Prof. Ken Duffy
Hamilton Institute
Maynooth University
Ken Duffy 


I'm an applied mathematician whose main research interests are in probability and statistics, and their application to science and engineering.

November 2014 - An Algebra for Immune Responses

Lymphocytes, the key players in an adaptive immune response, have long since been known to need to receive multiple signals to mount an effective defense. That discovery led to the two signal theory of T cell activation. The principle being that two independent signals, antigen followed by costimulation, ensure that lymphocyte expansion is only initiated in response to genuine infection. Redundancy in this second signal has long since left a conundrum in the field.

A paper published today in the journal Science brings this theory up to date with a quantitative edge. The work was led by the Walter and Eliza Hall Institute of Medical Research's Philip D. Hodgkin and Susanne Heinzel, driven by Phil's Ph.D. student Julia M. Marchingo, in collaboration with Hodgkin lab. members Andrey Kan, Robyn M. Sutherland, Cameron J. Wellard, Mark R. Dowling, two other WEHI lab. heads Gabrielle T. Belz and Andrew M. Lew, and myself. While the signaling of antigen, costimulation and cytokines are complex and involved, clever experimentation in concert with computer modelling and mathematics revealed a simple additive algebra of T cell expansion; one that puts the old conundrum to bed and will, hopefully, provide a quantitative paradigm for therapeutically manipulating immune response strength.

September 2014 - Randomness, Determinism and Immune Responses

One of the difficult challenges in science is doing justice while framing other people's hypotheses. Steven L. Reiner, one of the main instigators behind investigating the role of asymmetric cell division in immunology, and William C. Adams have neatly laid out a deterministic description of adaptive immune responses in a fascinating opinion piece published in Nature Reviews Immunology. My Australian colleagues Philip D. Hodgkin, one of the main proponents of stochastic processes in immunology, Mark R. Dowling and I have written a brief response in defense of randomness in correspondence to the same journal. The community has not yet acquired the data required to answer how deterministic and stochastic processes interleave to build the complete immune response, but with so many different groups designing experiments and doing analysis based on distinct hypotheses, we're all looking forward to the time when that resolution occurs.

July 2014 - Forensic Analysis of DNA

The basis of a DNA fingerprint is the measurement of the number of repeats of microsatellites, short repeated sequenes of of DNA, at various locations in the genome. While the combination of given numbers of repeats are (largely) unique for individuals, in forensics applications often the number of repeats cannot be measured precisely. This occurs as the amount of DNA at a crime scene can be small, requiring amplification before the measurement takes place, which can create false signals or missing data, or the sample itself may be made up of a composite from several individuals in which case one gets a combined, mixed measurement.

My colleagues Catherine Grgicak, a Forensic Scientist at Boston University, Desmond Lun, a Computer Scientist at Rutgers, and Muriel Médard, an Electronic Engineer at MIT, have been working in an interdisciplinary team, which I have recently joined , to carefully investigate this topic. The first piece of work I have been involved in is a paper to be presented at Asilomar in November 2014 entitled A signal model for forensic DNA mixtures by members of Catherine and Muriel's labs: Ullrich J. Monich, Catherine Grgicak, Viveck Cadambe, Jason Yonglin Wu, Genevieve Wellner, Ken Duffy and Muriel Médard. The paper details fundamental statistics of DNA fingerprints created by standard forensics techniques in controlled circumstances in an attempt to build a descriptive model of noise generated in the process.

February 2014 - Cellular Barcoding and Inferring Lineage Pathways

There are many instances where one would wish to know the familial relationships of cells, to be able identify those that came from the same parent. For example, the purpose of a bone-marrow transplant is to place new hematopoietic stem cells into to the recipient so that their blood system can be rebuilt. Does each stem cell contribute equally to this rebuilding? Does chance play a role? Are some stem cells specialized? These are all questions that can only be tackled if one can identify cells from the same progenitor.

Cellular barcoding, which Ton N. Schumacher's lab. has been at the forefront of developing, is an experimental method in which small pieces of non-functional DNA into otherwise identical cells. Each small piece, a barcode, can then be found in the progeny of that cell and so those in a common family can be identified. This technique has led to many high-profile results in the past year in everything from immunology, cancer and blood system development, including several from Ton's lab.

In a data analytics contribution to the Cellular Barcoding technique, my colleagues Leïla Perié, Philip D. Hodgkin, Shalin H. Naik, Ton N. Schumacher, Rob J. de Boer, and I have published a paper in Cell Reports that develops a mathematical framework for the analysis of data that comes from cellular barcoding experiments. The framework is designed to draw inferences about the compatibility of potential lineage pathways with the data. As an exemplar, we study data from the blood system taken from one of these high-profile results, published in Nature by S. H. Naik, L. Perié, E. Swart, C. Gerlach, N. van Rooij, R. J. de Boer and Ton N. Schumacher. The analysis suggests the classical model of hematopoiesis is not consistent with the data and, inspired by in vitro deductions from the 80s, we propose an alternate lineage pathway that is consistent.

January 2014 - Integrated Random Walks and Large Busy Periods

Ever since an elegant paper by Venkat Anantharam in 1989, it's been known that the way a random walk becomes large, or how a big queue builds up, is by a piecewise linear path. The question of how a large integrated random walk, or how a large volume of work done during a busy period of a queue, occurs was first tackled by A. A. Borovkov , O. J. Boxma and Z. Palmowski in a paper in 2003.

In a paper to appear in Stochastic Systems, Sean Meyn and I establish that the most likely path to a large busy period is concave in general and, remarkably, has a simple form, following a rescaled, upside-down version of the scaled cumulant generating function. Moreover, the path starts and ends in the same fashion as the most likely path to a long queue as identified by Ayalvadi J. Ganesh and Neil O'Connell in 2002.

For 10 billion simulated paths with i.i.d. Gaussian increments, below is a graph of the random walk that hits the highest height and the prediction from Ganesh and O'Connell conditioned on the height. Also plotted is the largest integrated random walk and our prediction conditioned on the area under the curve. The predicted paths start and end with the same slope, but diverge in between.
Integrated Gaussian Random Walk

November 2013 - Guesswork, Wiretap and Erasure Channels

Devices such as door-card readers transmit passwords over wireless channels. Unintended receivers, eavesdroppers, are typically distant and observe the communication over a noisy channel that distorts it. How hard is it for the eavesdropper to fill in the erased gaps and how does it depend on the properties of the noisy channel?

Complementing earlier work with a distinct interpretation of guesswork and wiretap channels by Neri Merhav and Erdal Arikan, as well as Manjesh Kumar Hanawal and Rajesh Sundaresan, this is a subject that with our collaborators, Flávio du Pin Calmon and Muriel Médard at MIT, Mark Christiansen and I had a paper on at this year's Asilomar Conference on Signals, Systems & Computers. The main observation is that the average noise on the channel is not the determining factor in how difficult the task is for the eavesdropper, but instead another average of the noise, a moment, that is determined again by Rényi entropy.

October 2013 - Limits of Statistical Inference

paper, which was driven by our collaborators, Flávio du Pin Calmon at MIT and Mayank Varia, at the Lincoln Lab., was presented by Muriel Médard at this year's Allerton conference Communication, Control, and Computing. The work establishes bounds on how much can be inferred about a function of a random variable's value, if only a noisy observation of the variable is available.

September 2013 - Constraint Satisfaction and Rate Control

IEEE/ACM Transactions on Networking papers are like buses in that you wait for a long time for one and then two come at once. The work on Decentralized Constraint Satisfaction that is mentioned below has appeared in the August issue of the journal. The final paper from my ex-student Kaidi Huang's doctoral thesis also appears in that issue. The work, performed in collaboration with my Hamilton Institute colleague David Malone, addresses a practical problem in wireless local area networks: rate control. When transmitting packets, WiFi cards must select a physical-layer rate at which to transmit. In principle, the faster the rate, the less robust the transmission is to noise. For the poorly engineered rates of standard WiFi, this is not strictly true as Kaidi, David and I demonstrated in a paper published in IEEE Transactions on Wireless Communications in 2011. In the present work, we proposed a rate adaption scheme based on opportunity cost and Bayesian decision making that was, demonstrably thanks to hard work of Kaidi and David, implementable on standard hardware and outperforms the standard algorithms.

July 2013 - Information Theory, Guesswork & Computational Security

The late Kai Lai Chung purportedly said that there's only one theorem in Information Theory: the Asymptotic Equipartition Property. At the 2013 IEEE International Symposium on Information Theory, Mark Christiansen presented a preliminary version of a surprising result that we established in collaboration with our friends from M.I.T., Flávio du Pin Calmon and Muriel Médard. Namely, despite the fact that everything within a Typical Set has, by construction, approximately equal probability, it is exponentially easier as a function of word length to guess a word chosen from a Typical Set than a naive application of the AEP would have you deduce. This has ramifications for physical layer security.

January 2013 - Guesswork & Information Theory

How hard is it to guess someone's password? More importantly, how do you measure how hard it is to guess someone's password? It's a question that has recieved less attention than one would have expected. The recently deceased information theorist James Massey published a fascinating one page paper at ISIT on this question in 1994 and it is the topic of a paper that my student, Mark Christiansen, and I have had published in IEEE Transactions on Information Theory. Building on beautiful work of others, which began with Erdal Arikan and included a contribution from my Hamilton Institute colleague David Malone, that identified Rényi entropy as a key player in the quantification of guesswork, in the article we provide a direct estimate on the likelihood that it takes a given number of guesses to guess a randomly selected object.

January 2013 - Networking & Medium Access Control

There are two fundamental paradigms for sharing a resource such as wireless medium. One can empower a centralized controller who gathers everyone's requirements and sets out a schedule of who gets access to the resource when. This system is used, for example, for speakers at every conference and in cell phone networks. The alternate system is to not be prescriptive, but to listen before speaking and when the medium becomes silent, randomly interject. This is used, for example, in human conversation as well forming the basis for the standard access method, IEEE 802.11, for WiFi networks.

Both of these systems have advantages and disadvantages. If one knows exactly what who needs what resources and can agree on a central party to adjudicate, the former is most efficient. If one is uncertain about the number of users and their requirements, the latter is more robust to this uncertainty, but suffers from collisions - people talking over one another and thus wasting resources.

In a paper that has been published in the journal Wireless Networks, written with my ex-student Minyu Fang, and colleagues at the Hamilton Institute David Malone and Douglas J. Leith, we investigate a system that has the best of both worlds: a decentralized stochastic algorithm is used to obtain a collision-free schedule.

October 2012 - Constraint Satisfaction Problems

Is it possible to find a global solution to a problem, with everyone only having local information and being unable to see
the global picture? That's the topic of a paper that my colleagues, the French mathematician Charles Bordenave
and Douglas J. Leith, my Scottish engineering colleague from the Hamilton Institute, address in a paper that has
been accepted for publication in IEEE/ACM Transactions on Networking. Two undergraduates, TCD's John Roche
and National University of Ireland Maynooth's Tony Poole, created java applets that illustrate how the proposed approach, which provably works, solves
problems. They also illustrate that, in this setting, it is easier to agree to disagree than it is to find a consensus.

September 2012 - Immunology & Cell Biology

My Australian immunology collaborator Philip Hodgkin and I had a paper published in Trends in Cell Biology. It expands on the hypothesis we employed in our earlier 2012 paper, published in Science, to explain cell fate diversity. Artwork based on the paper, illustrated by WEHI's Jie H. S. Zhou, was used for the edition's cover:

January 2012 - Immunology & Cellular Biology

With colleagues from the Walter and Eliza Hall Institute of Medical Research led by Philip D. Hodgkin, I had a paper published in Science. The piece analyses data from an experiment that took my Australian colleagues, particularly Mark Dowling and John Markham, four years to perfect. It enabled us to directly observe the times at which an important class of cells in the immune system, the antibody secreting B lymphocytes, make decisions on when and how to combat a pathogen. Despite the data looking immensely complex, it is consistent with a remarkably simple, holistic hypothesis.



Journal papers by subject: Applied Probability; Biology; Networks.

Applied Probability

Conference papers

Book chapters

Other articles

Research Funding

Recent Funding

  • "Quantitative analysis of immune cell fate: stochastic competition and censorship",
    2013-2017, Science Foundation Ireland Investigator Grant.
  • "Quantitative T Cell Immunology",
    2013-2017, European Union Marie Curie FP7 ITN, co-PI with 16 other institutions, including industry, with co-ordinating co-PI Grant Lythe (Leeds).
  • "Indo-European Research Network in Mathematics for Health and Disease",
    2013-2017, European Union Marie Curie FP7 IRSES, co-PI with five other institution, co-ordinating co-PI Carmen Molina-Paris (Leeds),
  • "Single cell lineage tracing to understand hematopoietic development and differentiation",
    2012-2015, Human Frontier Science Program Research Grant, co-PI with Andrew Cohen (Drexel), Philip Hodgkin (WEHI), Shalin Naik (WEHI) and Ton Schumacher (NKI).
  • "FLAVIA: FLexible Architecture for Virtualizable future wireless Internet Access"
    2010-2013, European Union Framework 7 Strep Grant, collaborator, ten partner academic and industry grant led by Giuseppe Bianchi (U. Rome), and locally by David Malone.
  • "Mathematical modelling of lymphocyte proliferation and differentiation during an adaptive immune response",
    2009, Science Foundation Ireland Short-term Travel Fellowship to visit Philip Hodgkin.
  • "Using 802.11 medium access control layer measurements to understand and improve network performance",
    2007-2010, Science Foundation Ireland Research Frontiers Programme, PI, with David Malone as co-PI.


Current postdocs

  • Tom Weber.

Current graduate-students

Past graduate-students

  • Kaidi Huang, Ph.D. (National University of Ireland Maynooth), 2010.
  • Minyu Fang, M.Sc. (National University of Ireland Maynooth), 2010.
  • Anthony Paul Metcalfe, M.Sc. (University of Dublin), 2004.
  • Mark Rodgers-Lee, M.Sc. (University of Dublin), 2003.


  • Brendan Williamson, Mathematics (Duke University), 2014.
  • Conor Leonard, Mathematics (University College Cork), 2013.
  • Brendan Williamson, Mathematics (Dublin City University), 2012.
  • John Roche, Mathematics (University of Dublin), 2011.
  • Tony Poole, Science (National University of Ireland Maynooth), 2011.
  • Joshua Tobin, Mathematics (University of Dublin), 2010.

Scholarly Activity



Editorial Boards

Technical Programme Committees
  • 1st International Workshop on Artificial Immune Systems: Systems & Synthetic Immunology, Computational Immunology & Immune-Inspired Engineering, 2015 (AIS 2015).
  • Stochastic, statistical and computational approaches to immunology, 2013, (ICMS).
  • 25th International Teletraffic Congress, 2013, (ITC 25).
  • 11th International Conference on Artificial Immune Systems, 2012, (ICARIS 2012).
  • ACM CoNEXT, 2012, (CoNEXT 2012).
  • 24th International Teletraffic Congress, 2012, (ITC 24).
  • 1st International Workshop on Network Science, 2011, (Hamilton Institute-NS1).
  • IEEE International Conference on Computer Communications and Networks, 2011, (ICCCN 2011).
  • 23rd International Teletraffic Congress, 2011, (ITC 23).
  • 3rd International Workshop on Systems Biology, 2010, (Hamilton Institute-SB3).
  • 22nd International Teletraffic Congress, 2010, (ITC 22).
  • 3rd International Workshop on Performance Analysis and Enhancement of Wireless Networks, 2008, (PAEWN 08).
  • Valuetools workshop on interdisciplinary systems approach in performance evaluation and design of computer and communication systems, 2007, (Inter-Perf 2007).
  • 1st Hamilton Institute Workshop on Applied Probability, 2007, (Hamilton Institute-AP1).


I have had the unusual honour of having acknowledgments in at least sixteen diverse places.

  1. Christopher Fuchs thanks me for teaching him about magnalium in his article "On the Quantumness of a Hilbert Space", Quantum Information, Statistics, Probability: Dedicated to Alexander S. Holevo on the Occasion of His 60th Birthday, edited by O. Hirota (Rinton Press, Princeton, NJ, 2004).
  2. In their O'Reilly book IPv6 Network Administration, the authors, Niall Murphy and David Malone, say the following: "Ken Duffy managed to resist the temptation to edit our manuscript, and we can only admire his restraint."
  3. My academic grand-nephew, Stephen Wills thanks me for facilitating corrections to his paper "On the generators of operator Markovian cocycles", Markov Processes and Related Fields 13 (2007), 191-211.
  4. In his CUP Monograph Control Techniques for Complex Networks, Sean Meyn thanks me for graphs generated by code that I wrote while attending a meeting organized by Serguei Foss, Takis Konstantopoulos and Stan Zachary.
  5. In their article "Large deviation results on some estimators for stationary Gaussian processes", Statistics 44 (2), 129-144, 2010, Claudio Macci and Lea Petrella thank me for illustrating to them results from the literature on semi-exponential distributions.
  6. In their article "A single-cell pedigree analysis of alternative stochastic lymphocyte fates", Proceedings of the National Academy of Sciences, 11, 106(32), 13457-13462, 2009, E. D. Hawkins, J. F. Markham, L. P. McGuinness and P. D. Hodgkin thank me, along with others, for critical reading of the manuscript and helpful suggestions.
  7. In the article "The effect of correlations on the population dynamics of lymphocytes", Journal of Theoretical Biology, 264 (2), 443-449 2010, by C. Wellard, J. Markham, E.D. Hawkins, P.D. Hodgkin, Cameron thanks me for discussions.
  8. In the article "On Thresholds for Robust Goodness-of-Fit Tests", presented at the IEEE Information Theory Workshop, 2010 by Jayakrishnan Unnikrishnan, Sean P. Meyn and Venugopal V. Veeravalli, the authors thank me for for initial discussions that motivated the problem studied in the paper.
  9. In the article "Large Deviations Of Max-Weight Scheduling Policies On Convex Rate Regions", Mathematics of Operations Research 35(4), 881-910, 2010, my colleague Vijay G. Subramanian is overly kind in thanking me for suggestions that helped expand its scope.
  10. In the article "Many Sources Large Deviations of Max-Weight Scheduling", IEEE Transactions on Information Theory 57(4), 2151-2168, 2011, Vijay G. Subramanian T. Javidi and S. Kittipiyakul thank me, Ruth Williams, Milan Matejdes and the reviewers for "tremendously helpful suggestions". I don't know what Ruth, Milan and the reviewers said, but in my case you're overly generous.
  11. In the article "How neurons migrate: a dynamic in-silico model of neuronal migration in the developing cortex" by Yaki Setty, Chih-Chun Chen, Maria Secrier, Nikita Skoblov, Dimitris Kalamatianos and Stephen Emmott BMC Systems Biology 5:154, the authors thank me for proof-reading duties.
  12. In "A model for studying the hemostatic consumption or destruction of platelets" by Mark R. Dowling, Emma C. Josefsson, Katya J. Henley, Benjamin T. Kile and Philip Hodgkin, PLoS One 2013, the authors kindly thank me for discussions.
  13. In "Measuring pulsed interference in 802.11 links", Brad W. Zarikoff and Douglas J. Leith in IEEE/ACM Transactions on Networking thank myself and G. Bianchi, for comments.
  14. In the important paper "Diverse and heritable lineage imprinting of early haematopoietic progenitors", published in Nature 496 (7444), 229-232, 2013, Shalin H. Naik, Leila Perie, Erwin Swart, Carmen Gerlach, Nienke van Rooij, Rob J. de Boer and Ton N. Schumacher kindly thank myself and others for comments on the manuscript.
  15. My friend Richard Abadi and his collaborator Jonathan Murphy have a fascinating piece in Brain Research entitled "Phenomenology of the sound-induced flash illusion" that I had the pleasure of discussing with them.
  16. My long-standing friend, the University of Limerick's James Gleeson had a high-profile paper, with collaborators from the US and the UK, entitled ``A simple generative model of collective online behavior'' published in Proceedings of the National Academy of Sciences in 2014 that I had the pleasure of having some forewarning about; it's a lovely piece of work.