A practical understanding of topological sorting and ordering

The shortest path problem is pivotal in graph theory. It aims at discovering the “most efficient’ or ‘best’ way of moving from x to y, where x and y are both nodes in a given graph. The most efficient or best in this context is evaluated by the lowest sum of edge weights between the path of both vertices. The shortest or quickest path is arrived at by summing the lengths of the individual edges. A best-case scenario is a graph with edges having positive weights. There is also the concept of single-source shortest path problem with s as the source node. For clarity, the source node initiates the transversal within the graph.

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Exploring Breadth First Search and Depth First Search in a graph

You might have encountered the words, breadth and depth in real-world scenarios. Breadth implies the complete range of knowledge of any given subject or topic. On the other hand, depth in terms of learning touches on the degree to which a particular subject is magnified or explored. Let’s begin with the breadth first search or the BFS of a given graph. Now BFS does not refer to Best Friends from school but Breadth-First Search. 

Exploring Breadth First Search or Breadth First Traversal 

BFS is an algorithm that is designed to search for a graph or tree data formation. It usually travels in a breadthward motion and utilises a queue as a prompt to identify the next vertex to commence a traversal. If a roadblock is encountered or no adjacent node is found, the tree root or the source node is removed for the queue. The traversal of the graph usually begins with a ‘search key’ or the initialising node. Imagine a hotel with so many floors and rooms as nodes, a breadth-first traversal algorithm is like a cleaning staff that will clean rooms floor by floor. All neighbouring nodes at the current depth or floor with the example above will be visited to clean before moving to the vertices or rooms on the next floor. No node is expected to be revisited as one would not expect hotel staff to clean the same room twice in the same period. Once a room is cleaned, it is ticked on a sheet as a visited while with BFS, the neighbouring reversed node is enqueued or marked as visited, 

Rules are important to ensure procedures are followed or order is maintained. There are relevant rules for a BFS algorithm. 

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Exploring the different types of weights in a connected graph

Finding the shortest path in a network is not always the fewest nodes that need to be travelled through to a given destination. This is only applicable to an unweighted graph. For a weighted graph, the shortest path between two nodes is the least total of weight that exists between these nodes. The weights attributed to the edges of these graphs are numerical in nature and positive figures. Negative weights will prevent the determination of a minimal path. A weighted graph is viewed as a labeled graph due to the numbers assigned to each edge. 

In mathematics, a weight is used to express a set of multiplicative constants allocated in front of terms in an edge of a tree. For example, the weight of a graph at a point [n] is the maximum number of edges in a given branch [n]. The weights of a graph are computed and analysed in light of the problem investigated. Weighted graphs are believed to be present in different graph scenarios such as shortest path, centrality and travelling salesman problems.

The weight of a graph assigns value to either the node or the relationship between nodes. These values determine the shortest path or most central node in a given network. The nature or type of weight is more relevant in the type of network. For example, a weight type flow may be more applicable to traffic in a computer network, fluids in a water pipe, currents in an electrical circuit or demand movements. We will look into the flow network as it pertains to weight in the subsequent section. 

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Finding the mother vertex in a graph

Networks or graphs are pivotal in so many real-world applications such as fraud prevention systems. search engines, recommendation systems, social networks and a lot more. The search for the mother vertex of a graph aids in understanding the accessibility of a given vertex or collection of vertices. 

What is the meaning of the mother vertex in a given graph?

In a given graph G = (V, E), a mother vertex v has a pathway for all other vertices in the specified graph to enable the connection. All other vertices in the graph can be accessed via the mother vertex. A mother vertex is quite common in directed graphs but is also applicable in undirected networks. We will briefly explore mother vertices in different network examples

Directed graph: Directed graphs or DiGraphs do hold directed edges or nodes. These edges can be unidirectional or bidirectional. For DiGraphs, self-loops are permissible but parallel edges are not. Mother vertex exists in directed graphs and there can be multiple of these as shown below.  Based on the directed graph below, nodes [4] and [6] are the mother vertex. 

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The role of the betweenness centrality measure in networks

Ever wondered how to detect the most influential individual, station, motorway or node in a network? It is not a normal popularity test but a mathematical way for determining a node with the most impact in the flow of information within a network. A very good way of determining nodes that are great connectors for moving from one point of a graph to another. In a real-world situation, when these nodes are removed, the movement to other nodes in the graph becomes quite challenging. With betweenness centrality, the number of paths a node is a part of is also revealed. In a connected graph, the Betweenness Centrality algorithm calculates the shortest path between nodes in the given network. The weight between nodes is quite important in determining the shortest path as factors such as frequency, capacity, time, flow and influence determine these weights. 

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Exploring Dijkstra’s shortest path algorithm

Several graph algorithms can help reveal hidden patterns in connected data. These algorithms can be classified into several categories such as approximations (e.g clustering), assortativity (e,g average neighbour degree), communities (e.g K-Clique) and centrality (e.g shortest path). In this blog, we will be looking at one of the most popular shortest path algorithms known as the Dijkstra’s algorithm. Exploring an example table and code implementation for this algorithm. Shortest path algorithm can be relevant in a traffic network situation a user desires to discover the fastest way to move from a source to a destination. It is an iterative algorithm that provides us with the shortest path from an origin node to all other nodes in the graph. This algorithm can work in weighted and unweighted graph scenarios. 

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Different ways of representing Graphs

Graph data can be represented in different formats for onward computation. The choice of the graph representation hugely relies on the density of the graph, space required, speed and weight of edges. The main ways a graph can be represented are as an adjacency matrix, incidence matrix, adjacency list and incidence list.

Adjacency Matrix: This is one of the most popular ways a graph is represented. One of the core aims of this matrix is to assess if the pairs of vertices in a given graph are adjacent or not. In an adjacency matrix, row and columns represent vertices. The row sum equals the degree of the vertex that the row represents. It is advisable to use the adjacency matrix for weighted edges.  You replace the standard ‘1’ with the respective weight.  It is easier to represent directed graphs with edge weights through an adjacency matrix. Adjacency matrix works best with dense graphs. As dense graphs usually have twice the size of the edges to the given nodes.

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Introduction to a graph data structure in Python

Graphs play an important part in intelligent technologies such as recommendation engines, search engines, fraud detection applications, network mapping, customer journey applications, latency evaluation and dependency management. Graphs data structures are also powerful in social networks such as Twitter, Facebook, LinkedIn and a few others. Graphs in the context of the social graph are used to recommend friends, events, company pages, jobs and personalise the ad experience. These are some of the use cases of graph data structures and applications. 

A quick reminder of the definition, graphs are considered to be the composition of vertices (nodes) with respective pairs of edges(also known as links or arcs). In a nutshell, they are viewed to comprise a determined set of nodes and edges which connect these given nodes. 

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The rise of eight different types of graph

We are witnessing the rise and adoption of graph databases across different verticals. Gartner acknowledged the five different types of graphs as social, intent, consumption, mobile and interest. In a presentation titled: Graph All the things! Introduction to graph databases, the team from Neo4j captured Gartner’s graph classifications in the illustration below. There is a slight difference in how one of the types of a graph is named by Neo4j in comparison to Gartner. To Neo4j it is a payment graph while to Gartner it is the consumption graph.  There is the argument that a consumption graph is a better name as we do not necessarily pay for every consumption. We will now look at each graph and add some additional types

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The world’s leading graph databases

The increase in data has led to a growing need for graph databases or technologies.  With a graph database, the relationships that exist within the data can be stored, refined and queried properly. A graph database, therefore, is a database created to store data without restricting it to a pre-set model. The data in these graph-based technology expresses how each entity is related to others.

Nodes and edges are quite important when looking at graph databases as the later represents the relationship with the former. This nodes and edges setup, makes the retrieval and querying of relationships easier. Retrieving complex hierarchical structures is an advantage that these graph technologies have over relational databases. The software review forum G2 has a list of the top-rated graph databases in the market. The leading graph database technologies on G2 have had more reviews, a higher percentage of positive feedback, more data generated from other online networks and social platforms.

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