BOOKLET OF ABSTRACTS

**KEYNOTE SPEAKERS**

**Ginestra Bianconi**

Queen Mary University of London

**Complex Quantum Network Geometries**

A quantum description of a discrete geometry is a central problem in physics, since it is the necessary to unify Quantum Mechanics with General Relativity. Until now, models for quantum space have focused on the description of discrete homogeneous geometrical spaces. Here, using tools developed in the context of Statistical Mechanics of Complex Networks, we define Complex Network Geometries constructed starting from simplicial complexes and describing the Evolution of Quantum Network States.

These networks are geometric networks with energies of the links that grow according to a non-equilibrium dynamics. The evolution in time of the geometric networks is a classical evolution describing a given path of a path integral defining the quantum evolution of quantum network states. The quantum network states are characterized by quantum occupation numbers that can be mapped respectively to the nodes, links, and triangles incident to each link of the network. We call the geometric networks describing the evolution of quantum network states the quantum geometric networks.

The quantum geometric networks have many properties common to complex networks including small-world property, high clustering coefficient, high modularity, scale-free degree distribution.

Moreover the quantum geometric networks can be distinguished between the Fermi-Dirac Network and the Bose-Einstein Network obeying respectively the Fermi-Dirac and Bose-Einstein statistics. We show that these networks can undergo structural phase transitions where the geometrical properties of the networks change drastically.

Finally we comment on the relation between the Fermi-Dirac Network and spin networks.

**Marian Boguña Espinal**

Universitat de Barcelona

**Towards a cosmological theory of complex networks**

Structural and dynamical similarities of different real networks suggest that some universal laws might accurately describe the dynamics of these networks, albeit the nature and common origin of such laws remain elusive. Here we show that the causal network representing the large-scale structure of spacetime in our accelerating universe is a power-law graph with strong clustering, similar to many complex networks such as the Internet, social, or biological networks. We prove that this structural similarity is a consequence of the asymptotic equivalence between the large-scale growth dynamics of complex networks and causal networks. This equivalence suggests that unexpectedly similar laws govern the dynamics of complex networks and spacetime in the universe, with implications to network science and cosmology. However, our simple framework is unable to explain the emergence of community structure, a property that, along with scale-free degree distributions and strong clustering, is commonly found in real complex networks. Here we show how latent network geometry coupled with preferential attachment of nodes to this geometry fills this gap. We call this mechanism geometric preferential attachment (GPA), and validate it against the Internet. GPA gives rise to soft communities that provide a different perspective on the community structure in networks. The connections between GPA and cosmological models, including inflation, are also discussed.

[1] Popularity versus similarity in growing networks

Fragkiskos Papadopoulos, Maksim Kitsak, M. Ángeles Serrano, Marián Boguñá, and Dmitri Krioukov

Nature 489, 537-540 (2012)

[2] Network Cosmology

Dmitri Krioukov, Maksim Kitsak, Robert S. Sinkovits, David Rideout, David Meyer, Marián Boguñá

Scientific Reports 2, 793 (2012)

[3] Emergence of Soft Communities from Geometric Preferential Attachment

Konstantin Zuev, Marián Boguñá, Ginestra Bianconi, and Dmitri Krioukov

Scientific Reports 5, 9421 (2015)

**Giovanni Petri**

ISI Foundation, Turin, Italy

**Summarising robust features of complex networks with topological methods**

Over the last decade network theory has emerged as the foremost tool for the understanding of complex systems. This success is due the ability of networks to simplify and capture non-trivial interaction patterns, in terms of pairwise relations. Higher order interaction patterns however remain elusive under this lens. Topological methods for data analysis have recently attracted large attention due to their capacity to capture mesoscopic features which are lost under standard network technique.

I will focus on persistent homology, a technique able to identify robust topological features underlying the structure of high-dimensional data and complex dynamical systems (such as brain dynamics, molecular folding, distributed sensing).

In the first part, I will present a few applications application of persistent homology to a variety of datasets, e.g. a urban mobile phone activity dataset, complex weighted networks and brain functional data of patients in different states of consciousness.

Finally, I will show how the persistent topological features can be summarised in terms of the Laplacian structure of the filtration used to approximate the datasets, and discuss ongoing work regarding temporal datasets and dynamical processes

**CONTRIBUTED TALKS**

**E. Saenz de Cabezon**, H.Wynn

**Combinatorial Algebraic Measures of Network Robustness**

The interplay between monomial ideals in a polynomial ring and simplicial complexes has been extensively studied in the last two decades.

In particular, the relation of monomial ideals and graphs has been a fruitful line of research. In this paper we step forward in this area by using algebraic invariants of monomial ideals to study network robustness. Our key point is the connection between the primary decomposition of a monomial ideal and vertex coverings of the network's graph.

We define two quantities associated to each of the vertices of a simple graph, based on the collection of minimal vertex covers of the graph.

They are called covering degree and covering index. We use them to describe new strategies for measuring the robustness of a network.

We study the correlation between the dened quantities and other quantities used in the context of network attacks. Using the attack strategies

associated to these quantities we study their eect on the connectedness of several network models. We also consider the complexity of the computation of the defined quantities and use a computational commutative algebra approach for their actual computation.

**Brenton Walker**

**The Topological Structure of Geographically Distributed Network Coded Data**

We generalize work using topology to study the coverage of wireless and sensor networks to that of covering a geographic area with network coded information in an opportunistic data distribution network. While the problem of sensor network coverage has been essentially a geometric and statistical problem, the network coded data coverage problem has a multi-dimensional algebraic aspect that leads to some surprising structure and coverage-testing counterexamples.

**Florian Klimm**, Dane Taylor, Heather Harrington, Miroslav Kramar, Konstantin Mischaikow, Mason Porter and Peter Mucha

**Contagion dynamics for topological data analysis**

The spread of social and biological contagions can exhibit complicated dynamics and patterns, which can be influenced significantly by the spatial structure in networks on which they spread. For example, the Black Death spread as a wave through Europe, following the underlying manifold of the Earth’s surface. In modern contagions, however, long-range connections (e.g., from airline transportation) can lead to the emergence of clusters of contagion in spatially distant regions. Consequently, it is unclear how (and whether) contagion dynamics reflect a network's underlying manifold when such long-range edges are present. We study this phenomenon by examining the Watts threshold model (WTM) on both empirical and synthetic networks. We introduce the idea of contagion maps, which use multiple realisations of a contagion to map nodes into a high-dimensional point cloud. In particular, each contagion yields a filtration of a network, and the set of filtrations produces a metric that allows us to map the network nodes as a point cloud, which we analyse using methods from dimension reduction and persistent homology. For contagions that predominantly exhibit wavefront propagation, we show that contagion maps adhere to a network's underlying manifold, even when there are a large number of edges that are not embedded in this manifold. We examine whether we observe wavefront propagation and/or the appearance of new contagion clusters as a function of the WTM’s threshold parameter (which governs how difficult it is for a contagion to spread to a given node) and the ratio of edges that are spatially embedded to those that are not. We summarise these results as bifurcation diagrams. Our analysis reveals parameter regimes in which the manifolds that underly networks can be accurately recovered in the point clouds that result from contagion maps.

**Bruno Coutinho**, Sungryong Hong, Arjun Dey and Albert-László Barabási

**The Cosmic Web as a Network**

The cosmic web, conceptualizing the large-scale structure of the universe as a network is deeply embedded in cosmology [2-4]. Yet, it remains a metaphor, little being known in a quantitative manner about the properties of this network. Based on geometric graph theory and concepts used in cosmology, we developed an algorithm to construct a directed network representation of the cosmic web, relying on a single free parameter that enables us to control the average mean degree of the network and therefore the regime of connectivity. We applied our algorithm to the data from the Illustris simulations[4], and data from the Sloan Digital Sky Survey[1]. In both cases our data sample contains around 60 thousand galaxies, and for each galaxy the data contains information about its stellar mass, color, metallicity, star formation rate and other properties. We show that our network representation provides a richer characterization of the large-scale structure of the universe than the typical 2-point correlation function used in astronomy, as we find differences in the network structure for distributions of galaxies with the same 2-point correlation. Furthermore, this methodology allows to systematically identify correlation between the properties of nearby galaxies, in both simulations and real data. We found significant differences in the network representation of simulations and real data concerning the size of the giant component and the correlation between the star formation rate of nearby galaxies. These discrepancies are related to known limitations in the simulation, for example, the overestimation of the number of low mass galaxies. Overall, we show that a network representation of the cosmic web can be used as a tool to analyze and characterize the large-scale structure of the universe at different scales, being especially useful to compare simulations with real data.

[1] S. Genel, M. Vogelsberger, V. Springel, D. Sijacki, D. Nelson, G. Snyder, V. Rodriguez-Gomez, P. Torrey, and L. Hernquist. Introducing the Illustris project: the evolution of galaxy populations across cosmic time. mnras, 445:175–200, November 2014.

[2] Sergei Shandarin, Salman Habib, and Katrin Heitmann. Origin of the cosmic network: Nature vs nurture. Phys. Rev. D, 81:103006, May 2010.

[3] R. Brent Tully, Helene Courtois, Yehuda Hoffman, and Daniel Pomarede. The Laniakea supercluster of galaxies. Nature, 513(7516):71–73, September 2014.

[4] M. Vogelsberger, S. Genel, V. Springel, P. Torrey, D. Sijacki, D. Xu, G. Snyder, S. Bird, D. Nelson, and L. Hernquist. Properties of galaxies reproduced by a hydrodynamic simulation. Nature, 509(7499):177–182, May 2014.

**James Clough**and Tim Evans

**Lorentzian Geometry in Citation Networks**

Citation networks are constrained in time because documents can only cite older documents. In the same way the causal set approach to quantum gravity views space-time as a discrete set of elements with a particular causal structure. In both cases we have examples of directed acyclic graphs. In the quantum gravity context the discrete structure is sufficient to fix the properties of the large scale continuous space-time we experience. In particular the space-time dimension can be estimated using just the causal relationships between the discrete points. We will show how to adapt these estimates of manifold dimension in order to characterise the structure of different citation networks.

We then apply these measures to three different types of citation network: academic papers on arXiv, US patents, and US Supreme Court judgements. We find that independent estimates of dimension converge on a consistent value for a given citation network. However, networks that otherwise appear very similar in structure turn out to have significantly different dimensions. We will show that by using network analysis methods that take the causal constraints into account, our approach reveals interesting distinctions in the structure of these temporal networks.

In the same way that some networks have a natural embedding into a Euclidean space, where the distance between nodes using some metric determines the likelihood of a connection between them, we suggest that others may, like causal sets, be naturally embedded in a Lorentzian space, where time is a coordinate having an opposite sign in the metric signature to the spatial coordinates.

By taking advantage of the natural time coordinate and the order it enforces on the network (since we know when papers are published) we can move towards embedding citation networks and other directed acyclic graphs into Minkowski spacetimes or other more complex spacetimes.

**Antoine Allard**, Guillermo García-Pérez, M. Ángeles Serrano and Marian Boguña

**Unveiling the hidden geometry of weighted networks**

Many real world complex networks are embedded in metric spaces. In some cases, such as airport networks, the embedding is explicit, while it is hidden for other cases such as the Internet or metabolic networks. Including this hidden underlying topology in mathematical models has been shown to naturally reproduce the strong clustering and the self-similarity of real complex networks. It has also explained the navigability of the Internet without an explicit knowledge of its global structure, and has shed light on the hierarchical organization of biochemical pathways in human cells. These achievements have been obtained by considering a binary network representation, in which links whether exist or not. There exist, however, complex networks, such as transportation networks or the world trade web, for which the magnitude of the interactions between nodes (i.e., the weights) plays a central role in their organisation and dynamics. A shift towards a weighted networks paradigm is therefore in order to fully grasp the interplay between the structure and the macroscopic behaviour of these complex networks.

We report the results of the first attempt to unveil the underlying geometry of weighted complex networks. We propose a versatile model that reproduces all of the aforementioned properties, but that also reproduces the non-trivial correlations between the degree and the strength of nodes found in real complex networks. Moreover, the nontrivial disparity observed in many real complex networks can be seen as a consequence of the coupling between the density of weights and the underlying geometry. In other words, the disparity is somewhat the weighted counterpart of the clustering, which is the topological reflection of the triangle inequality in the hidden metric space. Our work paves the way towards the embedding of real complex weighted networks into metric spaces, and the study of the navigability of networks whose links have limited flow capacities.

Emanuela Merelli and

In the present work we characterize and perform the run-time verification of the behavior properties of idiotypic network during successive stimuli, namely successive antigen injections. The idiotypic network hypothesis, formulated by Niels Jerne in 1974, postulates that the antibodies (called Abi) elicited directly by the antigen are new proteins that elicit and suppress other antibodies. The behavior of the whole system is characterized by three main states: virgin, activation and immune memory. We model the system as an adaptive system able to capture, by run-time verification, the behavioral differences of the three main states of the system during its exposition to successive antigen injections, by observing the interactive computation of antibodies. The model entangles both global and local information in a unique high-order system whose behavioral semantics lays on persistent entropy, trace equivalences, topological invariants, and the monoidal homotopy. We use topological data analysis (TDA) and the Clique Weight Rank Persistent Homology algorithm to construct the model, and C-IMMSIM framework to make the simulation.

**Matteo Rucco****Topology driven run-time verification of behavioural properties of the immune system during successive stimuli**In the present work we characterize and perform the run-time verification of the behavior properties of idiotypic network during successive stimuli, namely successive antigen injections. The idiotypic network hypothesis, formulated by Niels Jerne in 1974, postulates that the antibodies (called Abi) elicited directly by the antigen are new proteins that elicit and suppress other antibodies. The behavior of the whole system is characterized by three main states: virgin, activation and immune memory. We model the system as an adaptive system able to capture, by run-time verification, the behavioral differences of the three main states of the system during its exposition to successive antigen injections, by observing the interactive computation of antibodies. The model entangles both global and local information in a unique high-order system whose behavioral semantics lays on persistent entropy, trace equivalences, topological invariants, and the monoidal homotopy. We use topological data analysis (TDA) and the Clique Weight Rank Persistent Homology algorithm to construct the model, and C-IMMSIM framework to make the simulation.

**Vladimir Itskov**and Chad Giusti

Combinatorial topology of one-layer feedforward networks and non-convex codes.

It is often hypothesized that a crucial role for recurrent connections in neural networks is to constrain the set of possible response patterns, thereby shaping the neural code. Given a complete set of binary response patterns, i.e. the code of a network, can one rule out that the network functions as a collection of disconnected discriminators (perceptrons)? Mathematically this translates into questions about the combinatorics of hyperplane arrangements and convex sets. We identify a large class of codes, the non-convex codes, that indeed cannot be shaped by the feedforward architecture alone. However, these codes are difficult to distinguish from codes that share the same sets of maximal activity patterns in the presence of noise. When we coarsened the notion of combinatorial neural code to keep track only of maximal patterns (the simplicial complex of the code), we found the surprising result that all such codes (simplicial complexes) can in fact be encoded by one-layer feedforward networks. Our proofs use mathematical tools from classical combinatorial topology, such as the nerve lemma and the existence of an inverse nerve of a simplicial complex.