Quantifying local structure effects in network dynamics

Abstract

Mutual information between the time series of two dynamical elements measures how well their activities are coordinated. In a network of interacting elements, the average mutual information over all pairs of elements I is a global measure of the correlation between the elements' dynamics. Local topological features in the network have been shown to affect I . Here we define a generalized clustering coefficient C_{p} and show that this quantity captures the effects of local structures on the global dynamics of networks. Using random Boolean networks (RBNs) as models of networks of interacting elements, we show that the variation of I ( I averaged over an ensemble of RBNs with the number of nodes N and average connectivity k ) with N and k is caused by the variation of C_{p} . Also, the variability of I between RBNs with equal N and k is due to their distinct values of C_{p} . Consequently, we propose a rewiring method to generate ensembles of BNs, from ordinary RBNs, with fixed values of C_{p} up to order 5, while maintaining in- and out-degree distributions. Using this methodology, the dependency of C_{p} on N and k and the variability of I for RBNs with equal N and k are shown to disappear in RBNs with C_{p} set to zero. The I of ensembles of RBNs with fixed, nonzero C_{p} values, also becomes almost independent of N and k . In addition, it is shown that C_{p} exhibits a power-law dependence on N in ordinary RBNs, suggesting that the C_{p} affects even relatively large networks. The method of generating networks with fixed C_{p} values is useful to generate networks with small N whose dynamics have the same properties as those of large scale networks, or to generate ensembles of networks with the same C_{p} as some specific network, and thus comparable dynamics. These results show how a system's dynamics is constrained by its local structure, suggesting that the local topology of biological networks might be shaped by selection, for example, towards optimizing the coordination between its components.

Probing an excited-state quantum phase transition in a quantum many-body system via an out-of-time-order correlator

“…As an extension of the ground state quantum phase transition (QPT) [1,2], an excited-state quantum phase transition (ESQPT) is identified as the singular behaviors of the excited state as the control parameter passes through the critical point [3][4][5][6]. A remark feature of ESQPTs is the divergence of the density of states at higher excitation energies [5][6][7][8]. In the last two decades, ESQPTs have been studied in various models, both theoretically (see, e.g., Ref.…”

Information propagation within the Genetic Network of Saccharomyces cerevisiae

“…Previous studies shown that increasing C p tends to strongly enhance I [21], thus explaining why S. cerevisiae core network exhibits much higher values of I than the Rand-Beta model.…”

“…This is known to enhance the ability of networks to propagate information [21]. However, another interpretation is possible for the high C p .…”

“…For this reason, the concept of clustering coefficient [20] was extended to C p , the generalized clustering coefficient, that accounts for any loops containing p nodes [21]. Let i be the node index, and ${\kappa }_{r_{1,}{r}_2}^i$ be the amount of connections between the nodes at path length distance r 1 and the nodes at distance r 2 from i .…”

Majorana representation for mixed states

“…The Hilbert space H s as a whole can be seen as a stratified manifold foliated by the SU (2) orbits of all the possible configurations of constellations [16]. In addition, the Majorana constellation has been useful in other applications, such as the classification of spinor Bose-Einstein-condensate phases [17][18][19] or the investigation of the thermodynamical limit in the Lipkin-Meshkov-Glick model [20,21]. This representation also plays a role in the Atiyah mapping related to his conjecture on 'configurations of points' [22].…”