Towards Complex Networks

Class 6

March 4th, 2020

Data Science: Techniques and Applications (DSTA)

introduction to Graphs

Basic concepts

The study of how objects(entities) connect to each other and the properties of their connection.

Possible understanding: a relationship istance.

Terminology

$G=\langle V, E\rangle$ where $E\subseteq V \times V$
$|V|=n$ , $|E|=m$ , density is $|E|/|V|^2$

vertex v is adjacent to u if $(u, v)\in E;$ $(v, v)\not\in E.$
neigh. of v, N(v): the set of adjacent vertices; $deg(v)=|N(v)|$

The adjacency matrix $A_{n\times n}$ of G: $(u,v)\in E \leftrightarrow a_{ij}=1$

[incidence matrix $I_{n\times m}$ of G:]

Paths, connectedness

A path $u\rightarrow v$ is a sequence of edges $\langle (u, c_1), (c_1, c_2), \dots (c_k, v)\rangle$
its lenght (k+1) is the cardinatility of the path.

Two vertices are connected if $\exists$ a path betw. them.
A graph is connected if all its vertices are.

Distances, I

Distance is the lenght of the (possibly non-unique) shortest path connecting them, $\infty$ otherwise.
The diameter of a graph is the maximum distance between any two pairs

Distances, II

Average distances are also important

[Facebook, circa 2011}

[…] the first world-scale social-network graph-distance computations, using the entire Facebook network of active users (~721 million users, ~69 billion friendship links). The average distance […] is 4.74, corresponding to 3.74 intermediaries or “degrees of separation.”

Weighted, Directed, Multiplex

$G=\langle V, E, w\rangle$ where $w: E\rightarrow \mathbb{R}$

Path lenght: sum of the weights of the arcs.

$G=\langle V, E\rangle$ where $v\in V$ are nodes and $\langle u, v\rangle\in E$ are arcs. $w: E\rightarrow \mathbb{R}$

Out-neigh. and In-neigh.

$G=\langle V, E, D\rangle$ where $V$ are nodes D are dimensions/layers and $\langle u, v, d\rangle\in E$ are arcs

Interesting questions

interesting questions I: paths and oflow

interesting questions II: centralities

interesting questions III: clustering

import networkx as nx

G=nx.karate_club_graph()

print("Node Degree")
for v in G:
    print('%s %s' % (v,G.degree(v)))

Visualization

Visualizing frequencies by graphs

More on Graphs/Networks

The code can be cloned from Github

Example lecture: complex networks

Ch. 1: Food Webs

Important concepts

load data and organise it in a networkx data structure

modeling tip: it is ok to have a special node representing “nature”
modeling tip: look for invariants

find the connected component (the bowtie):

source, connect and sink.

Method

study degree distribution
find properties of a network in terms of the degree organization

study clustering coefficient: why is it so much better than plain network density?

Ch. 2: Trade networks

Important concepts

directed weighted networks
the need to represent

-sorts, and their quantitites

-time

discover multiplex networks

Towards Clustering

reciprocity
assorativity

discover hidden structures

Towards Complex Networks Class 6 March 4th, 2020