We study complex networks with weights, $w_{ij}$, associated with each link connecting node $i$ and $j$. The weights are chosen to be correlated with the network topology in the form found in two real world examples, (a) the world-wide airport network, and (b) the {\it E. Coli} metabolic network. Here $w_{ij} \sim x_{ij} (k_i k_j)^α$, where $k_i$ and $k_j$ are the degrees of nodes $i$ and $j$, $x_{ij}$ is a random number and $α$ represents the strength of the correlations. The case $α> 0$ represents correlation between weights and degree, while $α< 0$ represents anti-correlation and the case $α= 0$ reduces to the case of no correlations. We study the scaling of the lengths of the optimal paths, $\ell_{\rm opt}$, with the system size $N$ in strong disorder for scale-free networks for different $α$. We calculate the robustness of correlated scale-free networks with different $α$, and find the networks with $α< 0$ to be the most robust networks when compared to the other values of $α$. We propose an analytical method to study percolation phenomena on networks with this kind of correlation. We compare our simulation results with the real world-wide airport network, and we find good agreement.
The structure of complex networks can be characterized by counting and analyzing network motifs. Motifs are small subgraphs that occur repeatedly in a network, such as triangles or chains. Recent work has generalized motifs to temporal and dynamic network data. However, existing techniques do not generalize to sequential or trajectory data, which represents entities moving through the nodes of a network, such as passengers moving through transportation networks. The unit of observation in these data is fundamentally different, since we analyze full observations of trajectories (e.g., a trip fr…
We discuss two well known network measures: the overlap weight of an edge and the clustering coefficient of a node. For both of them it turns out that they are not very useful for data analytic task to identify important elements (nodes or links) of a given network. The reason for this is that they attain their largest values on maximal subgraphs of relatively small size that are more probable to appear in a network than that of larger size. We show how the definitions of these measures can be corrected in such a way that they give the expected results. We illustrate the proposed corrected mea…
Empirical networks of weighted dyadic relations often contain noisy edges that alter the global characteristics of the network and obfuscate the most important structures therein. Graph pruning is the process of identifying the most significant edges according to a generative null model, and extracting the subgraph consisting of those edges. Here, we focus on integer-weighted graphs commonly arising when weights count the occurrences of an "event" relating the nodes. We introduce a simple and intuitive null model related to the configuration model of network generation, and derive two signific…
Here we report a comprehensive analysis of the robustness of five high-quality real-world complex weighted networks to errors and attacks of nodes and links. We analyze C. Elegans, Cargo ship, E. Coli, Us Airports and Human brain real-world complex weighted networks. We use measures of the network damage conceived for a binary (e.g. largest connected cluster LCC, and binary efficiency Effbin) or a weighted network structure (e.g. the efficiency Eff, and the total energy-information En). We find that removing a very small fraction of nodes and links with respectively higher strength and weight …
