Blind and myopic ants in heterogeneous networks

Universal cover-time distribution of heterogeneous random walks

The cover-time problem, i.e., the time to visit every site in a system, is one of the key issues of random walks with wide applications in natural, social, and engineered systems. Addressing the full distribution of cover times for random walk on complex structures has been a long-standing challenge and has attracted persistent efforts. Usually it is assumed that the random walk is noncompact, to facilitate theoretical treatments by neglecting the correlations between visits. The known results are essentially limited to noncompact and homogeneous systems, where different sites are on an equal footing and have identical or close mean first-passage times, such as random walks on a torus. In contrast, realistic random walks are prevailingly heterogeneous with diversified mean first-passage times. Does a universal distribution still exist? Here, by considering the most general situations of noncompact random walks, we uncover a generalized rescaling relation for the cover time, exploiting the diversified mean first-passage times that have not been accounted for before. This allows us to concretely establish a universal distribution of the rescaled cover times for heterogeneous noncompact random walks, which turns out to be the Gumbel universality class that is ubiquitous for a large family of extreme value statistics. Our analysis is based on the transfer matrix framework, which is generic in that, besides heterogeneity, it is also robust against biased protocols, directed links, and self-connecting loops. The finding is corroborated with extensive numerical simulations of diverse heterogeneous noncompact random walks on both model and realistic topological structures. Our technical ingredient may be exploited for other extreme value or ergodicity problems with nonidentical distributions.

Fluctuations in complex networks with variable dimensionality and heterogeneity

Synchronizing individual activities is essential for the stable functioning of diverse complex systems. Understanding the relation between dynamic fluctuations and the connection topology of substrates is therefore important, but it remains restricted to regular lattices. Here we investigate the fluctuation of loads, assigned to the locally least-loaded nodes, in the largest-connected components of heterogeneous networks while varying their link density and degree exponents. The load fluctuation becomes finite when the link density exceeds a finite threshold in weakly heterogeneous substrates, which coincides with the spectral dimension becoming larger than 2 as in the linear diffusion model. The fluctuation, however, diverges also in strongly heterogeneous networks with the spectral dimension larger than 2. This anomalous divergence is shown to be driven by large local fluctuations at hubs and their neighbors, scaling linearly with degree, which can give rise to diverging fluctuations at small-degree nodes. Our analysis framework can be useful for understanding and controlling fluctuations in real-world systems.