Presentation Details 

Ball, Frank 
Epidemics on random networks with tunable clustering, degree correlation and degree distribution 
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There has been considerable interest recently in models for random networks and in the behaviour of epidemics defined on such networks. In this talk, based on joint work with Tom Britton (Stockholm University) and David Sirl (University of Nottingham), a random network model which allows for tunable, quite general forms of clustering, degree correlation and degree distribution is described and the behaviour of SIR (susceptibleinfectiverecovered) epidemics on such a network is analysed. Asymptotic properties of both the network and the epidemic, as the number of nodes in the network tends to infinity, are derived: the degree distribution, degree correlation and clustering coefficient, as well as a threshold parameter, which determines whether or not an epidemic with few initial infectives can become established and lead to a major outbreak, the probability of a major outbreak and relative final size of such an outbreak. The theory is illustrated by Monte Carlo simulations and numerical examples. 
Bandyopadhyay, Antar 
Depreferential attachment random graphs 
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In this talk we will introduce a new model of a growing sequence of random graphs where a new vertex is less likely to join to an existing vertex with high degree and more likely to join to a vertex with low degree. In contrast to the well studied model of emph{preferential attachment random graphs} where higher degree vertices are preferred, we will call our model emph{depreferential attachment random graph model}. We will consider two types of depreferential attachment models, namely, emph{inverse depreferential}, where the attachment probabilities are inversely proportional to the degree and emph{linear depreferential}, where the attachment probabilities are proportional to $c$degree, where $c > 0$ is a constant. We will give asymptotic degree distribution for both the model and show that the limiting degree distribution has very thin tail. We will also show that for a fixed vertex $v$, the degree grows as $sqrt{log n}$ for the inverse depreferential case and as $log n$ for the linear case, for a graph with $n$ vertices. Some of the results will also be generalized when each new vertex joins to $m > 1$ existing vertices. [This is a joint work with Subhabrata Sen, Stanford University]. 
Cruise, James 
Electricity networks, novel challenges 
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A move towards the decarbonisation of power networks is leading to increased variability and uncertainty on the both the demand and supply side. In this talk we will introduce the audience the power network, and differences to other well studied networks. As part of this we will describe a set of open problems on a range of scales from microgrids to transmission networks. 
Davidson, Angus 
Steiner trees in the stochastic mean field model of distance 
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Consider the complete graph on n nodes with iid exponential weights of unit mean on the edges. A number of properties of this model have been investigated including first passage percolation, diameter, minimum spanning tree, etc. In particular, Janson showed that the typical distance between two nodes scales as (log n)/n, whereas the diameter (maximum distance between any two nodes) scales as 3(log n)/n. Bollobas et al. showed that, for any fixed k, the weight of the Steiner tree connecting k typical nodes scales as (k1)log n/n, which recovers Janson's result for k=2. We extend this result to show that the worst case kSteiner tree, over all choices of k nodes, has weight scaling as (2k1)log n/n. Further, Janson derived the limiting distribution of the typical distance between two nodes. We refine the result of Bollobas et al. and present a perhaps surprising result in this direction for the typical Steiner tree which has implications for its limiting shape. This is joint work with Ayalvadi Ganesh and Balint Toth. 
Dettmann, Carl 
Spatial networks with random connections 
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Many networks of current interest have a spatial structure, in that the nodes and/or links are located in physical space. Examples include climate, communications, infrastructure, nanowire, neuronal and transport networks. An early and still popular model of spatial networks is the random geometric graph, where nodes are located randomly and links formed between sufficiently close nodes. Recent studies have considered random connection models, in which there is a link probability depending on distance. Applying Laplace's method to relevant multidimensional integrals, we find that the overall connection probability can be estimated from just a few moments of the link probability function for a wide variety of domain geometries. Furthermore, there are qualitative differences as a result of the random connections. In particular, the more realistic model allows a more accurate estimation of connectivity and resilience than the original. 
Fountoulakis, Nikolaos 
Metastability phenomena in bootstrap percolation and power law degree distributions 
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In this talk, I will review some recent results on the evolution of bootstrap percolation processes on several classes of random graphs that exhibit a power law degree distribution. It turns out that there is a critical initial density of active nodes which is asymptotically vanishing above which an outbreak occurs. We will present a collection of random graph models where this occurs and we will give an (almost) characterisation for this phenomenon. 
Freeman, Nic 
Selection and dimension 
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I will discuss the interaction between natural selection and the dimension of space in which it takes place. The Spatial Lambda FlemingViot process models evolution in a spatial continuum, and gives rise to networks composed of branching and coalescing ancestral lines. The geometry of these networks greatly depends on the dimension of space, and we explore connections to branching Brownian motion, the Brownian net, travelling waves, and mean curvature flow. (joint work with Alison Etheridge, Daniel Straulino and Sarah Penington) 
Georgiou, Orestis 
Connectivity of cooperative wireless networks 
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Wireless network connectivity has been extensively studied over the past few years, from local observables, to global network properties. Similarly, cooperation on dynamic networks such as social, academic, and opinion formation networks has also been extensively studied. In this talk, I will attempt to overlay cooperative network dynamics on top of an ad hoc wireless network (essentially a random geometric network). Nodes are assumed selfish and a snowdrift type game will dictate the way nodes decide to allocate their cooperative resource efforts towards other nodes in the network. The dynamics are strongly coupled with the physical network causing the cooperation network topology to converge towards a stable equilibrium state, a global maximum of the total payoff. I will also discuss the analogies between our simple model and devicetodevice proximity based services such as LTEDirect. 
Jalan, Sarika 
Unraveling complexity of various cancers through combined framework of multilevel theory, network theory and spectral graph theory 
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One of the major goals of the postgenomic era is to understand the role of proteomics and genomics in human health and diseases. Even enormous research, diseases like Cancer, Diabetes, Hypertension, etc. remains complex to be understood at the fundamental level. We apply combined framework of network science, spectral graph theory and multilevel analysis to the proteinprotein interaction networks of seven different cancers and their normal counterpart. The analysis reveals that both the normal and disease networks for all the seven cancers exhibit overall similar behavior for various structural properties indicating normal and disease states to be complex and robust. The crucial differences between them, which are of potential importance, are revealed through the analysis of cliques structures and the networks spectra. While the shortrange correlations in the eigenvalues exhibit universality and provide evidence towards the importance of random connections in the underlying networks, the longrange correlations along with the localization properties and clique structures reveal insightful structural patterns involving functionally important proteins. In the talk I will discuss important methods and techniques, based spectral graph theory, which we have developed for understanding biological systems as well as results pertaining to the cancer networks.
References:
1. Aparna Rai, V. Menon and Sarika Jalan, Randomness and preserved patterns in cancer network. Nature Scientific Reports 4 6368 (2014).
2. Sanjiv K. Dwivedi and Sarika Jalan*. Emergence of clustering: Role of inhibition, Phys. Rev. E 90, 032803 (2014).
3. Sarika Jalan*, K. Kanhaiya, Aparna Rai, O. Bandapalli and A. Yadav. Network topologies decoding cervical cancer, PLoS ONE 10(8), e0135183 (2015).
4. Sarika Jalan*, Alok Yadav. Assortative and disassortative mixing investigated using the spectra of graphs. Phys. Rev. E 91, 012813 (2015).
5. Alok Yadav and Sarika Jalan*, Origin and implication of zero degeneracy. Chaos 25, 043110 (2015).
6. Pramod Shinde and Sarika Jalan, A multilayer proteinprotein interaction network analysis of different life stages, EPL 112 58001 (2015). 
Johnson, Samuel 
Learning from food webs: Stability and trophic structure of complex dynamical systems 
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Rainforests, coral reefs and other very large ecosystems seem to be the most stable in nature, but this has long been regarded as mathematically paradoxical. More generally, the relationship between structure and dynamics in complex systems is the subject of much debate. I will discuss how 'trophic coherence', a recently identified property of food webs and other biological networks, is key to understanding many dynamical and structural features of complex systems. In particular, it allows networks to become more stable with increasing size and complexity, determines whether a given system will be in a regime of high or of negligible feedback, and influences spreading processes such as epidemics or cascades of neural activity. 
Jordan, Jonathan 
Preferential attachment with choice 
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We consider the degree distributions of preferential attachment random graph models with choice similar to those considered in recent work by Malyshkin and Paquette and Krapivsky and Redner. In these models a new vertex chooses $r$ vertices according to a preferential rule and connects to the vertex in the selection with the $s$th highest degree. For meek choice, where $s>1$, we show that both double exponential decay of the degree distribution and condensationlike behaviour are possible, and provide a criterion to distinguish between them. For greedy choice, where $s=1$, we confirm that the degree distribution asymptotically follows a power law with logarithmic correction when $r=2$ and shows condensationlike behaviour when $r>2$. 
Lestas, Ioannis 
Noise in gene regulatory networks and some fundamental limits for the suppression of molecular fluctuations 
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Biochemical reaction networks at the molecular level are known to be intrinsically stochastic systems, due to the spontaneous births and deaths of individual molecules, and are also in general poorly characterized. We discuss in this talk how various generic features in such processes, such as the presence of restricted signaling rates and delays can lead to hard bounds for noise suppression that hold for arbitrary feedback policies. These limits challenge conventional beliefs about biochemical accuracy, and provide an explanation to some interesting experimental observations where a huge amount of resources are used within the cell for noise suppression. The analysis is based on combinations of control and information theory which can often be exploited to rigorously analyze poorly characterized biological systems. 
Luczak, Malwina 
SIR epidemics on random graphs with a given degree sequence 
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We study the susceptibleinfectiverecovered (SIR) epidemic on a random graph chosen uniformly subject to having given vertex degrees. In this model, infective vertices infect each of their susceptible neighbours, and recover, at a constant rate. Suppose that initially there are only a few infective vertices. We prove that there is a threshold for a parameter involving the rates and vertex degrees below which only a small number of infections occur. Above the threshold a large outbreak may occur. We prove that, conditional on a large outbreak, the evolutions of certain quantities of interest, such as the fraction of infective vertices, converge to deterministic functions of time. In contrast to earlier results for this model, our results only require basic regularity conditions and a uniformly bounded second moment of the degree of a random vertex. We also study the regime just above the threshold: we determine the probability that a large epidemic occurs and the size of a large epidemic. This is joint work with Svante Janson and Peter Windridge. 
Mailler, Cecile 
Nonextensive condensation in reinforced branching processes 
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In this joint work with Steffen Dereich and Peter Mörters, we introduce and study a branching process with both a reinforcement and a mutation dynamics. Our model incompasses as a particular case the Bianconi and Barab'asi's preferential attachment graph with fitnesses but also a selection and mutation population branching process. We prove that this model exhibits, under some conditions on the parameters, a condensation phenomenon, and more precisely that ``the winner does not take it all'', disproving a claim made in the physics literature about the Bianconi and Barabasi's model. 
Markose, Sheri 
Financial network modelling for systemic risk management and macroprudential policy 
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Traditional models on the transmission of financial stress primarily rely on a correlations based approach using market price based indicators. These have been found to run into the volatility paradox in that they underestimate systemic risk and fail to provide early warning during conditions when debt is growing on balance sheets. Network models of bilateral balance sheet based interconnections between financial entities nationally and in a cross border setting are essential to monitor the risk of instability in these systems and one that can provide early warning of tipping points. A spectral approach to network instability is given along with network centralities for systemic importance and vulnerability. Applications to cross border financial crisis and also to a multilayer network setting for global derivatives markets are given. 
Penrose, Mathew 
Isolated points for inhomogeneous random graphs 
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Consider a random graph with n vertices placed at random in some space (for example, Euclidean space) and each pair of points connected with a probability that depends on their location (for example, via the distance between them). We discuss Poisson approximation for the number of isolated vertices in this model as n becomes large with the connection function function varying with n. One reason to be interested is because absence of isolated points is necessary for connectivity and is often asymptotically sufficient, in probability. 
Reinert, Gesine 
Network comparison 
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Networks are routinely used to represent large data sets, making comparison of networks a tantalizing research question in many areas. Techniques for such analysis vary from simply comparing network summary statistics to sophisticated but computationally expensive alignmentbased approaches. Most existing methods either do not generalize well to different types of networks or do not provide a quantitative similarity score between networks. In contrast, alignmentfree topology based network similarity scores such as Netdis empower us to analyze large sets of networks containing different types and sizes of data. Netdis defines network similarity through the counts of small subgraphs in the local neighbourhood of all nodes. Here, we introduce a subsampling procedure based on neighbourhoods which links naturally with the Netdis framework. Our theoretical arguments justify basing the Netdis statistic on a sample of similarsized neighbourhoods. Our tests on empirical and synthetic datasets indicate that often only 10% of the mediumsized neighbourhoods of a network suffice for optimal performance, leading to a drastic reduction in computational requirements. The sampling procedure is applicable even when only a small part of the network is known, and thus provides a novel tool for network comparison of very large and potentially datasets. 
Saha, Kumarjit 
Drainage networks in river basin modelling 
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Drainage networks can be loosely defined as follows. Start with a random set of vertices and fix a direction. Each vertex connects to a unique vertex along that direction selected on the basis of a fixed set of rules. The number of outgoing edge for each vertex is only one. It follows that the generated graph can not have any cycle with each component being a directed infinite tree. Various questions related to the geometry of these networks can be asked, e.g., whether this graph is connected or not, the number of topological ends and so on. In this talk we describe some well known drainage network models and some recent results on their geometry. Understanding the geometry reveals that it represents a tributary structure and can be used in river basin modelling. It is believed that various empirically observed scaling relations can be understood well if we can understand scaling limits of these drainage network models. We describe what do we mean by scaling limits of these models and how it can be used for justification of "Hack's law", a celebrated scaling relation in river basin study. This is joint work with Rahul Roy and Anish Sarkar. 
Sahasrabudhe, Neeraja 
Synchronization and fluctuation theorems for interacting Friedman urns. 
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We consider a model of N interacting twocolour Friedman urns. The interaction model considered is such that the reinforcement of each urn depends on the fraction of balls of a particular colour in that urn as well as the overall fraction of balls of that colour in all the urns combined together. We will prove an almost sure synchronization result, where by synchronization we mean that the fraction of balls of each colour, in each urn, converges to a common limit. Further, we will discuss some limit theorems for fluctuations around the synchronization limit. These results are, in spirit, similar to those known for a single Friedman urn model. 
Sinha, Sitabhra 
Mesoscale networks in neuroscience: dynamical models for the collective activity of brain regions 
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Nonlinear dynamics of interactions between clusters of neurons via complex networks lie at the base of all brain activity. How such communication between brain regions gives rise to the rich behavioral repertoire of the organism has been a longstanding question. In this talk, we will explore this question by looking at the simulations of collective dynamics of a detailed network of cortical areas in the Macaque brain recently compiled from the CoCoMac database. To understand the largescale dynamics of the brain, we simulate it at the mesoscopic level with each node representing a local region of cortex (the node dynamics being described using a phenomenological model comprising a pool of excitatory neurons coupled to a pool of inhibitory neurons resulting in oscillations over a large range of parameter values). Connecting these regions according to the Macaque cortical network produces activation patterns strikingly similar to those observed in recordings from the brain. The collective dynamics for even a simplified connection topology (viz., a globally coupled network) of such "oscillators" exhibits a rich variety of dynamical patterns emerging via spontaneous breaking of permutation or translational symmetry. Including realistic details such as the Macaque brain connection topology or communication delays between different regions do not significantly alter the observed phenomena. 
Sinha, Sudeshna 
Dynamics of rewired networks 
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We will show how spatiotemporal chaos in networks with strongly chaotic nodal dynamics can be tamed by dynamically changing links. Specifically, we will illustrate the results in examples ranging from neuronal networks to disease spreading models. Further we will show how random links can prevent blowups in coupled nonlinear systems suffering from unbounded growth. 
Skerman, Fiona 
Modularity of random graphs 
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An important problem in network analysis is to identify highly connected components or `communities'. Most popular clustering algorithms work by approximately optimising modularity. Given a graph $G$, the modularity of a partition of the vertex set measures the extent to which edge density is higher within parts than between parts; and the maximum modularity $q^*(G)$ of $G$ is the maximum of the modularity over all partitions of $V(G)$ and takes a value in the interval $[0,1)$ where larger values indicates a more clustered graph. Knowledge of the maximum modularity of random graphs helps determine the significance of a division into communities/vertex partition of a real network. We investigate the maximum modularity of ErdH{o}sR'enyi random graphs and find there are three different phases of the likely maximum modularity. For $n^2pto infty$ and $np leq 1+o(1)$ the maximum modularity is $1+o(1)$ whp and for $np toinfty$ the maximum modularity is $o(1)$ whp. For $np=c$ with $c>1$ a constant, functions are constructed with $0<a(c)<b(c)<1$ and $b(c) to 0$ as $c to infty$ such that whp the maximum modularity is bounded between these functions. Concentration of the maximum modularity about its expectation and structural properties of an optimal partition are also established. This is joint work with Prof. Colin McDiarmid. 
Smolyarenko, Igor 
Models of random growing networks with intrinsic node fitness 
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Network growth models with attachment rules governed by intrinsic node fitness are considered. Both direct and inverse problems of matching the growth rules to node degree distribution and correlation functions are given analytical solutions. It is found that the node degree distribution is generically broader than the distribution of fitness, saturating at power laws. The saturation mechanism is analysed using a feedback model with dynamically updated fitness distribution, leading to a nontrivial fixed point with a unique asymptotically powerlaw degree distribution. Applications of fieldtheoretic methods to network growth models are also discussed. 
Sundaresan, Rajesh 
Epidemics in a multicommunity network 
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We consider a multipopulation epidemic model with several isolated communities and one set of mobile nodes that can transfer the content between the communities. We first derive a multidimensional ODE (ordinary differential equation) which is a meanfield fluid approximation to the process of the number of infected nodes, appropriately scaled, and show that the approximation becomes tight as the sizes of the communities grow. We then use a singular perturbation approach which allows us to reduce the dimension of the ODE and to derive the structure of the system. We show that the epidemic may grow in time in all communities even if when in absence of the mobile users it would die quickly out within each one of the isolated communities. 