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We investigate the electrical current and flow (number of parallel paths) between two sets of n sources and n sinks in complex networks. We derive analytical formulas for the average current and flow as a function of n. We show that for small n, increasing n improves the total transport in the network, while for large n bottlenecks begin to form. For the case of flow, this leads to an optimal n* above which the transport is less efficient. For current, the typical decrease in the length of the connecting paths for large n compensates for the effect of the bottlenecks. We also derive an expression for the average flow as a function of n under the common limitation that transport takes place between specific pairs of sources and sinks.
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In systems of multiple agents, identifying the cause of observed agent dynamics is challenging. Often, these agents operate in diverse, non-stationary environments, where models rely on hand-crafted environment-specific features to infer influential regions in the systems surroundings. To overcome the limitations of these inflexible models, we present GP-LAPLACE, a technique for locating sources and sinks from trajectories in time-varying fields. Using Gaussian processes, we jointly infer a spatio-temporal vector field, as well as canonical vector calculus operations on that field. Notably, we do this from only agent trajectories without requiring knowledge of the environment, and also obtain a metric for denoting the significance of inferred causal features in the environment by exploiting our probabilistic method. To evaluate our approach, we apply it to both synthetic and real-world GPS data, demonstrating the applicability of our technique in the presence of multiple agents, as well as its superiority over existing methods.
Identifying the infection sources in a network, including the index cases that introduce a contagious disease into a population network, the servers that inject a computer virus into a computer network, or the individuals who started a rumor in a social network, plays a critical role in limiting the damage caused by the infection through timely quarantine of the sources. We consider the problem of estimating the infection sources and the infection regions (subsets of nodes infected by each source) in a network, based only on knowledge of which nodes are infected and their connections, and when the number of sources is unknown a priori. We derive estimators for the infection sources and their infection regions based on approximations of the infection sequences count. We prove that if there are at most two infection sources in a geometric tree, our estimator identifies the true source or sources with probability going to one as the number of infected nodes increases. When there are more than two infection sources, and when the maximum possible number of infection sources is known, we propose an algorithm with quadratic complexity to estimate the actual number and identities of the infection sources. Simulations on various kinds of networks, including tree networks, small-world networks and real world power grid networks, and tests on two real data sets are provided to verify the performance of our estimators.
We investigate the validity of the non-Hermitian Hamiltonian approach in describing quantum transport in disordered tight-binding networks connected to external environments, acting as sinks. Usually, non-Hermitian terms are added, on a phenomenological basis, to such networks to summarize the effects of the coupling to the sinks. Here we consider a paradigmatic model of open quantum network for which we derive a non-Hermitian effective model, discussing its limit of validity by a comparison with the analysis of the full Hermitian model. Specifically, we consider a ring of sites connected to a central one-dimensional lead. The lead acts as a sink which absorbs the excitation initially present in the ring. The coupling strength to the lead controls the opening of the ring system. This model has been widely discussed in literature in the context of light-harvesting systems. We analyze the effectiveness of the non-Hermitian description both in absence and in presence of static disorder on the ring. In both cases, the non-Hermitian model is valid when the energy range determined by the eigenvalues of the ring Hamiltonian is smaller than the energy band in the lead. Under such condition, we show that results about the interplay of opening and disorder, previously obtained within the non-Hermitian Hamiltonian approach, remain valid when the full Hermitian model in presence of disorder is considered. The results of our analysis can be extended to generic networks with sinks, leading to the conclusion that the non-Hermitian approach is valid when the energy dependence of the coupling to the external environments is sufficiently smooth in the energy range spanned by the eigenstates of the network.
Extreme events are emergent phenomena in multi-particle transport processes on complex networks. In practice, such events could range from power blackouts to call drops in cellular networks to traffic congestion on roads. All the earlier studies of extreme events on complex networks have focused only on the nodal events. If random walks are used to model transport process on a network, it is known that degree of the nodes determines the extreme event properties. In contrast, in this work, it is shown that extreme events on the edges display a distinct set of properties from that of the nodes. It is analytically shown that the probability for the occurrence of extreme events on an edge is independent of the degree of the nodes linked by the edge and is dependent only on the total number of edges on the network and the number of walkers on it. Further, it is also demonstrated that non-trivial correlations can exist between the extreme events on the nodes and the edges. These results are in agreement with the numerical simulations on synthetic and real-life networks.
We prove that the Abelian sandpile model on a random binary and binomial tree, as introduced in cite{rrs}, is not critical for all branching probabilities $p<1$; by estimating the tail of the annealed survival time of a random walk on the binary tree with randomly placed traps, we obtain some more information about the exponential tail of the avalanche radius. Next we study the sandpile model on $mathbb{Z}^d$ with some additional dissipative sites: we provide examples and sufficient conditions for non-criticality; we also make a connection with the parabolic Anderson model. Finally we initiate the study of the sandpile model with both sources and sinks and give a sufficient condition for non-criticality in the presence of a finite number of sources, using a connection with the homogeneous pinning model.
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