Large-Scale Topology Optimization Using Parameterized Boolean Networks

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.

Discrimination of singleton and periodic attractors in Boolean networks

Random Boolean networks have been proposed as a basic model of genetic regulatory networks for more than four decades. Attractors have been considered as the best way to represent the long-term behaviors of random Boolean networks. Most studies on attractors are made with random topologies. However, the real regulatory networks have been found to be modular or more complex topologies. In this work, we extend classical robustness and entropy analysis of random Boolean networks to random modular Boolean networks. We firstly focus on the robustness of the attractor to perturbations with different parameters. Then, we investigate and calculate how the amount of information propagated between the nodes when on an attractor, as quantified by the average pairwise mutual information. The results can be used to study the capability of genetic information propagation of different types of genetic regulatory networks.

We propose quantum Boolean networks, which can be classified as deterministic reversible asynchronous Boolean networks. This model is based on the previously developed concept of quantum Boolean functions. A quantum Boolean network is a Boolean network where the functions associated with the nodes are quantum Boolean functions. We study some properties of this novel model and, using a quantum simulator, we study how the dynamics change in function of connectivity of the network and the set of operators we allow. For some configurations, this model resembles the behavior of reversible Boolean networks, while for other configurations a more complex dynamic can emerge. For example, cycles larger than 2N were observed. Additionally, using a scheme akin to one used previously with random Boolean networks, we computed the average entropy and complexity of the networks. As opposed to classic random Boolean networks, where “complex” dynamics are restricted mainly to a connectivity close to a phase transition, quantum Boolean networks can exhibit stable, complex, and unstable dynamics independently of their connectivity.

We propose quantum Boolean networks, which can be classified as deterministic reversible asynchronous Boolean networks. This model is based on the previously developed concept of quantum Boolean functions. A quantum Boolean network is a Boolean network where the functions associated with the nodes are quantum Boolean functions. We study some properties of this novel model and, using a quantum simulator, we study how the dynamics change in function of connectivity of the network and the set of operators we allow. For some configurations, this model resembles the behavior of reversible Boolean networks, while for other configurations a more complex dynamic can emerge. For example, cycles larger than 2N were observed. Additionally, using a scheme akin to one used previously with random Boolean networks, we computed the average entropy and complexity of the networks. As opposed to classic random Boolean networks, where “complex” dynamics are restricted mainly to a connectivity close to a phase transition, quantum Boolean networks can exhibit stable, complex, and unstable dynamics independently of their connectivity.

The Boolean network (BN) is a mathematical model of genetic networks. It is known that detecting a singleton attractor, which is also called a fixed point, is NP-hard even for AND/OR BNs (i.e., BNs consisting of AND/OR nodes), where singleton attractors correspond to steady states. Though a naive algorithm can detect a singleton attractor for an AND/OR BN in O(n2 n ) time, no O((2 ― ∈) n ) (∈ > 0) time algorithm was known even for an AND/OR BN with non-restricted indegree, where n is the number of nodes in a BN. In this paper, we present an O(1.787 n ) time algorithm for detecting a singleton attractor of a given AND/OR BN, along with related results. We also show that detection of a singleton attractor in a BN with maximum indegree two is NP-hard and can be polynomially reduced to a satisfiability problem.

A genetic regulatory network (GRN) is a dynamic system to describe interactions among a large number of different substances in a living biological cell. There have been numerous attempts to model the dynamical behaviour of genetic regulatory networks. Boolean models are a class of discrete models for modelling genetic regulatory networks, which are conditioned on the premise that genes interact with each other through Boolean logic. As mentioned previously, Boolean models include Boolean Networks, Instantaneously Random Probabilistic Boolean Networks and Context-Sensitive Boolean Networks. There are a large number of issues surrounding Boolean models, including Boolean Networks and Probabilistic Boolean Networks. The model inference is the procedure of determining whether the modelled networks are consistent with the given data sample, and choosing one model, from among many, that makes more sense for the data. This paper has reviewed the research into modelling gene regulatory networks, especially upon Boolean models.

The chordless cycle sizes of spatially embedded networks are demonstrated to follow an exponential growth law similar to random graphs if the number of nodes Nx is below a critical value N*. For covalent polymer networks, increasing the network size, as measured by the number of cross-link nodes, beyond N* results in a crossover to a new regime in which the characteristic size of the chordless cycles h* no longer increases. From this result, the onset and intensity of finite-size effects can be predicted from measurement of h* in large networks. Although such information is largely inaccessible with experiments, the agreement of simulation results from molecular dynamics, Metropolis Monte Carlo, and kinetic Monte Carlo suggests the crossover is a fundamental physical feature which is insensitive to the details of the network generation. These results show random graphs as a promising model to capture structural differences in confined physical networks.

This paper considers a simple Boolean network with N nodes, each node's state at time t being determined by a certain number k of parent nodes, which is fixed for all nodes. The nodes, with randomly assigned neighborhoods, are updated based on various asynchronous schemes. We make use of a Boolean rule that is a generalization of rule 126 of elementary cellular automata. We provide formulas for the probability of finding a node in state 1 at a time t for the class of asynchronous random Boolean networks (ARBN) in which only one node is updated at every time step, and for the class of generalized ARBNs (GARBN) in which a random number of nodes can be updated at each time point. We use simulation methods to generate consecutive states of the network for both the real system and the models under the various schemes. The results match well. We study the dynamics of the models through sensitivity of the orbits to initial values, bifurcation diagrams, and fixed point analysis. We show, both theoretically and by example, that the ARBNs generate an ordered behavior regardless of the updating scheme used, whereas the GARBNs have behaviors that range from order to chaos depending on the type of random variable used to determine the number of nodes to be updated and the parameter combinations.

In this dissertation, we consider wireless multi-hop networks, where the nodes are randomly placed. We are particularly interested in their asymptotic properties when the number of nodes tends to infinity. We use percolation theory as our main tool of analysis. As a first model, we assume that nodes have a fixed connectivity range, and can establish wireless links to all nodes within this range, but no other (Boolean model). We compute for one-dimensional networks the probability that two nodes are connected, given the distance between them. We show that this probability tends exponentially to zero when the distance increases, proving that pure multi-hopping does not work in large networks. In two dimensions however, an unbounded cluster of connected nodes forms if the node density is above a critical threshold (super-critical phase). This is known as the percolation phenomenon. This cluster contains a positive fraction of the nodes that depends on the node density, and remains constant as the network size increases. Furthermore, the fraction of connected nodes tends rapidly to one when the node density is above the threshold. We compare this partial connectivity to full connectivity, and show that the requirement for full connectivity leads to vanishing throughput when the network size increases. In contrast, partial connectivity is perfectly scalable, at the cost of a tiny fraction of the nodes being disconnected. We consider two other connectivity models. The first one is a signal-to-interference- plus-noise-ratio based connectivity graph (STIRG). In this model, we assume deterministic attenuation of the signals as a function of distance. We prove that percolation occurs in this model in a similar way as in the previous model, and study in detail the domain of parameters where it occurs. We show in particular that the assumptions on the attenuation function dramatically impact the results: the commonly used power-law attenuation leads to particular symmetry properties. However, physics imposes that the received signal cannot be stronger than the emitted signal, implying a bounded attenuation function. We observe that percolation is harder to achieve in most cases with such an attenuation function. The second model is an information theoretic view on connectivity, where two arbitrary nodes are considered connected if it is possible to transmit data from one to the other at a given rate. We show that in this model the same partial connectivity can be achieved in a scalable way as in the Boolean model. This result is however a pure connectivity result in the sense that there is no competition and interferences between data flows. We also look at the other extreme, the Gupta and Kumar scenario, where all nodes want to transmit data simultaneously. We show first that under point-to-point communication and bounded attenuation function the total transport capacity of a fixed area network is bounded from above by a constant, whatever the number of nodes may be. However, if the network area increases linearly with the number of nodes (constant density), or if we assume power-law attenuation function, a throughput per node of order 1/√n can be achieved. This latter result improves the existing results about random networks by a factor (log n)1/2. In the last part of this dissertation, we address two problems related to latency. The first one is an intruder detection scenario, where a static sensor network has to detect an intruder that moves with constant speed along a straight line. We compute an upper bound to the time needed to detect the intruder, under the assumption that detection by disconnected sensors does not count. In the second scenario, sensors switch off their radio device for random periods, in order to save energy. This affects the delivery of alert messages, since they may have to wait for relays to turn on their radio to move further. We show that asymptotically, alert messages propagate with constant, deterministic speed in such networks.

We derive analytically the scaling behavior in the thermodynamic limit of the number of nonfrozen and relevant nodes in the most general class of critical Kauffman networks for any number of inputs per node, and for any choice of the probability distribution for the Boolean functions. By defining and analyzing a stochastic process that determines the frozen core we can prove that the mean number of nonfrozen nodes in any critical network with more than one input per node scales with the network size N as N2/3, with only N1/3 nonfrozen nodes having two nonfrozen inputs and the number of nonfrozen nodes with more than two inputs being finite in the thermodynamic limit. Using these results we can conclude that the mean number of relevant nodes increases for large N as N1/3, with only a finite number of relevant nodes having two relevant inputs, and a vanishing fraction of nodes having more than three of them. It follows that all relevant components apart from a finite number are simple loops, and that the mean number and length of attractors increases faster than any power law with network size.

Intrinsic properties of Boolean dynamics in complex networks

Singleton attractor (also called fixed point) detection is known to be NP-hard even for AND/OR Boolean networks (AND/OR BNs in short, i.e., BNs consisting of AND/OR nodes), where BN is a mathematical model of genetic networks and singleton attractors correspond to steady states. In our recent paper, we developed an O(1.787n) time algorithm for detecting a singleton attractor of a given AND/OR BN where n is the number of nodes. In this paper, we present an O(1.757n) time algorithm with which we succeeded in improving the above algorithm. We also show that this problem can be solved in \(2^{O(({\rm log} \, n)\sqrt{n})}\) time, which is less than O((1 + ∈)n) for any positive constant ∈, when a BN is planar.

This paper studies the dynamic single-source shortest paths SSSP in Erdos-Renyi random graphs generated by Gn,i¾?p model. In 2014, Ding and Lin AAIM 2014, LNCS 8546, 197---207 first considered the dynamic SSSP in general digraphs with arbitrary positive weights, and devised a nontrivial local search algorithm named DSPI which takes at most $$On\cdot \max \{1, n\log n / m\}$$Oni¾?max{1,nlogn/m} expected update time to handle a single weight increase, where n is the number of nodes and m is the number of edges in the digraph. DSPI also works on undirected graphs. This paper analyzes the expected update time of DSPI dealing with edge weight increases or edge deletions in Erdos-Renyi a.k.a., Gn,i¾?p random graphs. For weighted Gn,i¾?p random graphs with arbitrary positive edge weights, DSPI takes at most $$OhT_s$$OhTs expected update time to deal with a single edge weight increase as well as $$Opn^2 hT_s$$Opn2hTs total update time, where $$hT_s$$hTs is the height of input SSSP tree $$T_s$$Ts. For Gn,i¾?p random graphs, DSPI takes $$O\ln n$$Olnn expected update time to handle a single edge deletion as well as $$Opn^2 \ln n$$Opn2lnn total update time when $$20\ln n / n \le p \sqrt{2\ln n / n}$$p>2lnn/n. Specifically, DSPI takes the least total update time of $$On\ln n hT_s$$OnlnnhTs for weighted Gn,i¾?p random graphs with $$p = c\ln n / n, c > 1$$p=clnn/n,c>1 as well as $$On^{3/2}\ln n^{1/2}$$On3/2lnn1/2 for Gn,i¾?p random graphs with $$p = c\sqrt{\ln n / n}, c > \sqrt{2}$$p=clnn/n,c>2.

Collective decision making is crucial in human organizations and societies. When a collective is working on exploration of problem space and/or ideation for creative solutions, the evolutionary perspective is useful for conceptualizing and modeling collective decision making, where populations of ideas spread and evolve on a social network habitat via continual applications of evolutionary operators by human individuals (Sayama & Dionne 2015). Using an evolutionary approach to model collective decision making, we conducted agent-based simulations to investigate how collective decision making would be affected by the size and topology of social network structure (Sayama, Dionne & Yammarino 2013). In our model, each agent has its own utility function defined over a multi-dimensional problem space, which is marginally different from the “true” utility function that is not accessible from any agent. Each agent can memorize multiple ideas in mind, and iteratively applies evolutionary operators (e.g., replication, mutation, recombination, elimination) to the idea population it has. The outcomes of evolutionary operations are stored in the agent’s mind, and also sent to the neighbors to which the agent is connected. This allows the spread of ideas through social ties. Each simulation was run for a fixed number of iterations, and then the level of idea convergence at a social level and the true utility value of the most supported idea were measured as the performance metrics of collective decision making. The former metric characterizes the ability for the society to form consensus, while the latter characterizes its ability to find the true best solution. The size of the network (number of agents or nodes) was varied from 5 to 640 logarithmically. Three network topologies with the same average node degree were tested: Random, small-world and scale-free. For more details of the model and the simulation settings, see Sayama, Dionne & Yammarino (2013). Simulation results indicated that expanding the size of the social network generally improved the quality of ideas at the cost of decision convergence. Simulations with different social network topologies further revealed that collective decision making on small-world networks with high local clustering tended to achieve highest decision quality more often than on random or scale-free networks (Fig. 1). This can be understood in that local clustering helps agents in different regions in a network maintain their respective focus areas and engage in different local search, possibly enhancing the effective parallelism of collective decision making and therefore resulting in a greater number of successful decisions. In contrast, the links in random and scale-free networks are all “global”, mixing discussions prematurely and therefore reducing the effective parallelism of collective decision making. These observations have an interesting contrast with the fact that random and scale-free networks are highly efficient in information dissemination because of their global connectedness. Our results indicate that such efficiency of information dissemination may not necessarily imply the same for collective decision making. This work may also offer an evolutionary explanation for the nontrivial relationship between network clustering and problem solving, a problem that is actively debated in organizational science (Shore, Bernstein & Lazer 2014).