Cmpe 588- Modeling of Internet Emergence of Scale-Free Network with Chaotic Units Pulin Gong, Cees van Leeuwen by Oya Ünlü Instructor: Haluk Bingöl

Cmpe 588- Modeling of Internet Outline Introduction Random & Scale-Free Networks The Model for Network with Dynamic Units Algorithm Evolution of the Network Robust feature of Spatiotemporal Dynamics

Cmpe 588- Modeling of Internet Introduction A diversity of the systems such as world-wide web, social networks, protein interactions network and metabolic reactions in a cell, belong to complex systems. Distinctive feature of complex systems is emergent order resulting from their many interacting elements. The structure emerges spontaneously rather than by design. For many years science treated all complex networks as being completely random. However, Barabasi and Albert presented a scale-free model and showed that most of these networks are scale-free.

Cmpe 588- Modeling of Internet Random vs. Scale-Free

Cmpe 588- Modeling of Internet Random vs. Scale-Free (cont’d) In random networks: Most of the nodes have approximately equal number of links. Distribution of node linkages will follow a bell-shaped curve. In scale-free networks: Most nodes have just a few connections where some have a tremendous number of links and system has no scale. Distribution of node linkages follows a power law.

Cmpe 588- Modeling of Internet Scale-Free Networks Scale-free network model based on two mechanisms: Growth: Cumulative addition of new units. Preferential Attachment: Probability of attachment of a newly created node is proportional to the connected degree of target nodes. (Richly connected nodes tend to get richer). More recent models allows nodes to age so they no longer accept new links or a node acquire new links with other monotonically increasing function not limited to linear preferential attachment.

Cmpe 588- Modeling of Internet Oscillatory Activity Many of the realistic features are included in the researches. However, in the studies dynamics of the system haven’t been used to explain the emergence of the network structure. In real biological systems such as neuronal, genetic, metabolic network models etc., units with oscillatory activity is very common. Subject of the study: Growth and adaptive rewiring according to dynamical coherence is studied in order to explain the emergence of scale-free network structure.

Cmpe 588- Modeling of Internet The Model for Network with Dynamic Units Proposed a model for growth combined with adaptive rewiring according to the oscillatory dynamics of the network units. Studies showed that scale-free network with a high clustering coefficient is obtained if the oscillatory behavior is chaotic. Chaos: The disordered formless matter supposed to have existed before the ordered universe. Stochastic (random) behavior occurring in a deterministic system.

Cmpe 588- Modeling of Internet The Model for Network with Dynamic Units (cont’d) Spatiotemporal Dynamics: dynamics observed both in space and time in systems. Chaotic logistic maps are used to model the dynamics of the network:

Cmpe 588- Modeling of Internet The Model for Network with Dynamic Units (cont’d) Parameters: x i (n) = Activity of the i th unit at the n th time step. N = Total number of units in the current network. B(i) = Set of all neighbors of the unit i. M i = Number of units in the current set B(i) a = System parameter which controls the dynamics of each unit. d ij = Coherence between unit i and j. ε = Coupling strength

Cmpe 588- Modeling of Internet Algorithm Starting from a sparsely, fully connected small random network with M 0 total number of units and L 0 number of connections. 1. Add a new node i n with m connections to m different nodes in the current network randomly. 2. Choose random initial activation values in the range(-1,1) for all units of the new network. Calculate the state of the system according to second equation and discard an initial transient time T. 3. Calculate d inj (T+1) for the newly added node i n. Obtain the value j=j1 where the value d inj1 (T+1) is minimum among all other units.

Cmpe 588- Modeling of Internet Algorithm 4. Obtain the value j=j2 where the d inj2 (T+1) value is maximum among the neighbors of the new unit i n. 5. If j1 is a neighbor of unit i n then no change in the system. Otherwise connetion between i n and j2 is replaced by a connection between i n and j1. 6. Go to step 2 and repeat the algorithm for K 0 times. 7. Go to step 1 and repeat adding a new node.

Cmpe 588- Modeling of Internet Evolution of the Network By choosing the parameter a, system parameter that controls the dynamics of the units, within a chaotic range a network with (M 0 +t) nodes is formed.

Cmpe 588- Modeling of Internet Evolution of the Network The network has evolved into a scale-free state with the probability a node has k connections, following a power-law p(k)=k -γ with γ=3.09±0.17. The clustering coefficient is calculated as 0.15 and average path length is It is known that in real networks clustering coefficient is much larger than random networks. To compare the corresponding random graph with the model of the self-organized scale-free network, a random graph is formed by connecting nodes randomly. (Same number of nodes and connections). Clustering coefficient=1.7*10 -2 and avg. shortest path length=2.56.

Cmpe 588- Modeling of Internet Evolution of the Network When we compare the random graph with the model: Shortest path length is closer to random graph. Clustering coefficient is much larger than random graph. So: A growing network with chaotic units according to the model, produces a scale-free network with the characteristics of small-world networks.

Cmpe 588- Modeling of Internet Evolution of the Network To see the preferential attachment: Ki, the number of connections for the set of nodes at time t is calculated. Also at time t+Δt, Δt<<t, the number of connections,Ji, is measured. So increasing number of connections Δki= Ji-Ki is found.

Cmpe 588- Modeling of Internet Chaotic vs. Stochastic vs. Periodic In order to investigate the mechanisms responsible for the emergence of the scale-free network, some variants of the model is studied. The first variant: Dynamic of units: Periodic. To make the model periodic, parameter a is chosen 0.51 for equation 2, making the units period-1 state. Second variant: Dynamic of units: Stochastic. To make the model stochastic, a random generator is used instead of the logistic functions. For both of the models, same small random network is chosen and the algortihm is used for the same number of times as in chaotic model.

Cmpe 588- Modeling of Internet Chaotic vs. Stochastic vs. Periodic

Cmpe 588- Modeling of Internet Chaotic vs. Stochastic vs. Periodic For periodic units, distribution doesn’t have scale-free characteristics. Clustering coefficient is close to random network. For stochastic units, the clustering coefficient is close to random network. However, the distribution of connections has at least a scale-free part with a cut-off.

Cmpe 588- Modeling of Internet Robust feature of Spatiotemporal Dynamics The dynamics of the scale-free network of coupled maps is investigated for its robustness in terms of the change of average shortest path length under lesioning. The spatiotemporal dynamics characteristics of the networks are of central importance because they provide the functional significance of the networks. For the scale-free network the size of each dynamical cluster (the number of participating units) is calculated for every iteration. Distribution of the sizes of the dynamical clusters are calculated over a longtime period.

Cmpe 588- Modeling of Internet Robust feature of Spatiotemporal Dynamics

Cmpe 588- Modeling of Internet Robust feature of Spatiotemporal Dynamics The distribution has a power-law part followed by an exponential cut. For the scale-free network with dynamical units, the spatiotemporal dynamics of the complex network is robust. Also a small number of nodes with large numbers of connections certainly play a very important role in the spatiotemporal dynamics.

Cmpe 588- Modeling of Internet Conclusion The development of networks with chaotic units is investigated. Adding new nodes one by one and adaptive rewiring of the connections according to the dynamical coherence of the nodes, the network self-organizes into a scale-free network. Self-organized network has a high clustering coefficient and small characteristic path length. Adaptive rewiring without growth does not produce a scale-free network but a small-world network.

Cmpe 588- Modeling of Internet Conclusion Chaotic dynamical behaviors, growth, and adaptive rewiring according to the dynamical coherence of the nodes is needed for scale-free networks. The model can help to understand the real systems with dynamic units such as neural systems, or some complex systems such as the brain and visual system.

Cmpe 588- Modeling of Internet Questions ?