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Self-organized Model for Modular Complex Networks: Division and Independence
We introduce a minimal network model which generates a modular structure in a self-organized way. To this end, we modify the Barabasi-Albert model into the one evolving under the principle of division and independence as well as growth and preferential attachment (PA). A newly added vertex chooses one of the modules composed of existing vertices, and attaches edges to vertices belonging to that module following the PA rule. When the module size reaches a proper size, the module is divided into two, and a new module is created. The karate club network studied by Zachary is a prototypical example. We find that the model can reproduce successfully the behavior of the hierarchical clustering coefficient of a vertex with degree k, C(k), in good agreement with empirical measurements of real world networks
The Traveling Salesman Problem: Low-Dimensionality Implies a Polynomial Time Approximation Scheme
The Traveling Salesman Problem (TSP) is among the most famous NP-hard
optimization problems. We design for this problem a randomized polynomial-time
algorithm that computes a (1+eps)-approximation to the optimal tour, for any
fixed eps>0, in TSP instances that form an arbitrary metric space with bounded
intrinsic dimension.
The celebrated results of Arora (A-98) and Mitchell (M-99) prove that the
above result holds in the special case of TSP in a fixed-dimensional Euclidean
space. Thus, our algorithm demonstrates that the algorithmic tractability of
metric TSP depends on the dimensionality of the space and not on its specific
geometry. This result resolves a problem that has been open since the
quasi-polynomial time algorithm of Talwar (T-04)
Resolving structural variability in network models and the brain
Large-scale white matter pathways crisscrossing the cortex create a complex
pattern of connectivity that underlies human cognitive function. Generative
mechanisms for this architecture have been difficult to identify in part
because little is known about mechanistic drivers of structured networks. Here
we contrast network properties derived from diffusion spectrum imaging data of
the human brain with 13 synthetic network models chosen to probe the roles of
physical network embedding and temporal network growth. We characterize both
the empirical and synthetic networks using familiar diagnostics presented in
statistical form, as scatter plots and distributions, to reveal the full range
of variability of each measure across scales in the network. We focus on the
degree distribution, degree assortativity, hierarchy, topological Rentian
scaling, and topological fractal scaling---in addition to several summary
statistics, including the mean clustering coefficient, shortest path length,
and network diameter. The models are investigated in a progressive, branching
sequence, aimed at capturing different elements thought to be important in the
brain, and range from simple random and regular networks, to models that
incorporate specific growth rules and constraints. We find that synthetic
models that constrain the network nodes to be embedded in anatomical brain
regions tend to produce distributions that are similar to those extracted from
the brain. We also find that network models hardcoded to display one network
property do not in general also display a second, suggesting that multiple
neurobiological mechanisms might be at play in the development of human brain
network architecture. Together, the network models that we develop and employ
provide a potentially useful starting point for the statistical inference of
brain network structure from neuroimaging data.Comment: 24 pages, 11 figures, 1 table, supplementary material
Compact Routing on Internet-Like Graphs
The Thorup-Zwick (TZ) routing scheme is the first generic stretch-3 routing
scheme delivering a nearly optimal local memory upper bound. Using both direct
analysis and simulation, we calculate the stretch distribution of this routing
scheme on random graphs with power-law node degree distributions, . We find that the average stretch is very low and virtually
independent of . In particular, for the Internet interdomain graph,
, the average stretch is around 1.1, with up to 70% of paths
being shortest. As the network grows, the average stretch slowly decreases. The
routing table is very small, too. It is well below its upper bounds, and its
size is around 50 records for -node networks. Furthermore, we find that
both the average shortest path length (i.e. distance) and width of
the distance distribution observed in the real Internet inter-AS graph
have values that are very close to the minimums of the average stretch in the
- and -directions. This leads us to the discovery of a unique
critical quasi-stationary point of the average TZ stretch as a function of
and . The Internet distance distribution is located in a
close neighborhood of this point. This observation suggests the analytical
structure of the average stretch function may be an indirect indicator of some
hidden optimization criteria influencing the Internet's interdomain topology
evolution.Comment: 29 pages, 16 figure
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