Competition between intra-community and inter-community synchronization and relevance in brain cortical networks

In this paper the effects of inter-community links on the synchronization performance of community networks, especially on the competition between individual community and the whole network, are studied in detail. The study is organized from two aspects: the number or portion of inter-community links and the connection strategy of inter-community links between different communities. A critical point is found in the competition of global network and individual communities. Increasing the number of inter-community links will enhance the global synchronizability but degrade the synchronization performance of individual community before this point. After that the individual community will synchronize better and better as part of the whole network because the community structure is not so prominent. The critical point represents a balance region where the individual community is maximally independent while the information transmission remains effective between different communities. Among various connection strategies, connecting nodes belonging to different communities randomly rather than connecting nodes with larger degrees are the most efficient way to enhance global synchronization of the network. However, the dynamical modularity is the reverse case. A preferential connection scheme linking most of the hubs from the communities will allow rather efficient global synchronization while maintaining strong dynamical clustering of the communities. Interestingly, the observations are found to be relevant in a realistic network of cat cortex. The synchronization state is just at the critical point, which shows a reasonable combination of segregated function in individual communities and coordination among them. Our work sheds light on principles underlying the emergence of modular architectures in real network systems and provides guidance for the manipulation of synchronization in community networks.

Figure 1

The synchronization performance of global network $C_N$ (a), of individual community $C_C$ (b), and dynamical modularity $Q_D$ (c) as functions of the coupling strength $\sigma$ at different topological modularity $Q_T$. (d) $C_N$, $C_C$, and $Q_D$ at very small coupling strength for $Q_T=0.734$. There are four communities in the network; each one is an ER random network with 100 nodes, and the average degree in each community is 16. Each plot is obtained after averaging over 50 network configurations and 10 initial states of each configuration.

Figure 2

The change of state $x$ with time $t$ at coupling strength (a) 0.2, (b) 0.3, and (c) 0.6. The two bold curves and two regular curves present nodes belonging to the two different communities separately; the two solid curves represent nodes connected by inter-community link, and the two dashed curves present two nodes that are connected neither to each other nor to the other two nodes. (d) Change of $d_E$ (square line) and $d_I$ (circle line) with the coupling strength. The topological modularity of the adopted network is 0.746, and the other network parameters are the same as in Fig. 1.

Figure 3

The change of $C_N$ (a), $C_C$ (b), and $Q_D$ (c) with $Q_T$ at various coupling strength $\sigma$ as indicated. The other network parameters are the same as in Fig. 1.

Figure 4

The change of $C_C$, $C_N$, and $Q_D$ with $\sigma$ at $L_E=$ 64 ($Q_T=0.46$) and 196 ($Q_T=0.38$) for different rewiring strategies. Parameter $\alpha =4.0$ for preferential connection. There are two communities in the network, each one is a BA network with 100 nodes, and the average degree is 16. Each plot is obtained after the averaging of 50 network configurations and 10 initial states of each configuration.

Figure 5

The change of $C_C$, $C_N$, and $Q_D$ with $\sigma$ for different rewiring strategies. The other network parameters are the same as in Fig. 4.

Figure 6

(a) The cortical network of cats. A node represents a functional region of the cortex, and a link represents the existence of fiber projection between two regions. The different symbols represent different connection weight: 1 ($\circ$ sparse), 2 ($•$ intermediate), and 3 ($*$ dense). The communities and functional subdivision V, A, SM, and FL of the network are indicated by the solid lines. (b) The change of synchronization performance of global network ($C_N$) and of the four communities ($C_V$, $C_A$, $C_{\mathrm{SM}}$, and $C_{\mathrm{FL}}$) vs the coupling strength $\sigma$. Each plot is obtained after the averaging of 10 initial states.

Figure 7

The change of $C_C$ vs $Q_T$ at $\sigma =$ 20.0, 30.0, and 40.0. The stars present the results of the original cat cortical network. Each plot is obtained after averaging over 50 network configurations and 10 initial states of each configuration.