Critical Phenomena In Networks

We study complex contagion on random networks with high clustering. We focus on the homogeneous contagion rule, in which a node is infected if at least two neighbours become infected. This case is representative of higher order interactions, and demonstrates the significant effect of clustering on the contagion. Since complex contagion is frequently used to model social adoption, and social networks are characterised by high clustering, this is an important consideration.In locally treelike networks, this process corresponds to bootstrap percolation. In such networks one observes the continuous appearance of a giant component of infected nodes, which may then be followed by a second discontinuous hybrid transition with a large jump in contagion. However the initial infection required may be large. However, when there are triangles present in the network which share an edge, the contagion spreads directly between them. Clusters of triangles connected in this way thus facilitate the spread of the contagion.We explored this complex contagion on random highly clustered networks, generated using the fSTC random network model. This model closes triads at random in a backbone network (which can be chosen to be locally tree-like). Networks generated in this way contain small motifs in similar distributions to real-world networks, and in particular contain clusters of edge-adjacent triangles. The use of the fSTC random network model allows a theoretical treatment, so we can calculate the size of the contagion, the giant component and the critical thresholds using self-consistency equation techniques.While the critical phenomena remain qualitatively the same, we find the presence of clustering may move the network from a region without a discontinuous transition to one where it is present. Most importantly, with significant clustering, a large contagion may arise from a significantly smaller initial infection than would occur in a comparable non-clustered network.

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Standard percolation theory assumes that two active nodes belong to the same cluster if there is a path of adjacent active nodes between them. More general definitions are possible, with percolation clusters not necessarily coinciding with topologically connected components. In Extended-Range Percolation the path between two nodes in the same cluster may contain up to R-1 inactive consecutive nodes.A generating function approach allows to solve exactly site percolation on random locally treelike networks for small values of R, finding that not only the percolation threshold depends on R, but that, on strongly heterogeneous networks, also critical exponents are R-dependent. The combinatorial complexity of the approach increases rapidly, because two active nodes may be in the same cluster even if they are at distance larger than R, due to the presence of other active nodes, acting as ``bridges'', located off the shortest path. This makes the approach unsuitable for large R.The problem is solved by a message-passing approach which, while reproducing the results already found for small values of R, allows for a straightforward generalization to any value of R, taming the rapidly increasing combinatorial complexity of the previous approach. The same message passing framework allows to solve Extended-Range Percolation on more general substrates. The application to multilayer networks leads to a rich phenomenology already on homogeneous networks, due to the interplay between the extended range, which tends to increaserobustness (it reduces the percolation threshold) and interdependence, which tends to decrease robustness (it increases the percolation threshold). In particular, if active nodes are interdependent but inactive nodes are not, Extended-Range percolation in multilayer networks exhibits a phase-diagram characterized by two discontinuous hybrid transitions with a reentrant nonpercolating phase.

Self-organized criticality has often been depicted as a one-dimensional process where one parameter is tuned to its critical value. This can severely restrict the range of possible behaviors that an adaptive system can support. However, the number of parameters is vast in most real-world systems such as the human brain, where self-organized criticality has been suggested to play a role. Therefore, the states of the system corresponding to criticality can be expected to form a manifold in a high-dimensional parameter space. Hence, an adaptive system may move along this critical manifold and explore its parameter space without compromising the benefits of criticality.In our work (PRL 130, 188401), we demonstrated for the first time the phenomenon of self-organized critical drift, where simple adaptive rules drive a self-organizing system to drift on the critical manifold. We illustrated the phenomenon in a neuro-inspired adaptive network, where the network topology is shaped by the dynamics taking place on the network. Our results show that the criticality hypothesis is more flexible than previously thought – a system can remain at criticality while its structure continues to change.After this initial discovery, the next step is to explore further the properties of these manifolds. As an adaptive network moves along the manifold, it may drift to a dynamically different regime, and if the critical manifolds associated with different phase transitions intersect, the system may become critical in multiple different ways at once. In our current work, we study an adaptive network self-organizing to two different phase transitions (Hopf and Turing) and analyze how the system properties change as the system drifts along these intersecting manifolds. Overall, our work provides a novel perspective on how self-organizing systems move between different topological and dynamical regimes while remaining critical, effectively resolving many previous challenges in the field.

We examine bond percolation in a dynamic network model for the evolution of melt ponds on sea ice, where catchments for ponds are nodes and water fluxes between these are edges. Fluxes along edges can change between states of no flow, uni-directional flow, or bi-directional flow. Three different types of dynamic bond percolation are identified, and, when analysed, all are found to be examples of anomalous percolation, with calculated percolation thresholds that deviate substantially from those of random percolation. One occurs during initial formation of ponds and flooding of the floe. Another occurs during drainage, and a final one can occur post drainage as parts of the ice surface are submerged below sea level. Anomalous percolation occurs in each because the choices of edge changes are not independent. Analysis of how independence is broken helps to develop an understanding of the behaviour of the model and of pond systems. We find that ponds that overflow into others before joining together help to spread water across the floe and delay percolation. We find that percolation during drainage is strongly dependent on the network topology, displaying hysteresis and a delayed breakup of the network for lower levels of connectivity compared to classical percolation, but hastened break-up for an underlying network with higher connectivity. Our approach demonstrates how anomalous percolation can be used not just as a descriptor, but as an analysis tool, providing insight into the mathematics of processes on network models, and into the physics of the systems that these models represent.