Zoltán Toroczkai

Modelling disease outbreaks in realistic urban social networks

Most mathematical models for the spread of disease use differential equations based on uniform mixing assumptions or ad hoc models for the contact process. Here we explore the use of dynamic bipartite graphs to model the physical contact patterns that result from movements of individuals between specific locations. The graphs are generated by large-scale individual-based urban traffic simulations built on actual census, land-use and population-mobility data. We find that the contact network among people is a strongly connected small-world-like graph with a well-defined scale for the degree distribution. However, the locations graph is scale-free, which allows highly efficient outbreak detection by placing sensors in the hubs of the locations network. Within this large-scale simulation framework, we then analyse the relative merits of several proposed mitigation strategies for smallpox spread. Our results suggest that outbreaks can be contained by a strategy of targeted vaccination combined with early detection without resorting to mass vaccination of a population.

A Weighted and Directed Interareal Connectivity Matrix for Macaque Cerebral Cortex

Retrograde tracer injections in 29 of the 91 areas of the macaque cerebral cortex revealed 1,615 interareal pathways, a third of which have not previously been reported. A weight index (extrinsic fraction of labeled neurons [FLNe]) was determined for each area-to-area pathway. Newly found projections were weaker on average compared with the known projections; nevertheless, the 2 sets of pathways had extensively overlapping weight distributions. Repeat injections across individuals revealed modest FLNe variability given the range of FLNe values (standard deviation <1 log unit, range 5 log units). The connectivity profile for each area conformed to a lognormal distribution, where a majority of projections are moderate or weak in strength. In the G29 × 29 interareal subgraph, two-thirds of the connections that can exist do exist. Analysis of the smallest set of areas that collects links from all 91 nodes of the G29 × 91 subgraph (dominating set analysis) confirms the dense (66%) structure of the cortical matrix. The G29 × 29 subgraph suggests an unexpectedly high incidence of unidirectional links. The directed and weighted G29 × 91 connectivity matrix for the macaque will be valuable for comparison with connectivity analyses in other species, including humans. It will also inform future modeling studies that explore the regularities of cortical networks.

Cortical High-Density Counterstream Architectures

Background The cerebral cortex is divisible into many individual areas, each exhibiting distinct connectivity profiles, architecture, and physiological characteristics. Interactions among cortical areas underlie higher sensory, motor, and cognitive functions. Graph theory provides an important framework for understanding network properties of the interareal weighted and directed connectivity matrix reported in recent studies. Density and topology of the cortical graph. (Left) The 66% density of the cortical matrix (black triangle) is considerably greater than in previous reports (colored points) and is inconsistent with a small-world network. (Right) A bow-tie representation of the high-density cortical matrix. The high-efficiency cortical core has defined relations with the cortical periphery in the two fans. Advances We derive an exponential distance rule that predicts many binary and weighted features of the cortical network, including efficiency of information transfer, the high specificity of long-distance compared to short-distance connections, wire length minimization, and the existence of a highly interconnected cortical core. We propose a bow-tie representation of the cortex, which combines these features with hierarchical processing. Outlook The exponential distance rule has important implications for understanding scaling properties of the cortex and developing future large-scale dynamic models of the cortex. Small-world networks provide an appealing description of cortical architecture owing to their capacity for integration and segregation combined with an economy of connectivity. Previous reports of low-density interareal graphs and apparent small-world properties are challenged by data that reveal high-density cortical graphs in which economy of connections is achieved by weight heterogeneity and distance-weight correlations. These properties define a model that predicts many binary and weighted features of the cortical network including a core-periphery, a typical feature of self-organizing information processing systems. Feedback and feedforward pathways between areas exhibit a dual counterstream organization, and their integration into local circuits constrains cortical computation. Here, we propose a bow-tie representation of interareal architecture derived from the hierarchical laminar weights of pathways between the high-efficiency dense core and periphery.

Weight Consistency Specifies Regularities of Macaque Cortical Networks

To what extent cortical pathways show significant weight differences and whether these differences are consistent across animals (thereby comprising robust connectivity profiles) is an important and unresolved neuroanatomical issue. Here we report a quantitative retrograde tracer analysis in the cynomolgus macaque monkey of the weight consistency of the afferents of cortical areas across brains via calculation of a weight index (fraction of labeled neurons, FLN). Injection in 8 cortical areas (3 occipital plus 5 in the other lobes) revealed a consistent pattern: small subcortical input (1.3% cumulative FLN), high local intrinsic connectivity (80% FLN), high-input form neighboring areas (15% cumulative FLN), and weak long-range corticocortical connectivity (3% cumulative FLN). Corticocortical FLN values of projections to areas V1, V2, and V4 showed heavy-tailed, lognormal distributions spanning 5 orders of magnitude that were consistent, demonstrating significant connectivity profiles. These results indicate that 1) connection weight heterogeneity plays an important role in determining cortical network specificity, 2) high investment in local projections highlights the importance of local processing, and 3) transmission of information across multiple hierarchy levels mainly involves pathways having low FLN values.

A Predictive Network Model of Cerebral Cortical Connectivity Based on a Distance Rule

Recent advances in neuroscience have engendered interest in large-scale brain networks. Using a consistent database of cortico-cortical connectivity, generated from hemisphere-wide, retrograde tracing experiments in the macaque, we analyzed interareal weights and distances to reveal an important organizational principle of brain connectivity. Using appropriate graph theoretical measures, we show that although very dense (66%), the interareal network has strong structural specificity. Connection weights exhibit a heavy-tailed lognormal distribution spanning five orders of magnitude and conform to a distance rule reflecting exponential decay with interareal separation. A single-parameter random graph model based on this rule predicts numerous features of the cortical network: (1) the existence of a network core and the distribution of cliques, (2) global and local binary properties, (3) global and local weight-based communication efficiencies modeled as network conductance, and (4) overall wire-length minimization. These findings underscore the importance of distance and weight-based heterogeneity in cortical architecture and processing.

Structural bottlenecks for communication in networks

Sameet Sreenivasan, Reuven Cohen, Eduardo López, Zoltán Toroczkai, and H. Eugene Stanley Center for Polymer Studies and Department of Physics, Boston University, Boston, MA 02215 2 Center for Nonlinear Studies, Los Alamos National Laboratory, MS B258, Los Alamos, NM 87545 3 Laboratory of Networking and Information Systems and Department of Electrical and Computer Engineering, Boston University, Boston, MA 02215 4 Theoretical Division, Los Alamos National Laboratory, MS B258, Los Alamos, NM 87545 (Dated: February 1, 2008)

Suppressing Roughness of Virtual Times in Parallel Discrete-Event Simulations

In a parallel discrete-event simulation (PDES) scheme, tasks are distributed among processing elements (PEs) whose progress is controlled by a synchronization scheme. For lattice systems with short-range interactions, the progress of the conservative PDES scheme is governed by the Kardar-Parisi-Zhang equation from the theory of nonequilibrium surface growth. Although the simulated (virtual) times of the PEs progress at a nonzero rate, their standard deviation (spread) diverges with the number of PEs, hindering efficient data collection. We show that weak random interactions among the PEs can make this spread nondivergent. The PEs then progress at a nonzero, near-uniform rate without requiring global synchronizations.

The role of long-range connections on the specificity of the macaque interareal cortical network.

We investigated the influence of interareal distance on connectivity patterns in a database obtained from the injection of retrograde tracers in 29 areas distributed over six regions (occipital, temporal, parietal, frontal, prefrontal, and limbic). One-third of the 1,615 pathways projecting to the 29 target areas were reported only recently and deemed new-found projections (NFPs). NFPs are predominantly long-range, low-weight connections. A minimum dominating set analysis (a graph theoretic measure) shows that NFPs play a major role in globalizing input to small groups of areas. Randomization tests show that (i) NFPs make important contributions to the specificity of the connectivity profile of individual cortical areas, and (ii) NFPs share key properties with known connections at the same distance. We developed a similarity index, which shows that intraregion similarity is high, whereas the interregion similarity declines with distance. For area pairs, there is a steep decline with distance in the similarity and probability of being connected. Nevertheless, the present findings reveal an unexpected binary specificity despite the high density (66%) of the cortical graph. This specificity is made possible because connections are largely concentrated over short distances. These findings emphasize the importance of long-distance connections in the connectivity profile of an area. We demonstrate that long-distance connections are particularly prevalent for prefrontal areas, where they may play a prominent role in large-scale communication and information integration.

Efficient and Exact Sampling of Simple Graphs with Given Arbitrary Degree Sequence

Uniform sampling from graphical realizations of a given degree sequence is a fundamental component in simulation-based measurements of network observables, with applications ranging from epidemics, through social networks to Internet modeling. Existing graph sampling methods are either link-swap based (Markov-Chain Monte Carlo algorithms) or stub-matching based (the Configuration Model). Both types are ill-controlled, with typically unknown mixing times for link-swap methods and uncontrolled rejections for the Configuration Model. Here we propose an efficient, polynomial time algorithm that generates statistically independent graph samples with a given, arbitrary, degree sequence. The algorithm provides a weight associated with each sample, allowing the observable to be measured either uniformly over the graph ensemble, or, alternatively, with a desired distribution. Unlike other algorithms, this method always produces a sample, without back-tracking or rejections. Using a central limit theorem-based reasoning, we argue, that for large , and for degree sequences admitting many realizations, the sample weights are expected to have a lognormal distribution. As examples, we apply our algorithm to generate networks with degree sequences drawn from power-law distributions and from binomial distributions.

Complexity of the International Agro-Food Trade Network and Its Impact on Food Safety

With the world’s population now in excess of 7 billion, it is vital to ensure the chemical and microbiological safety of our food, while maintaining the sustainability of its production, distribution and trade. Using UN databases, here we show that the international agro-food trade network (IFTN), with nodes and edges representing countries and import-export fluxes, respectively, has evolved into a highly heterogeneous, complex supply-chain network. Seven countries form the core of the IFTN, with high values of betweenness centrality and each trading with over 77% of all the countries in the world. Graph theoretical analysis and a dynamic food flux model show that the IFTN provides a vehicle suitable for the fast distribution of potential contaminants but unsuitable for tracing their origin. In particular, we show that high values of node betweenness and vulnerability correlate well with recorded large food poisoning outbreaks.