Comparison of the macroscopic behavior simulated by the approximated and non-approximated stochastic version of the model.

<p>Percentage of nodes in each of the states is displayed for each time step. The left panel shows the comparison results on a lattice network with periodic boundary conditions of 16384 () nodes. The right panel shows the comparison results on a power grid network of 4941 nodes and 13188 links, whose highest degree is 19. The results were produced by averaging over 1000 executions. The markers display the execution results of the approximated version, while the lines display the results of the non-approximated version, i.e. the actual model simulations. For brevity only 40 markers are displayed for each state.</p

Comparison of the macroscopic fixed point values produced by the approximated and non-approximated version of the discrete-time generalization of the model.

<p>The norm of the error vector, whose components are the differences between the macroscopic fixed point values (percentage of nodes in each state) of both versions, is calculated for each combination of the parameters and . and parameters are fixed at and , respectively. Three different graphs are examined with three random initial state assignments of the nodes: a complete and a star graph with 6 nodes, and a lattice graph with periodic boundary conditions of 9 nodes. Our approximated version produces the same fixed points as the non-approximated version, except for the line that depicts the area where there is no clear winner.</p

State diagram of a node for the model.

<p>The diagram shows the dynamics of a single node. Curvy arrows depict state change due to contact with the neighbors, while less curvy arrows depict spontaneous state change.</p

Modeling the Spread of Multiple Concurrent Contagions on Networks

<div><p>Many contagions spread over various types of communication networks and their spreading dynamics have been extensively studied in the literature. Here we propose a general model for the concurrent spread of an arbitrary number of contagions in complex networks. The model is stochastic and runs in discrete time, and includes two widely used mechanisms by which a node can change its state. The first, termed the spontaneous state change mechanism, describes spontaneous transition to another state, while the second, termed the contact-induced state change mechanism, describes acquiring other contagions due to contact with the neighbors. We consider reactive discrete-time spreading processes of multiple concurrent contagions where time steps are of finite size without neglecting the possibility of multiple infecting events in a single time step. An essential element for making the model numerically tractable is the use of an approximation for the probability that a node transits to a specific state given any set of neighboring states. Different transmission probabilities may be present between each pair of states. We also derive corresponding continuous–time equations that are simple and intuitive. The model includes many well-known epidemic and rumor spreading models as a special case and it naturally captures spreading processes in multiplex networks.</p></div

A multiplex network.

<p>Different link types correspond to different layers in the multiplex network. We assume that transmission probabilities depend on the link type, i.e. each contagion or state propagates differently over each layer. This is depicted by coloring the contact-induced transition mechanism links differently for each separate layer. In the adaptation of the model for multiplex networks both contagions spread only on their respective layers. Hence, we have and .</p

An illustration of the two mechanisms of state change of a node.

<p>The number of different states that exist in the network is . Solid colored arrows indicate successful state transmissions, i.e. infectious links, and dashed lines indicate an unsuccessful state transmission. The probabilities of the realized transmission events are depicted next to each line. Solid gray lines indicate that the nodes have not been in contact at the given time step; a spontaneous transition has taken place instead. From top to bottom panel, descriptions go as follows. Panel 1: node changes its state spontaneously to state 2 after previously having been in state 1. The probability of state change with this mechanism is . Panel 2: Node does not make a spontaneous transition, and changes its state as a result of getting infected with state 3 from its neighbors. Note that a neighbor in state 2 also makes successful transmission, however, node chooses state 3 transmitted from one of the other two successful transmissions. The probability of state change with this mechanism is , where . Panel 3: node changes its state as a result of getting infected with state 4 from its neighbors, a contact which stimulates it to adopt state 2. The probability of state change with this mechanism is , where and . Panel 4: node maintains its state since none of the two mechanisms of state change caused it to make a transition. The probability of this event is the product of the probability that no spontaneous transition occurs and no state is transmitted upon contact with the neighbors , where .</p

Snapshots of one sample execution of the model on a lattice network with periodic boundary conditions of 65536 () nodes.

<p>The snapshots were taken at different time steps, as indicated below each individual snapshot. Cyclic-like behavior is clearly seen.</p

State diagram of a node for the altered model that operates only with the contact-based mechanism.

<p>The diagram shows the dynamics of a single node. Curvy arrows depict state change due to contact with the neighbors.</p

State diagram for the SIS model.

<p>The diagram shows the dynamics of a single node. A susceptible node can become infected by contacting its infected neighbors, with probability . On the other hand, infected nodes spontaneously recover with probability , and become susceptible again. The contact-induced transition mechanism is represented by a curvy arrow, whereas the spontaneous transition mechanism is represented by a less curvy arrow.</p

State diagram for the Maki-Thompson model of rumor spreading.

<p>The diagram shows the dynamics of a single node. An ignorant node can become a spreader by contacting its neighbors that spread the rumor, with rate . On the other hand, spreader nodes become stiflers by contacting other spreaders or stiflers with rate , or by spontaneously transitioning to the stifler state with rate . The contact-induced transition mechanism is represented by a curvy arrow, whereas the spontaneous transition mechanism is represented by a less curvy arrow. The dashed arrow denotes that , i.e. spreader becomes stifler by contacting other spreaders with rate .</p