Systemic risk measured by the resiliency of the system to initial shocks.

The analysis of systemic risk often revolves around examining various measures utilized by practitioners and policymakers. These measures typically focus on assessing the extent to which external events can impact a financial system, without delving into the nature of the initial shock. In contrast, our approach takes a symmetrical standpoint and introduces a set of measures centered on the quantity of external shock that the system can absorb before experiencing deterioration. To achieve this, we employ a linearized version of DebtRank, which facilitates a clear depiction of the onset of financial distress, thereby enabling accurate estimation of systemic risk. Through the utilization of spectral graph theory, we explicitly compute localized and uniform exogenous shocks, elucidating their behavior. Additionally, we expand the analysis to encompass heterogeneous shocks, necessitating computation via Monte Carlo simulations. We firmly believe that our approach is both comprehensive and intuitive, enabling a standardized assessment of failure risk in financial systems.

Causal motifs and existence of endogenous cascades in directed networks with application to company defaults.

Motivated by the problem of detection of cascades of defaults in economy, we developed a detection framework for an endogenous spreading based on causal motifs we define in this paper. We assume that the change of state of a vertex can be triggered either by an endogenous (related to the network) or an exogenous (unrelated to the network) event, that the underlying network is directed and that times when vertices changed their states are available. After simulating default cascades driven by different stochastic processes on different synthetic networks, we show that some of the smallest causal motifs can robustly detect endogenous spreading events. Finally, we apply the method to the data of defaults of Croatian companies and observe the time window in which an endogenous cascade was likely happening.

Controlling systemic risk: Network structures that minimize it and node properties to calculate it.

Evaluation of systemic risk in networks of financial institutions in general requires information of interinstitution financial exposures. In the framework of the DebtRank algorithm, we introduce an approximate method of systemic risk evaluation which requires only node properties, such as total assets and liabilities, as inputs. We demonstrate that this approximation captures a large portion of systemic risk measured by DebtRank. Furthermore, using Monte Carlo simulations, we investigate network structures that can amplify systemic risk. Indeed, while no topology in general sense is a priori more stable if the market is liquid (i.e., the price of transaction creation is small) [T. Roukny et al., Sci. Rep. 3, 2759 (2013)10.1038/srep02759], a larger complexity is detrimental for the overall stability [M. Bardoscia et al., Nat. Commun. 8, 14416 (2017)10.1038/ncomms14416]. Here we find that the measure of scalar assortativity correlates well with level of systemic risk. In particular, network structures with high systemic risk are scalar assortative, meaning that risky banks are mostly exposed to other risky banks. Network structures with low systemic risk are scalar disassortative, with interactions of risky banks with stable banks.

Bi-stability of SUDR+K model of epidemics and test kits applied to COVID-19.

Motivated by the many diverse responses of different countries to the COVID-19 emergency, here we develop a toy model of the dependence of the epidemics spreading on the availability of tests for disease. Our model, that we call SUDR+K, grounds on the usual SIR model, with the difference of splitting the total fraction of infected individuals in two components: patients that are still undetected and patients that have been already detected through tests. Moreover, we assume that available tests increase at a constant rate from the beginning of epidemics but are consumed to detect infected individuals. Strikingly, we find a bi-stable behavior between a phase with a giant fraction of infected and a phase with a very small fraction. We show that the separation between these two regimes is governed by a match between the rate of testing and a rate of infection spread at given time. We also show that the existence of two phases does not depend on the mathematical choice of the form of the term describing the rate at which undetected individuals are tested and detected. Presented research implies that a vigorous early testing activity, before the epidemics enters its giant phase, can potentially keep epidemics under control, and that even a very small change of the testing rate around the bi-stable point can determine a fluctuation of the size of the whole epidemics of various orders of magnitude. For the real application of realistic model to ongoing epidemics, we would gladly collaborate with field epidemiologists in order to develop quantitative models of testing process.

Color-avoiding percolation.

Many real world networks have groups of similar nodes which are vulnerable to the same failure or adversary. Nodes can be colored in such a way that colors encode the shared vulnerabilities. Using multiple paths to avoid these vulnerabilities can greatly improve network robustness, if such paths exist. Color-avoiding percolation provides a theoretical framework for analyzing this scenario, focusing on the maximal set of nodes which can be connected via multiple color-avoiding paths. In this paper we extend the basic theory of color-avoiding percolation that was published in S. M. Krause et al. [Phys. Rev. X 6, 041022 (2016)]2160-330810.1103/PhysRevX.6.041022. We explicitly account for the fact that the same particular link can be part of different paths avoiding different colors. This fact was previously accounted for with a heuristic approximation. Here we propose a better method for solving this problem which is substantially more accurate for many avoided colors. Further, we formulate our method with differentiated node functions, either as senders and receivers, or as transmitters. In both functions, nodes can be explicitly trusted or avoided. With only one avoided color we obtain standard percolation. Avoiding additional colors one by one, we can understand the critical behavior of color-avoiding percolation. For unequal color frequencies, we find that the colors with the largest frequencies control the critical threshold and exponent. Colors of small frequencies have only a minor influence on color-avoiding connectivity, thus allowing for approximations.

Reduction of Systemic Risk by Means of Pigouvian Taxation.

We analyze the possibility of reduction of systemic risk in financial markets through Pigouvian taxation of financial institutions, which is used to support the rescue fund. We introduce the concept of the cascade risk with a clear operational definition as a subclass and a network related measure of the systemic risk. Using financial networks constructed from real Italian money market data and using realistic parameters, we show that the cascade risk can be substantially reduced by a small rate of taxation and by means of a simple strategy of the money transfer from the rescue fund to interbanking market subjects. Furthermore, we show that while negative effects on the return on investment (ROI) are direct and certain, an overall positive effect on risk adjusted return on investments (ROIRA) is visible. Please note that the taxation is introduced as a monetary/regulatory, not as a _scal measure, as the term could suggest. The rescue fund is implemented in a form of a common reserve fund.

Extraction of temporal networks from term co-occurrences in online textual sources.

A stream of unstructured news can be a valuable source of hidden relations between different entities, such as financial institutions, countries, or persons. We present an approach to continuously collect online news, recognize relevant entities in them, and extract time-varying networks. The nodes of the network are the entities, and the links are their co-occurrences. We present a method to estimate the significance of co-occurrences, and a benchmark model against which their robustness is evaluated. The approach is applied to a large set of financial news, collected over a period of two years. The entities we consider are 50 countries which issue sovereign bonds, and which are insured by Credit Default Swaps (CDS) in turn. We compare the country co-occurrence networks to the CDS networks constructed from the correlations between the CDS. The results show relatively small, but significant overlap between the networks extracted from the news and those from the CDS correlations.

Geometric origin of scaling in large traffic networks.

Large scale traffic networks are an indispensable part of contemporary human mobility and international trade. Networks of airport travel and cargo ship movements are invaluable for the understanding of human mobility patterns [R. Guimera et al., Proc. Natl. Acad. Sci. U.S.A. 102, 7794 (2005))], epidemic spreading [V. Colizza et al., Proc. Natl. Acad. Sci. U.S.A. 103, 2015 (2006)], global trade [International Maritime Organization, http://www.imo.org/], and spread of invasive species [G. M. Ruiz et al., Nature (London) 408, 49 (2000)]. Different studies [M. Barthelemy, Phys. Rept. 499, 1 (2011)] point to the universal character of some of the exponents measured in such networks. Here we show that exponents which relate (i) the strength of nodes to their degree and (ii) weights of links to degrees of nodes that they connect have a geometric origin. We present a simple robust model which exhibits the observed power laws and relates exponents to the dimensionality of 2D space in which traffic networks are embedded. We show that the relation between weight strength and degree is s(k)~k(3/2), the relation between distance strength and degree is s(d)(k)~k(3/2), and the relation between weight of link and degrees of linked nodes is w(ij)~(k(i)k(j))(1/2) on the plane 2D surface. We further analyze the influence of spherical geometry, relevant for the whole planet, on exact values of these exponents. Our model predicts that these exponents should be found in future studies of port networks and it imposes constraints on more refined models of port networks.

Topologically biased random walk and community finding in networks.

We present an approach of topology biased random walks for undirected networks. We focus on a one-parameter family of biases, and by using a formal analogy with perturbation theory in quantum mechanics we investigate the features of biased random walks. This analogy is extended through the use of parametric equations of motion to study the features of random walks vs parameter values. Furthermore, we show an analysis of the spectral gap maximum associated with the value of the second eigenvalue of the transition matrix related to the relaxation rate to the stationary state. Applications of these studies allow ad hoc algorithms for the exploration of complex networks and their communities.

Hypergraph topological quantities for tagged social networks.

Recent years have witnessed the emergence of a new class of social networks, which require us to move beyond previously employed representations of complex graph structures. A notable example is that of the folksonomy, an online process where users collaboratively employ tags to resources to impart structure to an otherwise undifferentiated database. In a recent paper, we proposed a mathematical model that represents these structures as tripartite hypergraphs and defined basic topological quantities of interest. In this paper, we extend our model by defining additional quantities such as edge distributions, vertex similarity and correlations as well as clustering. We then empirically measure these quantities on two real life folksonomies, the popular online photo sharing site Flickr and the bookmarking site CiteULike. We find that these systems share similar qualitative features with the majority of complex networks that have been previously studied. We propose that the quantities and methodology described here can be used as a standard tool in measuring the structure of tagged networks.

Random hypergraphs and their applications.

In the last few years we have witnessed the emergence, primarily in online communities, of new types of social networks that require for their representation more complex graph structures than have been employed in the past. One example is the folksonomy, a tripartite structure of users, resources, and tags-labels collaboratively applied by the users to the resources in order to impart meaningful structure on an otherwise undifferentiated database. Here we propose a mathematical model of such tripartite structures that represents them as random hypergraphs. We show that it is possible to calculate many properties of this model exactly in the limit of large network size and we compare the results against observations of a real folksonomy, that of the online photography website Flickr. We show that in some cases the model matches the properties of the observed network well, while in others there are significant differences, which we find to be attributable to the practice of multiple tagging, i.e., the application by a single user of many tags to one resource or one tag to many resources.

Influence of reciprocal edges on degree distribution and degree correlations.

Reciprocal edges represent the lowest-order cycle possible to find in directed graphs without self-loops. Representing also a measure of feedback between vertices, it is interesting to understand how reciprocal edges influence other properties of complex networks. In this paper, we focus on the influence of reciprocal edges on vertex degree distribution and degree correlations. We show that there is a fundamental difference between properties observed on the static network compared to the properties of networks, which are obtained by simple evolution mechanism driven by reciprocity. We also present a way to statistically infer the portion of reciprocal edges, which can be explained as a consequence of feedback process on the static network. In the rest of the paper, the influence of reciprocal edges on a model of growing network is also presented. It is shown that our model of growing network nicely interpolates between Barabási-Albert (BA) model for undirected and the BA model for directed networks.

Reciprocity of networks with degree correlations and arbitrary degree sequences.

Although most of the real networks contain a mixture of directed and bidirectional (reciprocal) connections, the reciprocity r has received little attention as a subject of theoretical understanding. We study the expected reciprocity of networks with arbitrary input and output degree sequences and given 2-node degree correlations by means of statistical ensemble approach. We demonstrate that degree correlations are crucial to understand the reciprocity in real networks and a hierarchy of correlation contributions to r is revealed. Numerical experiments using network randomization methods show very good agreement to our analytical estimations.