The critical value of <i>p</i>_<i>c</i> in correlated bimodal networks of coupled oscillators.

<p>The system size is <i>N</i> = 3000. (a)-(c) The results for networks where <i>k</i>₁ = 600, <i>k</i>₂ = 150, <i>N</i>_<i>k</i>₁ = 600, and <i>N</i>_<i>k</i>₂ = 2400: (a) random inactivation; (b) targeted inactivation of high-degree nodes; (c) targeted inactivation of low-degree nodes. (d)-(f) The results similar to (a)-(c), but for another set of networks where <i>k</i>₁ = 450, <i>k</i>₂ = 150, <i>N</i>_<i>k</i>₁ = 900, and <i>N</i>_<i>k</i>₂ = 2100.</p

The histogram of the degrees of the neighboring nodes in the disassortative networks.

<p>The network is given by scale-free network with size <i>N</i> = 1000 and assortativity coefficient <i>r</i> = −0.48. The number of neighboring nodes with indices in the range [100<i>m</i> + 1,100(<i>m</i> + 1) + 1) (<i>m</i> = 1,…,9) is plotted for the nodes with index 800, 900, and 1000. (a) The network obtained by the GER method. (b) The network obtained by the SER method.</p

Adjacency matrices of scale-free networks.

<p>The dots located at (<i>i</i>, <i>j</i>) indicate the presence of the edges between node <i>i</i> and node <i>j</i>. (a) An uncorrelated network with <i>r</i> ≈ 0. (b) An assortative network with <i>r</i> = 0.4, generated by the GER method. (c) The same as (b), but generated by the SER method. (d) A disassortative network with <i>r</i> = −0.48, generated by the GER method. (e) The same as (d), but generated by the SER method.</p

Network reshuffling methods for changing the network assortativity.

<p>(a) The greedy edge-rewiring (GER) method [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0123722#pone.0123722.ref036" target="_blank">36</a>]. The remaining degrees of the two connected node pairs, (<i>j</i>₁, <i>k</i>₁) and (<i>j</i>₂, <i>k</i>₂), are sorted in the descending order and relabeled as <i>l</i>₁, <i>l</i>₂, <i>l</i>₃, and <i>l</i>₄ so that <i>l</i>₁ ≥ <i>l</i>₂ ≥ <i>l</i>₃ ≥ <i>l</i>₄. The size of the node corresponds to its remaining degree. When making the network assortative, Case I is chosen if the current state is Case II or III. When making the network disassortative, Case III is chosen if the current state is Case I or II. (b) The stochastic edge-rewiring (SER) method [<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0123722#pone.0123722.ref008" target="_blank">8</a>]. The acceptance probability for edge rewiring is given by <math><mrow>min<mrow><mo stretchy="true">{</mo><mn>1</mn><mo>,</mo><mrow><mi>E</mi><mo stretchy="false">(</mo><msub><mi>j</mi><mn>1</mn></msub><mo>,</mo><msub><mi>j</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mi>E</mi><mo stretchy="false">(</mo><msub><mi>k</mi><mn>1</mn></msub><mo>,</mo><msub><mi>k</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow><mrow><mi>E</mi><mo stretchy="false">(</mo><msub><mi>j</mi><mn>1</mn></msub><mo>,</mo><msub><mi>k</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mi>E</mi><mo stretchy="false">(</mo><msub><mi>j</mi><mn>2</mn></msub><mo>,</mo><msub><mi>k</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow><mo stretchy="true">}</mo></mrow></mrow></math> where <i>E</i>(<i>j</i>, <i>k</i>) is the joint probability distribution for the remaining degrees of the two nodes in the end of a randomly chosen edge.</p

Dynamical robustness of coupled SL oscillator networks.

<p>(a) Limit-cycle oscillation produced by the single isolated active oscillator. (b) Damping oscillation produced by the single isolated inactive oscillator. (c) The global oscillatory behavior in an uncorrelated network of active and inactive oscillators. The time evolutions of the state variables of some active (red) and inactive (blue) oscillator nodes are plotted. The parameters are set at <i>N</i> = 3000, <i>p</i> = 0.4, and <i>r</i> = 0. (d) The order parameter |<i>Z</i>| plotted against the fraction <i>p</i> of the inactive oscillators in uncorrelated (<i>r</i> = 0), assortative (<i>r</i> = 0.48), and disassortative (<i>r</i> = −0.7) networks with power-law degree distributions. The correlated networks were generated by the GER method.</p

Growth model.

<p>(A) Schematic representation of how cells get connected. At DIV 0, 4 cells of radius are located at random positions. The first iteration of the algorithm, DIV 1, assigns to each cell a disk of radius (green shade). At the next iteration, DIV 2, the disk's growth rate decreases, , and a link between two cells is established when their disks intersect (DIV 3). This process continues until steps. (B) Force diagram explaining cell migration and clustering. Tension forces , , and are acting on the central cluster composed of two cells, whose vector sum (red arrow) exceeds the adhesion to the substrate (green arrow). As a result, a new equilibrium state is produced with new tension forces , , and , being the central cluster pulled in the direction of the net force approaching the largest cluster.</p

Comparison between model and experiment.

<p>Legends in each panel clarifies on the topological quantities measured in experiments (dashed curves), and the corresponding trends of the simulated networks (solid curves). Simulation parameters are the same as in the caption of <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0085828#pone-0085828-g007" target="_blank">Fig. 7</a>, and each point is the ensemble average over 50 independent runs of the growth algorithm.</p

Extraction of the adjacency matrix defining the neural network connectivity.

<p>(A) Image cut taken from a 6 DIV culture and (B) the layer on top showing the identification of neurons and clusters of neurons (red), neurites connecting them (green) and neurite branching points (blue). (C) Mapping of the neuronal network into a graph where blue dots represent the nodes and green lines the links of the graph.</p

Schematic illustration of the network self-organization at three different instants of the automata generations.

<p>(A) DIV 3, (B) DIV 6, and (C) DIV 12. Simulation parameters: , , and .</p

Degree distribution and degree-degree correlation.

<p>(A) Cumulative node degree distributions on a semi-log scale for the state of the same culture at DIVs 3, 6, 7, and 12 (see legend for the symbol coding). Solid lines correspond to the best exponential fitting , with the mean degrees at DIV 3, 6, 7, and 12 respectively. (B) Degree correlation exponent (blue circles) measuring the network assortativity and the corresponding Pearson coefficient (red squares). Both quantities are averaged for the set of 6 cultures at each day of measure (DIV) and error bars represent the sem.</p