Absorbing and shattered fragmentation transitions in multilayer coevolution

Abstract

We introduce a coevolution voter model in a multilayer by coupling a fraction of nodes across two network layers (the degree of multiplexing) and allowing each layer to evolve according to its own topological temporal scale. When these time scales are the same, the time evolution equations can be mapped to a coevolution voter model in a single layer with an effective average degree. Thus the dynamics preserve the absorbing-fragmentation transition at a critical value that increases with the degree of multiplexing. When the two layers have different topological time scales, we find an anomalous transition, named shattered fragmentation, in which the network in one layer splits into two large components in opposite states and a multiplicity of isolated nodes. We identify the growth of the number of components as a signature of this anomalous transition. We also find the critical level of interlayer coupling needed to prevent the fragmentation in a layer connected to a layer that does not fragment.

Dynamical origins of the community structure of an online multi-layer society

“…This is the hallmark of a shattered fragmentation; topologies with at least one giant components and a multiplicity of components of smaller size. Shattered fragmentation was first observed in the CVM in multiplex networks where not all nodes are present in each layer (MCVM) [33] and for the CVM on a single-layer network with noise that targets a fixed subpopulation of nodes [34]. Here, we find the same phenomenology for re-wiring processes that are dominated by triadic closure processes.…”

“…This is an example of shattered fragmentation: topologies with one or two giant components and a multiplicity of components of negligible size. This was first observed in the CVM on a multiplex [29] for incomplete interlayer connectivity q < 1, and in [34] for the CVM on a monoplex with noise. Here we see that a similar effect can be caused by limiting the scope of rewiring.…”

Dynamical origins of the community structure of an online multi-layer society

“…This is the hallmark of a shattered fragmentation; topologies with at least one giant components and a multiplicity of components of smaller size. Shattered fragmentation was first observed in the CVM in multiplex networks where not all nodes are present in each layer (MCVM) [33] and for the CVM on a single-layer network with noise that targets a fixed subpopulation of nodes [34]. Here, we find the same phenomenology for re-wiring processes that are dominated by triadic closure processes.…”

“…This is an example of shattered fragmentation: topologies with one or two giant components and a multiplicity of components of negligible size. This was first observed in the CVM on a multiplex [29] for incomplete interlayer connectivity q < 1, and in [34] for the CVM on a monoplex with noise. Here we see that a similar effect can be caused by limiting the scope of rewiring.…”

Network coevolution drives segregation and enhances Pareto optimal equilibrium selection in coordination games

“…Nonetheless, for intermediate values of p we now have fragmentation in multiple components: first, only small components plus a large connected component and then two big components plus small components. This kind of shattered fragmentation has been observed as well in different versions of a Coevolving Voter Model 23 , 24 .…”

Multilayer coevolution dynamics of the nonlinear voter model

“…More details on the nature of these transitions were discussed elsewhere [36]. For the linear case q=1 [24], we also observe a delay of the fragmentation transition when increasing the degree of multiplexity. We have checked that the delay of the transition is a robust feature for networks with different average degree k 8, 16 á ñ = , as shown in figure 4(a), (b).…”

Dynamical origins of the community structure of an online multi-layer society

“…This is the hallmark of a shattered fragmentation; topologies with at least one giant components and a multiplicity of components of smaller size. Shattered fragmentation was first observed in the CVM in multiplex networks where not all nodes are present in each layer (MCVM) [33] and for the CVM on a single-layer network with noise that targets a fixed subpopulation of nodes [34]. Here, we find the same phenomenology for re-wiring processes that are dominated by triadic closure processes.…”

“…This is an example of shattered fragmentation: topologies with one or two giant components and a multiplicity of components of negligible size. This was first observed in the CVM on a multiplex [29] for incomplete interlayer connectivity q < 1, and in [34] for the CVM on a monoplex with noise. Here we see that a similar effect can be caused by limiting the scope of rewiring.…”

Network coevolution drives segregation and enhances Pareto optimal equilibrium selection in coordination games

“…Nonetheless, for intermediate values of p we now have fragmentation in multiple components: first, only small components plus a large connected component and then two big components plus small components. This kind of shattered fragmentation has been observed as well in different versions of a Coevolving Voter Model 23 , 24 .…”

Multilayer coevolution dynamics of the nonlinear voter model

“…More details on the nature of these transitions were discussed elsewhere [36]. For the linear case q=1 [24], we also observe a delay of the fragmentation transition when increasing the degree of multiplexity. We have checked that the delay of the transition is a robust feature for networks with different average degree k 8, 16 á ñ = , as shown in figure 4(a), (b).…”

Dynamical origins of the community structure of an online multi-layer society

“…This is the hallmark of a shattered fragmentation; topologies with at least one giant components and a multiplicity of components of smaller size. Shattered fragmentation was first observed in the CVM in multiplex networks where not all nodes are present in each layer (MCVM) [33] and for the CVM on a single-layer network with noise that targets a fixed subpopulation of nodes [34]. Here, we find the same phenomenology for re-wiring processes that are dominated by triadic closure processes.…”

“…This is an example of shattered fragmentation: topologies with one or two giant components and a multiplicity of components of negligible size. This was first observed in the CVM on a multiplex [29] for incomplete interlayer connectivity q < 1, and in [34] for the CVM on a monoplex with noise. Here we see that a similar effect can be caused by limiting the scope of rewiring.…”

Network coevolution drives segregation and enhances Pareto optimal equilibrium selection in coordination games

“…Nonetheless, for intermediate values of p we now have fragmentation in multiple components: first, only small components plus a large connected component and then two big components plus small components. This kind of shattered fragmentation has been observed as well in different versions of a Coevolving Voter Model 23 , 24 .…”

Multilayer coevolution dynamics of the nonlinear voter model

“…More details on the nature of these transitions were discussed elsewhere [36]. For the linear case q=1 [24], we also observe a delay of the fragmentation transition when increasing the degree of multiplexity. We have checked that the delay of the transition is a robust feature for networks with different average degree k 8, 16 á ñ = , as shown in figure 4(a), (b).…”