where the other .jar files are available (e.g. .\spark-3.0.2-bin-hadoop2.7\jars). With this change I could finally run my example, though it took me several tries to get this right. 

During Python's installation I had to change the value for the LongPathsEnabled setting from 0 to 1 via regedit to allow path lengths longer than 260 characters, as mentioned in the documentation. The setting is available via the following path:
HKEY_LOCAL_MACHINE\SYSTEM\CurrentControlSet\Control\FileSystem

In the process I also tried installing ‘pyspark’ and ‘graphframes’ via the Anaconda tool with the following commands:

pip3 install --user pyspark
pip3 install --user graphframes

From Anaconda’s point of view the installation was correct, fact which pointed me to the missing 'graphframe' library.

It took me 4-5 hours of troubleshooting and searching until I got my environment setup. I still have two more warnings to solve, though I will look into this later:
WARN NativeCodeLoader: Unable to load native-hadoop library for your platform... using builtin-java classes where applicable
WARN ProcfsMetricsGetter: Exception when trying to compute pagesize, as a result reporting of ProcessTree metrics is stopped

Notes:
Spaces in the folder's names might creates issues. Therefore, I used 'Programs' instead of 'Program Files' as main folder. 
There seem to be some confusion what environment variables are needed and how they need to be configured.
Unfortunately, the troubleshooting involved in setting up an environment and getting a simple example to work seems to be a recurring story over the years. Same situation was with the programming languages from 15-20 years ago. 

🔬Data Science: Recurrent Neural Network [RNN] (Definitions)

"A neural net with feedback connections, such as a BAM, Hopfield net, Boltzmann machine, or recurrent backpropagation net. In contrast, the signal in a feedforward neural net passes from the input units (through any hidden units) to the output units." (Laurene V Fausett, "Fundamentals of Neural Networks: Architectures, Algorithms, and Applications", 1994)

"A neural network topology where the units are connected so that inputs signals flow back and forth between the neural processing units until the neural network settles down. The outputs are then read from the output units." (Joseph P Bigus, "Data Mining with Neural Networks: Solving Business Problems from Application Development to Decision Support", 1996)

"Networks with feedback connections from neurons in one layer to neurons in a previous layer." (Nikola K Kasabov, "Foundations of Neural Networks, Fuzzy Systems, and Knowledge Engineering", 1996)

"RNN topology involves backward links from output to the input and hidden layers." (Siddhartha Bhattacharjee et al, "Quantum Backpropagation Neural Network Approach for Modeling of Phenol Adsorption from Aqueous Solution by Orange Peel Ash", 2013)

"Neural network whose feedback connections allow signals to circulate within it." (Terrence J Sejnowski, "The Deep Learning Revolution", 2018)

"An RNN is a special kind of neural network used for modeling sequential data." (Alex Thomas, "Natural Language Processing with Spark NLP", 2020)

"A recurrent neural network (RNN) is a class of artificial neural networks where connections between nodes form a directed graph along a temporal sequence. This allows it to exhibit temporal dynamic behavior." (Udit Singhania & B. K. Tripathy, "Text-Based Image Retrieval Using Deep Learning", 2021)

"A RNN [Recurrent Neural Network] models sequential interactions through a hidden state, or memory. It can take up to N inputs and produce up to N outputs. For example, an input sequence may be a sentence with the outputs being the part-of-speech tag for each word (N-to-N). An input could be a sentence, and the output a sentiment classification of the sentence (N-to-1). An input could be a single image, and the output could be a sequence of words corresponding to the description of an image (1-to-N). At each time step, an RNN calculates a new hidden state ('memory') based on the current input and the previous hidden state. The 'recurrent' stems from the facts that at each step the same parameters are used and the network performs the same calculations based on different inputs." (Wild ML)

"Recurrent Neural Network (RNN) refers to a type of artificial neural network used to understand sequential information and predict follow-on probabilities. RNNs are widely used in natural language processing, with applications including language modeling and speech recognition." (Accenture)

🔬Data Science: Graph (Definitions)

"Informally, a graph is a finite set of dots called vertices (or nodes) connected by links called edges (or arcs). More formally: a simple graph is a (usually finite) set of vertices V and set of unordered pairs of distinct elements of V called edges." (Craig F Smith & H Peter Alesso, "Thinking on the Web: Berners-Lee, Gödel and Turing", 2008)

"A computation object that is used to model relationships among things. A graph is defined by two finite sets: a set of nodes and a set of edges. Each node has a label to identify it and distinguish it from other nodes. Edges in a graph connect exactly two nodes and are denoted by the pair of labels of nodes that are related." (Clay Breshears, "The Art of Concurrency", 2009)

"A graph in mathematics is a set of nodes and a set of edges between pairs of those nodes; the edges are ordered or nonordered pairs, or a relation, that defines the pairs of nodes for which the relation being examined is valid. […] The edges can either be undirected or directed; directed edges depict a relation that requires the nodes to be ordered while an undirected edge defines a relation in which no ordering of the edges is implied." (Dennis M Buede, "The Engineering Design of Systems: Models and methods", 2009)

[undirected graph:] "A graph in which the nodes of an edge are unordered. This implies that the edge can be thought of as a two-way path." (Clay Breshears, "The Art of Concurrency", 2009)

[directed graph:] "A graph whose edges are ordered pairs of nodes; this allows connections between nodes in one direction. When drawn, the edges of a directed graph are commonly shown as arrows to indicate the “direction” of the edge." (Clay Breshears, "The Art of Concurrency", 2009)

"1.Generally, a set of homogeneous nodes (vertices) and edges (arcs) between pairs of nodes." (DAMA International, "The DAMA Dictionary of Data Management", 2011)

[directed acyclic graph:] "A graph that defines a partial order so that nodes can be sorted into a linear sequence with references only going in one direction. A directed acyclic graph has, as its name suggests, directed edges and no cycles." (Michael McCool et al, "Structured Parallel Programming", 2012)

"A data structure that consists of a set of nodes and a set of edges that relate the nodes to each other" (Nell Dale & John Lewis, "Computer Science Illuminated" 6th Ed., 2015)

[directed graph:] "A directed graph is one in which the edges have a specified direction from one vertex to another." (Dan Sullivan, "NoSQL for Mere Mortals", 2015)

[directed graph (digraph):] "A graph in which each edge is directed from one vertex to another (or the same) vertex" (Nell Dale & John Lewis, "Computer Science Illuminated" 6th Ed., 2015)

[undirected graph:] "A graph in which the edges have no direction" (Nell Dale & John Lewis, "Computer Science Illuminated" 6th Ed., 2015)

[undirected graph:] "An undirected graph is one in which the edges do not indicate a direction (such as from-to) between two vertices." (Dan Sullivan, "NoSQL for Mere Mortals®", 2015)

"Like a tree, a graph consists of a set of nodes connected by edges. These edges may or may not have a direction. If they do, the graph is referred to as a 'directed graph'. If a graph is directed, it may be possible to start at a node and follow edges in a path that leads back to the starting node. Such a path is called a 'cycle'. If a directed graph has no cycles, it is referred to as an 'acyclic graph'." (Robert J Glushko, "The Discipline of Organizing: Professional Edition" 4th Ed., 2016)

"In a computer science or mathematics context, a graph is a set of nodes and edges that connect the nodes." (Alex Thomas, "Natural Language Processing with Spark NLP", 2020)

Undirected graph "A graph in which the edges have no direction" (Nell Dale et al, "Object-Oriented Data Structures Using Java" 4th Ed., 2016)

🔬Data Science: Semantic Network [SN] (Definitions)

"We define a semantic network as 'the collection of all the relationships that concepts have to other concepts, to percepts, to procedures, and to motor mechanisms' of the knowledge." (John F Sowa, "Conceptual Structures", 1984)

"A graph for knowledge representation where concepts are represented as nodes in a graph and the binary semantic relations between the concepts are represented by named and directed edges between the nodes. All semantic networks have a declarative graphical representation that can be used either to represent knowledge or to support automated systems for reasoning about knowledge." (László Kovács et al, "Ontology-Based Semantic Models for Databases", 2009)

"A graph structure useful to represent the knowledge of a domain. It is composed of a set of objects, the graph nodes, which represent the concepts of the domain, and relations among such objects, the graph arches, which represent the domain knowledge. The semantic networks are also a reasoning tool as it is possible to find relations among the concepts of a semantic network that do not have a direct relation among them. To this aim, it is enough 'to follow the arrows' of the network arches that exit from the considered nodes and find in which node the paths meet." (Mario Ceresa, "Clinical and Biomolecular Ontologies for E-Health", Handbook of Research on Distributed Medical Informatics and E-Health, 2009)

"A form of visualization consisting of vertices (concepts) and directed or undirected edges (relationships)." (DAMA International, "The DAMA Dictionary of Data Management", 2011)

"A term used in computer language processing and in RF and OWL to refer to concepts linked by relationships. Memory maps are an informal example of a semantic network." (Kate Taylor, "A Common Sense Approach to Interoperability", 2011)

"nodes, encapsulating data and information, are connected by edges which include information about how these nodes are related to one another." (Simon Boese et al, "Semantic Document Networks to Support Concept Retrieval", 2014)

"A knowledge representation technique that represents the relationships among objects" (Nell Dale & John Lewis, "Computer Science Illuminated" 6th Ed., 2015)

"A knowledge base that represents semantic relations between concepts. Formally, the underlying representation model is a directed graph consisting of nodes, which represent concepts, and links, which represent semantic relations between concepts, mapping or connecting semantic fields." (Dmitry Korzun et al, "Semantic Methods for Data Mining in Smart Spaces", 2019)

"A knowledge base that represents semantic relations between concepts in a network. The model of knowledge representation is based on a directed or undirected graph consisting of vertices, which represent concepts, and edges, which represent semantic relations between concepts, mapping or connecting semantic fields." (Svetlana E Yalovitsyna et al, "Smart Museum: Semantic Approach to Generation and Presenting Information of Museum Collections", 2020)

🔭Data Science: Graphs (Just the Quotes)

"A semantic network or net represents knowledge as a net-like graph. An idea, event, situation or object almost always has a composite structure; this is represented in a semantic network by a corresponding structure of nodes (drawn as circles or boxes) representing conceptual units, and directed links (drawn as arrows between the nodes) representing the relations between the units. […] An abstract (graph-theoretic) network can be diagrammed, defined mathematically, programmed in a computer, or hard-wired electronically. It becomes semantic when you assign a meaning to each node and link. Unlike specialized networks and diagrams, semantic networks aim to represent any kind of knowledge which can be described in natural language. A semantic network system includes not only the explicitly stored net structure but also methods for automatically deriving from that a much larger structure or body of implied knowledge." (Fritz Lehman, "Semantic Networks", Computers & Mathematics with Applications Vol. 23 (2-5), 1992)

"The essential idea of semantic networks is that the graph-theoretic structure of relations and. abstractions can be used for inference as well as understanding. […] A semantic network is a discrete structure as is any linguistic description. Representation of the continuous 'outside world' with such a structure is necessarily incomplete, and requires decisions as to which information is kept and which is lost." (Fritz Lehman, "Semantic Networks",  Computers & Mathematics with Applications Vol. 23 (2-5), 1992)

"A graph enables us to visualize a relation over a set, which makes the characteristics of relations such as transitivity and symmetry easier to understand. […] Notions such as paths and cycles are key to understanding the more complex and powerful concepts of graph theory. There are many degrees of connectedness that apply to a graph; understanding these types of connectedness enables the engineer to understand the basic properties that can be defined for the graph representing some aspect of his or her system. The concepts of adjacency and reachability are the first steps to understanding the ability of an allocated architecture of a system to execute properly." (Dennis M Buede, "The Engineering Design of Systems: Models and methods", 2009)

"Graphs can embed complex semantic representations in a compact form. As such, modeling data as networks of related entities is a powerful mechanism for analytics, both for visual analyses and machine learning. Part of this power comes from performance advantages of using a graph data structure, and the other part comes from an inherent human ability to intuitively interact with small networks." (Benjamin Bengfort et al, "Applied Text Analysis with Python: Enabling Language-Aware Data Products with Machine Learning", 2018)

"In Exploiting semantic knowledge graphs can support interpretability and explainability of nearly all AI model types (including DL models) by discovering and depicting semantic and non-obvious relationships or depicting an ML model in a simplified and more readable, explainable way., a Data Mesh solution organizes data around business domain owners and transforms relevant data assets (data sources) to data products that can be consumed by distributed business users from various business domains or functions. These data products are created, governed, and used in an autonomous, decentralized, and self-service manner. Self-service capabilities, which we have already referenced as a Data Fabric capability, enable business organizations to entertain a data marketplace with shopping-for-data characteristics." (Eberhard Hechler et al, "Data Fabric and Data Mesh Approaches with AI", 2023)

"[...] a graph is a mathematical model that is used for describing relationships between entities. However, each complex network presents intrinsic properties. Such properties can be measured by particular metrics, and each measure may characterize one or several local and global aspects of the graph." (Aldo Marzullo et al, "Graph Machine Learning" 2nd Ed., 2025)

"Although creating simple subgraphs and merging them is a way to generate new graphs of increasing complexity, networks may also be generated by means of probabilistic models   and/or generative models that let a graph grow by itself. Such graphs usually share   interesting properties with real networks and have long been used to create benchmarks and synthetic graphs, especially in times when the amount of data available was not as overwhelming as today." (Aldo Marzullo et al, "Graph Machine Learning" 2nd Ed., 2025)

"As with many other deep learning-based approaches, another major challenge is in interpretability. While knowledge graphs provide a structured and transparent way to store relationships, LLMs operate as a black box, making it difficult to understand how specific outputs are generated. [...] Data alignment is also a key issue, as structured knowledge graphs and unstructured text data must be carefully preprocessed to ensure consistency.  Differences in data formats, ontology mismatches, and information redundancy can create inefficiencies when integrating these two paradigms. Developing robust pipelines that seamlessly connect graph-based insights with LLM-generated text remains an open challenge." (Aldo Marzullo et al, "Graph Machine Learning" 2nd Ed., 2025)

"Graph analytics is generally very effective in clustering users, merchants, and communities to provide an effective implementation of behavior analytics. On the other hand, second-party fraud can be identified with the implementation of monitoring employee behavior, as well as compliance checks. Graph analytics can indeed be useful for these use cases. Similar to the first-party models, employee behavior can also be analyzed using graph machine learning, although the dataset may need to encode a number of other sources of information besides transactional data. From a compliance standpoint, process mining techniques that still rely on a graph representation of the various procedural steps/pathways can be effective in identifying fraudulent behavior or non-compliant processes. Finally, third-party fraud, especially in the form of phishing attacks, can also be addressed using graph machine learning. In this context, understanding the network from which the phishing attack comes as well as the URLs being used (which can also benefit from a graph representation) can be critical for building an effective phishing classifier." (Aldo Marzullo et al, "Graph Machine Learning" 2nd Ed., 2025)

"The concept of temporal graphs is useful in all the real-world problems that can be represented as a graph, where the nodes and edges of the graph may change over time. For example, temporal graphs are extensively applied in modeling social networks. By capturing the evolving relationships between individuals, temporal graphs enable a more accurate representation of social dynamics. This is particularly useful for predicting changes in friendships, community structures, and the information diffusion over time." (Aldo Marzullo et al, "Graph Machine Learning" 2nd Ed., 2025)

🎓Knowledge Management: Concept Map (Definitions)

"Concept maps are built of nodes connected by connectors, which have written descriptions called linking phrases instead of polarity of strength. Concept maps can be used to describe conceptual structures and relations in them and the concept maps suit also aggregation and preservation of knowledge" (Hannu Kivijärvi et al, "A Support System for the Strategic Scenario Process", 2008) 

"A hierarchal picture of a mental map of knowledge." (Gregory MacKinnon, "Concept Mapping as a Mediator of Constructivist Learning", 2009)

"A tool that assists learners in the understanding of the relationships of the main idea and its attributes, also used in brainstorming and planning." (Diane L Judd, "Constructing Technology Integrated Activities that Engage Elementary Students in Learning", 2009)

"Concept maps are graphical knowledge representations that are composed to two components: (1) Nodes: represent the concepts, and (2) Links: connect concepts using a relationship." (Faisal Ahmad et al, "New Roles of Digital Libraries", 2009)

"A concept map is a diagram that depicts concepts and their hierarchical relationships." (Wan Ng & Ria Hanewald, "Concept Maps as a Tool for Promoting Online Collaborative Learning in Virtual Teams with Pre-Service Teachers", 2010)

"A diagram that facilitates organization, presentation, processing and acquisition of knowledge by showing relationships among concepts as node-link networks. Ideas in a concept map are represented as nodes and connected to other ideas/nodes through link labels." (Olusola O Adesope & John C Nesbit, "A Systematic Review of Research on Collaborative Learning with Concept Maps", 2010)

"A visual construct composed of encircled concepts (nodes) that are meaningfully inter-connected by descriptive concept links either directly, by branch-points (hierarchies), or indirectly by cross-links (comparisons). The construction of a concept map can serve as a tool for enhancing communication, either between an author and a student for a reading task, or between two or more students engaged in problem solving. (Dawndra Meers-Scott, "Teaching Critical Thinking and Team Based Concept Mapping", 2010)

"Are graphical ways of working with ideas and presenting information. They reveal patterns and relationships and help students to clarify their thinking, and to process, organize and prioritize. The visual representation of information through word webs or diagrams enables learners to see how the ideas are connected and understand how to group or organize information effectively." (Robert Z Zheng & Laura B Dahl, "Using Concept Maps to Enhance Students' Prior Knowledge in Complex Learning", 2010)

"Concept maps are hierarchical trees, in which concepts are connected with labelled, graphical links, most general at the top." (Alexandra Okada, "Eliciting Thinking Skills with Inquiry Maps in CLE", 2010)

"One powerful knowledge presentation format, devised by Novak, to visualize conceptual knowledge as graphs in which the nodes represent the concepts, and the links between the nodes are the relationships between these concepts." (Diana Pérez-Marín et al, "Adaptive Computer Assisted Assessment", 2010)

"A form of visualization showing relationships among concepts as arrows between labeled boxes, usually in a downward branching hierarchy." (DAMA International, "The DAMA Dictionary of Data Management", 2011)

"A graphical depiction of relationships ideas, principals, and activities leading to one major theme." (Carol A Brown, "Using Logic Models for Program Planning in K20 Education", 2013)

"A diagram that presents the relationships between concepts." (Gwo-Jen Hwang, "Mobile Technology-Enhanced Learning", 2015)

"A graphical two-dimensional display of knowledge. Concepts, usually presented within boxes or circles, are connected by directed arcs that encode, as linking phrases, the relationships between the pairs of concepts." (Anna Ursyn, "Visualization as Communication with Graphic Representation", 2015)

"A graphical tool for representing knowledge structure in a form of a graph whose nodes represent concepts, while arcs between nodes correspond to interrelations between them." (Yigal Rosen & Maryam Mosharraf, "Evidence-Centered Concept Map in Computer-Based Assessment of Critical Thinking", 2016) 

"Is a directed graph that shows the relationship between the concepts. It is used to organize and structure knowledge." (Anal Acharya & Devadatta Sinha, "A Web-Based Collaborative Learning System Using Concept Maps: Architecture and Evaluation", 2016)

"A graphic depiction of brainstorming, which starts with a central concept and then includes all related ideas." (Carolyn W Hitchens et al, "Studying Abroad to Inform Teaching in a Diverse Society", 2017)

"A graphic visualization of the connections between ideas in which concepts (drawn as nodes or boxes) are linked by explanatory phrases (on arrows) to form a network of propositions that depict the quality of the mapper’s understanding" (Ian M Kinchin, "Pedagogic Frailty and the Ecology of Teaching at University: A Case of Conceptual Exaptation", 2019)

"A diagram in which related concepts are linked to each other." (Steven Courchesne &Stacy M Cohen, "Using Technology to Promote Student Ownership of Retrieval Practice", 2020)

🎓Knowledge Management: Cognitive Map (Definitions)

"A cognitive map is a specific way of representing a person's assertions about some limited domain, such as a policy problem. It is designed to capture the structure of the person's causal assertions and to generate the consequences that follow front this structure." (Robert M Axelrod, "Structure of Decision: The cognitive maps of political elites", 1976)

"A cognitive map is the representation of thinking about a problem that follows from the process of mapping." (Colin Eden, "Analyzing cognitive maps to help structure issues or problems", 2002)

"A mental representation of a portion of the physical environment and the relative locations of points within it." (Andrew M Colman, "A Dictionary of Psychology" 3rd Ed, 2008)

"A mental model (or map) of the external environment which may be constructed following exploratory behaviour." (Michael Allaby, "A Dictionary of Zoology" 3rd Ed., 2009)

"An FCM [Fuzzy Cognitive Map] is a directed graph with concepts like policies, events etc. as nodes and causalities as edges. It represents causal relationship between concepts." (Florentin Smarandache &  W B Vasantha Kandasamy, "Fuzzy Cognitive Maps and Neutrosophic Cognitive Maps", 2014)

"A conceptual tool that provides a representation of particular natural or social environments in the form of a model." (Evangelos C Papakitsos et al, "The Challenges of Work-Based Learning via Systemic Modelling in the European Union", 2020)

"A representation of the conceptualization that the subject constructs of the system in which he evolves. The set of cognitive representations that emerge make it possible to understand his actions, the links between the factors structuring the cognitive patterns dictating his behaviors." (Henda E Karray & Souhaila Kammoun, "Strategic Orientation of the Managers of a Tunisian Family Group Before and After the Revolution", 2020)

"A cognitive map is a type of mental representation which serves an individual to acquire, code, store, recall, and decode information about the relative locations and attributes of phenomena in their everyday or metaphorical spatial environment." (Wikipedia) [source]

📉Graphical Representation: Line Graphs (Just the Quotes)

"Except in some of the simplest cases where the line connecting the plotted data is straight, it will generally be possible to fit a number of very different forms of equation to the same curve, none of them exactly, but all agreeing with the original about equally well. Interpolation on any of these curves will usually give results within the desired degree of accuracy. The greatest caution, however, should be observed in exterpolation, or the use of the equation outside of the limits of the observations." (John B Peddle, "The Construction of Graphical Charts", 1910)

"A series ot quantities or values can be most simply and often best shown by a series of corresponding lines or bars. All bars being drawn against one and the same scale, their lengths vary with the amounts which they represent." (Karl G Karsten, "Charts and Graphs", 1925)

"Graphs showing time changes, or the increases and decreases in the amount of something over a period of time, are generally of two kinds: (1) vertical bar graphs, and (2) broken- or smooth-line graphs. Both kinds differ from the categorical charts [...] in that they have two scales instead of only one; that is why it is preferable to call them graphs rather than charts, although these terms are used rather freely and interchangeably, and there is no standard convention." (William L Schaaf, "Mathematics For Everyday Use", 1942)

"When statistical data are of such a nature that it is permissible to assume that 'in-between values' vary continuously and uniformly (or very nearly so) from one observed or measured value to the next, a modification of the broken-line graph may be used. Instead of connecting the plotted points with straightline segments, a 'smooth' curved line is drawn between the points [...]. Such curvedline graphs may be drawn either 'free hand' or with the aid of drafting instruments known as French curves." (William L Schaaf, "Mathematics For Everyday Use", 1942)

"In line charts the grid structure plays a controlling role in interpreting facts. The number of vertical rulings should be sufficient to indicate the frequency of the plottings, facilitate the reading of the time values on the horizontal scale. and indicate the interval or subdivision of time." (Anna C Rogers, "Graphic Charts Handbook", 1961)

"Data should not be forced into an uncomfortable or improper mold. For example, data that is appropriate for line graphs is not usually appropriate for circle charts and in any case not without some arithmetic transformation. Only graphs that are designed to fit the data can be used profitably." (Cecil H Meyers, "Handbook of Basic Graphs: A modern approach", 1970)

"The numerous design possibilities include several varieties of line graphs that are geared to particular types of problems. The design of a graph should be adapted to the type of data being structured. The data might be percentages, index numbers, frequency distributions, probability distributions, rates of change, numbers of dollars, and so on. Consequently, the designer must be prepared to structure his graph accordingly." (Cecil H Meyers, "Handbook of Basic Graphs: A modern approach", 1970)

"While circle charts are not likely to present especially new or creative ideas, they do help the user to visualize relationships. The relationships depicted by circle charts do not tend to be very complex, in contrast to those of some line graphs. Normally, the circle chart is used to portray a common type of relationship (namely. part-to-total) in an attractive manner and to expedite the message transfer from designer to user." (Cecil H Meyers, "Handbook of Basic Graphs: A modern approach", 1970)

"There are several uses for which the line graph is particularly relevant. One is for a series of data covering a long period of time. Another is for comparing several series on the same graph. A third is for emphasizing the movement of data rather than the amount of the data. It also can be used with two scales on the vertical axis, one on the right and another on the left, allowing different series to use different scales, and it can be used to present trends and forecasts." (Anker V Andersen, "Graphing Financial Information: How accountants can use graphs to communicate", 1983)

"In the case of graphs, the number of lines which can be included on any one illustration will depend largely on how close the lines are and how often they cross one another. Three or four is likely to be the maximum acceptable number. In some instances, there may be an argument for using several graphs with one line each as opposed to one graph with multiple lines. It has been shown that these two arrangements are equally satisfactory if the user wishes to read off the value of specific points; if, however, he wishes to compare the lines, than the single multi-line graph is superior." (Linda Reynolds & Doig Simmonds, "Presentation of Data in Science" 4th Ed, 1984)

"A connected graph is appropriate when the time series is smooth, so that perceiving individual values is not important. A vertical line graph is appropriate when it is important to see individual values, when we need to see short-term fluctuations, and when the time series has a large number of values; the use of vertical lines allows us to pack the series tightly along the horizontal axis. The vertical line graph, however, usually works best when the vertical lines emanate from a horizontal line through the center of the data and when there are no long-term trends in the data." (William S Cleveland, "The Elements of Graphing Data", 1985)

"A bar graph typically presents either averages or frequencies. It is relatively simple to present raw data (in the form of dot plots or box plots). Such plots provide much more information. and they are closer to the original data. If the bar graph categories are linked in some way - for example, doses of treatments - then a line graph will be much more informative. Very complicated bar graphs containing adjacent bars are very difficult to grasp. If the bar graph represents frequencies. and the abscissa values can be ordered, then a line graph will be much more informative and will have substantially reduced chart junk." (Gerald van Belle, "Statistical Rules of Thumb", 2002)

"The biggest difference between line graphs and sparklines is that a sparkline is compact with no grid lines. It isnʼt meant to give precise values; rather, it should be considered just like any other word in the sentence. Its general shape acts as another term and lends additional meaning in its context. The driving forces behind these compact sparklines are speed and convenience." (Brian Suda, "A Practical Guide to Designing with Data", 2010)

"As with dot plots, the scale on line charts has a lot to do with how the message is conveyed. For example, using too large a scale runs the risk that viewers may gloss over a very important story in the data. However, using too small a scale might lead you to overemphasize minor fluctuations. As with dot plots, designers should plot all of the data points so that the line chart takes up two-thirds of the y-axis’s total scale." (Jason Lankow et al, "Infographics: The power of visual storytelling", 2012)

"The ability to see meaningful shapes in the data represents the highest level of data visualization, because it represents the highest level of data integration and a richer graphical landscape. Line charts and scatter plots are frequently used for this shape visualization." (Jorge Camões, "Data at Work: Best practices for creating effective charts and information graphics in Microsoft Excel", 2016)

"The law of continuity states that we interpret images so as not to generate abrupt transitions or otherwise create images that are more complex. […] we can arbitrarily fill in the missing elements to complete a pattern. It’s also the case of time series, in which we assume that data points in the future will be a smooth continuation of the past. […] In a line chart, those series with a similar slope (that is, they appear to follow the same direction) are understood as belonging to the same group." (Jorge Camões, "Data at Work: Best practices for creating effective charts and information graphics in Microsoft Excel", 2016)

"A difficulty with combined bivariate visualizations is that the connection between the individual displays has to be established by the observer mentally. That is, as the eyes move from one bivariate display to the next, the observer has to keep track of the visited dots in order to form a complete understanding of data tuples. Visualization techniques based on polylines aim to tackle this difficulty. The basic strategy is to create m axes, one for each attribute, and n polylines, one for each data tuple. The polyline of an m-variate data tuple is constructed as follows. For each attribute value of the data tuple, a position is computed at the corresponding attribute axis. The m positions that we obtain are then connected to form the polyline that represents the entire tuple." (Christian Tominski & Heidrun Schumann, "Interactive Visual Data Analysis", 2019)

"Look beyond the subject and you will see analytical and design choices that are just as applicable to you and your work: a line chart showing political forecasts involves the same thought process as would a line chart showing stock prices changing or average global temperatures rising. A line chart is a line chart, regardless of the subject matter." (Andy Kirk, "Data Visualisation: A Handbook for Data Driven Design" 2nd Ed., 2019)

"Researchers have studied how accurately people can read information displayed in different types of plots. They have found the following ordering, from most to leasta ccurately judged (•) Positions along a common scale, like in a rug plot, strip plot, or dot plot (•) Positions on identical, nonaligned scales, like in a bar plot (•) Length, like in a stacked bar plot (•) Angle and slope, like in a pie chart (•) Area, like in a stacked line plot or bubble chart (•) Volume and density, like in a three-dimensional bar plot (•) Color saturation and hue, like when overplotting with semitransparent points."  (Sam Lau et al, "Learning Data Science: Data Wrangling, Exploration, Visualization, and Modeling with Python", 2023)

"A line graph looks similar to a scatterplot, but each point is connected to form a wiggly line that runs from left to right. The values on the x-axis are either ordinal or numerical data that tell us the order of each data point. The connections between each point make it easier to see how much the values on the y-axis change from one point to the next. Because line charts show data in a particular order, a line in a line chart can only have one point for each value on the x-axis." (Nancy Organ, "Data Visualization for People of All Ages", 2024)

"Line charts are useful for identifying patterns and trends in a one‑dimensional sequence of univariate data, that is, continuous data over time with a single value per data item. They map the sequence data (e.g., time) to one dimension, typically the x‑axis, and the data value to another dimension, typically the y‑axis, forming a line; or to the color of a mark or region along the spatial axis, forming a bar. The data is adjusted in size to be within the limits of the display attribute." (Leandro N de Castro, "Exploratory Data Analysis: Descriptive Analysis, Visualization, and Dashboard Design", 2025)

📉Graphical Representation: Graphs (Just the Quotes)

"Graphs are all inclusive. No fact is too slight or too great to plot to a scale suited to the eye. Graphs may record the path of an ion or the orbit of the sun, the rise of a civilization, or the acceleration of a bullet, the climate of a century or the varying pressure of a heart beat, the growth of a business, or the nerve reactions of a child." (Henry D Hubbard [foreword to Willard C Brinton, "Graphic Presentation", 1939)])

"Graphs carry the message home. A universal language, graphs convey information directly to the mind. Without complexity there is imaged to the eye a magnitude to be remembered. Words have wings, but graphs interpret. Graphs are pure quantity, stripped of verbal sham, reduced to dimension, vivid, unescapable." (Henry D Hubbard [foreword to Willard C Brinton, "Graphic Presentation", 1939]) 

"The graphic language is modern. We are learning its alphabet. That it will develop a lexicon and a literature marvelous for its vividness and the variety of application is inevitable. Graphs are dynamic, dramatic. They may epitomize an epoch, each dot a fact, each slope an event, each curve a history. Wherever there are data to record, inferences to draw, or facts to tell, graphs furnish the unrivalled means whose power we are just beginning to realize and to apply."  (Henry D Hubbard [foreword to Willard C Brinton, "Graphic Presentation", 1939)])

"A type of picture-graph less commonly used than formerly is the pictorial representation of an object which has been arbitrarily subdivided to show certain numerical relationships; as, for example, the pictorial representation of the food values of beefsteak. This is a very poor type of graphic representation, and should definitely be avoided. The irregular outline of the picture as a whole, and of each of the shaded areas, makes a comparison of the areas difficult, if not altogether impossible; the shading only to the confusion." (William L Schaaf, "Mathematics For Everyday Use", 1942)

"Graphs showing time changes, or the increases and decreases in the amount of something over a period of time, are generally of two kinds: (1) vertical bar graphs, and (2) broken- or smooth-line graphs. Both kinds differ from the categorical charts [...] in that they have two scales instead of only one; that is why it is preferable to call them graphs rather than charts, although these terms are used rather freely and interchangeably, and there is no standard convention." (William L Schaaf, "Mathematics For Everyday Use", 1942)

"When statistical data are of such a nature that it is permissible to assume that 'in-between values' vary continuously and uniformly (or very nearly so) from one observed or measured value to the next, a modification of the broken-line graph may be used. Instead of connecting the plotted points with straightline segments, a 'smooth' curved line is drawn between the points [...]. Such curvedline graphs may be drawn either 'free hand' or with the aid of drafting instruments known as French curves." (William L Schaaf, "Mathematics For Everyday Use", 1942)

"You may expect to find graphs anywhere: in books, in periodicals, in newspapers, in pamphlets, on show cards in advertisements, in business reports, and so on. Their use, however, is sometimes limited. For one thing, they are of necessity less accurate than the figures on which they are based, which, of course, doesn’t matter too much in many cases. In the second place, they are sometimes misleading, which may or may not be intentional. It is also possible that the reader of a chart or graph may misinterpret it." (William L Schaaf, "Mathematics For Everyday Use", 1942)

"If one technique of data analysis were to be exalted above all others for its ability to be revealing to the mind in connection with each of many different models, there is little doubt which one would be chosen. The simple graph has brought more information to the data analyst’s mind than any other device. It specializes in providing indications of unexpected phenomena." (John W Tukey, "The Future of Data Analysis", Annals of Mathematical Statistics Vol. 33 (1), 1962)

"There is no more reason to expect one graph to ‘tell all’ than to expect one number to do the same." (John W Tukey, "Exploratory Data Analysis", 1977)

"[...] exploratory data analysis is an attitude, a state of flexibility, a willingness to look for those things that we believe are not there, as well as for those we believe might be there. Except for its emphasis on graphs, its tools are secondary to its purpose." (John W Tukey, [comment] 1979)

"We would wish ‘numerate’ to imply the possession of two attributes. The first of these is an ‘at-homeness’ with numbers and an ability to make use of mathematical skills which enable an individual to cope with the practical mathematical demands of his everyday life. The second is ability to have some appreciation and understanding of information which is presented in mathematical terms, for instance in graphs, charts or tables or by reference to percentage increase or decrease." (Cockcroft Committee, "Mathematics Counts: A Report into the Teaching of Mathematics in Schools", 1982)

"We would wish ‘numerate’ to imply the possession of two attributes. The first of these is an ‘at-homeness’ with numbers and an ability to make use of mathematical skills which enable an individual to cope with the practical mathematical demands of his everyday life. The second is ability to have some appreciation and understanding of information which is presented in mathematical terms, for instance in graphs, charts or tables or by reference to percentage increase or decrease." (Cockcroft Committee, "Mathematics Counts: A Report into the Teaching of Mathematics in Schools", 1982)

"Iteration and experimentation are important for all of data analysis, including graphical data display. In many cases when we make a graph it is immediately clear that some aspect is inadequate and we regraph the data. In many other cases we make a graph, and all is well, but we get an idea for studying the data in a different way with a different graph; one successful graph often suggests another." (William S Cleveland, "The Elements of Graphing Data", 1985)

"There are some who argue that a graph is a success only if the important information in the data can be seen within a few seconds. While there is a place for rapidly-understood graphs, it is too limiting to make speed a requirement in science and technology, where the use of graphs ranges from, detailed, in-depth data analysis to quick presentation." (William S Cleveland, "The Elements of Graphing Data", 1985)

"A first analysis of experimental results should, I believe, invariably be conducted using flexible data analytical techniques – looking at graphs and simple statistics – that so far as possible allow the data to ‘speak for themselves’. The unexpected phenomena that such a approach often uncovers can be of the greatest importance in shaping and sometimes redirecting the course of an ongoing investigation." (George Box, "Signal to Noise Ratios, Performance Criteria, and Transformations", Technometrics 30, 1988) 

"We are not saying that the primary purpose of a graph is to convey numbers with as many decimal places as possible. We agree with Ehrenberg (1975) that if this were the only goal, tables would be better. The power of a graph is its ability to enable one to take in the quantitative information, organize it, and see patterns and structure not readily revealed by other means of studying the data." (William Cleveland & Robert McGill, "Graphical Perception: Theory, Experimentation, and Application to the Development of Graphical Models", Journal of the American Statistical Association 79, 1984)

"It’s not easy to select more than a few clearly distinct colors. Also, 'distinct' is context-dependent, because: What will be the spatial relationships of the different colors in your output? You can successfully have fairly similar colors adjacent to each other, since the contrast is more obvious when they’re adjacent. However, if you want to use colors to track identity and difference across scattered points or patches, then you need bigger separations between colors, since you want to be able to see easily that patch 'A' here is of the same kind as patch 'A' there and different from patch 'B' somewhere else, when mingled with patches of other kinds. And size matters. Big patches of similar color (as on a map) can look quite distinct, while the same colors used to plot filled circular blobs on a graph might be barely distinguishable, and totally indistinguishable if used to plot colored '.'s or '+'s. [...] It’s all very psycho-visual and success usually requires experimentation!" (Ted Harding, R-help mailing list, 2004)