Why is a Mathematician Speaking at a Workshop about Language, Concepts, and History?

Mason A. Porter (@masonporter)Mathematical InstituteUniversity of Oxford

What I Do

• I study networks, their structure, how networks change, dynamical processes on networks (and how structure affects them), ways of representing more complicated structures using networks to have more nuanced models, and so on.• Modeling (of networks, of dynamical processes, …)• Analytical calculations• Numerical calculations• Data analysis

Example of a Social Network

• Example: social connections among members of a karate club

Some Questions That One Can Ask

• Who are the most important members of the karate club?• How should we measure this? What are different notions of importance? How do

the ways of measuring important members and social ties change with the size of networks?

• Which are the most important social ties?• How does information or a disease propagate in the network? Who

should we vaccinate to prevent disease spread through the whole network?• Are there dense “communities” of friends (which may not be obvious)?• How does the network change in time?

The Zachary Karate Club(W. W. Zachary, 1977)

• This network is so (in)famous that it has its own Wikipedia entry: https://en.wikipedia.org/wiki/Zachary%27s_karate_club • “Network Scientists with Karate Trophies”• http://networkkarate.tumblr.com/ • I am a member of the Zachary Karate Club Club.

Another Example: A Rabbit Warren(S. H. Lee, et al., PRE, 2014)

“Multilayer Networks”• Our review article: M. Kivelä et al., J. Cplx. Networks, 2014• One motivation: But just how are those social networks

among the karate-club members defined?• Not just one number but multi-relational: going to movies, going out

to eat, etc.

• One can keep track of different “layers” of connectivity• Keep more nuanced information and integrate different types of

information• Develop methods and tools to analyze them (very active research

area)

Example: Multiplex Networks(e.g. color the edges)

Some Current and Former Group Members

Example: Network of Networks(e.g. color the nodes; coupled infrastructures)

“Zachary Karate Club Club” Multilayer Network

How I Do It• Modeling (of networks, of dynamical processes, …)• Analytical calculations• Numerical calculations• Data analysis

Representing a Network as a Matrix

• Adjacency matrix A• This example: binary (“unweighted”)• Aij = 1 if there is a connection between nodes i and j• Aij = 0 if no connection

• How do we generalize this representation to weighted, directed, and multilayer examples?

Modeling• Example: Models of networks as different types of random graphs

• E.g., there is a uniform, independent probability p of an unweighted, undirected edge existing between each pair of nodes• Not realistic, but we can get much more sophisticated

• Calculate “average properties” of ensembles of random graphs• Mathematically: A random-graph ensemble is a probability distribution on the set of

graphs, and one can calculate expectations, etc.• Physically, an “ensemble” is an ensemble is the strict sense of statistical mechanics• Compare these properties to sample means in numerical simulations of numerous

instances of the model• Use appropriate random-graph models as “null models” against which to compare

real structures to examine what a given model can explain and what can depart from it

Modeling• Example: equations

for dynamical process on a network• E.g. Watts Threshold

Model for social influence (D. J. Watts, PNAS, 2002)• Numerous

generalizations

Analytical Calculations• One can do various calculations with exact or (more often) approximate

calculations.• One can find simpler (”lower-dimensional”) models that approximate certain

features of a dynamical process on a network.• For example, maybe all nodes with k friends behave the same way as time becomes

infinite, so now one has an equation for every possible value of k rather than for every node (if one is asking a question that allows one to consider the long-time limit).

• Sometimes one can rigorously prove mathematical statements about the structure and/or dynamical processes.

Numerical Calculations

• One can calculate numerical properties of the ensembles of random graphs (including very large networks, perhaps with tens of thousands or even millions of nodes or more).• One can directly simulate dynamical processes on networks

(“equations of motion”)• One can calculate structural quantities of networks (either from

empirical data or random graphs)• E.g. measures of importance (“centrality”) of nodes

Data Analysis• This is closely related to many of the numerical calculations,

and it also relates to work on development and application of algorithms for network analysis.• Areas like statistics, machine learning, optimization, and

time-series analysis can also be involved.

Example: Calculating Importance of Nodes Based on Being on Many Short Paths• Betweenness centrality: which

nodes (or edges) are on a lot of short paths?

• Geodesic node betweenness centrality is the number of shortest (“geodesic”) paths through node i divided by the total number of geodesic paths.

• Similar formula for geodesic edge betweenness

• One can also define notions of betweenness based on ideas like random walks (or by restricting to particle paths in useful ways).

Example: Important Mathematics Departments Using Hub and Authority Centrality• S. A. Meyer, P. J. Mucha, & MAP, “Mathematical genealogy and department

prestige”, Chaos, Vol. 21: 041104 (2011)• We consider Mathematical Genealogy Project data in the US from 1973–2010

(data from 10/09) • Example: Marty Golubitsky earned a PhD from MIT and later supervised

students at University of Houston and Ohio State.• Directed edge of unit weight from MIT to UH (and also from MIT to OSU)

• A university is a good authority if it hires students from good hubs, and a university is good hub if its students are hired by good authorities.

• Caveats• Our measurement has a time delay (only have the MITOSU edge after Marty supervises a PhD student there)• Hubs and authorities should change in time. (We developed a method to do this in D. Taylor et al., 2015.)

How Do Our Rankings Do?

Example: Algorithmic Detection of Dense “Communities” of Nodes• Survey article (very friendly!):

“Communities in Networks,” MAP, J.-P. Onnela & P. J. Mucha, Notices of the American Mathematical Society 56, 1082–1097 & 1164–1166 (2009).• Examples

• Gang collaborations in Los Angeles• Facebook friendships• Time-dependent voting communities in U.S.

Congress (e.g. obtain political realignments directly from the data)

• Many more …

Communities in Facebook Networks(assuming each node assigned to one community, but one need not assume this)

Numerous ways to do

this!

How Do Different Universities Organize?(A. L. Traud et al., Physica A, 2012)

All people

Women only

P. J. Mucha et al., Science, 2010

Family-tree record as a proxy of human migration due to marriage (in Korea, across a few hundred years)S. H. Lee et al., Physical Review X, 2014

What I am Interested from Others

• Collaborations• Interesting data sets and problems we could perhaps help

give insights into (narrow down some possibilities, which can then be explored in depth with domain expertise)

What I Think I Have to Offer to Others

• Powerful tools and methods that can offer insights onto a diverse set of problems• Enthusiasm

To Tantalize: A Current Project(note: not networks)

• Compare authors based on time-series analysis of punctuation patterns (no words).• Seeking students for a Masters

project or undergraduate project.• “Interviewing” a prospective student

next week.

• Exercise: What work/author is on the left, and what is on the right?• (From blog entry of Adam J. Calhoun)

Conclusions

• Networks provide a powerful representation of a diverse set of situations, and they have been very successful in yielding insights into them• We’re developing tools to do this in progressively more

nuanced ways (preserving multiple types of social ties, etc.) while retaining the power of abstraction and large-scale computations.