We interpret the subgraph centrality as the partition function of a network. The entropy, the internal energy and the Helmholtz free energy are defined for networks and molecular graphs on the basis of graph spectral theory. Various relations of these quantities to the structure and the dynamics of the complex networks are discussed. They include the cohesiveness of the network and the critical coupling of coupled phase oscillators. We explore several models of network growing/evolution as well as real-world networks, such as those representing metabolic and protein-protein interaction networks as well as the interaction between secondary structure elements in proteins.
We introduce a new centrality measure that characterizes the participation of each node in all subgraphs in a network. Smaller subgraphs are given more weight than larger ones, which makes this measure appropriate for characterizing network motifs. We show that the subgraph centrality (SC) can be obtained mathematically from the spectra of the adjacency matrix of the network. This measure is better able to discriminate the nodes of a network than alternate measures such as degree, closeness, betweenness and eigenvector centralities. We study eight real-world networks for which SC displays usef…
The discriminant power of centrality indices for the degree, eigenvector, closeness, betweenness and subgraph centrality is analyzed. It is defined by the number of graphs for which the standard deviation of the centrality of its nodes is zero. On the basis of empirical analysis it is concluded that the subgraph centrality displays better discriminant power than the rest of centralities. We also propose some new conjectures about the types of graphs for which the subgraph centrality does not discriminate among nonequivalent nodes.
Matrix-based centrality measures have enjoyed significant popularity in network analysis, in no small part due to our ability to rigorously analyze their behavior as parameters vary. Recent work has considered the relationship between subgraph centrality, which is defined using the matrix exponential $f(x) = \exp(x)$, and the walk structure of a network. In a walk-regular graph, the number of closed walks of each length must be the same for all nodes, implying uniform $f$-subgraph centralities for any $f$ (or maximum $f$-$\textit{walk entropy}$). We consider when non--walk-regular graphs can a…
We present some initial results from a case study in social media data harvesting and visualization utilizing the tools and analytical features of NodeXL applied to a degree asymmetric vertex graph set. We consider twitter graphs harvested for topics related to suicidal ideation, suicide attempts, self-harm and bullycide. While the twitter-sphere only captures a small and age biased sample of communications it is a readily available public database for a wealth of rich topics yielding a large sample set. All these topics gave rise to highly asymmetric vertex degree graphs and all shared the sa…
The notion of return probability -- explored most famously by George Pólya on d-dimensional lattices -- has potential as a measure for the analysis of networks. We present an efficient method for finding return probability distributions for connected undirected graphs. We argue that return probability has the same discriminatory power as existing k-step measures -- in particular, beta centrality (with negative beta), the graph-theoretical power index (GPI), and subgraph centrality. We compare the running time of our algorithm to beta centrality and subgraph centrality and find that it is signi…
Networks are versatile representations of the interactions between entities in complex systems. Cycles on such networks represent feedback processes which play a central role in system dynamics. In this work, we introduce a measure of the importance of any individual cycle, as the fraction of the total information flow of the network passing through the cycle. This measure is computationally cheap, numerically well-conditioned, induces a centrality measure on arbitrary subgraphs and reduces to the eigenvector centrality on vertices. We demonstrate that this measure accurately reflects the impa…
The calculation of centrality measures is common practice in the study of networks, as they attempt to quantify the importance of individual vertices, edges, or other components. Different centralities attempt to measure importance in different ways. In this paper, we examine a conjecture posed by E. Estrada regarding the ability of several measures to distinguish the vertices of networks. Estrada conjectured that if all vertices of a graph have the same subgraph centrality, then all vertices must also have the same degree, eigenvector, closeness, and betweenness centralities. We provide a cou…
This work deals with the issue of assessing the influence of a node in the entire network and in the subnetwork to which it belongs as well, adapting the classical idea of vertex centrality. We provide a general definition of relative vertex centrality measure with respect to the classical one, referred to the whole network. Specifically, we give a decomposition of the relative centrality measure by including also the relative influence of the single node with respect to a given subgraph containing it. The proposed measure of relative centrality is tested in the empirical networks generated by…
The spectrum of the non-backtracking matrix plays a crucial role in determining various structural and dynamical properties of networked systems, ranging from the threshold in bond percolation and non-recurrent epidemic processes, to community structure, to node importance. Here we calculate the largest eigenvalue of the non-backtracking matrix and the associated non-backtracking centrality for uncorrelated random networks, finding expressions in excellent agreement with numerical results. We show however that the same formulas do not work well for many real-world networks. We identify the mec…
Measures of complex network analysis, such as vertex centrality, have the potential to unveil existing network patterns and behaviors. They contribute to the understanding of networks and their components by analyzing their structural properties, which makes them useful in several computer science domains and applications. Unfortunately, there is a large number of distinct centrality measures and little is known about their common characteristics in practice. By means of an empirical analysis, we aim at a clear understanding of the main centrality measures available, unveiling their similariti…
Complex networks can be used to represent complex systems which originate in the real world. Here we study a transformation of these complex networks into simplicial complexes, where cliques represent the simplices of the complex. We extend the concept of node centrality to that of simplicial centrality and study several mathematical properties of degree, closeness, betweenness, eigenvector, Katz, and subgraph centrality for simplicial complexes. We study the degree distributions of these centralities at the different levels. We also compare and describe the differences between the centralitie…
In complex networks a common task is to identify the most important or "central" nodes. There are several definitions, often called centrality measures, which often lead to different results. Here we study extensively correlations between four local and global measures namely the degree, the shortest-path-betweenness, the random-walk betweenness and the subgraph centrality on different random-network models like Erdos-Renyi, Small-World and Barabasi-Albert as well as on different real networks like metabolic pathways, social collaborations and computer networks. Correlations are quite differen…
Many scale-free networks exhibit a rich club structure, where high degree vertices form tightly interconnected subgraphs. In this paper, we explore the emergence of rich clubs in the context of shortest path based centrality metrics. We term these subgraphs of connected high closeness or high betweeness vertices as rich centrality clubs (RCC). Our experiments on real world and synthetic networks highlight the inter-relations between RCCs, expander graphs, and the core-periphery structure of the network. We show empirically and theoretically that RCCs exist, if the core-periphery structure of t…
We introduce a new matrix function for studying graphs and real-world networks based on a double-factorial penalization of walks between nodes in a graph. This new matrix function is based on the matrix error function. We find a very good approximation of this function using a matrix hyperbolic tangent function. We derive a communicability function, a subgraph centrality and a double-factorial Estrada index based on this new matrix function. We obtain upper and lower bounds for the double-factorial Estrada index of graphs, showing that they are similar to those of the single-factorial Estrada …
Convexity in a network (graph) has been recently defined as a property of each of its subgraphs to include all shortest paths between the nodes of that subgraph. It can be measured on the scale [0, 1] with 1 being assigned to fully convex networks. The largest convex component of a graph that emerges after the removal of the least number of edges is called a convex skeleton. It is basically a tree of cliques, which has been shown to have many interesting features. In this article the notions of convexity and convex skeletons in the context of scientific collaboration networks are discussed. Mo…
Temporal networks are widely used to represent a vast diversity of systems, including in particular social interactions, and the spreading processes unfolding on top of them. The identification of structures playing important roles in such processes remains largely an open question, despite recent progresses in the case of static networks. Here, we consider as candidate structures the recently introduced concept of span-cores: the span-cores decompose a temporal network into subgraphs of controlled duration and increasing connectivity, generalizing the core-decomposition of static graphs. To a…
We introduce a new network statistic that measures diverse structural properties at the micro-, meso-, and macroscopic scales, while still being easy to compute and easy to interpret at a glance. Our statistic, the onion spectrum, is based on the onion decomposition, which refines the k-core decomposition, a standard network fingerprinting method. The onion spectrum is exactly as easy to compute as the k-cores: It is based on the stages at which each vertex gets removed from a graph in the standard algorithm for computing the k-cores. But the onion spectrum reveals much more information about …
