More fun in the multiplex (maths) than the multiplex (cinema)

“Some of you may have had occasion to run into mathematicians and to wonder therefore how they got that way, and here, in partial explanation perhaps, is the story of the great Russian mathematician Nicolai Ivanovich Lobachevsky  a blog post!

You have to stretch yourself, swim out of your depth on occasion. The seminar I went to today – at the Mitchell Centre for Social Network Analysis‘s “Multilayer Networks: Mulitplexity, networks of networks and all that jazz” [slideshare] the maths went WAY over my head. Why didn’t I feel patronised or bored?  Ten percent because I  mentally prepped myself for outadepthness, but 90 percent because you don’t feel that way when you are in the presence of someone (chap called Mason Porter from University of Oxford, in this instance) who clearly loves their subject, knows it inside out and can communicate it. The passion and joie de vivre pull you through. His Candide mash-ups didn’t hurt either.

I couldn’t, with a gun to my head, explain the maths and the graphs. There was good stuff on alternative ways of representing nodes in networks and how they interconnect (3D is better than 2D).  The ‘take home’ seems to be, entirely reasonable; if you are going to throw around terms like networks, nodes and so on, define your terms or you will create conceptual Gordian knots that some other poor sod has to unpick.

Here in lieu of a proper analysis, are the bullet points I wrote down, with subsequent googles.

Multiplex networks – “In a multiplex network, each type of interaction between the nodes is described by a single layer network and the different layers of networks describe the different modes of interaction. Indeed, the scientific interest in multiplex networks has recently seen a surge. However, a fundamental scientific language that can be used consistently and broadly across the many disciplines that are involved in complex systems research has to be developed. This absence is a major obstacle to further progress in this topical area of current interest. “

best exemplified by a geek game called “Munchkin quest.” People define these differently, which leads to all sorts of mayhem. “The literature is messy” said Porter, with the bonus hashtag #makeitstop. He then showed a neat (and probably laborious) parsing of the different definitions he and colleagues had encountered.

Barry Wellman – Professor Barry Wellman studies networks: community, communication, computer, and social. His research examines virtual community, the virtual workplace, social support, community, kinship, friendship, and social network theory and methods. Based at the University of Toronto, he directs NetLab, is the S.D. Clark Professor at the Department of Sociology, is a member of the Cities Centre, and the Knowledge Media Design Institute, and is a cross-appointed member of the Faculty of Information. He is the co-author of Networked: The New Social Operating System (with Lee Rainie, Director of the Pew Internet and American Life Project) published by MIT Press in Spring 2012.

Krackhardt (1987) cognitive social structures and distinction between advice networks and friendship networks (you may like hanging out with fur mommy, but wire mommy is where the nutrition is at)

There’s a famous paper by Wayne Zachary on a Karate Club which split when he was studying it in 1977.  The dataset is here.

And so now there is a group of people who use the Zachary paper and its implications/methodological tools to look at networks and social interactions and it is called – you guessed it – the Zachary Karate Club Club. They indulge in such iffy sounding pursuits as “algorithmic community detection,” have a trophy and a tumblr, and are of course part of the “community detection community” (someone phone Private Eye right this minute.)  Clearly gunning for an Ig_Nobel

Paper to read after I finish my PhD – Community Structure in Time Dependent, Multi-scale and Multiplex Networks

Words and terminology that l scribbled down to google.

Weighted networks– “A weighted network is a network where the ties among nodes have weights assigned to them. A network is a system whose elements are somehow connected (Wasserman and Faust, 1994).[1] The elements of a system are represented as nodes (also known as actors or vertices) and the connections among interacting elements are known as ties, edges, arcs, or links. The nodes might be neurons, individuals, groups, organisations, airports, or even countries, whereas ties can take the form of friendship, communication, collaboration, alliance, flow, or trade, to name a few.”

Markovianity

transitivity – “In mathematics, a binary relation R over a set X is transitive if whenever an element a is related to an element b, and b is in turn related to an element c, then a is also related to c. Transitivity is a key property of both partial order relations and equivalence relations.”

eigenvector centralitiesEigenvector centrality is a measure of the influence of a node in a network. It assigns relative scores to all nodes in the network based on the concept that connections to high-scoring nodes contribute more to the score of the node in question than equal connections to low-scoring nodes. Google‘s PageRank is a variant of the Eigenvector centrality measure.[19] Another closely related centrality measure is Katz centrality.”

preferential attachment models “A preferential attachment process is any of a class of processes in which some quantity, typically some form of wealth or credit, is distributed among a number of individuals or objects according to how much they already have, so that those who are already wealthy receive more than those who are not. “Preferential attachment” is only the most recent of many names that have been given to such processes. They are also referred to under the names “Yule process”, “cumulative advantage”, “the rich get richer”, and, less correctly, the “Matthew effect“. They are also related to Gibrat’s law. The principal reason for scientific interest in preferential attachment is that it can, under suitable circumstances, generate power law distributions.”

Markov process– “In probability theory and statistics, a Markov process or Markoff process, named after the Russian mathematician Andrey Markov, is a stochastic process that satisfies the Markov property. A Markov process can be thought of as ‘memoryless’: loosely speaking, a process satisfies the Markov property if one can make predictions for the future of the process based solely on its present state just as well as one could knowing the process’s full history. i.e., conditional on the present state of the system, its future and past are independent.”

Kuramoto model- “The Kuramoto model, first proposed by Yoshiki Kuramoto (蔵本 由紀 Kuramoto Yoshiki) [1] ,[2] is a mathematical model used to describe synchronization. More specifically, it is a model for the behavior of a large set of coupled oscillators [3] .[4] Its formulation was motivated by the behavior of systems of chemical and biological oscillators, and it has found widespread applications such as in neuroscience [5] [6] .[7] Kuramoto was quite surprised when the behavior of some physical systems, namely coupled arrays of Josephson junctions followed his model.[8]The model makes several assumptions, including that there is weak coupling, that the oscillators are identical or nearly identical, and that interactions depend sinusoidally on the phase difference between each pair of objects.”

adjacency tensors – wikipedia let me down. Found adjacency matrices instead. I think my brain is way past full anyway…

modularity measures for multiple networks “Modularity is one measure of the structure of networks or graphs. It was designed to measure the strength of division of a network into modules (also called groups, clusters or communities). Networks with high modularity have dense connections between the nodes within modules but sparse connections between nodes in different modules. Modularity is often used in optimization methods for detecting community structure in networks. However, it has been shown that modularity suffers a resolution limit and, therefore, it is unable to detect small communities. Biological networks, including animal brains, exhibit a high degree of modularity.”

And while googling that, stumbled on Brain Connectivity Toolbox

linear oscillator models. Er, this? Brain DEFINITELY full.

Digression – this guy clearly has a lot of fun. Xkcd is about and for people just like him!

I’ve never met Cosma Shalizi, but I bet he and Mason Porter would get along like a house on fire.

This summer they are going to sit around at the Lake Como School of Advanced Studies, and probably dream up some articles for the Journal of Complex Networks.

Verdict. WAAAAAY over my head. So what? I had a ball. May have learnt something too. Result!!!

Connections to my PhD? Who knows for sure. But there was something here, and it was a different (more confusing!) use of my time than this morning’s reading group on (so-called) distributed leadership in social movements. The connecting quote for the two events might be –

Meaning in Movement: An ideational analysis of Sheffield-based Protest Networks Contesting Globalisation and War” by Kevin Gillan.

By emphasising the network form McLeish argues that the flows of information and interaction between groups and individuals are more important that (sic) the points of convergence. The ‘nodes’ – the points at which multiple flows connect – may represent a key moment during a movement’s history but have a tendency to create ossified traditions, incapable of reacting to changing political opportunities. ‘Organisers thrown up by events, who find themselves serving or surfing these waves of history narcissistically imagine themselves their authors. Last year’s bright creative movement becomes a fossilized bureaucracy or an inert ritualistic subculture.”
page 279 “Meaning in Movement: An ideational analysis of Sheffield-based Protest Networks Contesting Globalisation and War” by Kevin Gillan.

And we started with Lehrer, so in the best Ouroborosian tradition, we’ll end with New Math. Base 8 is just like base 10 really… if you’re missing two fingers…

One thought on “More fun in the multiplex (maths) than the multiplex (cinema)

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  1. Math has its own way to describe networks of various kinds, which finance uses snippets from in describing the long chains of complex financial transactions employed within networked banking systems.

    The gordian knot is a good way of explaining what happens in finance (Fisher, R. W. 2013 ‘Horseshift’ – he discusses the FED’s gordian knot). However in terms of crisis literature, pre crisis bank risk is passed through long nodes and chains of counter-party risk connections, clever at the nodes (investment bank minds securitising sub prime mortgages) and dumb in the chains (investors buying things they fail to understand) – this failure to understand prevents sight of what is missing, and what can be found at the end of this so called dumb chain or gordian knot – as we saw in 2007, it was AIG and various re-insurance companies like Swiss Re who couldn’t pay out the complex derivative instruments and investors suffered as tax payer bailouts ensued.

    These chains need to be broken and networks with borders enabled so we can always have sight who lies behind the risk, who is holding the baby, and avoid the gordian knot.

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