Carlo Piccardi

C. Piccardi, An egonet-based approach to effective weighted network comparison, arXiv:2410.03252, 2024. [doi]

With the impressive growth of network models in practically every scientific and technological area, we are often faced with the need to compare graphs, i.e., to quantify their (dis)similarity using appropriate metrics. This is necessary, for example, to identify networks with comparable characteristics or to spot anomalous instants in a time sequence of graphs. While a large number of metrics are available for binary networks, the set of comparison methods capable of handling weighted graphs is much smaller. Yet, the strength of connections is often a key ingredient of the model, and ignoring this information could lead to misleading results. In this paper we introduce a family of dissimilarity measures to compare undirected weighted networks. They fall into the class of alignment-free metrics: as such, they do not require the correspondence of the nodes between the two graphs and can also compare networks of different sizes. In short, they are based on the distributions, on the graph, of a few egonet features which are easily defined and computed: the distance between two graphs is then the distance between the corresponding distributions. On a properly defined testbed with a pool of weighted network models with diversified characteristics, the proposed metrics are shown to achieve state-of-the-art performance in the model classification task. The effectiveness and applicability of the proposed metrics are then demonstrated on two examples. In the first, some ''filtering'' schemes -- designed to eliminate non-significant links while maintaining most of the total weight -- are evaluated in their ability to produce as output a graph faithful to the original, in terms of the local structure around nodes. In the second example, analyzing a timeline of stock market correlation graphs highlights anomalies associated with periods of financial instability.

Matlab code of the EgoDistW function and network data used in the paper are available here.

C. Piccardi, L. Tajoli, and R. Vitali, Patterns of variability in the structure of global value chains: a network analysis, Review of World Economics, 160, 1009-1036, 2024. [doi]

Global Value Chains (GVCs) are a feature of the organization of production in many sectors and countries and they deeply affect international trade patterns. How far the separation of production stages—generating increasingly widespread GVCs—can go, is currently a matter of debate. The main focus of this paper is to investigate GVCs at the country-industry level by modelling them through the construction of a specific network and using network analysis tools. In particular, the aim is to propose a network-based measure of GVCs length to assess whether the structure of GVCs has stretched or shrank over time. Analyzing the evolution of these structures is important to better understand the role played by countries in the production chain, with implications also for their fragility or resilience in presence of external shocks. Our measure allows to consider differently shaped GVCs, and the results show that there are relevant differences among sectors and countries in terms of the evolution of GVCs, especially considering direct or indirect links. Overall, we find a general stability over time of GVCs, confirming the importance of the “relational approach” in GVCs. But the shifts in the geographical patterns of the connections also support the view that firms organizing this complex form of production are ready to grasp better opportunities when they appear in the global markets.

P. Bolzern, A. Colombo, and C. Piccardi, Manipulating opinions in social networks with community structure, IEEE Transactions on Network Science and Engineering, 11(1), 185-196, 2024. [doi]

Online social media have been one of the greatest drivers of societal change of the past two decades, but are now being recognized as one of the major causes of opinion radicalization and one of the most effective tools for opinion manipulation. Starting from a class of stochastic models of opinion dynamics, and considering different structures of social networks with increasingly realistic features (including a snapshot of the Facebook friendship network), we develop a mathematical model of different forms of opinion manipulation. We then explore how network properties, and in particular degree distribution and community structure, interact with the attack to amplify or reduce its effect on the population, both globally and on specific subsets. We find, in particular, that degree heterogeneity is key to making online social media susceptible to very effective attacks, even with relatively little effort. Communities instead play a more complex role, acting both as barriers to the spread of manipulated opinions through the whole population and as amplifiers of manipulated opinions when the target of the attack is a community of the online social medium. The results of our study can help design effective strategies to prevent the manipulation of opinions through online social media.

A. Radici, J. Claudet, A. Ligas, I. Bitetto, G. Lembo, M.T. Spedicato, P. Sartor, C. Piccardi, and P. Melià, Assessing fish–fishery dynamics from a spatially explicit metapopulation perspective reveals winners and losers in fisheries management, Journal of Applied Ecology, 60, 2482-2493, 2023. [doi]

Sustainable management of living resources must reconcile biodiversity conservation and socioeconomic viability of human activities. In the case of fisheries, sustainable management design is made challenging by the complex spatiotemporal interactions between fish and fisheries. We develop a comprehensive metapopulation framework integrating data on species life-history traits, connectivity and habitat distribution to identify priority areas for fishing regulation and assess how management impacts are spatially distributed. We trial this approach on European hake fisheries in the north-western Mediterranean, where we assess area-based management scenarios in terms of stock status and fishery productivity to prioritize areas for protection. Model simulations show that local fishery closures have the potential to enhance both spawning stock biomass and landings on a regional scale compared to a status quo scenario, but that improving protection is easier than increasing productivity. Moreover, the interaction between metapopulation dynamics and the redistribution of fishing effort following local closures implies that benefits and drawbacks are heterogeneously distributed in space, the former being concentrated in the proximity of the protected site. A network analysis shows that priority areas for protection are those with the highest connectivity (as expressed by network metrics) if the objective is to improve the spawning stock, while no significant relationship emerges between connectivity and potential for increased landings. Synthesis and applications. Our framework provides a tool for (1) assessing area-based management measures aimed at improving fisheries outcomes in terms of both conservation and socioeconomic viability and (2) describing the spatial distribution of costs and benefits, which can help guide effective management and gain stakeholder support. Adult dispersal remains the main source of uncertainty that needs to be investigated to effectively apply our model to fisheries regulation.

C. Piccardi, Metrics for network comparison using egonet feature distribution, Scientific Reports, 13, 14657, 2023. [doi]

Identifying networks with similar characteristics in a given ensemble, or detecting pattern discontinuities in a temporal sequence of networks, are two examples of tasks that require an effective metric capable of quantifying network (dis)similarity. Here we propose a method based on a global portrait of graph properties built by processing local nodes features. More precisely, a set of dissimilarity measures is defined by elaborating the distributions, over the network, of a few egonet features, namely the degree, the clustering coefficient, and the egonet persistence. The method, which does not require the alignment of the two networks being compared, exploits the statistics of the three features to define one- or multi-dimensional distribution functions, which are then compared to define a distance between the networks. The effectiveness of the method is evaluated using a standard classification test, i.e., recognizing the graphs originating from the same synthetic model. Overall, the proposed distances have performances comparable to the best state-of-the-art techniques (graphlet-based methods) with similar computational requirements. Given its simplicity and flexibility, the method is proposed as a viable approach for network comparison tasks.

Matlab code of the EgoDist function and network data used in the paper are available at GitHub and at MathWorks File Exchange.

V.P. Hoang, C. Piccardi, L. Tajoli, Reshaping the structure of the World Trade Network: a pivotal role for China?, Applied Network Science, 8, 35, 2023. [doi]

In recent years, the global trade landscape has undergone significant changes, particularly in the aftermath of the 2008 financial crisis and more recently as a consequence of Covid-19 pandemic. To understand the structure of international trade and the impact of these changes, this study applies a combination of network analysis and causal inference techniques to the most extensive coverage of available data in terms of time span and spatial extension. The study is conducted in two phases. The first one explores the structure of international trade by providing a comprehensive analysis of the World Trade Network (WTN) from various perspectives, including the identification of key players and clusters of strongly interacting countries. The second phase investigates the impact of the rising role of China on the global structure of the WTN. Overall, the results highlight a structural change in the WTN, evidenced by a variety of network metrics, around China’s rapid growth years. Additionally, the reshaping of the WTN is not only accompanied by a significant increase in trade flows between China and its partners, but also by a corresponding decline in trade among non-China-partner countries. These results suggest that China played a pivotal role in the restructuring of the WTN in the first decades of this century. The findings of this study shed light on the interpretation of the rapidly changing landscape of global trade.

L. Tajoli, F. Airoldi, and C. Piccardi, The network of international trade in services, Applied Network Science, 6, 68, 2021. [doi]

While the share of services in international trade has been increasing very slowly over the years, oscillating around 20 per cent since the 1990s, their role has constantly gained importance. Trade in services certainly faces many more obstacles than trade in goods, but its impact on globalization and countries’ competitiveness is crucial, and it is therefore worth investigating its characteristics. The present work aims to analyse the networks of international trade in services and to unveil specific properties by exploiting a number of existing methodologies and algorithms. After describing the global properties of the networks of the various service classes, we investigate differences and similarities among them, and we highlight the most influential countries in the trade of specific services. We find that traded services display sharply different characteristics and they can be grouped in two different sets according to their network structures. Countries’ positions in these networks are diversified, with connections unevenly distributed, especially for some service categories. We discover that the structure of links, i.e. the topology of the networks, identifies the role of countries much more clearly than the sole amount of services traded. Overall, the results highlight important features, as well as changes over time, in the landscape of the international services.

P. Finotelli, C. Piccardi, E. Miglio, and P. Dulio, A graphlet-based topological characterization of the resting-state network in healthy people, Frontiers in Neuroscience, 15, 665544, 2021. [doi]

In this paper, we propose a graphlet-based topological algorithm for the investigation of the brain network at resting state (RS). To this aim, we model the brain as a graph, where (labeled) nodes correspond to specific cerebral areas and links are weighted connections determined by the intensity of the functional magnetic resonance imaging (fMRI). Then, we select a number of working graphlets, namely, connected and non-isomorphic induced subgraphs. We compute, for each labeled node, its Graphlet Degree Vector (GDV), which allows us to associate a GDV matrix to each one of the 133 subjects of the considered sample, reporting how many times each node of the atlas “touches” the independent orbits defined by the graphlet set. We focus on the 56 independent columns (i.e., non-redundant orbits) of the GDV matrices. By aggregating their count all over the 133 subjects and then by sorting each column independently, we obtain a sorted node table, whose top-level entries highlight the nodes (i.e., brain regions) most frequently touching each of the 56 independent graphlet orbits. Then, by pairwise comparing the columns of the sorted node table in the top-k entries for various values of k, we identify sets of nodes that are consistently involved with high frequency in the 56 independent graphlet orbits all over the 133 subjects. It turns out that these sets consist of labeled nodes directly belonging to the default mode network (DMN) or strongly interacting with it at the RS, indicating that graphlet analysis provides a viable tool for the topological characterization of such brain regions. We finally provide a validation of the graphlet approach by testing its power in catching network differences. To this aim, we encode in a Graphlet Correlation Matrix (GCM) the network information associated with each subject then construct a subject-to-subject Graphlet Correlation Distance (GCD) matrix based on the Euclidean distances between all possible pairs of GCM. The analysis of the clusters induced by the GCD matrix shows a clear separation of the subjects in two groups, whose relationship with the subject characteristics is investigated.

F. Pierri, C. Piccardi, and S. Ceri, A multi-layer approach to disinformation detection in US and Italian news spreading on Twitter, EPJ Data Science, 9, 35, 2020. [doi]

We tackle the problem of classifying news articles pertaining to disinformation vs mainstream news by solely inspecting their diffusion mechanisms on Twitter. This approach is inherently simple compared to existing text-based approaches, as it allows to by-pass the multiple levels of complexity which are found in news content (e.g. grammar, syntax, style). As we employ a multi-layer representation of Twitter diffusion networks where each layer describes one single type of interaction (tweet, retweet, mention, etc.), we quantify the advantage of separating the layers with respect to an aggregated approach and assess the impact of each layer on the classification. Experimental results with two large-scale datasets, corresponding to diffusion cascades of news shared respectively in the United States and Italy, show that a simple Logistic Regression model is able to classify disinformation vs mainstream networks with high accuracy (AUROC up to 94%). We also highlight differences in the sharing patterns of the two news domains which appear to be common in the two countries. We believe that our network-based approach provides useful insights which pave the way to the future development of a system to detect misleading and harmful information spreading on social media.

F. Pierri, C. Piccardi, and S. Ceri, Topology comparison of Twitter diffusion networks effectively reveals misleading information, Scientific Reports, 10, 1372, 2020. [doi]

In recent years, malicious information had an explosive growth in social media, with serious social and political backlashes. Recent important studies, featuring large-scale analyses, have produced deeper knowledge about this phenomenon, showing that misleading information spreads faster, deeper and more broadly than factual information on social media, where echo chambers, algorithmic and
human biases play an important role in diffusion networks. Following these directions, we explore the possibility of classifying news articles circulating on social media based exclusively on a topological analysis of their diffusion networks. To this aim we collected a large dataset of diffusion networks on Twitter pertaining to news articles published on two distinct classes of sources, namely outlets that convey mainstream, reliable and objective information and those that fabricate and disseminate various kinds of misleading articles, including false news intended to harm, satire intended to make people laugh, click-bait news that may be entirely factual or rumors that are unproven. We carried out an extensive comparison of these networks using several alignment-free approaches including basic network properties, centrality measures distributions, and network distances. We accordingly evaluated to what extent these techniques allow to discriminate between the networks associated to the aforementioned news domains. Our results highlight that the communities of users spreading mainstream news, compared to those sharing misleading news, tend to shape diffusion networks with subtle yet systematic differences which might be effectively employed to identify misleading and harmful information.

M. Tantardini, F. Ieva, L. Tajoli, and C. Piccardi, Comparing methods for comparing networks, Scientific Reports, 9, 17557, 2019. [doi]

With the impressive growth of available data and the flexibility of network modelling, the problem of devising effective quantitative methods for the comparison of networks arises. Plenty of such methods have been designed to accomplish this task: most of them deal with undirected and unweighted networks only, but a few are capable of handling directed and/or weighted networks too, thus properly exploiting richer information. In this work, we contribute to the effort of comparing the different methods for comparing networks and providing a guide for the selection of an appropriate one. First, we review and classify a collection of network comparison methods, highlighting the criteria they are based on and their advantages and drawbacks. The set includes methods requiring known node correspondence, such as DeltaCon and Cut Distance, as well as methods not requiring a priori known node-correspondence, such as alignment-based, graphlet-based, and spectral methods, and the recently proposed Portrait Divergence and NetLSD. We test the above methods on synthetic networks and we assess their usability and the meaningfulness of the results they provide. Finally, we apply the methods to two real-world datasets, the European Air Transportation Network and the FAO Trade Network, in order to discuss the results that can be drawn from this type of analysis.

A. Mechiche-Alami, C. Piccardi, K.A. Nicholas, J.W. Seaquist, Transnational land acquisitions beyond the food and financial crises, Environmental Research Letters, 14, 084021, 2019. [doi]

Large-scale land acquisitions (LSLA) in resource-rich countries came to global attention after the food and financial crises of 2008. Previous research has assessed the magnitude of these land investments in terms of land areas acquired. In this study,we analyze the trends in the evolution of LSLA by framing the latter as virtual land trade network with land transactions occurring between 2000 and 2015, in order to shed light on the development and evolution of this system. Based on an index we introduce to represent both the number of countries and size of deals,we discover three main phases of trade activity: a steady increase from2000 until 2007 (Phase 1) followed by a peak coinciding with the food and financial crises between 2008 and 2010 (Phase 2) and concluded by a decline from 2011 to 2015 (Phase 3).We identify 73 countries that remained active in land trading during all three phases and forma core of land traders much larger than previously thought. Using network analysis methods, we group countries with similar trade patterns into categories of competitive, preferential, diversified, and occasional importers or exporters. Finally, in exploring the changes in investors and their interests in land throughout the phases, we attribute the evolution of LSLA to the different stages in the globalization and financialization of different industries. By showing that land investments seem fully integrated as investment strategies across industries we argue for the urgency of better regulation of LSLA so that they also benefit local populations without damaging the environment regardless of their primary purpose.

F. Riva, E. Colombo, C. Piccardi, Towards modelling diffusion mechanisms for sustainable off-grid electricity planning, Energy for Sustainable Development, 52, 11-25, 2019. [doi]

The electrification-based literature reports a limited knowledge about the mechanisms of evolution of electricity demand in off-grid settings, especially in remote contexts of developing countries, due to the lack of robust and appropriate modelling frameworks. Such lack of understanding and modelling endeavour contributes to an inefficient allocation of resources for electrification projects and inappropriate off-grid sizing processes. As a first step towards the development of a more appropriate electricity demand model, we present a comparative study of two approaches for modelling different diffusion mechanisms of electricity connections: system-dynamics and agent-based models. The latter includes the modelling of social network archetypes in the simulation of diffusion processes. We model different scenarios of diffusion and we use them for evaluating the impact on the sizing process of an off-grid hydroelectric system. The results suggest that the structure of the social network can represent a crucial parameter that can impact on timing needed to complete the diffusion of electricity access - from few months to even >10 years. This affects the sizing process and the long-term sustainability of the power system, leading to variation of the hydroelectric capacity and the battery size up to around 55% and 100%, respectively. Our results indicate that the agent-based approach allows a more diversified representation of diffusion processes, but the limitations and scarcity of data can be an obstacle to their prompt application for energy application in unelectrified areas. On the contrary, system-dynamics can represent a more appropriate method since it requires less quantitative data and it provides a more structural and holistic modelling framework for conceptualising and formulating in a the determinants and complexities affecting the evolution of electricity demand in unelectrified areas.

F. Dercole, F. Della Rossa, C. Piccardi, Direct reciprocity and model-predictive rationality explain network reciprocity over social ties, Scientific Reports, 9, 5367, 2019. [doi]

Since M. A. Nowak & R. May's (1992) influential paper, limiting each agent's interactions to a few neighbors in a network of contacts has been proposed as the simplest mechanism to support the evolution of cooperation in biological and socio-economic systems. The network allows cooperative agents to self-assort into clusters, within which they reciprocate cooperation. This (induced) network reciprocity has been observed in several theoreticalmodels and shown to predict the fixation of cooperation under a simple rule: the benefit produced by an act of cooperation must outweigh the cost of cooperating with all neighbors. However, the experimental evidence among humans is controversial: though the rule seems to be confirmed, the underlying modeling assumptions are not. Specifically, models assume that agents update their strategies by imitating better performing neighbors, even though imitation lacks rationality when interactions are far from all-to-all. Indeed, imitation did not emerge in experiments. What did emerge is that humans are conditioned by their own mood and that, when in a cooperative mood, they reciprocate cooperation. To help resolve the controversy, we design a model in which we rationally confront the two main behaviors emerging from experiments - reciprocal cooperation and unconditional defection - in a networked prisoner's dilemma. Rationality is introduced by means of a predictive rule for strategy update and is bounded by the assumed model society. We show that both reciprocity and a multi-step predictive horizon are necessary to stabilize cooperation, and sufficient for its fixation, provided the game benefit-to-cost ratio is larger than a measure of network connectivity. We hence rediscover the rule of network reciprocity, underpinned however by a different evolutionary mechanism.

C. Piccardi and L. Tajoli, Complexity, centralization, and fragility in economic networks, PLoS One, 13(11), e0208265, 2018. [doi]

Trade networks, across which countries distribute their products, are crucial components of the globalized world economy. Their structure affects the mechanism of propagation of shocks from country to country, as observed in a very sharp way in the past decade, characterized by economic uncertainty in many parts of the world. Such trade structures are strongly heterogeneous across products, given the different features of the countries which buy and sell goods. By using a diversified pool of indicators from network science and product complexity theory, we quantitatively demonstrate that, overall, products with higher complexity - i.e., with larger technological content and/or number of components - are traded through more centralized networks - i.e., with a smaller number of countries concentrating most of the export flow. Since centralized networks are known to be more vulnerable, we argue that the current composition of production and trading is associated to high fragility at the level of the most complex - thus strategic - products.

M. Bastos, C. Piccardi, M. Levy, N. McRoberts, and M. Lubell, Core-periphery or decentralized? Topological shifts of specialized information on Twitter, Social Networks, 52, 282-293, 2018. [doi]

In this paper we investigate shifts in Twitter network topology resulting from the type of information being shared. We identified communities matching areas of agricultural expertise and measured the core-periphery centralization of network formations resulting from users sharing generic versus specialized information. We found that centralization increases when specialized information is shared and that the network adopts decentralized formations as conversations become more generic. The results are consistent with classical diffusion models positing that specialized information comes with greater centralization, but they also show that users favor decentralized formations, which can foster community cohesion, when spreading specialized information is secondary.

C. Piccardi, M. Riccaboni, L. Tajoli, and Zhen Zhu, Random walks on the world input-output network, Journal of Complex Networks, 6, 187-205, 2018. [doi]

Modern production is increasingly fragmented across countries.To disentangle the world production system at sector level, we use the World Input-Output Database to construct the World Input-Output Network (WION) where the nodes are the individual sectors in different countries and the edges are the transactions between them. In order to explore the features and dynamics of the WION, in this article we detect the communities in the WION and evaluate their significance using a random walk Markov chain approach. Our results contribute to the recent stream of literature analysing the role of global value chains in economic integration across countries, by showing global value chains as endogenously emerging communities in the world production system, and discussing how different perspectives produce different results in terms of the pattern of integration.

F. Calderoni, D. Brunetto, and C. Piccardi, Communities in criminal networks: A case study, Social Networks, 48, 116-125, 2017. [doi]

Criminal organizations tend to be clustered to reduce risks of detection and information leaks. Yet, the literature exploring the relevance of subgroups for their internal structure is so far very limited. The paper applies methods of community analysis to explore the structure of a criminal network representing the individuals' co-participation in meetings. It draws from a case study on a large law enforcement operation (``Operazione Infinito'') tackling the 'Ndrangheta, a mafia organization from Calabria, a southern Italian region. The results show that the network is indeed clustered and that communities are associated, in a non trivial way, with the internal organization of the 'Ndrangheta into different ``locali'' (similar to mafia families). Furthermore, the results of community analysis can improve the prediction of the ``locale'' membership of the criminals (up to two thirds of any random sample of nodes) and the leadership roles (above 90% precision in classifying nodes as either bosses or non-bosses). The implications of these findings on the interpretation of the structure and functioning of the criminal network are discussed.

G. Berlusconi, F. Calderoni, N. Parolini, M. Verani, and C. Piccardi, Link prediction in criminal networks: A tool for criminal intelligence analysis, PLoS ONE, 11(4): e0154244, 2016. [doi]

The problem of link prediction has recently received increasing attention from scholars in network science. In social network analysis, one of its aims is to recover missing links, namely connections among actors which are likely to exist but have not been reported because data are incomplete or subject to various types of uncertainty. In the field of criminal investigations, problems of incomplete information are encountered almost by definition, given the obvious anti-detection strategies set up by criminals and the limited investigative resources. In this paper, we work on a specific dataset obtained from a real investigation, and we propose a strategy to identify missing links in a criminal network on the basis of the topological analysis of the links classified as marginal, i.e. removed during the investigation procedure. The main assumption is that missing links should have opposite features with respect to marginal ones. Measures of node similarity turn out to provide the best characterization in this sense. The inspection of the judicial source documents confirms that the predicted links, in most instances, do relate actors with large likelihood of co-participation in illicit activities.

I. Cingolani, C. Piccardi, and L. Tajoli, Discovering preferential patterns in sectoral trade networks, PLoS ONE, 10(10), e0140951, 2015. [doi]

We analyze the patterns of import/export bilateral relations, with the aim of assessing the relevance and shape of "preferentiality" in countries' trade decisions. Preferentiality here is defined as the tendency to concentrate trade on one or few partners. With this purpose, we adopt a systemic approach through the use of the tools of complex network analysis. In particular, we apply a pattern detection
approach based on community and pseudocommunity analysis, in order to highlight the groups of countries within which most of members' trade occur. The method is applied to two intra-industry trade networks consisting of 221 countries, relative to the low-tech "Textiles and Textile Articles" and the high-tech "Electronics" sectors for the year 2006, to look at the structure of world trade before the start of the international financial crisis. It turns out that the two networks display some similarities and some differences in preferential trade patterns: they both include few significant communities that define narrow sets of countries trading with each other as preferential destinations markets or supply sources, and they are characterized by the presence of similar hierarchical structures, led by the largest economies. But there are also distinctive features due to the characteristics of the industries examined, in which the organization of production and the destination markets are different. Overall, the extent of preferentiality and partner selection at the sector level confirm the relevance of international trade costs still today, inducing countries to seek the highest efficiency in their trade patterns.

C. Piccardi, A. Colombo, and R. Casagrandi, Connectivity interplays with age in shaping contagion over networks with vital dynamics, Physical Review E, 91(2), 022809, 2015. [doi]

The effects of network topology on the emergence and persistence of infectious diseases have been broadly explored in recent years. However, the influence of the vital dynamics of the hosts (i.e., birth-death processes) on the network structure, and their effects on the pattern of epidemics, have received less attention in the scientific community. Here, we study Susceptible-Infected-Recovered(-Susceptible) [SIR(S)] contact processes in standard networks (of Erdos-Renyi and Barabasi-Albert type) that are subject to host demography. Accounting for the vital dynamics of hosts is far from trivial, and it causes the scale-free networks to lose their characteristic fat-tailed degree distribution.We introduce a broad class of models that integrate the birth and death of individuals (nodes) with the simplest mechanisms of infection and recovery, thus generating age-degree structured networks of hosts that interact in a complex manner. In our models, the epidemiological state of each individualmay depend both on the number of contacts (which changes through time because of the birth-death process) and on its age, paving the way for a possible age-dependent description of contagion and recovery processes.We study how the proportion of infected individuals scales with the number of contacts among them. Rather unexpectedly, we discover that the result of highly connected individuals at the highest risk of infection is not as general as commonly believed. In infections that confer permanent immunity to individuals of vital populations (SIR processes), the nodes that are most likely to be infected are those with intermediate degrees. Our age-degree structured models allow such findings to be deeply analyzed and interpreted, and they may aid in the development of effective prevention policies.

F. Della Rossa, M. Gobbi, G. Mastinu, C. Piccardi, and G. Previati, Bifurcation analysis of a car and driver model, Vehicle System Dynamics, 52, 142-156, 2014. [doi]

The bifurcation analysis of a simple mathematical model describing a road vehicle with a driver is presented. The mechanical model of the car has two degrees of freedom and the related equations of motion contain the nonlinear tyre characteristics. The driver is described by a well-known model proposed in the literature. The road vehicle model has been validated in a case study. Bifurcation analysis is adopted as the proper procedure for analysing both steady-state cornering and straight ahead motion at different speeds. The importance of properly computing steady-state equilibria is highlighted. The effect of a skilled driver is to broaden the basin of attraction of stable equilibria and, in some cases, to stabilise originally unstable behaviours.Asubcritical Hopf bifurcation is normally found which limits the forward speed of either understeering or oversteering vehicles. A three-parameter bifurcation analysis is performed to understand the influence on stability of driver gain, of driver prediction time, of vehicle speed. It turns out, as expected from practice, that an oversteering vehicle is more challenging to be controlled than an understeering one. The paper proposes an insight into vehicle-driver interaction. The stabilising or de-stabilising effect of the driver is ultimately explained referring to the existence of a Hopf bifurcation.

P. Landi and C. Piccardi, Community analysis in directed networks: In-, out-, and pseudo-communities, Physical Review E, 89(1), 012814, 2014. [doi]

When analyzing important classes of complex interconnected systems, link directionality can hardly be
neglected if a precise and effective picture of the structure and function of the system is needed. If community
analysis is performed, the notion of "community" itself is called into question, since the property of having a comparatively looser external connectivity could refer to the inbound or outbound links only or to both categories. In this paper, we introduce the notions of in-, out-, and in-/out-community in order to correctly classify the directedness of the interaction of a subnetwork with the rest of the system. Furthermore, we extend the scope of community analysis by introducing the notions of in-, out-, and in-/out-pseudocommunity. They are subnetworks having strong internal connectivity but also important interactions with the rest of the system, the latter taking place by means of a minority of its nodes only. The various types of (pseudo-)communities are qualified and distinguished by a suitable set of indicators and, on a given network, they can be discovered by using a "local" searching algorithm. The application to a broad set of benchmark networks and real-world examples proves that the proposed approach is able to effectively disclose the different types of structures above defined and to usefully classify the directionality of their interactions with the rest of the system.

Matlab code and data used in the paper are available here.

F. Della Rossa, F. Dercole, and C. Piccardi, Profiling core-periphery network structure by random walkers, Scientific Reports, 3, 1467, 2013. [doi]

Disclosing the main features of the structure of a network is crucial to understand a number of static and dynamic properties, such as robustness to failures, spreading dynamics, or collective behaviours. Among the possible characterizations, the core-periphery paradigm models the network as the union of a dense core with a sparsely connected periphery, highlighting the role of each node on the basis of its topological position. Here we show that the core-periphery structure can effectively be profiled by elaborating the behaviour of a random walker. A curve - the core-periphery profile - and a numerical indicator are derived, providing a global topological portrait. Simultaneously, a coreness value is attributed to each node, qualifying its position and role. The application to social, technological, economical, and biological
networks reveals the power of this technique in disclosing the overall network structure and the peculiar role
of some specific nodes.

Matlab code and data used in the paper are available here.

C. Piccardi and L. Tajoli, Existence and significance of communities in the World Trade Web, Physical Review E, 85(6), 066119, 2012. [doi]

TheWorld TradeWeb (WTW), which models the international transactions among countries, is a fundamental tool for studying the economics of trade flows, their evolution over time, and their implications for a number of phenomena, including the propagation of economic shocks among countries. In this respect, the possible existence of communities is a key point, because it would imply that countries are organized in groups of preferential partners. In this paper, we use four approaches to analyze communities in the WTW between 1962 and 2008, based, respectively, on modularity optimization, cluster analysis, stability functions, and persistence probabilities. Overall, the four methods agree in finding no evidence of significant partitions. A few weak communities emerge from the analysis, but they do not represent secluded groups of countries, as intercommunity linkages are also strong, supporting the view of a truly globalized trading system.

C. Piccardi, Finding and testing network communities by lumped Markov chains, PLoS ONE, 6(11), e27028, 2011. [doi]

Identifying communities (or clusters), namely groups of nodes with comparatively strong internal connectivity, is a
fundamental task for deeply understanding the structure and function of a network. Yet, there is a lack of formal criteria for defining communities and for testing their significance. We propose a sharp definition that is based on a quality threshold. By means of a lumped Markov chain model of a random walker, a quality measure called "persistence probability" is associated to a cluster, which is then defined as an "alpha-community" if such a probability is not smaller than a. Consistently, a partition composed of alpha-communities is an "alpha-partition". These definitions turn out to be very effective for finding and testing communities. If a set of candidate partitions is available, setting the desired a-level allows one to immediately select
the a-partition with the finest decomposition. Simultaneously, the persistence probabilities quantify the quality of each single community. Given its ability in individually assessing each single cluster, this approach can also disclose single well defined communities even in networks that overall do not possess a definite clusterized structure.

Matlab code and data used in the paper are available here.