An introduction

Why networks?

Network Data

Network Measures

Networks are everywhere!

The focus is not on individual attributes, but…

… on who is connected to whom.

AKA networks are about

**relationships**Interactions, or

flows between nodes (actors, places, …)

Social Network Analysis, Graph Theory, Network Science and Complexity Theory

Level (aggregation of linkages and nodes) | Geographical scale (network boundary) | |||
---|---|---|---|---|

1 – Local | 2 – Regional | 3 – Global | ||

A -- Macro | Cities | City as a whole entity | Central place system (Christaller 1966) | |

B -- Meso | Firms | Clusters (Porter 2000) | Regional economic intergration and functional polycentrism (e.g. Scott 2001: Boschma 2005) | Global inter-firm network (e.g. Cohen 1981) |

C – Micro | People | A neighbourly economic suport support network (e.g. Wellman 1979) | A local labou market (e.g. Granovetter 1974) | The global economic elite (Sassen 1991) |

Source: (Neal and Rozenblat 2021)

Source: Facebook.com

Source: http://www.opte.org/

Leonhard Euler (1707 – 1783)

Euler’s number \(e = 2.71828\)

First use of Graphs to solve mathematical problems

example: The Bridges of Königsberg

Different types and shapes

Basic building blocks

Directionality

Weights

Basic elements: nodes and links

Data is not arranged in vectors but on square matrices

1 and 0 represent the existence of a link

**Adjacency matrix**

A | B | C | D | |

A | 0 | 1 | 1 | 1 |

B | 0 | 0 | 1 | 0 |

C | 0 | 0 | 0 | 1 |

D | 0 | 0 | 0 | 0 |

Basic elements: nodes and links

Data is not arranged in vectors but on square matrices

1 and 0 represent the existence of a link

**Edge list**i j w A B 1 A C 1 A D 1 B C 1 C D 1 D C 1

Basic elements: nodes and links

Data is not arranged in vectors but on square matrices

1 and 0 represent the existence of a link

Symmetrical ties

Symmetrical adjacency matrices

**Adjacency matrix**

A | B | C | D | |

A | 0 | 1 | 1 | 1 |

B | 1 | 0 | 1 | 0 |

C | 1 | 1 | 0 | 1 |

D | 1 | 0 | 1 | 0 |

**Edge list**

i | j | w |
---|---|---|

A | B | 1 |

A | C | 1 |

A | D | 1 |

B | A | 1 |

B | C | 1 |

C | A | 1 |

C | B | 1 |

C | D | 1 |

D | A | 1 |

D | C | 1 |

Links are characterized by weights i.e. strength of a friendship, commuting flows, trade flows

**Adjacency matrix**

A | B | C | D | |

A | 0 | 3 | 1 | 5 |

B | 0 | 0 | 1 | 0 |

C | 0 | 0 | 0 | 6 |

D | 0 | 0 | 7 | 0 |

Links are characterized by weights i.e. strength of a friendship, commuting flows, trade flows

**Edge list**

i | j | w |
---|---|---|

A | B | 3 |

A | C | 1 |

A | D | 5 |

B | C | 1 |

C | D | 6 |

D | C | 7 |

Links are characterized by weights i.e. strength of a friendship, commuting flows, trade flows…

… and also have symmetrical matrices

**Adjacency matrix**

A | B | C | D | |

A | 0 | 3 | 1 | 5 |

B | 3 | 0 | 1 | 0 |

C | 1 | 1 | 0 | 6 |

D | 5 | 0 | 6 | 0 |

**Edge list**

i | j | w |
---|---|---|

A | B | 3 |

A | C | 1 |

A | D | 5 |

B | C | 1 |

B | A | 3 |

C | D | 6 |

C | A | 1 |

C | B | 1 |

D | C | 6 |

D | A | 5 |

dyads: pair of connected nodes

triads: three connected nodes

ego-networks

sub-graphs: selection of connected nodes

cliques: the maximum number of actors who have all possible ties present among themselves; a maximal complete sub-graph

4 possible combinations of network data

Directional and non-weighted (or binary)

Directional and weighted

Undirected and non-weighted (or binary)

Undirected and weighted

Nodes, aka vertices (singular: vertex)

Links or ties, edges or arcs

Conventional data: actors and attributes

Network data: actors and

**their relations**

\(k\) numbers of nodes

maximum number of links:

\(k(k-1)\) for directed

\(k(k-1)/2\) for undirected networks

Numbers of links / Max. number of links:

\(L/k(k-1)\) for directed

\(2L/k(k-1)\) for undirected networks

A macro characteristic of the network ➔ efficiency

Multiple

*paths*between two nodesA

*walk*is a path in which each node and each link are used once*Geodesic distance*: the number of relations in the shortest possible path from one node to another*Diameter*: the longest geodesic distance in the network

The extent to which ties are reciprocated (if A ➔ B, then B ➔ A)

At least two different approaches:

Reciprocated ties / maximum number of ties (0.33)

Reciprocated ties / number of existing ties (0.5)

A bias towards connections between network nodes with similar network characteristics

Assortativity coefficient is the Pearson correlation coefficient of the degree centralities of pairs of connected nodes

Large proportion of links are highly “clustered” into local neighborhoods ➔ small worlds

\(C_{i} = 2 E_{i} / k_{i}(k_{i}-1)\)

the ratio between the number of edges \(E_{i}\) that exist among \(i\)’s nearest neighbors and the maximum number of these edges, where \(k_{i}\) is the number of nodes in the sub-net

AKA Transitivity

Exploratory process

The emphasis is not on the behaviour of single entities, but on the information of who is connected to whom

“The problem of community detection requires the partition of a network into communities of densely connected nodes, with the nodes belonging to different communities being only sparsely connected” (Blondel et al. 2008, 2)

Clusters of nodes with dense connections

*inside*the clusters, but*not between*the clusters

Louvain method (Blondel et al. 2008): maximization of links inside the communities (modularity)

Network of communities extracted from a Belgian mobile phone network

Node size proportional to the number of individuals in the community

Colour represents the main language spoken in the community (red for French and green for Dutch)

What does the intermediate community of mixed colours between the two main clusters represent?

Redrawing the Map of Great Britain from a Network of Human Interactions (Ratti et al. 2010)

Do administrative boundaries follow the more natural ways people interact across space?

Large telecommunications database in Great Britain.

In social networks centrality is related with power, e.g. dominance and dependence

Micro and macro property

Position in the network

Many centrality measures

Source: schochastics.net

The simplest centrality measure

Who has the most connections?

The number of links per node

\(D_{i} = \sum_{j}^{N} x_{ij}= \sum_{j}^{N} x_{ji}\)

Indegree: the number of ingoing links per node \(D_{i} = \sum_{j}^{N} x_{ji}\)

Outdegree: the number of outgoing links per node \(D_{i} = \sum_{j}^{N} x_{ij}\)

\[D_{i} = \sum_{j}^{N} w_{ij} = \sum_{j}^{N} w_{ji}\]

\(w\) denotes the weight of the links

A | B | C | D | out-degree | |
---|---|---|---|---|---|

A | 0 | 3 | 1 | 5 | 9 |

B | 3 | 0 | 1 | 0 | 4 |

C | 2 | 4 | 0 | 9 | 15 |

D | 7 | 0 | 5 | 0 | 12 |

in-degree | 12 | 7 | 7 | 14 |

Which node has the shortest distance to other nodes

Instead of focusing on the number of links, the focus turns to the network distances

Different definitions:

Closeness, \(c_{i} = 1/\sum_{j} d_{ij}\)

Fareness, \(c_{i} = \sum_{j} d_{ij}\)

`igraph`

calculates closeness

Which nodes control information flows

How many geodesic links between actors \(i\) and \(j\) contain actor \(k\)?

\(B_{k} = \sum_{i \not=j \not=k} \frac{g_{ij}(k)}{g_{ij}}\)

Not all connections are equal

Connections to more central vertices are more important than connections to less central ones

It is still important to have a large number of connections, but a node with fewer and more important connections will end up being more central than a node with more and less important links

Global structure of the network: both direct links and shortest paths

Eigenvector of the adjacency matrix associated with the larger eigenvalue

The basis of Google search engine

Webpages that link to \(i\), and have high PageRank scores themselves, should be given more weight.

Webpages that link to \(i\), but link to a lot of other webpages in general, should be given less weight.

Directed networks

Iterative algorithm

While easy to identify in simple graphs, much more difficult in large, complex networks

Various other centrality measures exist (e.g. see

`igraph`

)Highlight different aspects of the distribution and sources of power

Source: Tranos, Gheasi, and Nijkamp (2015)

Two mode networks

Multilayer

Dynamic

Spatial

Multipirtite

See also Light and Moody (2020)

Blondel, Vincent D, Jean-Loup Guillaume, Renaud Lambiotte, and Etienne Lefebvre. 2008. “Fast Unfolding of Communities in Large Networks.” *Journal of Statistical Mechanics: Theory and Experiment* 2008 (10): P10008.

Christaller, Walter. 1966. *Central Places in Southern Germany*. Vol. 10. Prentice-Hall.

Hanneman, Robert A, and Mark Riddle. 2005. “Introduction to Social Network Methods.” University of California Riverside. http://faculty.ucr.edu/~hanneman/nettext/.

Light, Ryan, and James Moody. 2020. *The Oxford Handbook of Social Networks*. Oxford University Press.

Neal, Zachary P, and Céline Rozenblat. 2021. *Handbook of Cities and Networks*. Edward Elgar Publishing.

Ratti, Carlo, Stanislav Sobolevsky, Francesco Calabrese, Clio Andris, Jonathan Reades, Mauro Martino, Rob Claxton, and Steven H Strogatz. 2010. “Redrawing the Map of Great Britain from a Network of Human Interactions.” *PloS One* 5 (12): e14248.

Tranos, Emmanouil, Masood Gheasi, and Peter Nijkamp. 2015. “International Migration: A Global Complex Network.” *Environment and Planning B: Planning and Design* 42 (1): 4–22.