Detection of patient-zero can give new insights to the epidemiologists about the nature of first transmissions into a population. In this paper, we study the statistical inference problem of detecting the source of epidemics from a snapshot of spreading on an arbitrary network structure. By using exact analytic calculations and Monte Carlo estimators, we demonstrate the detectability limits for the SIR model, which primarily depend on the spreading process characteristics. Finally, we demonstrate the applicability of the approach in a case of a simulated sexually transmitted infection spreading over an empirical temporal network of sexual interactions.

In the study of disease spreading on empirical complex networks in SIR model, initially infected nodes can be ranked according to some measure of their epidemic impact. The highest ranked nodes, also referred to as "superspreaders", are associated to dominant epidemic risks and therefore deserve special attention. In simulations on studied empirical complex networks, it is shown that the ranking depends on the dynamical regime of the disease spreading. A possible mechanism leading to this dependence is illustrated in an analytically tractable example. In systems where the allocation of resources to counter disease spreading to individual nodes is based on their ranking, the dynamical regime of disease spreading is frequently not known before the outbreak of the disease. Therefore, we introduce a quantity called epidemic centrality as an average over all relevant regimes of disease spreading as a basis of the ranking. A recently introduced concept of phase diagram of epidemic spreading is used as a framework in which several types of averaging are studied. The epidemic centrality is compared to structural properties of nodes such as node degree, k-cores and betweenness. There is a growing trend of epidemic centrality with degree and k-cores values, but the variation of epidemic centrality is much smaller than the variation of degree or k-cores value. It is found that the epidemic centrality of the structurally peripheral nodes is of the same order of magnitude as the epidemic centrality of the structurally central nodes. The implications of these findings for the distributions of resources to counter disease spreading are discussed. Author SummaryStudies of disease spreading on complex networks have provided a deep insight into the conditions of onset, dynamics and prevention of epidemics in human populations and malicious software propagation in computer networks. Identifying nodes which, when initially infected, on average infect the largest part of the network and ranking them according to their epidemic impact (the portion of the network eventually infected) is a priority for public health policies. In the study of epidemic spreading on empirical complex networks in the Susceptible-Infected-Recovered model, we find that the required ranking depends on the disease spreading regime, i.e. on how fast the disease is transmitted between nodes and how fast the infected node recovers. A measure called epidemic centrality, averaging the epidemic impact over all possible disease spreading regimes, is introduced as a basis of epidemic ranking. We find the epidemic centrality of nodes which are structurally central, to be of the same order of magnitude as the epidemic centrality of structurally peripheral nodes. These findings point to the need to study if the impact of an epidemic starting at structurally peripheral nodes might be considerably underestimated. Network periphery should gain a more prominent role in the study of the allocation of resources in future epidemic preparedness plans.

The epidemic spreading on arbitrary complex networks is studied in SIR (Susceptible Infected Recovered) compartment model. We propose our implementation of a Naive SIR algorithm for epidemic simulation spreading on networks that uses data structures efficiently to reduce running time. The Naive SIR algorithm models full epidemic dynamics and can be easily upgraded to parallel version. We also propose novel algorithm for epidemic simulation spreading on networks called the FastSIR algorithm that has better average case running time than the Naive SIR algorithm. The FastSIR algorithm uses novel approach to reduce average case running time by constant factor by using probability distributions of the number of infected nodes. Moreover, the FastSIR algorithm does not follow epidemic dynamics in time, but still captures all infection transfers. Furthermore, we also propose an efficient recursive method for calculating probability distributions of the number of infected nodes. Average case running time of both algorithms has also been derived and experimental analysis was made on five different empirical complex networks.

The disease spreading on complex networks is studied in SIR model. Simulations on empirical complex networks reveal two specific regimes of disease spreading: local containment and epidemic outbreak. The variables measuring the extent of disease spreading are in general characterized by a bimodal probability distribution. Phase diagrams of disease spreading for empirical complex networks are introduced. A theoretical model of disease spreading on m-ary tree is investigated both analytically and in simulations. It is shown that the model reproduces qualitative features of phase diagrams of disease spreading observed in empirical complex networks. The role of tree-like structure of complex networks in disease spreading is discussed.

In this paper we introduce a statistical inference framework for estimating the contagion source from a partially observed contagion spreading process on an arbitrary network structure. The framework is based on a maximum likelihood estimation of a partial epidemic realization and involves large scale simulation of contagion spreading processes from the set of potential source locations. We present a number of different likelihood estimators that are used to determine the conditional probabilities associated to observing partial epidemic realization with particular source location candidates. This statistical inference framework is also applicable for arbitrary compartment contagion spreading processes on networks. We compare estimation accuracy of these approaches in a number of computational experiments performed with the SIR (susceptible-infected-recovered), SI (susceptible-infected) and ISS (ignorant-spreading-stifler) contagion spreading models on synthetic and real-world complex networks.The structure of vast majority of biological networks (biochemical, ecological), technological networks (internet, transportation, power grids), social networks and information networks (citation, WWW) can be represented by complex networks [16], [7], [3]. Epidemic or contagion processes are amongst the most prevalent type of dynamic processes of interest characteristic for these real-life complex networks and they include disease epidemics, computer virus spreading, information and rumor propagation [23]. Different mathematical frameworks have been used to study epidemic spreading. We can divide them into two major categories based upon assumptions they make: the homogeneous mixing framework and the heterogeneous mixing framework. The homogeneous mixing framework assumes that all individuals in a population have an equal probability of contact. This is a traditional mathematical framework [12], [10] in which differential equations are used to model epidemic dynamics. The heterogeneous mixing framework assumes * Corresponding author. Adress: Rudjer Bošković Institute,

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We propose a novel statistical node embedding of directed graphs, which is based on a global minimization of pairwise relative entropy and graph geodesics in a non-linear way. Each node is encoded with a probability density function over a measurable real n-dimensional space. Furthermore, we analyze the connection to the geometrical properties of such embedding and characterize the curvature of the statistical manifolds. Extensive experiments show that our proposed embedding is better preserving the global geodesic information of graphs, as well as outperforming existing embedding models on directed graphs in a variety of evaluation metrics, in an unsupervised setting.Preprint. Under review.

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