Diffusion processes on complex networks
Janusz Szwabiński
Contact data
● office hours (C-11 building, room 5.16): – Monday, 13.00-15.00 – Thursday, 14.00-16.00 – preferably make an appointment via email, providing details of your problem
● http://prac.im.pwr.wroc.pl/~szwabin/index
Course overview
● Introduction to complex networks (5 weeks)
● Diffusion and random walks (1 week)
● Epidemic spreading in population networks (3 weeks)
● Rumor and information spreading (1 week)
● Opinion formation processes (2 weeks)
● Diffusion of innovation (3 weeks)
Bibliography
● M. E. J. Newman, “The structure and function of complex networks”, SIAM Review 45, 167-256 (2003), https://arxiv.org/abs/cond-mat/0303516
● R. Albert and A.-L. Barabasi, “Statistical mechanics of complex networks”, Reviews of Modern Physics 74, 47-97 (2002), https://arxiv.org/abs/cond-mat/0106096
● A. Barrat, M. Barthelemy and A. Vespignani, “Dynamical Processes on complex networks”
● R. Pastor-Satorras, C. Castellano, P. Van Mieghem and A. Vespignani, “Epidemic procesess on complex networks”, Reviews of Modern Physics 87, 925-979 (2015), https://arxiv.org/abs/1408.2701
● D. Easley and J. Kleinberg, “Networks, Crowds and Markets: Reasoning about highly connected world”
Tools
● Python as the course programming language
● useful packages: – NetworkX (https://networkx.github.io/) – Stanford Network Analysis Platform (SNAP, http://snap.stanford.edu/) – igraph (http://igraph.org/)
Assessment
● each assignment in the lab will be graded on a 100 point basis – submissions that do not run will receive at most 20% of the points – in case of not meeting the deadline for the assignment, the score will by reduced by 10% for each day of the delay Average score Grade X < 65 2.0 65 ≤ X < 70 3.0 70 ≤ X < 75 3.5 75 ≤ X < 85 4.0 85 ≤ X < 95 4.5 95 ≤ X ≤ 100 5.0
Assessment
● final project – find a scientific paper on diffusion processes on networks (virus spreading, opinion spreading, diffusion of innovation etc.) – redo simulations for the given model – introduce a small modification to the model and simulate again – present results in form of a poster – groups of 2-3 people allowed – short presentations in the last two lectures
● final grade grade = 0.5*L + 0.5*P
Outlook of today’s talk
● Definition of a graph
● Beginnings of the graph theory
● Examples of real complex networks
● Short introduction into NetworkX
Definition of a graph
A graph G is a set V of vertices and a set E of edges that connect the vertices.
In other words, a graph is an ordered pair of the sets V and E,
G = (V,E)
Vocabulary
● vertex (pl. vertices) – a fundamental unit of a graph; it may also be called: – a site (physics) – a node (computer science) – an actor (social science)
● edge – a line connecting two vertices; it may also be called: – a bond (physics) – a link (computer science) – a tie (social science)
Vocabulary
● undirected graph – every edge e is an unordered pair of vertices
e = (v,w) = (w,v)
● digraph (directed graph) – every edge e is directed from a vertex v to some vertex w
v w
e = (v,w) ≠ (w,v)
origin destination
Vocabulary
● complex network – a realization of an abstract graph structure, consists of items somehow connected with each other
● degree – the number of edges connected to a vertex
Since there may be more than one edge between any two vertices, the degree is not necessarily equal to the number of vertices adjacent to a vertex.
A digraph has both an in-degree and an out-degree for each vertex.
Vocabulary
● component to which a vertex belongs is that set of vertices that can be reached from it by paths running along edges of the graph
– in-component in a digraph – vertices from which a vertex can be reached – out-component – vertices that can be reached from a vertex
Vocabulary
● geodesic path is the shortest path through the network from one vertex to another (there may be more geodesic paths between 2 vertices)
● diameter – the length (in the number of edges) of the longest geodesic path between any two vertices in a graph
Seven Bridges of Königsberg
Source: Wikipedia
Seven Bridges of Königsberg
Leonhard Euler (1736):
Source: Wikipedia
Seven Bridges of Königsberg
Source: Wikipedia
Examples of networks http://www.lx97.com/maps/ http://en.wikipedia.org/wiki/File:UnitedStatesPowerGrid.jpg http://blaauw.eecs.umich.edu/project.php?id=14&sid=nnxdqhqhnoszrgfg
http://www.fisherycrisis.com/coral7.html http://www.funpecrp.com.br/gmr/year2005/vol3-4/wob01_full_text.htm
Short introduction into NetworkX
Demo