Correlated edge overlaps in multiplex networks

We develop the theory of sparse multiplex networks with partially overlapping links based on their local treelikeness. This theory enables us to find the giant mutually connected component in a two-layer multiplex network with arbitrary correlations between connections of different types. We find that correlations between the overlapping and nonoverlapping links markedly change the phase diagram of the system, leading to multiple hybrid phase transitions. For assortative correlations we observe recurrent hybrid phase transitions.

Figure 1

(a) Graphical representation of the expression for the size of the mutual component $S$, Eq. (1), and (c) of the equations for the five probabilities, Eq. (2), using notations (b) for five probabilities.

Figure 2

Phase diagrams in the $\tilde{c}$ vs. $c$ plane for the uncorrelated and assortatively correlated cases. The appearance of the giant mutually connected component with a discontinuous hybrid transition is shown as the heavy black line. (a) Symmetric uncorrelated joint degree distribution. [(b)–(e)] Assortative correlations of Eq. (5) for $d=1$, $0.25$, $0.1$, and 0, respectively. For comparison, the dashed line shows the boundary between phases in the uncorrelated case. The plots also show the values of the probabilities $x=y$, $u=v$, and $w$ immediately above the discontinuous transition. (f) Multiple transitions of $S$ for $d=0.25$, fixed $\tilde{c}=0.45$, and varying $c$, i.e., along the dotted line of panel (c).

Figure 3

Phase diagrams in the $\tilde{c}$ vs. $c$ plane for the disassortatively correlated casedefined by Eq. (8). The appearance of the giant mutually connected component with a discontinuous hybrid transition is shown as the heavy black line. [(a)–(e)] Disassortative correlations of Eq. (8) for $q_{\text{cut}}=10$, $5.7132$, $5.5$, 5, and 3. For $q_{\text{cut}}<5.7131\cdots$ instead of a single line of transitions there are two branches, and for $q_{\text{cut}}<3$ the right-hand side branch extends to $\tilde{c}=\infty$. The dashed line is the phase boundary in the uncorrelated case. The plots also show the values of the probabilities immediately above the discontinuous transition. (f) Multiple transitions of $S$ for $q_{\text{cut}}=5$, fixed $\tilde{c}=0.56$, and $c$ varying along the dotted line of panel (c).

Figure 4

Phase diagrams in the $\tilde{c}$ vs. $c$ plane for the disassortatively correlated case defined by Eq. (12). The appearance of the giant mutually connected component with a discontinuous hybrid transition is shown as the heavy black line. [(a)–(e)] Disassortative correlations of Eq. (8) with $q_{\text{cut}}=Bc$, for $B=7$, $3.7166$, $3.5$, $1.6$, and $1.497$. For $B<3.7165\cdots$, instead of a single line of transitions, there are two branches, and for $B<1.497\cdots$, the right-hand side branch extends to $\tilde{c}=\infty$. The dot-dashed curve is a line of continuous phase transitions that take place at $e^{-2c}$ for $c<1/B$. The dashed line is the phase boundary in the uncorrelated case. The plots also show the values of the probabilities immediately above the discontinuous transition. (f) Multiple transitions of $S$ for $B=3$, fixed $\tilde{c}=0.575$, and $c$ varying along the dotted line of panel (c).

Figure 5

Size of the largest mutually connected component $S$ as a function of mean nonoverlapped degree $c$ for correlated two layer multiplex with degree distribution of the form Eq. (4). Each point corresponds to a single network realisation of size $N={10}^5$ (orange plusses), ${10}^6$ (red diamonds), ${10}^7$ (green triangles), or ${10}^8$ (blue circles). (a) Assortative correlations between overlapped and nonoverlapped degree of the form Eq. (5). (b) Disassortative correlations as in Eq. (8). (c) Disassortative correlations as in Eq. (12). Theoretical results are also plotted (continuous black line), compare Figs. 2, 3, and 4, which show theoretical results for the same parameter choices.