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The Network and Criminality Workshop will explore the capacity of mathematics and computation to extract insight on network structures relevant to crime, riots, terrorism, etc. It will include presentations on current work (both application-oriented and on methods that can be applied in the future) and active discussion on how to address existing challenges.

Invited speakers (in alphabetical order) are as follows:

Prof. Alex Arenas, Professor of Computer Science & Mathematics, URV, http://deim.urv.cat/~alexandre.arenas/

Prof. Henri Berestycki, Professor of Mathematics, EHESS, http://en.wikipedia.org/wiki/Henri_Berestycki

Prof. Andrea Bertozzi, Professor of Mathematics, UCLA, http://www.math.ucla.edu/~bertozzi/

Dr. Paolo Campana, Research Fellow, Oxford, http://www.sociology.ox.ac.uk/academic-staff/paolo-campana.html

Toby Davies, Graduate Student,  UCL, http://www.bartlett.ucl.ac.uk/casa/people/mphil-phd-students/Toby_Davies

Dr. Hannah Fry, Lecturer in the mathematics of cities, UCL, https://iris.ucl.ac.uk/iris/browse/profile?upi=HMFRY30

Dr. Yves van Gennip, Lecturer in Mathematics, Nottingham, http://www.nottingham.ac.uk/mathematics/people/y.vangennip

Prof. Sandra González-Bailón, Assistant Professor at UPenn, http://dimenet.asc.upenn.edu/people/sgonzalezbailon/

Prof. Federico Varese, Professor of Criminology, Oxford, http://www.law.ox.ac.uk/profile/federico.vareserecep

If you are interested in attending this workshop, please register by following this link: https://www.maths.ox.ac.uk/node/13764/.

There is a well established body of research on quadratic optimization problems based on reformulations of the original problem as a conic program over the cone of completely positive matrices, or its conic dual, the cone of copositive matrices. As a result of this reformulation approach, novel solution schemes for quadratic polynomial optimization problems have been designed by drawing on conic programming tools, and the extensively studied cones of completely positive and of copositive matrices. In particular, this approach has been applied to address key combinatorial optimization problems. Along this line of research, we consider quadratically constrained quadratic programs and provide sufficient and necessary conditions for
this type of problems to be reformulated as a conic program over the cone of completely positive matrices. Thus, recent related results for quadratic problems can be further strengthened. Moreover, these results can be generalized to optimization problems involving higher order polynomias.

A key question to develop our understanding of turbulence in shear flows is: what is the smallest perturbation to the laminar flow that causes a transition to turbulence, and how does this change with the Reynolds number, R?  Finding this so-called ``minimal seed'' is as yet unachievable in direct numerical simulations of the Navier-Stokes equations. We search for the minimal seed in a low-dimensional model analogue to the full Navier-Stokes in plane sinusoidal flow, developed by Waleffe (1997). A previous such calculation found the minimal seed as the least distance (energy norm) from the origin (laminar flow) to the basin of attraction of another fixed point (turbulent attractor).  However, using a non-linear optimization technique, we found an internal boundary of the basin of attraction of the origin that separates flows which directly relaminarize from flows which undergo transient turbulence. It is this boundary which contains the minimal seed, and we find it to be smaller than the previously calculated minimal seed. We present results over a range of Reynolds numbers up to 2000 and find an R^{-1} scaling law fits reasonably well. We propose a new scaling law which asymptotes to R^{-1} for large R but, using some additional information, matches the minimal seed scaling better at low R.

In today's interconnected world, the dissemination of an idea, a trend, a rumor through social networks, as well as the propagation of information or cyber-viruses through digital networks are all common phenomena. They are conceptually similar to the spread of infectious diseases among hosts, as common to all these phenomena is the dissemination of a spreading agent on a networked system. A large body of research has been produced in recent years to characterize the spread of epidemics on static connectivity patterns in a wide range of biological and socio-technical systems. In particular, understanding the mechanisms and conditions for widespread dissemination represents a crucial step for its prevention and control (e.g. in the case of diseases) or for its enhancement (e.g. in the case of viral marketing). This task is however further hindered by the temporal nature characterizing the activation of the connections shaping the networked system, for which data has recently become available. As an example, in networks of proximity contacts among individuals, connections represent sequences of contacts that are active for given periods of time. The time variation of contacts in a networked system may fundamentally alter the properties of spreading processes occurring on it, with respect to static networks, and affect the condition at which epidemics become possible. In this talk I will present a novel theoretical framework adopting a multi-layer perspective for the analytical understanding of the interplay between temporal networks and spreading dynamics. The framework is tested on a set of time-varying network models and empirical networks.

Although not all complex networks are embedded into physical spaces, it is possible to find an abstract Euclidean space in which they are embedded. This Euclidean space naturally arises from the use of the concept of network communicability. In this talk I will introduce the basic concepts of communicability, communicability distance and communicability angles. Both, analytic and computational evidences will be provided that shows that the average communicability angle represents a measure of the spatial efficiency of a network. We will see how this abstract spatial efficiency is related to the real-world efficiency with which networks uses the available physical space for classes of networks embedded into physical spaces. More interesting, we will show how this abstract concept give important insights about properties of networks not embedded in physical spaces.

I will present a survey of the main results about first and second order models of swarming where repulsion and attraction are modeled through pairwise potentials. We will mainly focus on the stability of the fascinating patterns that you get by random data particle simulations, flocks and mills, and their qualitative behavior.
 

This talk covers two topics: (1) Phenotype change, where we consider the steady-fitness states, in a model developed by Korobeinikov and Dempsey (2014), in which the phenotype is modelled on a continuous scale providing a structured variable to quantify the phenotype state. This enables thresholds for survival/extinction to be established in terms of fitness.

Topic (2) looks at the steady-size distribution of an evolving cohort of cells, such as tumour cells in vitro, and therein establishes thresholds for growth or decay of the cohort. This is established using a new class of non-local (but linear) singular eigenvalue problems which have point spectra, like the traditional Sturm-Liouville problems.  The first eigenvalue gives the threshold required. But these problems are first order unless dispersion is added to incorporate random perturbations. But the same idea will apply here also.  Current work involves binary asymmetrical division of cells, simultaneous with growth. It has implications to cancer biology, helping biologists to conceptualise non-local effects and the part they may play in cancer. This is developed in Zaidi et al (2015).

Acknowledgement. The support of Gravida (NCGD) is gratefully acknowledged.

References

Korobeinikov A & Dempsey C. A continuous phenotype space model of RNA virus evolution within a host. Mathematical Biosciences and Engineering 11, (2014), 919-927.

Zaidi AA, van-Brunt B, & Wake GC. A model for asymmetrical cell division Mathematical Biosciences and Engineering (June 2015).